



CFRG                                                    A. Faz-Hernandez
Internet-Draft                                          Cloudflare, Inc.
Intended status: Informational                                  S. Scott
Expires: 17 December 2022                                   Cornell Tech
                                                             N. Sullivan
                                                        Cloudflare, Inc.
                                                              R.S. Wahby
                                                     Stanford University
                                                               C.A. Wood
                                                        Cloudflare, Inc.
                                                            15 June 2022


                       Hashing to Elliptic Curves
                    draft-irtf-cfrg-hash-to-curve-16

Abstract

   This document specifies a number of algorithms for encoding or
   hashing an arbitrary string to a point on an elliptic curve.  This
   document is a product of the Crypto Forum Research Group (CFRG) in
   the IRTF.

Discussion Venues

   This note is to be removed before publishing as an RFC.

   Discussion of this document takes place on the Crypto Forum Research
   Group mailing list (cfrg@ietf.org), which is archived at
   https://mailarchive.ietf.org/arch/search/?email_list=cfrg.

   Source for this draft and an issue tracker can be found at
   https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 1]

Internet-Draft                hash-to-curve                    June 2022


   This Internet-Draft will expire on 17 December 2022.

Copyright Notice

   Copyright (c) 2022 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   5
     1.1.  Requirements Notation . . . . . . . . . . . . . . . . . .   6
   2.  Background  . . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.1.  Elliptic curves . . . . . . . . . . . . . . . . . . . . .   6
     2.2.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   8
       2.2.1.  Mappings  . . . . . . . . . . . . . . . . . . . . . .   8
       2.2.2.  Encodings . . . . . . . . . . . . . . . . . . . . . .   9
       2.2.3.  Random oracle encodings . . . . . . . . . . . . . . .   9
       2.2.4.  Serialization . . . . . . . . . . . . . . . . . . . .  10
       2.2.5.  Domain separation . . . . . . . . . . . . . . . . . .  10
   3.  Encoding byte strings to elliptic curves  . . . . . . . . . .  11
     3.1.  Domain separation requirements  . . . . . . . . . . . . .  13
   4.  Utility functions . . . . . . . . . . . . . . . . . . . . . .  14
     4.1.  The sgn0 function . . . . . . . . . . . . . . . . . . . .  16
   5.  Hashing to a finite field . . . . . . . . . . . . . . . . . .  17
     5.1.  Efficiency considerations in extension fields . . . . . .  18
     5.2.  hash_to_field implementation  . . . . . . . . . . . . . .  19
     5.3.  expand_message  . . . . . . . . . . . . . . . . . . . . .  20
       5.3.1.  expand_message_xmd  . . . . . . . . . . . . . . . . .  20
       5.3.2.  expand_message_xof  . . . . . . . . . . . . . . . . .  22
       5.3.3.  Using DSTs longer than 255 bytes  . . . . . . . . . .  23
       5.3.4.  Defining other expand_message variants  . . . . . . .  24
   6.  Deterministic mappings  . . . . . . . . . . . . . . . . . . .  25
     6.1.  Choosing a mapping function . . . . . . . . . . . . . . .  25
     6.2.  Interface . . . . . . . . . . . . . . . . . . . . . . . .  25
     6.3.  Notation  . . . . . . . . . . . . . . . . . . . . . . . .  26
     6.4.  Sign of the resulting point . . . . . . . . . . . . . . .  26
     6.5.  Exceptional cases . . . . . . . . . . . . . . . . . . . .  26
     6.6.  Mappings for Weierstrass curves . . . . . . . . . . . . .  27
       6.6.1.  Shallue-van de Woestijne method . . . . . . . . . . .  27



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 2]

Internet-Draft                hash-to-curve                    June 2022


       6.6.2.  Simplified Shallue-van de Woestijne-Ulas method . . .  28
       6.6.3.  Simplified SWU for AB == 0  . . . . . . . . . . . . .  29
     6.7.  Mappings for Montgomery curves  . . . . . . . . . . . . .  31
       6.7.1.  Elligator 2 method  . . . . . . . . . . . . . . . . .  31
     6.8.  Mappings for twisted Edwards curves . . . . . . . . . . .  32
       6.8.1.  Rational maps from Montgomery to twisted Edwards
               curves  . . . . . . . . . . . . . . . . . . . . . . .  32
       6.8.2.  Elligator 2 method  . . . . . . . . . . . . . . . . .  33
   7.  Clearing the cofactor . . . . . . . . . . . . . . . . . . . .  33
   8.  Suites for hashing  . . . . . . . . . . . . . . . . . . . . .  34
     8.1.  Implementing a hash-to-curve suite  . . . . . . . . . . .  37
     8.2.  Suites for NIST P-256 . . . . . . . . . . . . . . . . . .  37
     8.3.  Suites for NIST P-384 . . . . . . . . . . . . . . . . . .  38
     8.4.  Suites for NIST P-521 . . . . . . . . . . . . . . . . . .  39
     8.5.  Suites for curve25519 and edwards25519  . . . . . . . . .  40
     8.6.  Suites for curve448 and edwards448  . . . . . . . . . . .  41
     8.7.  Suites for secp256k1  . . . . . . . . . . . . . . . . . .  42
     8.8.  Suites for BLS12-381  . . . . . . . . . . . . . . . . . .  43
       8.8.1.  BLS12-381 G1  . . . . . . . . . . . . . . . . . . . .  43
       8.8.2.  BLS12-381 G2  . . . . . . . . . . . . . . . . . . . .  44
     8.9.  Defining a new hash-to-curve suite  . . . . . . . . . . .  45
     8.10. Suite ID naming conventions . . . . . . . . . . . . . . .  46
   9.  IANA considerations . . . . . . . . . . . . . . . . . . . . .  47
   10. Security considerations . . . . . . . . . . . . . . . . . . .  48
     10.1.  Properties of encodings  . . . . . . . . . . . . . . . .  48
     10.2.  Hashing passwords  . . . . . . . . . . . . . . . . . . .  49
     10.3.  Constant-time requirements . . . . . . . . . . . . . . .  49
     10.4.  encode_to_curve: output distribution and
            indifferentiability  . . . . . . . . . . . . . . . . . .  49
     10.5.  hash_to_field security . . . . . . . . . . . . . . . . .  50
     10.6.  expand_message_xmd security  . . . . . . . . . . . . . .  51
     10.7.  Domain separation for expand_message variants  . . . . .  51
     10.8.  Target security levels . . . . . . . . . . . . . . . . .  55
   11. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  55
   12. Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  55
   13. References  . . . . . . . . . . . . . . . . . . . . . . . . .  55
     13.1.  Normative References . . . . . . . . . . . . . . . . . .  55
     13.2.  Informative References . . . . . . . . . . . . . . . . .  56
   Appendix A.  Related work . . . . . . . . . . . . . . . . . . . .  65
   Appendix B.  Hashing to ristretto255  . . . . . . . . . . . . . .  67
   Appendix C.  Hashing to decaf448  . . . . . . . . . . . . . . . .  68
   Appendix D.  Rational maps  . . . . . . . . . . . . . . . . . . .  69
     D.1.  Generic Montgomery to twisted Edwards map . . . . . . . .  70
     D.2.  Weierstrass to Montgomery map . . . . . . . . . . . . . .  72
   Appendix E.  Isogeny maps for suites  . . . . . . . . . . . . . .  72
     E.1.  3-isogeny map for secp256k1 . . . . . . . . . . . . . . .  73
     E.2.  11-isogeny map for BLS12-381 G1 . . . . . . . . . . . . .  74
     E.3.  3-isogeny map for BLS12-381 G2  . . . . . . . . . . . . .  78



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 3]

Internet-Draft                hash-to-curve                    June 2022


   Appendix F.  Straight-line implementations of deterministic
           mappings  . . . . . . . . . . . . . . . . . . . . . . . .  80
     F.1.  Shallue-van de Woestijne method . . . . . . . . . . . . .  80
     F.2.  Simplified SWU method . . . . . . . . . . . . . . . . . .  81
       F.2.1.  sqrt_ratio subroutines  . . . . . . . . . . . . . . .  82
     F.3.  Elligator 2 method  . . . . . . . . . . . . . . . . . . .  86
   Appendix G.  Curve-specific optimized sample code . . . . . . . .  87
     G.1.  Interface and projective coordinate systems . . . . . . .  87
     G.2.  Elligator 2 . . . . . . . . . . . . . . . . . . . . . . .  88
       G.2.1.  curve25519 (q = 5 (mod 8), K = 1) . . . . . . . . . .  88
       G.2.2.  edwards25519  . . . . . . . . . . . . . . . . . . . .  89
       G.2.3.  curve448 (q = 3 (mod 4), K = 1) . . . . . . . . . . .  90
       G.2.4.  edwards448  . . . . . . . . . . . . . . . . . . . . .  91
       G.2.5.  Montgomery curves with q = 3 (mod 4)  . . . . . . . .  93
       G.2.6.  Montgomery curves with q = 5 (mod 8)  . . . . . . . .  95
     G.3.  Cofactor clearing for BLS12-381 G2  . . . . . . . . . . .  96
   Appendix H.  Scripts for parameter generation . . . . . . . . . .  98
     H.1.  Finding Z for the Shallue-van de Woestijne map  . . . . .  98
     H.2.  Finding Z for Simplified SWU  . . . . . . . . . . . . . .  99
     H.3.  Finding Z for Elligator 2 . . . . . . . . . . . . . . . . 100
   Appendix I.  sqrt and is_square functions . . . . . . . . . . . . 100
     I.1.  sqrt for q = 3 (mod 4)  . . . . . . . . . . . . . . . . . 101
     I.2.  sqrt for q = 5 (mod 8)  . . . . . . . . . . . . . . . . . 101
     I.3.  sqrt for q = 9 (mod 16) . . . . . . . . . . . . . . . . . 101
     I.4.  Constant-time Tonelli-Shanks algorithm  . . . . . . . . . 102
     I.5.  is_square for F = GF(p^2) . . . . . . . . . . . . . . . . 103
   Appendix J.  Suite test vectors . . . . . . . . . . . . . . . . . 104
     J.1.  NIST P-256  . . . . . . . . . . . . . . . . . . . . . . . 104
       J.1.1.  P256_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . . . . . 104
       J.1.2.  P256_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . . . . . 106
     J.2.  NIST P-384  . . . . . . . . . . . . . . . . . . . . . . . 108
       J.2.1.  P384_XMD:SHA-384_SSWU_RO_ . . . . . . . . . . . . . . 108
       J.2.2.  P384_XMD:SHA-384_SSWU_NU_ . . . . . . . . . . . . . . 110
     J.3.  NIST P-521  . . . . . . . . . . . . . . . . . . . . . . . 112
       J.3.1.  P521_XMD:SHA-512_SSWU_RO_ . . . . . . . . . . . . . . 112
       J.3.2.  P521_XMD:SHA-512_SSWU_NU_ . . . . . . . . . . . . . . 115
     J.4.  curve25519  . . . . . . . . . . . . . . . . . . . . . . . 117
       J.4.1.  curve25519_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . . 117
       J.4.2.  curve25519_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . . 119
     J.5.  edwards25519  . . . . . . . . . . . . . . . . . . . . . . 121
       J.5.1.  edwards25519_XMD:SHA-512_ELL2_RO_ . . . . . . . . . . 121
       J.5.2.  edwards25519_XMD:SHA-512_ELL2_NU_ . . . . . . . . . . 123
     J.6.  curve448  . . . . . . . . . . . . . . . . . . . . . . . . 125
       J.6.1.  curve448_XOF:SHAKE256_ELL2_RO_  . . . . . . . . . . . 125
       J.6.2.  curve448_XOF:SHAKE256_ELL2_NU_  . . . . . . . . . . . 128
     J.7.  edwards448  . . . . . . . . . . . . . . . . . . . . . . . 130
       J.7.1.  edwards448_XOF:SHAKE256_ELL2_RO_  . . . . . . . . . . 130
       J.7.2.  edwards448_XOF:SHAKE256_ELL2_NU_  . . . . . . . . . . 133



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 4]

Internet-Draft                hash-to-curve                    June 2022


     J.8.  secp256k1 . . . . . . . . . . . . . . . . . . . . . . . . 135
       J.8.1.  secp256k1_XMD:SHA-256_SSWU_RO_  . . . . . . . . . . . 135
       J.8.2.  secp256k1_XMD:SHA-256_SSWU_NU_  . . . . . . . . . . . 137
     J.9.  BLS12-381 G1  . . . . . . . . . . . . . . . . . . . . . . 139
       J.9.1.  BLS12381G1_XMD:SHA-256_SSWU_RO_ . . . . . . . . . . . 139
       J.9.2.  BLS12381G1_XMD:SHA-256_SSWU_NU_ . . . . . . . . . . . 141
     J.10. BLS12-381 G2  . . . . . . . . . . . . . . . . . . . . . . 143
       J.10.1.  BLS12381G2_XMD:SHA-256_SSWU_RO_  . . . . . . . . . . 143
       J.10.2.  BLS12381G2_XMD:SHA-256_SSWU_NU_  . . . . . . . . . . 147
   Appendix K.  Expand test vectors  . . . . . . . . . . . . . . . . 149
     K.1.  expand_message_xmd(SHA-256) . . . . . . . . . . . . . . . 150
     K.2.  expand_message_xmd(SHA-256) (long DST)  . . . . . . . . . 154
     K.3.  expand_message_xmd(SHA-512) . . . . . . . . . . . . . . . 158
     K.4.  expand_message_xof(SHAKE128)  . . . . . . . . . . . . . . 163
     K.5.  expand_message_xof(SHAKE128) (long DST) . . . . . . . . . 167
     K.6.  expand_message_xof(SHAKE256)  . . . . . . . . . . . . . . 171
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 175

1.  Introduction

   Many cryptographic protocols require a procedure that encodes an
   arbitrary input, e.g., a password, to a point on an elliptic curve.
   This procedure is known as hashing to an elliptic curve, where the
   hashing procedure provides collision resistance and does not reveal
   the discrete logarithm of the output point.  Prominent examples of
   cryptosystems that hash to elliptic curves include password-
   authenticated key exchanges [BM92] [J96] [BMP00] [p1363.2], Identity-
   Based Encryption [BF01], Boneh-Lynn-Shacham signatures [BLS01]
   [I-D.irtf-cfrg-bls-signature], Verifiable Random Functions [MRV99]
   [I-D.irtf-cfrg-vrf], and Oblivious Pseudorandom Functions [NR97]
   [I-D.irtf-cfrg-voprf].

   Unfortunately for implementors, the precise hash function that is
   suitable for a given protocol implemented using a given elliptic
   curve is often unclear from the protocol's description.  Meanwhile,
   an incorrect choice of hash function can have disastrous consequences
   for security.

   This document aims to bridge this gap by providing a comprehensive
   set of recommended algorithms for a range of curve types.  Each
   algorithm conforms to a common interface: it takes as input an
   arbitrary-length byte string and produces as output a point on an
   elliptic curve.  We provide implementation details for each
   algorithm, describe the security rationale behind each
   recommendation, and give guidance for elliptic curves that are not
   explicitly covered.  We also present optimized implementations for
   internal functions used by these algorithms.




Faz-Hernandez, et al.   Expires 17 December 2022                [Page 5]

Internet-Draft                hash-to-curve                    June 2022


   Readers wishing to quickly specify or implement a conforming hash
   function should consult Section 8, which lists recommended hash-to-
   curve suites and describes both how to implement an existing suite
   and how to specify a new one.

   This document does not cover rejection sampling methods, sometimes
   referred to as "try-and-increment" or "hunt-and-peck," because the
   goal is to describe algorithms that can plausibly be computed in
   constant time.  Use of these rejection methods is NOT RECOMMENDED,
   because they have been a perennial cause of side-channel
   vulnerabilities.  See Dragonblood [VR20] as one example of this
   problem in practice, and see Appendix A for a further description of
   rejection sampling methods.

   This document represents the consensus of the Crypto Forum Research
   Group (CFRG).

1.1.  Requirements Notation

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in
   BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

2.  Background

2.1.  Elliptic curves

   The following is a brief definition of elliptic curves, with an
   emphasis on important parameters and their relation to hashing to
   curves.  For further reference on elliptic curves, consult
   [CFADLNV05] or [W08].

   Let F be the finite field GF(q) of prime characteristic p > 3.  (This
   document does not consider elliptic curves over fields of
   characteristic 2 or 3.)  In most cases F is a prime field, so q = p.
   Otherwise, F is an extension field, so q = p^m for an integer m > 1.
   This document writes elements of extension fields in a primitive
   element or polynomial basis, i.e., as a vector of m elements of GF(p)
   written in ascending order by degree.  The entries of this vector are
   indexed in ascending order starting from 1, i.e., x = (x_1, x_2, ...,
   x_m).  For example, if q = p^2 and the primitive element basis is (1,
   I), then x = (a, b) corresponds to the element a + b * I, where x_1 =
   a and x_2 = b.  (Note that all choices of basis are isomorphic, but
   certain choices may result in a more efficient implementation; this
   document does not make any particular assumptions about choice of
   basis.)



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 6]

Internet-Draft                hash-to-curve                    June 2022


   An elliptic curve E is specified by an equation in two variables and
   a finite field F.  An elliptic curve equation takes one of several
   standard forms, including (but not limited to) Weierstrass,
   Montgomery, and Edwards.

   The curve E induces an algebraic group of order n, meaning that the
   group has n distinct elements.  (This document uses additive notation
   for the elliptic curve group operation.)  Elements of an elliptic
   curve group are points with affine coordinates (x, y) satisfying the
   curve equation, where x and y are elements of F.  In addition, all
   elliptic curve groups have a distinguished element, the identity
   point, which acts as the identity element for the group operation.
   On certain curves (including Weierstrass and Montgomery curves), the
   identity point cannot be represented as an (x, y) coordinate pair.

   For security reasons, cryptographic uses of elliptic curves generally
   require using a (sub)group of prime order.  Let G be such a subgroup
   of the curve of prime order r, where n = h * r.  In this equation, h
   is an integer called the cofactor.  An algorithm that takes as input
   an arbitrary point on the curve E and produces as output a point in
   the subgroup G of E is said to "clear the cofactor."  Such algorithms
   are discussed in Section 7.

   Certain hash-to-curve algorithms restrict the form of the curve
   equation, the characteristic of the field, or the parameters of the
   curve.  For each algorithm presented, this document lists the
   relevant restrictions.

   The table below summarizes quantities relevant to hashing to curves:






















Faz-Hernandez, et al.   Expires 17 December 2022                [Page 7]

Internet-Draft                hash-to-curve                    June 2022


         +========+=====================+=======================+
         | Symbol | Meaning             | Relevance             |
         +========+=====================+=======================+
         | F,q,p  | A finite field F of | For prime fields, q = |
         |        | characteristic p    | p; otherwise, q = p^m |
         |        | and #F = q = p^m.   | and m>1.              |
         +--------+---------------------+-----------------------+
         |   E    | Elliptic curve.     | E is specified by an  |
         |        |                     | equation and a field  |
         |        |                     | F.                    |
         +--------+---------------------+-----------------------+
         |   n    | Number of points on | n = h * r, for h and  |
         |        | the elliptic curve  | r defined below.      |
         |        | E.                  |                       |
         +--------+---------------------+-----------------------+
         |   G    | A prime-order       | Destination group to  |
         |        | subgroup of the     | which byte strings    |
         |        | points on E.        | are encoded.          |
         +--------+---------------------+-----------------------+
         |   r    | Order of G.         | r is a prime factor   |
         |        |                     | of n (usually, the    |
         |        |                     | largest such factor). |
         +--------+---------------------+-----------------------+
         |   h    | Cofactor, h >= 1.   | An integer satisfying |
         |        |                     | n = h * r.            |
         +--------+---------------------+-----------------------+

            Table 1: Summary of symbols and their definitions.

2.2.  Terminology

   In this section, we define important terms used throughout the
   document.

2.2.1.  Mappings

   A mapping is a deterministic function from an element of the field F
   to a point on an elliptic curve E defined over F.

   In general, the set of all points that a mapping can produce over all
   possible inputs may be only a subset of the points on an elliptic
   curve (i.e., the mapping may not be surjective).  In addition, a
   mapping may output the same point for two or more distinct inputs
   (i.e., the mapping may not be injective).  For example, consider a
   mapping from F to an elliptic curve having n points: if the number of
   elements of F is not equal to n, then this mapping cannot be
   bijective (i.e., both injective and surjective) since the mapping is
   defined to be deterministic.



Faz-Hernandez, et al.   Expires 17 December 2022                [Page 8]

Internet-Draft                hash-to-curve                    June 2022


   Mappings may also be invertible, meaning that there is an efficient
   algorithm that, for any point P output by the mapping, outputs an x
   in F such that applying the mapping to x outputs P.  Some of the
   mappings given in Section 6 are invertible, but this document does
   not discuss inversion algorithms.

2.2.2.  Encodings

   Encodings are closely related to mappings.  Like a mapping, an
   encoding is a function that outputs a point on an elliptic curve.  In
   contrast to a mapping, however, the input to an encoding is an
   arbitrary-length byte string.

   This document constructs deterministic encodings by composing a hash
   function Hf with a deterministic mapping.  In particular, Hf takes as
   input an arbitrary string and outputs an element of F.  The
   deterministic mapping takes that element as input and outputs a point
   on an elliptic curve E defined over F.  Since Hf takes arbitrary-
   length byte strings as inputs, it cannot be injective: the set of
   inputs is larger than the set of outputs, so there must be distinct
   inputs that give the same output (i.e., there must be collisions).
   Thus, any encoding built from Hf is also not injective.

   Like mappings, encodings may be invertible, meaning that there is an
   efficient algorithm that, for any point P output by the encoding,
   outputs a string s such that applying the encoding to s outputs P.
   The instantiation of Hf used by all encodings specified in this
   document (Section 5) is not invertible.  Thus, the encodings are also
   not invertible.

   In some applications of hashing to elliptic curves, it is important
   that encodings do not leak information through side channels.  [VR20]
   is one example of this type of leakage leading to a security
   vulnerability.  See Section 10.3 for further discussion.

2.2.3.  Random oracle encodings

   A random-oracle encoding satisfies a strong property: it can be
   proved indifferentiable from a random oracle [MRH04] under a suitable
   assumption.

   Both constructions described in Section 3 are indifferentiable from
   random oracles [MRH04] when instantiated following the guidelines in
   this document.  The constructions differ in their output
   distributions: one gives a uniformly random point on the curve, the
   other gives a point sampled from a nonuniform distribution.





Faz-Hernandez, et al.   Expires 17 December 2022                [Page 9]

Internet-Draft                hash-to-curve                    June 2022


   A random-oracle encoding with a uniform output distribution is
   suitable for use in many cryptographic protocols proven secure in the
   random oracle model.  See Section 10.1 for further discussion.

2.2.4.  Serialization

   A procedure related to encoding is the conversion of an elliptic
   curve point to a bit string.  This is called serialization, and is
   typically used for compactly storing or transmitting points.  The
   inverse operation, deserialization, converts a bit string to an
   elliptic curve point.  For example, [SEC1] and [p1363a] give standard
   methods for serialization and deserialization.

   Deserialization is different from encoding in that only certain
   strings (namely, those output by the serialization procedure) can be
   deserialized.  In contrast, this document is concerned with encodings
   from arbitrary strings to elliptic curve points.  This document does
   not cover serialization or deserialization.

2.2.5.  Domain separation

   Cryptographic protocols proven secure in the random oracle model are
   often analyzed under the assumption that the random oracle only
   answers queries associated with that protocol (including queries made
   by adversaries) [BR93].  In practice, this assumption does not hold
   if two protocols use the same function to instantiate the random
   oracle.  Concretely, consider protocols P1 and P2 that query a random
   oracle RO: if P1 and P2 both query RO on the same value x, the
   security analysis of one or both protocols may be invalidated.

   A common way of addressing this issue is called domain separation,
   which allows a single random oracle to simulate multiple, independent
   oracles.  This is effected by ensuring that each simulated oracle
   sees queries that are distinct from those seen by all other simulated
   oracles.  For example, to simulate two oracles RO1 and RO2 given a
   single oracle RO, one might define

   RO1(x) := RO("RO1" || x)
   RO2(x) := RO("RO2" || x)

   where || is the concatenation operator.  In this example, "RO1" and
   "RO2" are called domain separation tags; they ensure that queries to
   RO1 and RO2 cannot result in identical queries to RO, meaning that it
   is safe to treat RO1 and RO2 as independent oracles.

   In general, domain separation requires defining a distinct injective
   encoding for each oracle being simulated.  In the above example,
   "RO1" and "RO2" have the same length and thus satisfy this



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 10]

Internet-Draft                hash-to-curve                    June 2022


   requirement when used as prefixes.  The algorithms specified in this
   document take a different approach to ensuring injectivity; see
   Section 5.3 and Section 10.7 for more details.

3.  Encoding byte strings to elliptic curves

   This section presents a general framework and interface for encoding
   byte strings to points on an elliptic curve.  The constructions in
   this section rely on three basic functions:

   *  The function hash_to_field hashes arbitrary-length byte strings to
      a list of one or more elements of a finite field F; its
      implementation is defined in Section 5.

      hash_to_field(msg, count)

      Inputs:
      - msg, a byte string containing the message to hash.
      - count, the number of elements of F to output.

      Outputs:
      - (u_0, ..., u_(count - 1)), a list of field elements.

      Steps: defined in Section 5.

   *  The function map_to_curve calculates a point on the elliptic curve
      E from an element of the finite field F over which E is defined.
      Section 6 describes mappings for a range of curve families.

      map_to_curve(u)

      Input: u, an element of field F.
      Output: Q, a point on the elliptic curve E.
      Steps: defined in Section 6.

   *  The function clear_cofactor sends any point on the curve E to the
      subgroup G of E.  Section 7 describes methods to perform this
      operation.

      clear_cofactor(Q)

      Input: Q, a point on the elliptic curve E.
      Output: P, a point in G.
      Steps: defined in Section 7.







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 11]

Internet-Draft                hash-to-curve                    June 2022


   The two encodings (Section 2.2.2) defined in this section have the
   same interface and are both random-oracle encodings (Section 2.2.3).
   Both are implemented as a composition of the three basic functions
   above.  The difference between the two is that their outputs are
   sampled from different distributions:

   *  encode_to_curve is a nonuniform encoding from byte strings to
      points in G.  That is, the distribution of its output is not
      uniformly random in G: the set of possible outputs of
      encode_to_curve is only a fraction of the points in G, and some
      points in this set are more likely to be output than others.
      Section 10.4 gives a more precise definition of encode_to_curve's
      output distribution.

      encode_to_curve(msg)

      Input: msg, an arbitrary-length byte string.
      Output: P, a point in G.

      Steps:
      1. u = hash_to_field(msg, 1)
      2. Q = map_to_curve(u[0])
      3. P = clear_cofactor(Q)
      4. return P

   *  hash_to_curve is a uniform encoding from byte strings to points in
      G.  That is, the distribution of its output is statistically close
      to uniform in G.

      This function is suitable for most applications requiring a random
      oracle returning points in G, when instantiated with any of the
      map_to_curve functions described in Section 6.  See Section 10.1
      for further discussion.

      hash_to_curve(msg)

      Input: msg, an arbitrary-length byte string.
      Output: P, a point in G.

      Steps:
      1. u = hash_to_field(msg, 2)
      2. Q0 = map_to_curve(u[0])
      3. Q1 = map_to_curve(u[1])
      4. R = Q0 + Q1              # Point addition
      5. P = clear_cofactor(R)
      6. return P





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 12]

Internet-Draft                hash-to-curve                    June 2022


   Each hash-to-curve suite in Section 8 instantiates one of these
   encoding functions for a specifc elliptic curve.

3.1.  Domain separation requirements

   All uses of the encoding functions defined in this document MUST
   include domain separation (Section 2.2.5) to avoid interfering with
   other uses of similar functionality.

   Applications that instantiate multiple, independent instances of
   either hash_to_curve or encode_to_curve MUST enforce domain
   separation between those instances.  This requirement applies both in
   the case of multiple instances targeting the same curve and in the
   case of multiple instances targeting different curves.  (This is
   because the internal hash_to_field primitive (Section 5) requires
   domain separation to guarantee independent outputs.)

   Domain separation is enforced with a domain separation tag (DST),
   which is a byte string constructed according to the following
   requirements:

   1.  Tags MUST be supplied as the DST parameter to hash_to_field, as
       described in Section 5.

   2.  Tags MUST have nonzero length.  A minimum length of 16 bytes is
       RECOMMENDED to reduce the chance of collisions with other
       applications.

   3.  Tags SHOULD begin with a fixed identification string that is
       unique to the application.

   4.  Tags SHOULD include a version number.

   5.  For applications that define multiple ciphersuites, each
       ciphersuite's tag MUST be different.  For this purpose, it is
       RECOMMENDED to include a ciphersuite identifier in each tag.

   6.  For applications that use multiple encodings, either to the same
       curve or to different curves, each encoding MUST use a different
       tag.  For this purpose, it is RECOMMENDED to include the
       encoding's Suite ID (Section 8) in the domain separation tag.
       For independent encodings based on the same suite, each tag
       SHOULD also include a distinct identifier, e.g., "ENC1" and
       "ENC2".

   As an example, consider a fictional application named Quux that
   defines several different ciphersuites, each for a different curve.
   A reasonable choice of tag is "QUUX-V<xx>-CS<yy>-<suiteID>", where



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 13]

Internet-Draft                hash-to-curve                    June 2022


   <xx> and <yy> are two-digit numbers indicating the version and
   ciphersuite, respectively, and <suiteID> is the Suite ID of the
   encoding used in ciphersuite <yy>.

   As another example, consider a fictional application named Baz that
   requires two independent random oracles to the same curve.
   Reasonable choices of tags for these oracles are "BAZ-V<xx>-CS<yy>-
   <suiteID>-ENC1" and "BAZ-V<xx>-CS<yy>-<suiteID>-ENC2", respectively,
   where <xx>, <yy>, and <suiteID> are as described above.

   The example tags given above are assumed to be ASCII-encoded byte
   strings without null termination, which is the RECOMMENDED format.
   Other encodings can be used, but in all cases the encoding as a
   sequence of bytes MUST be specified unambiguously.

4.  Utility functions

   Algorithms in this document use the utility functions described
   below, plus standard arithmetic operations (addition, multiplication,
   modular reduction, etc.) and elliptic curve point operations (point
   addition and scalar multiplication).

   For security, implementations of these functions SHOULD be constant
   time: in brief, this means that execution time and memory access
   patterns SHOULD NOT depend on the values of secret inputs,
   intermediate values, or outputs.  For such constant-time
   implementations, all arithmetic, comparisons, and assignments MUST
   also be implemented in constant time.  Section 10.3 briefly discusses
   constant-time security issues.

   Guidance on implementing low-level operations (in constant time or
   otherwise) is beyond the scope of this document; readers should
   consult standard reference material [MOV96] [CFADLNV05].

   *  CMOV(a, b, c): If c is False, CMOV returns a, otherwise it returns
      b.  For constant-time implementations, this operation must run in
      time independent of the value of c.

   *  AND, OR, NOT, and XOR are standard bitwise logical operators.  For
      constant-time implementations, short-circuit operators MUST be
      avoided.

   *  is_square(x): This function returns True whenever the value x is a
      square in the field F.  By Euler's criterion, this function can be
      calculated in constant time as

      is_square(x) := { True,  if x^((q - 1) / 2) is 0 or 1 in F;
                      { False, otherwise.



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 14]

Internet-Draft                hash-to-curve                    June 2022


      In certain extension fields, is_square can be computed in constant
      time more quickly than by the above exponentiation.  [AR13] and
      [S85] describe optimized methods for extension fields.
      Appendix I.5 gives an optimized straight-line method for GF(p^2).

   *  sqrt(x): The sqrt operation is a multi-valued function, i.e.,
      there exist two roots of x in the field F whenever x is square
      (except when x = 0).  To maintain compatibility across
      implementations while allowing implementors leeway for
      optimizations, this document does not require sqrt() to return a
      particular value.  Instead, as explained in Section 6.4, any
      function that calls sqrt also specifies how to determine the
      correct root.

      The preferred way of computing square roots is to fix a
      deterministic algorithm particular to F.  We give several
      algorithms in Appendix I.

   *  sgn0(x): This function returns either 0 or 1 indicating the "sign"
      of x, where sgn0(x) == 1 just when x is "negative".  (In other
      words, this function always considers 0 to be positive.)
      Section 4.1 defines this function and discusses its
      implementation.

   *  inv0(x): This function returns the multiplicative inverse of x in
      F, extended to all of F by fixing inv0(0) == 0.  A straightforward
      way to implement inv0 in constant time is to compute

      inv0(x) := x^(q - 2).

      Notice that on input 0, the output is 0 as required.  Certain
      fields may allow faster inversion methods; detailed discussion of
      such methods is beyond the scope of this document.

   *  I2OSP and OS2IP: These functions are used to convert a byte string
      to and from a non-negative integer as described in [RFC8017].
      (Note that these functions operate on byte strings in big-endian
      byte order.)

   *  a || b: denotes the concatenation of byte strings a and b.  For
      example, "ABC" || "DEF" == "ABCDEF".

   *  substr(str, sbegin, slen): for a byte string str, this function
      returns the slen-byte substring starting at position sbegin;
      positions are zero indexed.  For example, substr("ABCDEFG", 2, 3)
      == "CDE".





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 15]

Internet-Draft                hash-to-curve                    June 2022


   *  len(str): for a byte string str, this function returns the length
      of str in bytes.  For example, len("ABC") == 3.

   *  strxor(str1, str2): for byte strings str1 and str2, strxor(str1,
      str2) returns the bitwise XOR of the two strings.  For example,
      strxor("abc", "XYZ") == "9;9" (the strings in this example are
      ASCII literals, but strxor is defined for arbitrary byte strings).
      In this document, strxor is only applied to inputs of equal
      length.

4.1.  The sgn0 function

   This section defines a generic sgn0 implementation that applies to
   any field F = GF(p^m).  It also gives simplified implementations for
   the cases F = GF(p) and F = GF(p^2).

   The definition of the sgn0 function for extension fields relies on
   the polynomial basis or vector representation of field elements, and
   iterates over the entire vector representation of the input element.
   As a result, sgn0 depends on the primitive polynomial used to define
   the polynomial basis; see Section 8 for more information about this
   basis, and see Section 2.1 for a discussion of representing elements
   of extension fields as vectors.

   sgn0(x)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.
   - p, the characteristic of F (see immediately above).
   - m, the extension degree of F, m >= 1 (see immediately above).

   Input: x, an element of F.
   Output: 0 or 1.

   Steps:
   1. sign = 0
   2. zero = 1
   3. for i in (1, 2, ..., m):
   4.   sign_i = x_i mod 2
   5.   zero_i = x_i == 0
   6.   sign = sign OR (zero AND sign_i) # Avoid short-circuit logic ops
   7.   zero = zero AND zero_i
   8. return sign

   When m == 1, sgn0 can be significantly simplified:






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 16]

Internet-Draft                hash-to-curve                    June 2022


   sgn0_m_eq_1(x)

   Input: x, an element of GF(p).
   Output: 0 or 1.

   Steps:
   1. return x mod 2

   The case m == 2 is only slightly more complicated:

   sgn0_m_eq_2(x)

   Input: x, an element of GF(p^2).
   Output: 0 or 1.

   Steps:
   1. sign_0 = x_0 mod 2
   2. zero_0 = x_0 == 0
   3. sign_1 = x_1 mod 2
   4. s = sign_0 OR (zero_0 AND sign_1) # Avoid short-circuit logic ops
   5. return s

5.  Hashing to a finite field

   The hash_to_field function hashes a byte string msg of arbitrary
   length into one or more elements of a field F.  This function works
   in two steps: it first hashes the input byte string to produce a
   uniformly random byte string, and then interprets this byte string as
   one or more elements of F.

   For the first step, hash_to_field calls an auxiliary function
   expand_message.  This document defines two variants of
   expand_message: one appropriate for hash functions like SHA-2
   [FIPS180-4] or SHA-3 [FIPS202], and another appropriate for
   extendable-output functions such as SHAKE128 [FIPS202].  Security
   considerations for each expand_message variant are discussed below
   (Section 5.3.1, Section 5.3.2).

   Implementors MUST NOT use rejection sampling to generate a uniformly
   random element of F, to ensure that the hash_to_field function is
   amenable to constant-time implementation.  The reason is that
   rejection sampling procedures are difficult to implement in constant
   time, and later well-meaning "optimizations" may silently render an
   implementation non-constant-time.  This means that any hash_to_field
   function based on rejection sampling would be incompatible with
   constant-time implementation.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 17]

Internet-Draft                hash-to-curve                    June 2022


   The hash_to_field function is also suitable for securely hashing to
   scalars.  For example, when hashing to the scalar field for an
   elliptic curve (sub)group with prime order r, it suffices to
   instantiate hash_to_field with target field GF(r).

   The hash_to_field function is designed to be indifferentiable from a
   random oracle [MRH04] when expand_message (Section 5.3) is modeled as
   a random oracle (see Section 10.5 for details about its
   indifferentiability).  Ensuring indifferentiability requires care; to
   see why, consider a prime p that is close to 3/4 * 2^256.  Reducing a
   random 256-bit integer modulo this p yields a value that is in the
   range [0, p / 3] with probability roughly 1/2, meaning that this
   value is statistically far from uniform in [0, p - 1].

   To control bias, hash_to_field instead uses random integers whose
   length is at least ceil(log2(p)) + k bits, where k is the target
   security level for the suite in bits.  Reducing such integers mod p
   gives bias at most 2^-k for any p; this bias is appropriate when
   targeting k-bit security.  For each such integer, hash_to_field uses
   expand_message to obtain L uniform bytes, where

   L = ceil((ceil(log2(p)) + k) / 8)

   These uniform bytes are then interpreted as an integer via OS2IP.
   For example, for a 255-bit prime p, and k = 128-bit security, L =
   ceil((255 + 128) / 8) = 48 bytes.

   Note that k is an upper bound on the security level for the
   corresponding curve.  See Section 10.8 for more details, and
   Section 8.9 for guidelines on choosing k for a given curve.

5.1.  Efficiency considerations in extension fields

   The hash_to_field function described in this section is inefficient
   for certain extension fields.  Specifically, when hashing to an
   element of the extension field GF(p^m), hash_to_field requires
   expanding msg into m * L bytes (for L as defined above).  For
   extension fields where log2(p) is significantly smaller than the
   security level k, this approach is inefficient: it requires
   expand_message to output roughly m * log2(p) + m * k bits, whereas m
   * log2(p) + k bytes suffices to generate an element of GF(p^m) with
   bias at most 2^-k.  In such cases, applications MAY use an
   alternative hash_to_field function, provided it meets the following
   security requirements:

   *  The function MUST output field element(s) that are uniformly
      random except with bias at most 2^-k.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 18]

Internet-Draft                hash-to-curve                    June 2022


   *  The function MUST NOT use rejection sampling.

   *  The function SHOULD be amenable to straight line implementations.

   For example, Pornin [P20] describes a method for hashing to
   GF(9767^19) that meets these requirements while using fewer output
   bits from expand_message than hash_to_field would for that field.

5.2.  hash_to_field implementation

   The following procedure implements hash_to_field.

   The expand_message parameter to this function MUST conform to the
   requirements given in Section 5.3.  Section 3.1 discusses the
   REQUIRED method for constructing DST, the domain separation tag.
   Note that hash_to_field may fail (abort) if expand_message fails.

   hash_to_field(msg, count)

   Parameters:
   - DST, a domain separation tag (see Section 3.1).
   - F, a finite field of characteristic p and order q = p^m.
   - p, the characteristic of F (see immediately above).
   - m, the extension degree of F, m >= 1 (see immediately above).
   - L = ceil((ceil(log2(p)) + k) / 8), where k is the security
     parameter of the suite (e.g., k = 128).
   - expand_message, a function that expands a byte string and
     domain separation tag into a uniformly random byte string
     (see Section 5.3).

   Inputs:
   - msg, a byte string containing the message to hash.
   - count, the number of elements of F to output.

   Outputs:
   - (u_0, ..., u_(count - 1)), a list of field elements.

   Steps:
   1. len_in_bytes = count * m * L
   2. uniform_bytes = expand_message(msg, DST, len_in_bytes)
   3. for i in (0, ..., count - 1):
   4.   for j in (0, ..., m - 1):
   5.     elm_offset = L * (j + i * m)
   6.     tv = substr(uniform_bytes, elm_offset, L)
   7.     e_j = OS2IP(tv) mod p
   8.   u_i = (e_0, ..., e_(m - 1))
   9. return (u_0, ..., u_(count - 1))




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 19]

Internet-Draft                hash-to-curve                    June 2022


5.3.  expand_message

   expand_message is a function that generates a uniformly random byte
   string.  It takes three arguments:

   1.  msg, a byte string containing the message to hash,

   2.  DST, a byte string that acts as a domain separation tag, and

   3.  len_in_bytes, the number of bytes to be generated.

   This document defines the following two variants of expand_message:

   *  expand_message_xmd (Section 5.3.1) is appropriate for use with a
      wide range of hash functions, including SHA-2 [FIPS180-4], SHA-3
      [FIPS202], BLAKE2 [RFC7693], and others.

   *  expand_message_xof (Section 5.3.2) is appropriate for use with
      extendable-output functions (XOFs) including functions in the
      SHAKE [FIPS202] or BLAKE2X [BLAKE2X] families.

   These variants should suffice for the vast majority of use cases, but
   other variants are possible; Section 5.3.4 discusses requirements.

5.3.1.  expand_message_xmd

   The expand_message_xmd function produces a uniformly random byte
   string using a cryptographic hash function H that outputs b bits.
   For security, H MUST meet the following requirements:

   *  The number of bits output by H MUST be b >= 2 * k, for k the
      target security level in bits, and b MUST be divisible by 8.  The
      first requirement ensures k-bit collision resistance; the second
      ensures uniformity of expand_message_xmd's output.

   *  H MAY be a Merkle-Damgaard hash function like SHA-2.  In this
      case, security holds when the underlying compression function is
      modeled as a random oracle [CDMP05].  (See Section 10.6 for
      discussion.)

   *  H MAY be a sponge-based hash function like SHA-3 or BLAKE2.  In
      this case, security holds when the inner function is modeled as a
      random transformation or as a random permutation [BDPV08].

   *  Otherwise, H MUST be a hash function that has been proved
      indifferentiable from a random oracle [MRH04] under a reasonable
      cryptographic assumption.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 20]

Internet-Draft                hash-to-curve                    June 2022


   SHA-2 [FIPS180-4] and SHA-3 [FIPS202] are typical and RECOMMENDED
   choices.  As an example, for the 128-bit security level, b >= 256
   bits and either SHA-256 or SHA3-256 would be an appropriate choice.

   The hash function H is assumed to work by repeatedly ingesting fixed-
   length blocks of data.  The length in bits of these blocks is called
   the input block size (s).  As examples, s = 1024 for SHA-512
   [FIPS180-4] and s = 576 for SHA3-512 [FIPS202].  For correctness, H
   requires b <= s.

   The following procedure implements expand_message_xmd.

   expand_message_xmd(msg, DST, len_in_bytes)

   Parameters:
   - H, a hash function (see requirements above).
   - b_in_bytes, b / 8 for b the output size of H in bits.
     For example, for b = 256, b_in_bytes = 32.
   - s_in_bytes, the input block size of H, measured in bytes (see
     discussion above). For example, for SHA-256, s_in_bytes = 64.

   Input:
   - msg, a byte string.
   - DST, a byte string of at most 255 bytes.
     See below for information on using longer DSTs.
   - len_in_bytes, the length of the requested output in bytes,
     not greater than the lesser of (255 * b_in_bytes) or 2^16-1.

   Output:
   - uniform_bytes, a byte string.

   Steps:
   1.  ell = ceil(len_in_bytes / b_in_bytes)
   2.  ABORT if ell > 255 or len_in_bytes > 65535 or len(DST) > 255
   3.  DST_prime = DST || I2OSP(len(DST), 1)
   4.  Z_pad = I2OSP(0, s_in_bytes)
   5.  l_i_b_str = I2OSP(len_in_bytes, 2)
   6.  msg_prime = Z_pad || msg || l_i_b_str || I2OSP(0, 1) || DST_prime
   7.  b_0 = H(msg_prime)
   8.  b_1 = H(b_0 || I2OSP(1, 1) || DST_prime)
   9.  for i in (2, ..., ell):
   10.    b_i = H(strxor(b_0, b_(i - 1)) || I2OSP(i, 1) || DST_prime)
   11. uniform_bytes = b_1 || ... || b_ell
   12. return substr(uniform_bytes, 0, len_in_bytes)

   Note that the string Z_pad (step 6) is prefixed to msg before
   computing b_0 (step 7).  This is necessary for security when H is a
   Merkle-Damgaard hash, e.g., SHA-2 (see Section 10.6).  Hashing this



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 21]

Internet-Draft                hash-to-curve                    June 2022


   additional data means that the cost of computing b_0 is higher than
   the cost of simply computing H(msg).  In most settings this overhead
   is negligible, because the cost of evaluating H is much less than the
   other costs involved in hashing to a curve.

   It is possible, however, to entirely avoid this overhead by taking
   advantage of the fact that Z_pad depends only on H, and not on the
   arguments to expand_message_xmd.  To do so, first precompute and save
   the internal state of H after ingesting Z_pad.  Then, when computing
   b_0, initialize H using the saved state.  Further details are
   implementation dependent, and beyond the scope of this document.

5.3.2.  expand_message_xof

   The expand_message_xof function produces a uniformly random byte
   string using an extendable-output function (XOF) H.  For security, H
   MUST meet the following criteria:

   *  The collision resistance of H MUST be at least k bits.

   *  H MUST be an XOF that has been proved indifferentiable from a
      random oracle under a reasonable cryptographic assumption.

   The SHAKE [FIPS202] XOF family is a typical and RECOMMENDED choice.
   As an example, for 128-bit security, SHAKE128 would be an appropriate
   choice.

   The following procedure implements expand_message_xof.























Faz-Hernandez, et al.   Expires 17 December 2022               [Page 22]

Internet-Draft                hash-to-curve                    June 2022


   expand_message_xof(msg, DST, len_in_bytes)

   Parameters:
   - H(m, d), an extendable-output function that processes
              input message m and returns d bytes.

   Input:
   - msg, a byte string.
   - DST, a byte string of at most 255 bytes.
     See below for information on using longer DSTs.
   - len_in_bytes, the length of the requested output in bytes.

   Output:
   - uniform_bytes, a byte string.

   Steps:
   1. ABORT if len_in_bytes > 65535 or len(DST) > 255
   2. DST_prime = DST || I2OSP(len(DST), 1)
   3. msg_prime = msg || I2OSP(len_in_bytes, 2) || DST_prime
   4. uniform_bytes = H(msg_prime, len_in_bytes)
   5. return uniform_bytes

5.3.3.  Using DSTs longer than 255 bytes

   The expand_message variants defined in this section accept domain
   separation tags of at most 255 bytes.  If applications require a
   domain separation tag longer than 255 bytes, e.g., because of
   requirements imposed by an invoking protocol, implementors MUST
   compute a short domain separation tag by hashing, as follows:

   *  For expand_message_xmd using hash function H, DST is computed as

      DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST)

   *  For expand_message_xof using extendable-output function H, DST is
      computed as

      DST = H("H2C-OVERSIZE-DST-" || a_very_long_DST, ceil(2 * k / 8))

   Here, a_very_long_DST is the DST whose length is greater than 255
   bytes, "H2C-OVERSIZE-DST-" is a 17-byte ASCII string literal, and k
   is the target security level in bits.









Faz-Hernandez, et al.   Expires 17 December 2022               [Page 23]

Internet-Draft                hash-to-curve                    June 2022


5.3.4.  Defining other expand_message variants

   When defining a new expand_message variant, the most important
   consideration is that hash_to_field models expand_message as a random
   oracle.  Thus, implementors SHOULD prove indifferentiability from a
   random oracle under an appropriate assumption about the underlying
   cryptographic primitives; see Section 10.5 for more information.

   In addition, expand_message variants:

   *  MUST give collision resistance commensurate with the security
      level of the target elliptic curve.

   *  MUST be built on primitives designed for use in applications
      requiring cryptographic randomness.  As examples, a secure stream
      cipher is an appropriate primitive, whereas a Mersenne twister
      pseudorandom number generator [MT98] is not.

   *  MUST NOT use rejection sampling.

   *  MUST give independent values for distinct (msg, DST, length)
      inputs.  Meeting this requirement is subtle.  As a simplified
      example, hashing msg || DST does not work, because in this case
      distinct (msg, DST) pairs whose concatenations are equal will
      return the same output (e.g., ("AB", "CDEF") and ("ABC", "DEF")).
      The variants defined in this document use a suffix-free encoding
      of DST to avoid this issue.

   *  MUST use the domain separation tag DST to ensure that invocations
      of cryptographic primitives inside of expand_message are domain
      separated from invocations outside of expand_message.  For
      example, if the expand_message variant uses a hash function H, an
      encoding of DST MUST be added either as a prefix or a suffix of
      the input to each invocation of H.  Adding DST as a suffix is the
      RECOMMENDED approach.

   *  SHOULD read msg exactly once, for efficiency when msg is long.

   In addition, each expand_message variant MUST specify a unique
   EXP_TAG that identifies that variant in a Suite ID.  See Section 8.10
   for more information.










Faz-Hernandez, et al.   Expires 17 December 2022               [Page 24]

Internet-Draft                hash-to-curve                    June 2022


6.  Deterministic mappings

   The mappings in this section are suitable for implementing either
   nonuniform or uniform encodings using the constructions in Section 3.
   Certain mappings restrict the form of the curve or its parameters.
   For each mapping presented, this document lists the relevant
   restrictions.

   Note that mappings in this section are not interchangeable: different
   mappings will almost certainly output different points when evaluated
   on the same input.

6.1.  Choosing a mapping function

   This section gives brief guidelines on choosing a mapping function
   for a given elliptic curve.  Note that the suites given in Section 8
   are recommended mappings for the respective curves.

   If the target elliptic curve is a Montgomery curve (Section 6.7), the
   Elligator 2 method (Section 6.7.1) is recommended.  Similarly, if the
   target elliptic curve is a twisted Edwards curve (Section 6.8), the
   twisted Edwards Elligator 2 method (Section 6.8.2) is recommended.

   The remaining cases are Weierstrass curves.  For curves supported by
   the Simplified SWU method (Section 6.6.2), that mapping is the
   recommended one.  Otherwise, the Simplified SWU method for AB == 0
   (Section 6.6.3) is recommended if the goal is best performance, while
   the Shallue-van de Woestijne method (Section 6.6.1) is recommended if
   the goal is simplicity of implementation.  (The reason for this
   distinction is that the Simplified SWU method for AB == 0 requires
   implementing an isogeny map in addition to the mapping function,
   while the Shallue-van de Woestijne method does not.)

   The Shallue-van de Woestijne method (Section 6.6.1) works with any
   curve, and may be used in cases where a generic mapping is required.
   Note, however, that this mapping is almost always more
   computationally expensive than the curve-specific recommendations
   above.

6.2.  Interface

   The generic interface shared by all mappings in this section is as
   follows:

       (x, y) = map_to_curve(u)






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 25]

Internet-Draft                hash-to-curve                    June 2022


   The input u and outputs x and y are elements of the field F.  The
   affine coordinates (x, y) specify a point on an elliptic curve
   defined over F.  Note, however, that the point (x, y) is not a
   uniformly random point.

6.3.  Notation

   As a rough guide, the following conventions are used in pseudocode:

   *  All arithmetic operations are performed over a field F, unless
      explicitly stated otherwise.

   *  u: the input to the mapping function.  This is an element of F
      produced by the hash_to_field function.

   *  (x, y), (s, t), (v, w): the affine coordinates of the point output
      by the mapping.  Indexed variables (e.g., x1, y2, ...) are used
      for candidate values.

   *  tv1, tv2, ...: reusable temporary variables.

   *  c1, c2, ...: constant values, which can be computed in advance.

6.4.  Sign of the resulting point

   In general, elliptic curves have equations of the form y^2 = g(x).
   The mappings in this section first identify an x such that g(x) is
   square, then take a square root to find y.  Since there are two
   square roots when g(x) != 0, this may result in an ambiguity
   regarding the sign of y.

   When necessary, the mappings in this section resolve this ambiguity
   by specifying the sign of the y-coordinate in terms of the input to
   the mapping function.  Two main reasons support this approach: first,
   this covers elliptic curves over any field in a uniform way, and
   second, it gives implementors leeway in optimizing square-root
   implementations.

6.5.  Exceptional cases

   Mappings may have exceptional cases, i.e., inputs u on which the
   mapping is undefined.  These cases must be handled carefully,
   especially for constant-time implementations.








Faz-Hernandez, et al.   Expires 17 December 2022               [Page 26]

Internet-Draft                hash-to-curve                    June 2022


   For each mapping in this section, we discuss the exceptional cases
   and show how to handle them in constant time.  Note that all
   implementations SHOULD use inv0 (Section 4) to compute multiplicative
   inverses, to avoid exceptional cases that result from attempting to
   compute the inverse of 0.

6.6.  Mappings for Weierstrass curves

   The mappings in this section apply to a target curve E defined by the
   equation

       y^2 = g(x) = x^3 + A * x + B

   where 4 * A^3 + 27 * B^2 != 0.

6.6.1.  Shallue-van de Woestijne method

   Shallue and van de Woestijne [SW06] describe a mapping that applies
   to essentially any elliptic curve.  (Note, however, that this mapping
   is more expensive to evaluate than the other mappings in this
   document.)

   The parameterization given below is for Weierstrass curves; its
   derivation is detailed in [W19].  This parameterization also works
   for Montgomery (Section 6.7) and twisted Edwards (Section 6.8) curves
   via the rational maps given in Appendix D: first evaluate the
   Shallue-van de Woestijne mapping to an equivalent Weierstrass curve,
   then map that point to the target Montgomery or twisted Edwards curve
   using the corresponding rational map.

   Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B.

   Constants:

   *  A and B, the parameter of the Weierstrass curve.

   *  Z, a non-zero element of F meeting the below criteria.
      Appendix H.1 gives a Sage [SAGE] script that outputs the
      RECOMMENDED Z.

      1.  g(Z) != 0 in F.

      2.  -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.

      3.  -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.

      4.  At least one of g(Z) and g(-Z / 2) is square in F.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 27]

Internet-Draft                hash-to-curve                    June 2022


   Sign of y: Inputs u and -u give the same x-coordinate for many values
   of u.  Thus, we set sgn0(y) == sgn0(u).

   Exceptions: The exceptional cases for u occur when (1 + u^2 * g(Z)) *
   (1 - u^2 * g(Z)) == 0.  The restrictions on Z given above ensure that
   implementations that use inv0 to invert this product are exception
   free.

   Operations:

   1. tv1 = u^2 * g(Z)
   2. tv2 = 1 + tv1
   3. tv1 = 1 - tv1
   4. tv3 = inv0(tv1 * tv2)
   5. tv4 = sqrt(-g(Z) * (3 * Z^2 + 4 * A))    # can be precomputed
   6. If sgn0(tv4) == 1, set tv4 = -tv4        # sgn0(tv4) MUST equal 0
   7. tv5 = u * tv1 * tv3 * tv4
   8. tv6 = -4 * g(Z) / (3 * Z^2 + 4 * A)      # can be precomputed
   9.  x1 = -Z / 2 - tv5
   10. x2 = -Z / 2 + tv5
   11. x3 = Z + tv6 * (tv2^2 * tv3)^2
   12. If is_square(g(x1)), set x = x1 and y = sqrt(g(x1))
   13. Else If is_square(g(x2)), set x = x2 and y = sqrt(g(x2))
   14. Else set x = x3 and y = sqrt(g(x3))
   15. If sgn0(u) != sgn0(y), set y = -y
   16. return (x, y)

   Appendix F.1 gives an example straight-line implementation of this
   mapping.

6.6.2.  Simplified Shallue-van de Woestijne-Ulas method

   The function map_to_curve_simple_swu(u) implements a simplification
   of the Shallue-van de Woestijne-Ulas mapping [U07] described by Brier
   et al.  [BCIMRT10], which they call the "simplified SWU" map.  Wahby
   and Boneh [WB19] generalize and optimize this mapping.

   Preconditions: A Weierstrass curve y^2 = x^3 + A * x + B where A != 0
   and B != 0.

   Constants:

   *  A and B, the parameters of the Weierstrass curve.

   *  Z, an element of F meeting the below criteria.  Appendix H.2 gives
      a Sage [SAGE] script that outputs the RECOMMENDED Z.  The criteria
      are:




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 28]

Internet-Draft                hash-to-curve                    June 2022


      1.  Z is non-square in F,

      2.  Z != -1 in F,

      3.  the polynomial g(x) - Z is irreducible over F, and

      4.  g(B / (Z * A)) is square in F.

   Sign of y: Inputs u and -u give the same x-coordinate.  Thus, we set
   sgn0(y) == sgn0(u).

   Exceptions: The exceptional cases are values of u such that Z^2 * u^4
   + Z * u^2 == 0.  This includes u == 0, and may include other values
   depending on Z.  Implementations must detect this case and set x1 = B
   / (Z * A), which guarantees that g(x1) is square by the condition on
   Z given above.

   Operations:

   1. tv1 = inv0(Z^2 * u^4 + Z * u^2)
   2.  x1 = (-B / A) * (1 + tv1)
   3.  If tv1 == 0, set x1 = B / (Z * A)
   4. gx1 = x1^3 + A * x1 + B
   5.  x2 = Z * u^2 * x1
   6. gx2 = x2^3 + A * x2 + B
   7.  If is_square(gx1), set x = x1 and y = sqrt(gx1)
   8.  Else set x = x2 and y = sqrt(gx2)
   9.  If sgn0(u) != sgn0(y), set y = -y
   10. return (x, y)

   Appendix F.2 gives a general and optimized straight-line
   implementation of this mapping.  For more information on optimizing
   this mapping, see [WB19] Section 4 or the example code found at
   [hash2curve-repo].

6.6.3.  Simplified SWU for AB == 0

   Wahby and Boneh [WB19] show how to adapt the simplified SWU mapping
   to Weierstrass curves having A == 0 or B == 0, which the mapping of
   Section 6.6.2 does not support.  (The case A == B == 0 is excluded
   because y^2 = x^3 is not an elliptic curve.)

   This method applies to curves like secp256k1 [SEC2] and to pairing-
   friendly curves in the Barreto-Lynn-Scott [BLS03], Barreto-Naehrig
   [BN05], and other families.

   This method requires finding another elliptic curve E' given by the
   equation



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 29]

Internet-Draft                hash-to-curve                    June 2022


       y'^2 = g'(x') = x'^3 + A' * x' + B'

   that is isogenous to E and has A' != 0 and B' != 0.  (See [WB19],
   Appendix A, for one way of finding E' using [SAGE].)  This isogeny
   defines a map iso_map(x', y') given by a pair of rational functions.
   iso_map takes as input a point on E' and produces as output a point
   on E.

   Once E' and iso_map are identified, this mapping works as follows: on
   input u, first apply the simplified SWU mapping to get a point on E',
   then apply the isogeny map to that point to get a point on E.

   Note that iso_map is a group homomorphism, meaning that point
   addition commutes with iso_map.  Thus, when using this mapping in the
   hash_to_curve construction of Section 3, one can effect a small
   optimization by first mapping u0 and u1 to E', adding the resulting
   points on E', and then applying iso_map to the sum.  This gives the
   same result while requiring only one evaluation of iso_map.

   Preconditions: An elliptic curve E' with A' != 0 and B' != 0 that is
   isogenous to the target curve E with isogeny map iso_map from E' to
   E.

   Helper functions:

   *  map_to_curve_simple_swu is the mapping of Section 6.6.2 to E'

   *  iso_map is the isogeny map from E' to E

   Sign of y: for this map, the sign is determined by
   map_to_curve_simple_swu.  No further sign adjustments are necessary.

   Exceptions: map_to_curve_simple_swu handles its exceptional cases.
   Exceptional cases of iso_map are inputs that cause the denominator of
   either rational function to evaluate to zero; such cases MUST return
   the identity point on E.

   Operations:

   1. (x', y') = map_to_curve_simple_swu(u)    # (x', y') is on E'
   2.   (x, y) = iso_map(x', y')               # (x, y) is on E
   3. return (x, y)

   See [hash2curve-repo] or [WB19] Section 4.3 for details on
   implementing the isogeny map.






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 30]

Internet-Draft                hash-to-curve                    June 2022


6.7.  Mappings for Montgomery curves

   The mapping defined in this section applies to a target curve M
   defined by the equation

       K * t^2 = s^3 + J * s^2 + s

6.7.1.  Elligator 2 method

   Bernstein, Hamburg, Krasnova, and Lange give a mapping that applies
   to any curve with a point of order 2 [BHKL13], which they call
   Elligator 2.

   Preconditions: A Montgomery curve K * t^2 = s^3 + J * s^2 + s where J
   != 0, K != 0, and (J^2 - 4) / K^2 is non-zero and non-square in F.

   Constants:

   *  J and K, the parameters of the elliptic curve.

   *  Z, a non-square element of F.  Appendix H.3 gives a Sage [SAGE]
      script that outputs the RECOMMENDED Z.

   Sign of t: this mapping fixes the sign of t as specified in [BHKL13].
   No additional adjustment is required.

   Exceptions: The exceptional case is Z * u^2 == -1, i.e., 1 + Z * u^2
   == 0.  Implementations must detect this case and set x1 = -(J / K).
   Note that this can only happen when q = 3 (mod 4).

   Operations:

   1.  x1 = -(J / K) * inv0(1 + Z * u^2)
   2.  If x1 == 0, set x1 = -(J / K)
   3. gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2
   4.  x2 = -x1 - (J / K)
   5. gx2 = x2^3 + (J / K) * x2^2 + x2 / K^2
   6.  If is_square(gx1), set x = x1, y = sqrt(gx1) with sgn0(y) == 1.
   7.  Else set x = x2, y = sqrt(gx2) with sgn0(y) == 0.
   8.   s = x * K
   9.   t = y * K
   10. return (s, t)

   Appendix F.3 gives an example straight-line implementation of this
   mapping.  Appendix G.2 gives optimized straight-line procedures that
   apply to specific classes of curves and base fields.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 31]

Internet-Draft                hash-to-curve                    June 2022


6.8.  Mappings for twisted Edwards curves

   Twisted Edwards curves (a class of curves that includes Edwards
   curves) are given by the equation

       a * v^2 + w^2 = 1 + d * v^2 * w^2

   with a != 0, d != 0, and a != d [BBJLP08].

   These curves are closely related to Montgomery curves (Section 6.7):
   every twisted Edwards curve is birationally equivalent to a
   Montgomery curve ([BBJLP08], Theorem 3.2).  This equivalence yields
   an efficient way of hashing to a twisted Edwards curve: first, hash
   to an equivalent Montgomery curve, then transform the result into a
   point on the twisted Edwards curve via a rational map.  This method
   of hashing to a twisted Edwards curve thus requires identifying a
   corresponding Montgomery curve and rational map.  We describe how to
   identify such a curve and map immediately below.

6.8.1.  Rational maps from Montgomery to twisted Edwards curves

   There are two ways to select a Montgomery curve and rational map for
   use when hashing to a given twisted Edwards curve.  The selected
   Montgomery curve and rational map MUST be specified as part of the
   hash-to-curve suite for a given twisted Edwards curve; see Section 8.

   1.  When hashing to a standardized twisted Edwards curve for which a
       corresponding Montgomery form and rational map are also
       standardized, the standard Montgomery form and rational map
       SHOULD be used to ensure compatibility with existing software.

       In certain cases, e.g., edwards25519 [RFC7748], the sign of the
       rational map from the twisted Edwards curve to its corresponding
       Montgomery curve is not given explicitly.  In this case, the sign
       MUST be fixed such that applying the rational map to the twisted
       Edwards curve's base point yields the Montgomery curve's base
       point with correct sign.  (For edwards25519, see [RFC7748] and
       [EID4730].)

       When defining new twisted Edwards curves, a Montgomery equivalent
       and rational map SHOULD also be specified, and the sign of the
       rational map SHOULD be stated explicitly.

   2.  When hashing to a twisted Edwards curve that does not have a
       standardized Montgomery form or rational map, the map given in
       Appendix D SHOULD be used.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 32]

Internet-Draft                hash-to-curve                    June 2022


6.8.2.  Elligator 2 method

   Preconditions: A twisted Edwards curve E and an equivalent Montgomery
   curve M meeting the requirements in Section 6.8.1.

   Helper functions:

   *  map_to_curve_elligator2 is the mapping of Section 6.7.1 to the
      curve M.

   *  rational_map is a function that takes a point (s, t) on M and
      returns a point (v, w) on E, as defined in Section 6.8.1.

   Sign of t (and v): for this map, the sign is determined by
   map_to_curve_elligator2.  No further sign adjustments are required.

   Exceptions: The exceptions for the Elligator 2 mapping are as given
   in Section 6.7.1.  The exceptions for the rational map are as given
   in Section 6.8.1.  No other exceptions are possible.

   The following procedure implements the Elligator 2 mapping for a
   twisted Edwards curve.  (Note that the output point is denoted (v, w)
   because it is a point on the target twisted Edwards curve.)

   map_to_curve_elligator2_edwards(u)

   Input: u, an element of F.
   Output: (v, w), a point on E.

   1. (s, t) = map_to_curve_elligator2(u)      # (s, t) is on M
   2. (v, w) = rational_map(s, t)              # (v, w) is on E
   3. return (v, w)

7.  Clearing the cofactor

   The mappings of Section 6 always output a point on the elliptic
   curve, i.e., a point in a group of order h * r (Section 2.1).
   Obtaining a point in G may require a final operation commonly called
   "clearing the cofactor," which takes as input any point on the curve
   and produces as output a point in the prime-order (sub)group G
   (Section 2.1).

   The cofactor can always be cleared via scalar multiplication by h.
   For elliptic curves where h = 1, i.e., the curves with a prime number
   of points, no operation is required.  This applies, for example, to
   the NIST curves P-256, P-384, and P-521 [FIPS186-4].





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 33]

Internet-Draft                hash-to-curve                    June 2022


   In some cases, it is possible to clear the cofactor via a faster
   method than scalar multiplication by h.  These methods are equivalent
   to (but usually faster than) multiplication by some scalar h_eff
   whose value is determined by the method and the curve.  Examples of
   fast cofactor clearing methods include the following:

   *  For certain pairing-friendly curves having subgroup G2 over an
      extension field, Scott et al.  [SBCDK09] describe a method for
      fast cofactor clearing that exploits an efficiently-computable
      endomorphism.  Fuentes-Castaneda et al.  [FKR11] propose an
      alternative method that is sometimes more efficient.  Budroni and
      Pintore [BP17] give concrete instantiations of these methods for
      Barreto-Lynn-Scott pairing-friendly curves [BLS03].  This method
      is described for the specific case of BLS12-381 in Appendix G.3.

   *  Wahby and Boneh ([WB19], Section 5) describe a trick due to Scott
      for fast cofactor clearing on any elliptic curve for which the
      prime factorization of h and the structure of the elliptic curve
      group meet certain conditions.

   The clear_cofactor function is parameterized by a scalar h_eff.
   Specifically,

       clear_cofactor(P) := h_eff * P

   where * represents scalar multiplication.  When a curve does not
   support a fast cofactor clearing method, h_eff = h and the cofactor
   MUST be cleared via scalar multiplication.

   When a curve admits a fast cofactor clearing method, clear_cofactor
   MAY be evaluated either via that method or via scalar multiplication
   by the equivalent h_eff; these two methods give the same result.
   Note that in this case scalar multiplication by the cofactor h does
   not generally give the same result as the fast method, and MUST NOT
   be used.

8.  Suites for hashing

   This section lists recommended suites for hashing to standard
   elliptic curves.

   A hash-to-curve suite fully specifies the procedure for hashing byte
   strings to points on a specific elliptic curve group.  Section 8.1
   describes how to implement a suite.  Applications that require
   hashing to an elliptic curve should use either an existing suite or a
   new suite specified as described in Section 8.9.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 34]

Internet-Draft                hash-to-curve                    June 2022


   All applications using a hash-to-curve suite MUST choose a domain
   separation tag (DST) in accordance with the guidelines in
   Section 3.1.  In addition, applications whose security requires a
   random oracle that returns uniformly random points on the target
   curve MUST use a suite whose encoding type is hash_to_curve; see
   Section 3 and immediately below for more information.

   A hash-to-curve suite comprises the following parameters:

   *  Suite ID, a short name used to refer to a given suite.
      Section 8.10 discusses the naming conventions for suite IDs.

   *  encoding type, either uniform (hash_to_curve) or nonuniform
      (encode_to_curve).  See Section 3 for definitions of these
      encoding types.

   *  E, the target elliptic curve over a field F.

   *  p, the characteristic of the field F.

   *  m, the extension degree of the field F.  If m > 1, the suite MUST
      also specify the polynomial basis used to represent extension
      field elements.

   *  k, the target security level of the suite in bits.  (See
      Section 10.8 for discussion.)

   *  L, the length parameter for hash_to_field (Section 5).

   *  expand_message, one of the variants specified in Section 5.3 plus
      any parameters required for the specified variant (for example, H,
      the underlying hash function).

   *  f, a mapping function from Section 6.

   *  h_eff, the scalar parameter for clear_cofactor (Section 7).

   In addition to the above parameters, the mapping f may require
   additional parameters Z, M, rational_map, E', or iso_map.  When
   applicable, these MUST be specified.











Faz-Hernandez, et al.   Expires 17 December 2022               [Page 35]

Internet-Draft                hash-to-curve                    June 2022


   The below table lists suites RECOMMENDED for some elliptic curves.
   The corresponding parameters are given in the following subsections.
   Applications instantiating cryptographic protocols whose security
   analysis relies on a random oracle that outputs points with a uniform
   distribution MUST NOT use a nonuniform encoding.  Moreover,
   applications that use a nonuniform encoding SHOULD carefully analyze
   the security implications of nonuniformity.  When the required
   encoding is not clear, applications SHOULD use a uniform encoding for
   security.

      +==============+===================================+=========+
      | E            | Suites                            | Section |
      +==============+===================================+=========+
      | NIST P-256   | P256_XMD:SHA-256_SSWU_RO_         | Section |
      |              | P256_XMD:SHA-256_SSWU_NU_         | 8.2     |
      +--------------+-----------------------------------+---------+
      | NIST P-384   | P384_XMD:SHA-384_SSWU_RO_         | Section |
      |              | P384_XMD:SHA-384_SSWU_NU_         | 8.3     |
      +--------------+-----------------------------------+---------+
      | NIST P-521   | P521_XMD:SHA-512_SSWU_RO_         | Section |
      |              | P521_XMD:SHA-512_SSWU_NU_         | 8.4     |
      +--------------+-----------------------------------+---------+
      | curve25519   | curve25519_XMD:SHA-512_ELL2_RO_   | Section |
      |              | curve25519_XMD:SHA-512_ELL2_NU_   | 8.5     |
      +--------------+-----------------------------------+---------+
      | edwards25519 | edwards25519_XMD:SHA-512_ELL2_RO_ | Section |
      |              | edwards25519_XMD:SHA-512_ELL2_NU_ | 8.5     |
      +--------------+-----------------------------------+---------+
      | curve448     | curve448_XOF:SHAKE256_ELL2_RO_    | Section |
      |              | curve448_XOF:SHAKE256_ELL2_NU_    | 8.6     |
      +--------------+-----------------------------------+---------+
      | edwards448   | edwards448_XOF:SHAKE256_ELL2_RO_  | Section |
      |              | edwards448_XOF:SHAKE256_ELL2_NU_  | 8.6     |
      +--------------+-----------------------------------+---------+
      | secp256k1    | secp256k1_XMD:SHA-256_SSWU_RO_    | Section |
      |              | secp256k1_XMD:SHA-256_SSWU_NU_    | 8.7     |
      +--------------+-----------------------------------+---------+
      | BLS12-381 G1 | BLS12381G1_XMD:SHA-256_SSWU_RO_   | Section |
      |              | BLS12381G1_XMD:SHA-256_SSWU_NU_   | 8.8     |
      +--------------+-----------------------------------+---------+
      | BLS12-381 G2 | BLS12381G2_XMD:SHA-256_SSWU_RO_   | Section |
      |              | BLS12381G2_XMD:SHA-256_SSWU_NU_   | 8.8     |
      +--------------+-----------------------------------+---------+

             Table 2: Suites for hashing to elliptic curves.






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 36]

Internet-Draft                hash-to-curve                    June 2022


8.1.  Implementing a hash-to-curve suite

   A hash-to-curve suite requires the following functions.  Note that
   some of these require utility functions from Section 4.

   1.  Base field arithmetic operations for the target elliptic curve,
       e.g., addition, multiplication, and square root.

   2.  Elliptic curve point operations for the target curve, e.g., point
       addition and scalar multiplication.

   3.  The hash_to_field function; see Section 5.  This includes the
       expand_message variant (Section 5.3) and any constituent hash
       function or XOF.

   4.  The suite-specified mapping function; see the corresponding
       subsection of Section 6.

   5.  A cofactor clearing function; see Section 7.  This may be
       implemented as scalar multiplication by h_eff or as a faster
       equivalent method.

   6.  The desired encoding function; see Section 3.  This is either
       hash_to_curve or encode_to_curve.

8.2.  Suites for NIST P-256

   This section defines ciphersuites for the NIST P-256 elliptic curve
   [FIPS186-4].

   P256_XMD:SHA-256_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + A * x + B, where

      -  A = -3

      -  B = 0x5ac635d8aa3a93e7b3ebbd55769886bc651d06b0cc53b0f63bce3c3e2
         7d2604b

   *  p: 2^256 - 2^224 + 2^192 + 2^96 - 1

   *  m: 1

   *  k: 128

   *  expand_message: expand_message_xmd (Section 5.3.1)



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 37]

Internet-Draft                hash-to-curve                    June 2022


   *  H: SHA-256

   *  L: 48

   *  f: Simplified SWU method (Section 6.6.2)

   *  Z: -10

   *  h_eff: 1

   P256_XMD:SHA-256_SSWU_NU_ is identical to P256_XMD:SHA-256_SSWU_RO_,
   except that the encoding type is encode_to_curve (Section 3).

   An optimized example implementation of the Simplified SWU mapping to
   P-256 is given in Appendix F.2.

8.3.  Suites for NIST P-384

   This section defines ciphersuites for the NIST P-384 elliptic curve
   [FIPS186-4].

   P384_XMD:SHA-384_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + A * x + B, where

      -  A = -3

      -  B = 0xb3312fa7e23ee7e4988e056be3f82d19181d9c6efe8141120314088f5
         013875ac656398d8a2ed19d2a85c8edd3ec2aef

   *  p: 2^384 - 2^128 - 2^96 + 2^32 - 1

   *  m: 1

   *  k: 192

   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-384

   *  L: 72

   *  f: Simplified SWU method (Section 6.6.2)

   *  Z: -12




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 38]

Internet-Draft                hash-to-curve                    June 2022


   *  h_eff: 1

   P384_XMD:SHA-384_SSWU_NU_ is identical to P384_XMD:SHA-384_SSWU_RO_,
   except that the encoding type is encode_to_curve (Section 3).

   An optimized example implementation of the Simplified SWU mapping to
   P-384 is given in Appendix F.2.

8.4.  Suites for NIST P-521

   This section defines ciphersuites for the NIST P-521 elliptic curve
   [FIPS186-4].

   P521_XMD:SHA-512_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + A * x + B, where

      -  A = -3

      -  B = 0x51953eb9618e1c9a1f929a21a0b68540eea2da725b99b315f3b8b4899
         18ef109e156193951ec7e937b1652c0bd3bb1bf073573df883d2c34f1ef451f
         d46b503f00

   *  p: 2^521 - 1

   *  m: 1

   *  k: 256

   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-512

   *  L: 98

   *  f: Simplified SWU method (Section 6.6.2)

   *  Z: -4

   *  h_eff: 1

   P521_XMD:SHA-512_SSWU_NU_ is identical to P521_XMD:SHA-512_SSWU_RO_,
   except that the encoding type is encode_to_curve (Section 3).

   An optimized example implementation of the Simplified SWU mapping to
   P-521 is given in Appendix F.2.



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 39]

Internet-Draft                hash-to-curve                    June 2022


8.5.  Suites for curve25519 and edwards25519

   This section defines ciphersuites for curve25519 and edwards25519
   [RFC7748].  Note that these ciphersuites MUST NOT be used when
   hashing to ristretto255 [I-D.irtf-cfrg-ristretto255-decaf448].  See
   Appendix B for information on how to hash to that group.

   curve25519_XMD:SHA-512_ELL2_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: K * t^2 = s^3 + J * s^2 + s, where

      -  J = 486662

      -  K = 1

   *  p: 2^255 - 19

   *  m: 1

   *  k: 128

   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-512

   *  L: 48

   *  f: Elligator 2 method (Section 6.7.1)

   *  Z: 2

   *  h_eff: 8

   edwards25519_XMD:SHA-512_ELL2_RO_ is identical to curve25519_XMD:SHA-
   512_ELL2_RO_, except for the following parameters:

   *  E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where

      -  a = -1

      -  d = 0x52036cee2b6ffe738cc740797779e89800700a4d4141d8ab75eb4dca1
         35978a3

   *  f: Twisted Edwards Elligator 2 method (Section 6.8.2)

   *  M: curve25519 defined in [RFC7748], Section 4.1



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 40]

Internet-Draft                hash-to-curve                    June 2022


   *  rational_map: the birational map defined in [RFC7748], Section 4.1

   curve25519_XMD:SHA-512_ELL2_NU_ is identical to curve25519_XMD:SHA-
   512_ELL2_RO_, except that the encoding type is encode_to_curve
   (Section 3).

   edwards25519_XMD:SHA-512_ELL2_NU_ is identical to
   edwards25519_XMD:SHA-512_ELL2_RO_, except that the encoding type is
   encode_to_curve (Section 3).

   Optimized example implementations of the above mappings are given in
   Appendix G.2.1 and Appendix G.2.2.

8.6.  Suites for curve448 and edwards448

   This section defines ciphersuites for curve448 and edwards448
   [RFC7748].  Note that these ciphersuites MUST NOT be used when
   hashing to decaf448 [I-D.irtf-cfrg-ristretto255-decaf448].  See
   Appendix C for information on how to hash to that group.

   curve448_XOF:SHAKE256_ELL2_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: K * t^2 = s^3 + J * s^2 + s, where

      -  J = 156326

      -  K = 1

   *  p: 2^448 - 2^224 - 1

   *  m: 1

   *  k: 224

   *  expand_message: expand_message_xof (Section 5.3.2)

   *  H: SHAKE256

   *  L: 84

   *  f: Elligator 2 method (Section 6.7.1)

   *  Z: -1

   *  h_eff: 4




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 41]

Internet-Draft                hash-to-curve                    June 2022


   edwards448_XOF:SHAKE256_ELL2_RO_ is identical to
   curve448_XOF:SHAKE256_ELL2_RO_, except for the following parameters:

   *  E: a * v^2 + w^2 = 1 + d * v^2 * w^2, where

      -  a = 1

      -  d = -39081

   *  f: Twisted Edwards Elligator 2 method (Section 6.8.2)

   *  M: curve448, defined in [RFC7748], Section 4.2

   *  rational_map: the 4-isogeny map defined in [RFC7748], Section 4.2

   curve448_XOF:SHAKE256_ELL2_NU_ is identical to
   curve448_XOF:SHAKE256_ELL2_RO_, except that the encoding type is
   encode_to_curve (Section 3).

   edwards448_XOF:SHAKE256_ELL2_NU_ is identical to
   edwards448_XOF:SHAKE256_ELL2_RO_, except that the encoding type is
   encode_to_curve (Section 3).

   Optimized example implementations of the above mappings are given in
   Appendix G.2.3 and Appendix G.2.4.

8.7.  Suites for secp256k1

   This section defines ciphersuites for the secp256k1 elliptic curve
   [SEC2].

   secp256k1_XMD:SHA-256_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + 7

   *  p: 2^256 - 2^32 - 2^9 - 2^8 - 2^7 - 2^6 - 2^4 - 1

   *  m: 1

   *  k: 128

   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-256

   *  L: 48



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 42]

Internet-Draft                hash-to-curve                    June 2022


   *  f: Simplified SWU for AB == 0 (Section 6.6.3)

   *  Z: -11

   *  E': y'^2 = x'^3 + A' * x' + B', where

      -  A': 0x3f8731abdd661adca08a5558f0f5d272e953d363cb6f0e5d405447c01
         a444533

      -  B': 1771

   *  iso_map: the 3-isogeny map from E' to E given in Appendix E.1

   *  h_eff: 1

   secp256k1_XMD:SHA-256_SSWU_NU_ is identical to secp256k1_XMD:SHA-
   256_SSWU_RO_, except that the encoding type is encode_to_curve
   (Section 3).

   An optimized example implementation of the Simplified SWU mapping to
   the curve E' isogenous to secp256k1 is given in Appendix F.2.

8.8.  Suites for BLS12-381

   This section defines ciphersuites for groups G1 and G2 of the
   BLS12-381 elliptic curve [BLS12-381].  The curve parameters in this
   section match the ones listed in
   [I-D.irtf-cfrg-pairing-friendly-curves], Appendix C.

8.8.1.  BLS12-381 G1

   BLS12381G1_XMD:SHA-256_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + 4

   *  p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f
      6241eabfffeb153ffffb9feffffffffaaab

   *  m: 1

   *  k: 128

   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-256




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 43]

Internet-Draft                hash-to-curve                    June 2022


   *  L: 64

   *  f: Simplified SWU for AB == 0 (Section 6.6.3)

   *  Z: 11

   *  E': y'^2 = x'^3 + A' * x' + B', where

      -  A' = 0x144698a3b8e9433d693a02c96d4982b0ea985383ee66a8d8e8981aef
         d881ac98936f8da0e0f97f5cf428082d584c1d

      -  B' = 0x12e2908d11688030018b12e8753eee3b2016c1f0f24f4070a0b9c14f
         cef35ef55a23215a316ceaa5d1cc48e98e172be0

   *  iso_map: the 11-isogeny map from E' to E given in Appendix E.2

   *  h_eff: 0xd201000000010001

   BLS12381G1_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G1_XMD:SHA-
   256_SSWU_RO_, except that the encoding type is encode_to_curve
   (Section 3).

   Note that the h_eff values for these suites are chosen for
   compatibility with the fast cofactor clearing method described by
   Scott ([WB19] Section 5).

   An optimized example implementation of the Simplified SWU mapping to
   the curve E' isogenous to BLS12-381 G1 is given in Appendix F.2.

8.8.2.  BLS12-381 G2

   BLS12381G2_XMD:SHA-256_SSWU_RO_ is defined as follows:

   *  encoding type: hash_to_curve (Section 3)

   *  E: y^2 = x^3 + 4 * (1 + I)

   *  base field F is GF(p^m), where

      -  p: 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6
         b0f6241eabfffeb153ffffb9feffffffffaaab

      -  m: 2

      -  (1, I) is the basis for F, where I^2 + 1 == 0 in F

   *  k: 128




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 44]

Internet-Draft                hash-to-curve                    June 2022


   *  expand_message: expand_message_xmd (Section 5.3.1)

   *  H: SHA-256

   *  L: 64

   *  f: Simplified SWU for AB == 0 (Section 6.6.3)

   *  Z: -(2 + I)

   *  E': y'^2 = x'^3 + A' * x' + B', where

      -  A' = 240 * I

      -  B' = 1012 * (1 + I)

   *  iso_map: the isogeny map from E' to E given in Appendix E.3

   *  h_eff: 0xbc69f08f2ee75b3584c6a0ea91b352888e2a8e9145ad7689986ff0315
      08ffe1329c2f178731db956d82bf015d1212b02ec0ec69d7477c1ae954cbc06689
      f6a359894c0adebbf6b4e8020005aaa95551

   BLS12381G2_XMD:SHA-256_SSWU_NU_ is identical to BLS12381G2_XMD:SHA-
   256_SSWU_RO_, except that the encoding type is encode_to_curve
   (Section 3).

   Note that the h_eff values for these suites are chosen for
   compatibility with the fast cofactor clearing method described by
   Budroni and Pintore ([BP17], Section 4.1), and summarized in
   Appendix G.3.

   An optimized example implementation of the Simplified SWU mapping to
   the curve E' isogenous to BLS12-381 G2 is given in Appendix F.2.

8.9.  Defining a new hash-to-curve suite

   For elliptic curves not listed elsewhere in Section 8, a new hash-to-
   curve suite can be defined by:

   1.  E, F, p, and m are determined by the elliptic curve and its base
       field.

   2.  k is an upper bound on the target security level of the suite
       (Section 10.8).  A reasonable choice of k is ceil(log2(r) / 2),
       where r is the order of the subgroup G of the curve E
       (Section 2.1).





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 45]

Internet-Draft                hash-to-curve                    June 2022


   3.  Choose encoding type, either hash_to_curve or encode_to_curve
       (Section 3).

   4.  Compute L as described in Section 5.

   5.  Choose an expand_message variant from Section 5.3 plus any
       underlying cryptographic primitives (e.g., a hash function H).

   6.  Choose a mapping following the guidelines in Section 6.1, and
       select any required parameters for that mapping.

   7.  Choose h_eff to be either the cofactor of E or, if a fast
       cofactor clearing method is to be used, a value appropriate to
       that method as discussed in Section 7.

   8.  Construct a Suite ID following the guidelines in Section 8.10.

8.10.  Suite ID naming conventions

   Suite IDs MUST be constructed as follows:

   CURVE_ID || "_" || HASH_ID || "_" || MAP_ID || "_" || ENC_VAR || "_"

   The fields CURVE_ID, HASH_ID, MAP_ID, and ENC_VAR are ASCII-encoded
   strings of at most 64 characters each.  Fields MUST contain only
   ASCII characters between 0x21 and 0x7E (inclusive) except that
   underscore (i.e., 0x5f) is not allowed.

   As indicated above, each field (including the last) is followed by an
   underscore ("_", ASCII 0x5f).  This helps to ensure that Suite IDs
   are prefix free.  Suite IDs MUST include the final underscore and
   MUST NOT include any characters after the final underscore.

   Suite ID fields MUST be chosen as follows:

   *  CURVE_ID: a human-readable representation of the target elliptic
      curve.

   *  HASH_ID: a human-readable representation of the expand_message
      function and any underlying hash primitives used in hash_to_field
      (Section 5).  This field MUST be constructed as follows:

        EXP_TAG || ":" || HASH_NAME

      EXP_TAG indicates the expand_message variant:

      -  "XMD" for expand_message_xmd (Section 5.3.1).




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 46]

Internet-Draft                hash-to-curve                    June 2022


      -  "XOF" for expand_message_xof (Section 5.3.2).

      HASH_NAME is a human-readable name for the underlying hash
      primitive.  As examples:

      1.  For expand_message_xof (Section 5.3.2) with SHAKE128, HASH_ID
          is "XOF:SHAKE128".

      2.  For expand_message_xmd (Section 5.3.1) with SHA3-256, HASH_ID
          is "XMD:SHA3-256".

      Suites that use an alternative hash_to_field function that meets
      the requirements in Section 5.1 MUST indicate this by appending a
      tag identifying that function to the HASH_ID field, separated by a
      colon (":", ASCII 0x3A).

   *  MAP_ID: a human-readable representation of the map_to_curve
      function as defined in Section 6.  These are defined as follows:

      -  "SVDW" for or Shallue and van de Woestijne (Section 6.6.1).

      -  "SSWU" for Simplified SWU (Section 6.6.2, Section 6.6.3).

      -  "ELL2" for Elligator 2 (Section 6.7.1, Section 6.8.2).

   *  ENC_VAR: a string indicating the encoding type and other
      information.  The first two characters of this string indicate
      whether the suite represents a hash_to_curve or an encode_to_curve
      operation (Section 3), as follows:

      -  If ENC_VAR begins with "RO", the suite uses hash_to_curve.

      -  If ENC_VAR begins with "NU", the suite uses encode_to_curve.

      -  ENC_VAR MUST NOT begin with any other string.

      ENC_VAR MAY also be used to encode other information used to
      identify variants, for example, a version number.  The RECOMMENDED
      way to do so is to add one or more subfields separated by colons.
      For example, "RO:V02" is an appropriate ENC_VAR value for the
      second version of a uniform encoding suite, while
      "RO:V02:FOO01:BAR17" might be used to indicate a variant of that
      suite.

9.  IANA considerations

   This document has no IANA actions.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 47]

Internet-Draft                hash-to-curve                    June 2022


10.  Security considerations

   This section contains additional security considerations about the
   hash-to-curve mechanisms described in this document.

10.1.  Properties of encodings

   Each encoding type (Section 3) accepts an arbitrary byte string and
   maps it to a point on the curve sampled from a distribution that
   depends on the encoding type.  It is important to note that using a
   nonuniform encoding or directly evaluating one of the mappings of
   Section 6 produces an output that is easily distinguished from a
   uniformly random point.  Applications that use a nonuniform encoding
   SHOULD carefully analyze the security implications of nonuniformity.
   When the required encoding is not clear, applications SHOULD use a
   uniform encoding.

   Both encodings given in Section 3 can output the identity element of
   the group G.  The probability that either encoding function outputs
   the identity element is roughly 1/r for a random input, which is
   negligible for cryptographically useful elliptic curves.  Further, it
   is computationally infeasible to find an input to either encoding
   function whose corresponding output is the identity element.  (Both
   of these properties hold when the encoding functions are instantiated
   with a hash_to_field function that follows all guidelines in
   Section 5.)  Protocols that use these encoding functions SHOULD NOT
   add a special case to detect and "fix" the identity element.

   When the hash_to_curve function (Section 3) is instantiated with a
   hash_to_field function that is indifferentiable from a random oracle
   (Section 5), the resulting function is indifferentiable from a random
   oracle ([MRH04], [BCIMRT10], [FFSTV13], [LBB19], [H20]).  In many
   cases such a function can be safely used in cryptographic protocols
   whose security analysis assumes a random oracle that outputs
   uniformly random points on an elliptic curve.  As Ristenpart et al.
   discuss in [RSS11], however, not all security proofs that rely on
   random oracles continue to hold when those oracles are replaced by
   indifferentiable functionalities.  This limitation should be
   considered when analyzing the security of protocols relying on the
   hash_to_curve function.











Faz-Hernandez, et al.   Expires 17 December 2022               [Page 48]

Internet-Draft                hash-to-curve                    June 2022


10.2.  Hashing passwords

   When hashing passwords using any function described in this document,
   an adversary who learns the output of the hash function (or
   potentially any intermediate value, e.g., the output of
   hash_to_field) may be able to carry out a dictionary attack.  To
   mitigate such attacks, it is recommended to first execute a more
   costly key derivation function (e.g., PBKDF2 [RFC2898], scrypt
   [RFC7914], or Argon2 [I-D.irtf-cfrg-argon2]) on the password, then
   hash the output of that function to the target elliptic curve.  For
   collision resistance, the hash underlying the key derivation function
   should be chosen according to the guidelines listed in Section 5.3.1.

10.3.  Constant-time requirements

   Constant-time implementations of all functions in this document are
   STRONGLY RECOMMENDED for all uses, to avoid leaking information via
   side channels.  It is especially important to use a constant-time
   implementation when inputs to an encoding are secret values; in such
   cases, constant-time implementations are REQUIRED for security
   against timing attacks (e.g., [VR20]).  When constant-time
   implementations are required, all basic operations and utility
   functions must be implemented in constant time, as discussed in
   Section 4.  In some applications (e.g., embedded systems), leakage
   through other side channels (e.g., power or electromagnetic side
   channels) may be pertinent.  Defending against such leakage is
   outside the scope of this document, because the nature of the leakage
   and the appropriate defense depend on the application.

10.4.  encode_to_curve: output distribution and indifferentiability

   The encode_to_curve function (Section 3) returns points sampled from
   a distribution that is statistically far from uniform.  This
   distribution is bounded roughly as follows: first, it includes at
   least one eighth of the points in G, and second, the probability of
   points in the distribution varies by at most a factor of four.  These
   bounds hold when encode_to_curve is instantiated with any of the
   map_to_curve functions in Section 6.

   The bounds above are derived from several works in the literature.
   Specifically:

   *  Shallue and van de Woestijne [SW06] and Fouque and Tibouchi [FT12]
      derive bounds on the Shallue-van de Woestijne mapping
      (Section 6.6.1).

   *  Fouque and Tibouchi [FT10] and Tibouchi [T14] derive bounds for
      the Simplified SWU mapping (Section 6.6.2, Section 6.6.3).



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 49]

Internet-Draft                hash-to-curve                    June 2022


   *  Bernstein et al.  [BHKL13] derive bounds for the Elligator 2
      mapping (Section 6.7.1, Section 6.8.2).

   Indifferentiability of encode_to_curve follows from an argument
   similar to the one given by Brier et al.  [BCIMRT10]; we briefly
   sketch.  Consider an ideal random oracle Hc() that samples from the
   distribution induced by the map_to_curve function called by
   encode_to_curve, and assume for simplicity that the target elliptic
   curve has cofactor 1 (a similar argument applies for non-unity
   cofactors).  Indifferentiability holds just if it is possible to
   efficiently simulate the "inner" random oracle in encode_to_curve,
   namely, hash_to_field.  The simulator works as follows: on a fresh
   query msg, the simulator queries Hc(msg) and receives a point P in
   the image of map_to_curve (if msg is the same as a prior query, the
   simulator just returns the value it gave in response to that query).
   The simulator then computes the possible preimages of P under
   map_to_curve, i.e., elements u of F such that map_to_curve(u) == P
   (Tibouchi [T14] shows that this can be done efficiently for the
   Shallue-van de Woestijne and Simplified SWU maps, and Bernstein et
   al. show the same for Elligator 2).  The simulator selects one such
   preimage at random and returns this value as the simulated output of
   the "inner" random oracle.  By hypothesis, Hc() samples from the
   distribution induced by map_to_curve on a uniformly random input
   element of F, so this value is uniformly random and induces the
   correct point P when passed through map_to_curve.

10.5.  hash_to_field security

   The hash_to_field function defined in Section 5 is indifferentiable
   from a random oracle [MRH04] when expand_message (Section 5.3) is
   modeled as a random oracle.  By composability of indifferentiability
   proofs, this also holds when expand_message is proved
   indifferentiable from a random oracle relative to an underlying
   primitive that is modeled as a random oracle.  When following the
   guidelines in Section 5.3, both variants of expand_message defined in
   that section meet this requirement (see also Section 10.6).

   We very briefly sketch the indifferentiability argument for
   hash_to_field.  Notice that each integer mod p that hash_to_field
   returns (i.e., each element of the vector representation of F) is a
   member of an equivalence class of roughly 2^k integers of length
   log2(p) + k bits, all of which are equal modulo p.  For each integer
   mod p that hash_to_field returns, the simulator samples one member of
   this equivalence class at random and outputs the byte string returned
   by I2OSP.  (Notice that this is essentially the inverse of the
   hash_to_field procedure.)





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 50]

Internet-Draft                hash-to-curve                    June 2022


10.6.  expand_message_xmd security

   The expand_message_xmd function defined in Section 5.3.1 is
   indifferentiable from a random oracle [MRH04] when one of the
   following holds:

   1.  H is indifferentiable from a random oracle,

   2.  H is a sponge-based hash function whose inner function is modeled
       as a random transformation or random permutation [BDPV08], or

   3.  H is a Merkle-Damgaard hash function whose compression function
       is modeled as a random oracle [CDMP05].

   For cases (1) and (2), the indifferentiability of expand_message_xmd
   follows directly from the indifferentiability of H.

   For case (3), i.e., for H a Merkle-Damgaard hash function,
   indifferentiability follows from [CDMP05], Theorem 3.5.  In
   particular, expand_message_xmd computes b_0 by prefixing the message
   with one block of 0-bytes plus auxiliary information (length,
   counter, and DST).  Then, each of the output blocks b_i, i >= 1 in
   expand_message_xmd is the result of invoking H on a unique, prefix-
   free encoding of b_0.  This is true, first, because the length of the
   input to all such invocations is equal and fixed by the choice of H
   and DST, and second, because each such input has a unique suffix
   (because of the inclusion of the counter byte I2OSP(i, 1)).

   The essential difference between the construction of [CDMP05] and
   expand_message_xmd is that the latter hashes a counter appended to
   strxor(b_0, b_(i - 1)) (step 10) rather than to b_0.  This approach
   increases the Hamming distance between inputs to different
   invocations of H, which reduces the likelihood that nonidealities in
   H affect the distribution of the b_i values.

   We note that expand_message_xmd can be used to instantiate a general-
   purpose indifferentiable functionality with variable-length output
   based on any hash function meeting one of the above criteria.
   Applications that use expand_message_xmd outside of hash_to_field
   should ensure domain separation by picking a distinct value for DST.

10.7.  Domain separation for expand_message variants

   As discussed in Section 2.2.5, the purpose of domain separation is to
   ensure that security analyses of cryptographic protocols that query
   multiple independent random oracles remain valid even if all of these
   random oracles are instantiated based on one underlying function H.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 51]

Internet-Draft                hash-to-curve                    June 2022


   The expand_message variants in this document (Section 5.3) ensure
   domain separation by appending a suffix-free-encoded domain
   separation tag DST_prime to all strings hashed by H, an underlying
   hash or extendable-output function.  (Other expand_message variants
   that follow the guidelines in Section 5.3.4 are expected to behave
   similarly, but these should be analyzed on a case-by-case basis.)
   For security, applications that use the same function H outside of
   expand_message should enforce domain separation between those uses of
   H and expand_message, and should separate all of these from uses of H
   in other applications.

   This section suggests four methods for enforcing domain separation
   from expand_message variants, explains how each method achieves
   domain separation, and lists the situations in which each is
   appropriate.  These methods share a high-level structure: the
   application designer fixes a tag DST_ext distinct from DST_prime and
   augments calls to H with DST_ext.  Each method augments calls to H
   differently, and each may impose additional requirements on DST_ext.

   These methods can be used to instantiate multiple domain separated
   functions (e.g., H1 and H2) by selecting distinct DST_ext values for
   each (e.g., DST_ext1, DST_ext2).

   1.  (Suffix-only domain separation.)  This method is useful when
       domain separating invocations of H from expand_message_xmd or
       expand_message_xof.  It is not appropriate for domain separating
       expand_message from HMAC-H [RFC2104]; for that purpose, see
       method 4.

       To instantiate a suffix-only domain separated function Hso,
       compute

      Hso(msg) = H(msg || DST_ext)

       DST_ext should be suffix-free encoded (e.g., by appending one
       byte encoding the length of DST_ext) to make it infeasible to
       find distinct (msg, DST_ext) pairs that hash to the same value.

       This method ensures domain separation because all distinct
       invocations of H have distinct suffixes, since DST_ext is
       distinct from DST_prime.

   2.  (Prefix-suffix domain separation.)  This method can be used in
       the same cases as the suffix-only method.

       To instantiate a prefix-suffix domain separated function Hps,
       compute




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 52]

Internet-Draft                hash-to-curve                    June 2022


      Hps(msg) = H(DST_ext || msg || I2OSP(0, 1))

       DST_ext should be prefix-free encoded (e.g., by adding a one-byte
       prefix that encodes the length of DST_ext) to make it infeasible
       to find distinct (msg, DST_ext) pairs that hash to the same
       value.

       This method ensures domain separation because appending the byte
       I2OSP(0, 1) ensures that inputs to H inside Hps are distinct from
       those inside expand_message.  Specifically, the final byte of
       DST_prime encodes the length of DST, which is required to be
       nonzero (Section 3.1, requirement 2), and DST_prime is always
       appended to invocations of H inside expand_message.

   3.  (Prefix-only domain separation.)  This method is only useful for
       domain separating invocations of H from expand_message_xmd.  It
       does not give domain separation for expand_message_xof or HMAC-H.

       To instantiate a prefix-only domain separated function Hpo,
       compute

      Hpo(msg) = H(DST_ext || msg)

       In order for this method to give domain separation, DST_ext
       should be at least b bits long, where b is the number of bits
       output by the hash function H.  In addition, at least one of the
       first b bits must be nonzero.  Finally, DST_ext should be prefix-
       free encoded (e.g., by adding a one-byte prefix that encodes the
       length of DST_ext) to make it infeasible to find distinct (msg,
       DST_ext) pairs that hash to the same value.

       This method ensures domain separation as follows.  First, since
       DST_ext contains at least one nonzero bit among its first b bits,
       it is guaranteed to be distinct from the value Z_pad
       (Section 5.3.1, step 4), which ensures that all inputs to H are
       distinct from the input used to generate b_0 in
       expand_message_xmd.  Second, since DST_ext is at least b bits
       long, it is almost certainly distinct from the values b_0 and
       strxor(b_0, b_(i - 1)), and therefore all inputs to H are
       distinct from the inputs used to generate b_i, i >= 1, with high
       probability.

   4.  (XMD-HMAC domain separation.)  This method is useful for domain
       separating invocations of H inside HMAC-H (i.e., HMAC [RFC2104]
       instantiated with hash function H) from expand_message_xmd.  It
       also applies to HKDF-H [RFC5869], as discussed below.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 53]

Internet-Draft                hash-to-curve                    June 2022


       Specifically, this method applies when HMAC-H is used with a non-
       secret key to instantiate a random oracle based on a hash
       function H (note that expand_message_xmd can also be used for
       this purpose; see Section 10.6).  When using HMAC-H with a high-
       entropy secret key, domain separation is not necessary; see
       discussion below.

       To choose a non-secret HMAC key DST_key that ensures domain
       separation from expand_message_xmd, compute

      DST_key_preimage = "DERIVE-HMAC-KEY-" || DST_ext || I2OSP(0, 1)
      DST_key = H(DST_key_preimage)

       Then, to instantiate the random oracle Hro using HMAC-H, compute

      Hro(msg) = HMAC-H(DST_key, msg)

       The trailing zero byte in DST_key_preimage ensures that this
       value is distinct from inputs to H inside expand_message_xmd
       (because all such inputs have suffix DST_prime, which cannot end
       with a zero byte as discussed above).  This ensures domain
       separation because, with overwhelming probability, all inputs to
       H inside of HMAC-H using key DST_key have prefixes that are
       distinct from the values Z_pad, b_0, and strxor(b_0, b_(i - 1))
       inside of expand_message_xmd.

       For uses of HMAC-H that instantiate a private random oracle by
       fixing a high-entropy secret key, domain separation from
       expand_message_xmd is not necessary.  This is because, similarly
       to the case above, all inputs to H inside HMAC-H using this
       secret key almost certainly have distinct prefixes from all
       inputs to H inside expand_message_xmd.

       Finally, this method can be used with HKDF-H [RFC5869] by fixing
       the salt input to HKDF-Extract to DST_key, computed as above.
       This ensures domain separation for HKDF-Extract by the same
       argument as for HMAC-H using DST_key.  Moreover, assuming that
       the IKM input to HKDF-Extract has sufficiently high entropy (say,
       commensurate with the security parameter), the HKDF-Expand step
       is domain separated by the same argument as for HMAC-H with a
       high-entropy secret key (since PRK is exactly that).










Faz-Hernandez, et al.   Expires 17 December 2022               [Page 54]

Internet-Draft                hash-to-curve                    June 2022


10.8.  Target security levels

   Each ciphersuite specifies a target security level (in bits) for the
   underlying curve.  This parameter ensures the corresponding
   hash_to_field instantiation is conservative and correct.  We stress
   that this parameter is only an upper bound on the security level of
   the curve, and is neither a guarantee nor endorsement of its
   suitability for a given application.  Mathematical and cryptographic
   advancements may reduce the effective security level for any curve.

11.  Acknowledgements

   The authors would like to thank Adam Langley for his detailed writeup
   of Elligator 2 with Curve25519 [L13]; Dan Boneh, Christopher Patton,
   Benjamin Lipp, and Leonid Reyzin for educational discussions; and
   David Benjamin, Daniel Bourdrez, Frank Denis, Sean Devlin, Justin
   Drake, Bjoern Haase, Mike Hamburg, Dan Harkins, Daira Hopwood, Thomas
   Icart, Andy Polyakov, Thomas Pornin, Mamy Ratsimbazafy, Michael
   Scott, Filippo Valsorda, and Mathy Vanhoef for helpful reviews and
   feedback.

12.  Contributors

   *  Sharon Goldberg, Boston University (goldbe@cs.bu.edu)

   *  Ela Lee, Royal Holloway, University of London
      (Ela.Lee.2010@live.rhul.ac.uk)

   *  Michele Orru (michele.orru@ens.fr)

13.  References

13.1.  Normative References

   [EID4730]  Langley, A., "RFC 7748, Errata ID 4730", July 2016,
              <https://www.rfc-editor.org/errata/eid4730>.

   [I-D.irtf-cfrg-pairing-friendly-curves]
              Sakemi, Y., Kobayashi, T., Saito, T., and R. S. Wahby,
              "Pairing-Friendly Curves", Work in Progress, Internet-
              Draft, draft-irtf-cfrg-pairing-friendly-curves-10, 30 July
              2021, <https://datatracker.ietf.org/doc/html/draft-irtf-
              cfrg-pairing-friendly-curves-10>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://doi.org/10.17487/RFC2119>.



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 55]

Internet-Draft                hash-to-curve                    June 2022


   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://doi.org/10.17487/RFC7748>.

   [RFC8017]  Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
              "PKCS #1: RSA Cryptography Specifications Version 2.2",
              RFC 8017, DOI 10.17487/RFC8017, November 2016,
              <https://doi.org/10.17487/RFC8017>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://doi.org/10.17487/RFC8174>.

13.2.  Informative References

   [AFQTZ14]  Aranha, D.F., Fouque, P.A., Qian, C., Tibouchi, M., and
              J.C. Zapalowicz, "Binary Elligator squared",
              DOI 10.1007/978-3-319-13051-4_2, pages 20-37, In Selected
              Areas in Cryptography - SAC 2014, 2014,
              <https://doi.org/10.1007/978-3-319-13051-4_2>.

   [AR13]     Adj, G. and F. Rodriguez-Henriquez, "Square Root
              Computation over Even Extension Fields",
              DOI 10.1109/TC.2013.145, pages 2829-2841, In IEEE
              Transactions on Computers. vol 63 issue 11, November 2014,
              <https://doi.org/10.1109/TC.2013.145>.

   [BBJLP08]  Bernstein, D.J., Birkner, P., Joye, M., Lange, T., and C.
              Peters, "Twisted Edwards curves",
              DOI 10.1007/978-3-540-68164-9_26, pages 389-405,
              In AFRICACRYPT 2008, 2008,
              <https://doi.org/10.1007/978-3-540-68164-9_26>.

   [BCIMRT10] Brier, E., Coron, J-S., Icart, T., Madore, D., Randriam,
              H., and M. Tibouchi, "Efficient Indifferentiable Hashing
              into Ordinary Elliptic Curves",
              DOI 10.1007/978-3-642-14623-7_13, pages 237-254,
              In Advances in Cryptology - CRYPTO 2010, 2010,
              <https://doi.org/10.1007/978-3-642-14623-7_13>.

   [BDPV08]   Bertoni,, G., Daemen, J., Peeters, M., and G. Van Assche,
              "On the Indifferentiability of the Sponge Construction",
              DOI 10.1007/978-3-540-78967-3_11, pages 181-197,
              In Advances in Cryptology - EUROCRYPT 2008, 2008,
              <https://doi.org/10.1007/978-3-540-78967-3_11>.






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 56]

Internet-Draft                hash-to-curve                    June 2022


   [BF01]     Boneh, D. and M. Franklin, "Identity-based encryption from
              the Weil pairing", DOI 10.1007/3-540-44647-8_13,
              pages 213-229, In Advances in Cryptology - CRYPTO 2001,
              August 2001, <https://doi.org/10.1007/3-540-44647-8_13>.

   [BHKL13]   Bernstein, D.J., Hamburg, M., Krasnova, A., and T. Lange,
              "Elligator: elliptic-curve points indistinguishable from
              uniform random strings", DOI 10.1145/2508859.2516734,
              pages 967-980, In Proceedings of the 2013 ACM SIGSAC
              Conference on Computer and Communications Security,
              November 2013, <https://doi.org/10.1145/2508859.2516734>.

   [BLAKE2X]  Aumasson, J-P., Neves, S., Wilcox-O'Hearn, Z., and C.
              Winnerlein, "BLAKE2X", December 2016,
              <https://blake2.net/blake2x.pdf>.

   [BLMP19]   Bernstein, D.J., Lange, T., Martindale, C., and L. Panny,
              "Quantum circuits for the CSIDH: optimizing quantum
              evaluation of isogenies", DOI 10.1007/978-3-030-17656-3,
              In Advances in Cryptology - EUROCRYPT 2019, 2019,
              <https://doi.org/10.1007/978-3-030-17656-3>.

   [BLS01]    Boneh, D., Lynn, B., and H. Shacham, "Short signatures
              from the Weil pairing", DOI 10.1007/s00145-004-0314-9,
              pages 297-319, In Journal of Cryptology, vol 17, July
              2004, <https://doi.org/10.1007/s00145-004-0314-9>.

   [BLS03]    Barreto, P., Lynn, B., and M. Scott, "Constructing
              Elliptic Curves with Prescribed Embedding Degrees",
              DOI 10.1007/3-540-36413-7_19, pages 257-267, In Security
              in Communication Networks, 2003,
              <https://doi.org/10.1007/3-540-36413-7_19>.

   [BLS12-381]
              Bowe, S., "BLS12-381: New zk-SNARK Elliptic Curve
              Construction", March 2017,
              <https://electriccoin.co/blog/new-snark-curve/>.

   [BM92]     Bellovin, S.M. and M. Merritt, "Encrypted key exchange:
              Password-based protocols secure against dictionary
              attacks", DOI 10.1109/RISP.1992.213269, pages 72-84,
              In IEEE Symposium on Security and Privacy - Oakland 1992,
              1992, <https://doi.org/10.1109/RISP.1992.213269>.








Faz-Hernandez, et al.   Expires 17 December 2022               [Page 57]

Internet-Draft                hash-to-curve                    June 2022


   [BMP00]    Boyko, V., MacKenzie, P.D., and S. Patel, "Provably secure
              password-authenticated key exchange using Diffie-Hellman",
              DOI 10.1007/3-540-45539-6_12, pages 156-171, In Advances
              in Cryptology - EUROCRYPT 2000, May 2000,
              <https://doi.org/10.1007/3-540-45539-6_12>.

   [BN05]     Barreto, P. and M. Naehrig, "Pairing-Friendly Elliptic
              Curves of Prime Order", DOI 10.1007/11693383_22,
              pages 319-331, In Selected Areas in Cryptography 2005,
              2006, <https://doi.org/10.1007/11693383_22>.

   [BP17]     Budroni, A. and F. Pintore, "Efficient hash maps to G2 on
              BLS curves", ePrint 2017/419, May 2017,
              <https://eprint.iacr.org/2017/419>.

   [BR93]     Bellare, M. and P. Rogaway, "Random oracles are practical:
              a paradigm for designing efficient protocols",
              DOI 10.1145/168588.168596, pages 62-73, In Proceedings of
              the 1993 ACM Conference on Computer and Communications
              Security, December 1993,
              <https://doi.org/10.1145/168588.168596>.

   [C93]      Cohen, H., "A Course in Computational Algebraic Number
              Theory", ISBN 9783642081422, publisher Springer-Verlag,
              1993, <https://doi.org/10.1007/978-3-662-02945-9>.

   [CDMP05]   Coron, J-S., Dodis, Y., Malinaud, C., and P. Puniya,
              "Merkle-Damgaard Revisited: How to Construct a Hash
              Function", DOI 10.1007/11535218_26, pages 430-448,
              In Advances in Cryptology - CRYPTO 2005, 2005,
              <https://doi.org/10.1007/11535218_26>.

   [CFADLNV05]
              Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T.,
              Nguyen, K., and F. Vercauteren, "Handbook of Elliptic and
              Hyperelliptic Curve Cryptography", ISBN 9781584885184,
              publisher Chapman and Hall / CRC, 2005,
              <https://www.crcpress.com/9781584885184>.

   [CK11]     Couveignes, J. and J. Kammerer, "The geometry of flex
              tangents to a cubic curve and its parameterizations",
              DOI 10.1016/j.jsc.2011.11.003, pages 266-281, In Journal
              of Symbolic Computation, vol 47 issue 3, 2012,
              <https://doi.org/10.1016/j.jsc.2011.11.003>.







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 58]

Internet-Draft                hash-to-curve                    June 2022


   [F11]      Farashahi, R.R., "Hashing into Hessian curves",
              DOI 10.1007/978-3-642-21969-6_17, pages 278-289,
              In AFRICACRYPT 2011, 2011,
              <https://doi.org/10.1007/978-3-642-21969-6_17>.

   [FFSTV13]  Farashahi, R.R., Fouque, P.A., Shparlinski, I.E.,
              Tibouchi, M., and J.F. Voloch, "Indifferentiable
              deterministic hashing to elliptic and hyperelliptic
              curves", DOI 10.1090/S0025-5718-2012-02606-8,
              pages 491-512, In Math. Comp. vol 82, 2013,
              <https://doi.org/10.1090/S0025-5718-2012-02606-8>.

   [FIPS180-4]
              National Institute of Standards and Technology (NIST),
              "Secure Hash Standard (SHS)", August 2015,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.180-4.pdf>.

   [FIPS186-4]
              National Institute of Standards and Technology (NIST),
              "FIPS Publication 186-4: Digital Signature Standard", July
              2013, <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.186-4.pdf>.

   [FIPS202]  National Institute of Standards and Technology (NIST),
              "SHA-3 Standard: Permutation-Based Hash and Extendable-
              Output Functions", August 2015,
              <https://nvlpubs.nist.gov/nistpubs/FIPS/
              NIST.FIPS.202.pdf>.

   [FJT13]    Fouque, P-A., Joux, A., and M. Tibouchi, "Injective
              encodings to elliptic curves",
              DOI 10.1007/978-3-642-39059-3_14, pages 203-218, In ACISP
              2013, 2013,
              <https://doi.org/10.1007/978-3-642-39059-3_14>.

   [FKR11]    Fuentes-Castaneda, L., Knapp, E., and F. Rodriguez-
              Henriquez, "Fast Hashing to G2 on Pairing-Friendly
              Curves", DOI 10.1007/978-3-642-28496-0_25, pages 412-430,
              In Selected Areas in Cryptography, 2011,
              <https://doi.org/10.1007/978-3-642-28496-0_25>.

   [FSV09]    Farashahi, R.R., Shparlinski, I.E., and J.F. Voloch, "On
              hashing into elliptic curves", DOI 10.1515/JMC.2009.022,
              pages 353-360, In Journal of Mathematical Cryptology, vol
              3 no 4, 2009, <https://doi.org/10.1515/JMC.2009.022>.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 59]

Internet-Draft                hash-to-curve                    June 2022


   [FT10]     Fouque, P-A. and M. Tibouchi, "Estimating the size of the
              image of deterministic hash functions to elliptic
              curves.", DOI 10.1007/978-3-642-14712-8_5, pages 81-91,
              In Progress in Cryptology - LATINCRYPT 2010, 2010,
              <https://doi.org/10.1007/978-3-642-14712-8_5>.

   [FT12]     Fouque, P-A. and M. Tibouchi, "Indifferentiable Hashing to
              Barreto-Naehrig Curves", DOI 10.1007/978-3-642-33481-8_1,
              pages 1-7, In Progress in Cryptology - LATINCRYPT 2012,
              2012, <https://doi.org/10.1007/978-3-642-33481-8_1>.

   [H20]      Hamburg, M., "Indifferentiable hashing from Elligator 2",
              2020, <https://eprint.iacr.org/2020/1513>.

   [hash2curve-repo]
              "Hashing to Elliptic Curves - GitHub repository", 2019,
              <https://github.com/cfrg/draft-irtf-cfrg-hash-to-curve>.

   [I-D.irtf-cfrg-argon2]
              Biryukov, A., Dinu, D., Khovratovich, D., and S.
              Josefsson, "Argon2 Memory-Hard Function for Password
              Hashing and Proof-of-Work Applications", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-argon2-13, 11 March 2021,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              argon2-13>.

   [I-D.irtf-cfrg-bls-signature]
              Boneh, D., Gorbunov, S., Wahby, R. S., Wee, H., and Z.
              Zhang, "BLS Signatures", Work in Progress, Internet-Draft,
              draft-irtf-cfrg-bls-signature-04, 10 September 2020,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              bls-signature-04>.

   [I-D.irtf-cfrg-ristretto255-decaf448]
              Valence, H. D., Grigg, J., Hamburg, M., Lovecruft, I.,
              Tankersley, G., and F. Valsorda, "The ristretto255 and
              decaf448 Groups", Work in Progress, Internet-Draft, draft-
              irtf-cfrg-ristretto255-decaf448-03, 25 February 2022,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              ristretto255-decaf448-03>.

   [I-D.irtf-cfrg-voprf]
              Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A.
              Wood, "Oblivious Pseudorandom Functions (OPRFs) using
              Prime-Order Groups", Work in Progress, Internet-Draft,
              draft-irtf-cfrg-voprf-09, 8 February 2022,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              voprf-09>.



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 60]

Internet-Draft                hash-to-curve                    June 2022


   [I-D.irtf-cfrg-vrf]
              Goldberg, S., Reyzin, L., Papadopoulos, D., and J. Vcelak,
              "Verifiable Random Functions (VRFs)", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-vrf-12, 26 May 2022,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              vrf-12>.

   [Icart09]  Icart, T., "How to Hash into Elliptic Curves",
              DOI 10.1007/978-3-642-03356-8_18, pages 303-316,
              In Advances in Cryptology - CRYPTO 2009, 2009,
              <https://doi.org/10.1007/978-3-642-03356-8_18>.

   [J96]      Jablon, D.P., "Strong password-only authenticated key
              exchange", DOI 10.1145/242896.242897, pages 5-26,
              In SIGCOMM Computer Communication Review, vol 26 issue 5,
              1996, <https://doi.org/10.1145/242896.242897>.

   [jubjub-fq]
              "zkcrypto/jubjub - fq.rs", 2019,
              <https://github.com/zkcrypto/jubjub/blob/master/src/
              fq.rs>.

   [KLR10]    Kammerer, J., Lercier, R., and G. Renault, "Encoding
              points on hyperelliptic curves over finite fields in
              deterministic polynomial time",
              DOI 10.1007/978-3-642-17455-1_18, pages 278-297,
              In PAIRING 2010, 2010,
              <https://doi.org/10.1007/978-3-642-17455-1_18>.

   [L13]      Langley, A., "Implementing Elligator for Curve25519",
              2013, <https://www.imperialviolet.org/2013/12/25/
              elligator.html>.

   [LBB19]    Lipp, B., Blanchet, B., and K. Bhargavan, "A Mechanised
              Proof of the WireGuard Virtual Private Network Protocol",
              In INRIA Research Report No. 9269, April 2019,
              <https://hal.inria.fr/hal-02100345/>.

   [MOV96]    Menezes, A.J., van Oorschot, P.C., and S.A. Vanstone,
              "Handbook of Applied Cryptography", ISBN 9780849385230,
              publisher CRC Press, 1996,
              <http://cacr.uwaterloo.ca/hac/>.









Faz-Hernandez, et al.   Expires 17 December 2022               [Page 61]

Internet-Draft                hash-to-curve                    June 2022


   [MRH04]    Maurer, U., Renner, R., and C. Holenstein,
              "Indifferentiability, impossibility results on reductions,
              and applications to the random oracle methodology",
              DOI 10.1007/978-3-540-24638-1_2, pages 21-39, In TCC 2004:
              Theory of Cryptography, February 2004,
              <https://doi.org/10.1007/978-3-540-24638-1_2>.

   [MRV99]    Micali, S., Rabin, M., and S. Vadhan, "Verifiable Random
              Functions", DOI 10.1109/SFFCS.1999.814584, In Symposium on
              the Foundations of Computer Science, October 1999,
              <https://doi.org/10.1109/SFFCS.1999.814584>.

   [MT98]     Matsumoto, M. and T. Nishimura, "Mersenne twister: A
              623-dimensionally equidistributed uniform pseudo-random
              number generator", DOI 10.1145/272991.272995, pages 3-30,
              In ACM Transactions on Modeling and Computer Simulation
              (TOMACS), Volume 8, Issue 1, January 1998,
              <https://doi.org/10.1145/272991.272995>.

   [NR97]     Naor, M. and O. Reingold, "Number-theoretic constructions
              of efficient pseudo-random functions",
              DOI 10.1109/SFCS.1997.646134, In Symposium on the
              Foundations of Computer Science, October 1997,
              <https://doi.org/10.1109/SFCS.1997.646134>.

   [p1363.2]  IEEE Computer Society, "IEEE Standard Specification for
              Password-Based Public-Key Cryptography Techniques",
              September 2008,
              <https://standards.ieee.org/standard/1363_2-2008.html>.

   [p1363a]   IEEE Computer Society, "IEEE Standard Specifications for
              Public-Key Cryptography---Amendment 1: Additional
              Techniques", March 2004,
              <https://standards.ieee.org/standard/1363a-2004.html>.

   [P20]      Pornin, T., "Efficient Elliptic Curve Operations On
              Microcontrollers With Finite Field Extensions", 2020,
              <https://eprint.iacr.org/2020/009>.

   [RCB16]    Renes, J., Costello, C., and L. Batina, "Complete addition
              formulas for prime order elliptic curves",
              DOI 10.1007/978-3-662-49890-3_16, pages 403-428,
              In Advances in Cryptology - EUROCRYPT 2016, May 2016,
              <https://doi.org/10.1007/978-3-662-49890-3_16>.







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 62]

Internet-Draft                hash-to-curve                    June 2022


   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,
              <https://doi.org/10.17487/RFC2104>.

   [RFC2898]  Kaliski, B., "PKCS #5: Password-Based Cryptography
              Specification Version 2.0", RFC 2898,
              DOI 10.17487/RFC2898, September 2000,
              <https://doi.org/10.17487/RFC2898>.

   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,
              <https://doi.org/10.17487/RFC5869>.

   [RFC7693]  Saarinen, M-J., Ed. and J-P. Aumasson, "The BLAKE2
              Cryptographic Hash and Message Authentication Code (MAC)",
              RFC 7693, DOI 10.17487/RFC7693, November 2015,
              <https://doi.org/10.17487/RFC7693>.

   [RFC7914]  Percival, C. and S. Josefsson, "The scrypt Password-Based
              Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
              August 2016, <https://doi.org/10.17487/RFC7914>.

   [RSS11]    Ristenpart, T., Shacham, H., and T. Shrimpton, "Careful
              with Composition: Limitations of the Indifferentiability
              Framework", DOI 10.1007/978-3-642-20465-4_27,
              pages 487-506, In Advances in Cryptology - EUROCRYPT 2011,
              May 2011, <https://doi.org/10.1007/978-3-642-20465-4_27>.

   [S05]      Skalba, M., "Points on elliptic curves over finite
              fields", DOI 10.4064/aa117-3-7, pages 293-301, In Acta
              Arithmetica, vol 117 no 3, 2005,
              <https://doi.org/10.4064/aa117-3-7>.

   [S85]      Schoof, R., "Elliptic Curves Over Finite Fields and the
              Computation of Square Roots mod p",
              DOI 10.1090/S0025-5718-1985-0777280-6, pages 483-494,
              In Mathematics of Computation vol 44 issue 170, April
              1985, <https://doi.org/10.1090/S0025-5718-1985-0777280-6>.

   [SAGE]     The Sage Developers, "SageMath, the Sage Mathematics
              Software System", 2019, <https://www.sagemath.org>.








Faz-Hernandez, et al.   Expires 17 December 2022               [Page 63]

Internet-Draft                hash-to-curve                    June 2022


   [SBCDK09]  Scott, M., Benger, N., Charlemagne, M., Dominguez Perez,
              L.J., and E.J. Kachisa, "Fast Hashing to G2 on Pairing-
              Friendly Curves", DOI 10.1007/978-3-642-03298-1_8,
              pages 102-113, In Pairing-Based Cryptography - Pairing
              2009, 2009, <https://doi.org/10.1007/978-3-642-03298-1_8>.

   [SEC1]     Standards for Efficient Cryptography Group (SECG), "SEC 1:
              Elliptic Curve Cryptography", May 2009,
              <http://www.secg.org/sec1-v2.pdf>.

   [SEC2]     Standards for Efficient Cryptography Group (SECG), "SEC 2:
              Recommended Elliptic Curve Domain Parameters", January
              2010, <http://www.secg.org/sec2-v2.pdf>.

   [SS04]     Schinzel, A. and M. Skalba, "On equations y^2 = x^n + k in
              a finite field.", DOI 10.4064/ba52-3-1, pages 223-226,
              In Bulletin Polish Acad. Sci. Math. vol 52, no 3, 2004,
              <https://doi.org/10.4064/ba52-3-1>.

   [SW06]     Shallue, A. and C. van de Woestijne, "Construction of
              rational points on elliptic curves over finite fields",
              DOI 10.1007/11792086_36, pages 510-524, In Algorithmic
              Number Theory. ANTS 2006., 2006,
              <https://doi.org/10.1007/11792086_36>.

   [T14]      Tibouchi, M., "Elligator squared: Uniform points on
              elliptic curves of prime order as uniform random strings",
              DOI 10.1007/978-3-662-45472-5_10, pages 139-156,
              In Financial Cryptography and Data Security - FC 2014,
              2014, <https://doi.org/10.1007/978-3-662-45472-5_10>.

   [TK17]     Tibouchi, M. and T. Kim, "Improved elliptic curve hashing
              and point representation", DOI 10.1007/s10623-016-0288-2,
              pages 161-177, In Designs, Codes, and Cryptography, vol
              82, 2017, <https://doi.org/10.1007/s10623-016-0288-2>.

   [U07]      Ulas, M., "Rational points on certain hyperelliptic curves
              over finite fields", DOI 10.4064/ba55-2-1, pages 97-104,
              In Bulletin Polish Acad. Sci. Math. vol 55, no 2, 2007,
              <https://doi.org/10.4064/ba55-2-1>.

   [VR20]     Vanhoef, M. and E. Ronen, "Dragonblood: Analyzing the
              Dragonfly Handshake of WPA3 and EAP-pwd", In IEEE
              Symposium on Security & Privacy (SP), 2020,
              <https://eprint.iacr.org/2019/383>.






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 64]

Internet-Draft                hash-to-curve                    June 2022


   [W08]      Washington, L.C., "Elliptic curves: Number theory and
              cryptography", ISBN 9781420071467, publisher Chapman and
              Hall / CRC, edition 2nd, 2008,
              <https://www.crcpress.com/9781420071467>.

   [W19]      Wahby, R.S., "An explicit, generic parameterization for
              the Shallue--van de Woestijne map", 2019,
              <https://github.com/cfrg/draft-irtf-cfrg-hash-to-
              curve/raw/master/doc/svdw_params.pdf>.

   [WB19]     Wahby, R.S. and D. Boneh, "Fast and simple constant-time
              hashing to the BLS12-381 elliptic curve",
              DOI 10.13154/tches.v2019.i4.154-179, ePrint 2019/403,
              issue 4, volume 2019, In IACR Trans. CHES, August 2019,
              <https://eprint.iacr.org/2019/403>.

Appendix A.  Related work

   The problem of mapping arbitrary bit strings to elliptic curve points
   has been the subject of both practical and theoretical research.
   This section briefly describes the background and research results
   that underly the recommendations in this document.  This section is
   provided for informational purposes only.

   A naive but generally insecure method of mapping a string msg to a
   point on an elliptic curve E having n points is to first fix a point
   P that generates the elliptic curve group, and a hash function Hn
   from bit strings to integers less than n; then compute Hn(msg) * P,
   where the * operator represents scalar multiplication.  The reason
   this approach is insecure is that the resulting point has a known
   discrete log relationship to P.  Thus, except in cases where this
   method is specified by the protocol, it must not be used; doing so
   risks catastrophic security failures.

   Boneh et al.  [BLS01] describe an encoding method they call
   MapToGroup, which works roughly as follows: first, use the input
   string to initialize a pseudorandom number generator, then use the
   generator to produce a value x in F.  If x is the x-coordinate of a
   point on the elliptic curve, output that point.  Otherwise, generate
   a new value x in F and try again.  Since a random value x in F has
   probability about 1/2 of corresponding to a point on the curve, the
   expected number of tries is just two.  However, the running time of
   this method, which is generally referred to as a probabilistic try-
   and-increment algorithm, depends on the input string.  As such, it is
   not safe to use in protocols sensitive to timing side channels, as
   was exemplified by the Dragonblood attack [VR20].





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 65]

Internet-Draft                hash-to-curve                    June 2022


   Schinzel and Skalba [SS04] introduce a method of constructing
   elliptic curve points deterministically, for a restricted class of
   curves and a very small number of points.  Skalba [S05] generalizes
   this construction to more curves and more points on those curves.
   Shallue and van de Woestijne [SW06] further generalize and simplify
   Skalba's construction, yielding concretely efficient maps to a
   constant fraction of the points on almost any curve.  Fouque and
   Tibouchi [FT12] give a parameterization of this mapping for Barreto-
   Naehrig pairing-friendly curves [BN05].

   Ulas [U07] describes a simpler version of the Shallue-van de
   Woestijne map, and Brier et al.  [BCIMRT10] give a further
   simplification, which the authors call the "simplified SWU" map.
   That simplified map applies only to fields of characteristic p = 3
   (mod 4); Wahby and Boneh [WB19] generalize to fields of any
   characteristic, and give further optimizations.

   Boneh and Franklin give a deterministic algorithm mapping to certain
   supersingular curves over fields of characteristic p = 2 (mod 3)
   [BF01].  Icart gives another deterministic algorithm which maps to
   any curve over a field of characteristic p = 2 (mod 3) [Icart09].
   Several extensions and generalizations follow this work, including
   [FSV09], [FT10], [KLR10], [F11], and [CK11].

   Following the work of Farashahi [F11], Fouque et al.  [FJT13]
   describe a mapping to curves over fields of characteristic p = 3 (mod
   4) having a number of points divisible by 4.  Bernstein et al.
   [BHKL13] optimize this mapping and describe a related mapping that
   they call "Elligator 2," which applies to any curve over a field of
   odd characteristic having a point of order 2.  This includes
   Curve25519 and Curve448, both of which are CFRG-recommended curves
   [RFC7748].  Bernstein et al.  [BLMP19] extend the Elligator 2 map to
   a class of supersingular curves over fields of characteristic p = 3
   (mod 4).

   An important caveat regarding all of the above deterministic mapping
   functions is that none of them map to the entire curve, but rather to
   some fraction of the points.  This means that they cannot be used
   directly to construct a random oracle that outputs points on the
   curve.

   Brier et al.  [BCIMRT10] give two solutions to this problem.  The
   first, which Brier et al. prove applies to Icart's method, computes
   f(H0(msg)) + f(H1(msg)) for two distinct hash functions H0 and H1
   from bit strings to F and a mapping f from F to the elliptic curve E.
   The second, which applies to essentially all deterministic mappings
   but is more costly, computes f(H0(msg)) + H2(msg) * P, for P a
   generator of the elliptic curve group and H2 a hash from bit strings



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 66]

Internet-Draft                hash-to-curve                    June 2022


   to integers modulo r, the order of the elliptic curve group.
   Farashahi et al.  [FFSTV13] improve the analysis of the first method,
   showing that it applies to essentially all deterministic mappings.
   Tibouchi and Kim [TK17] further refine the analysis and describe
   additional optimizations.

   Complementary to the problem of mapping from bit strings to elliptic
   curve points, Bernstein et al.  [BHKL13] study the problem of mapping
   from elliptic curve points to uniformly random bit strings, giving
   solutions for a class of curves including Montgomery and twisted
   Edwards curves.  Tibouchi [T14] and Aranha et al.  [AFQTZ14]
   generalize these results.  This document does not deal with this
   complementary problem.

Appendix B.  Hashing to ristretto255

   ristretto255 [I-D.irtf-cfrg-ristretto255-decaf448] provides a prime-
   order group based on Curve25519 [RFC7748].  This section describes
   hash_to_ristretto255, which implements a random-oracle encoding to
   this group that has a uniform output distribution (Section 2.2.3) and
   the same security properties and interface as the hash_to_curve
   function (Section 3).

   The ristretto255 API defines a one-way map
   ([I-D.irtf-cfrg-ristretto255-decaf448], Section 4.3.4); this section
   refers to that map as ristretto255_map.

   The hash_to_ristretto255 function MUST be instantiated with an
   expand_message function that conforms to the requirements given in
   Section 5.3.  In addition, it MUST use a domain separation tag
   constructed as described in Section 3.1, and all domain separation
   recommendations given in Section 10.7 apply when implementing
   protocols that use hash_to_ristretto255.


















Faz-Hernandez, et al.   Expires 17 December 2022               [Page 67]

Internet-Draft                hash-to-curve                    June 2022


   hash_to_ristretto255(msg)

   Parameters:
   - DST, a domain separation tag (see discussion above).
   - expand_message, a function that expands a byte string and
     domain separation tag into a uniformly random byte string
     (see discussion above).
   - ristretto255_map, the one-way map from the ristretto255 API.

   Input: msg, an arbitrary-length byte string.
   Output: P, an element of the ristretto255 group.

   Steps:
   1. uniform_bytes = expand_message(msg, DST, 64)
   2. P = ristretto255_map(uniform_bytes)
   3. return P

   Since hash_to_ristretto255 is not a hash-to-curve suite, it does not
   have a Suite ID.  If a similar identifier is needed, it MUST be
   constructed following the guidelines in Section 8.10, with the
   following parameters:

   *  CURVE_ID: "ristretto255"

   *  HASH_ID: as described in Section 8.10

   *  MAP_ID: "R255MAP"

   *  ENC_VAR: "RO"

   For example, if expand_message is expand_message_xmd using SHA-512,
   the REQUIRED identifier is:

   ristretto255_XMD:SHA-512_R255MAP_RO_

Appendix C.  Hashing to decaf448

   Similar to ristretto255, decaf448
   [I-D.irtf-cfrg-ristretto255-decaf448] provides a prime-order group
   based on Curve448 [RFC7748].  This section describes
   hash_to_decaf448, which implements a random-oracle encoding to this
   group that has a uniform output distribution (Section 2.2.3) and the
   same security properties and interface as the hash_to_curve function
   (Section 3).

   The decaf448 API defines a one-way map
   ([I-D.irtf-cfrg-ristretto255-decaf448], Section 5.3.4); this section
   refers to that map as decaf448_map.



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 68]

Internet-Draft                hash-to-curve                    June 2022


   The hash_to_decaf448 function MUST be instantiated with an
   expand_message function that conforms to the requirements given in
   Section 5.3.  In addition, it MUST use a domain separation tag
   constructed as described in Section 3.1, and all domain separation
   recommendations given in Section 10.7 apply when implementing
   protocols that use hash_to_decaf448.

   hash_to_decaf448(msg)

   Parameters:
   - DST, a domain separation tag (see discussion above).
   - expand_message, a function that expands a byte string and
     domain separation tag into a uniformly random byte string
     (see discussion above).
   - decaf448_map, the one-way map from the decaf448 API.

   Input: msg, an arbitrary-length byte string.
   Output: P, an element of the decaf448 group.

   Steps:
   1. uniform_bytes = expand_message(msg, DST, 112)
   2. P = decaf448_map(uniform_bytes)
   3. return P

   Since hash_to_decaf448 is not a hash-to-curve suite, it does not have
   a Suite ID.  If a similar identifier is needed, it MUST be
   constructed following the guidelines in Section 8.10, with the
   following parameters:

   *  CURVE_ID: "decaf448"

   *  HASH_ID: as described in Section 8.10

   *  MAP_ID: "D448MAP"

   *  ENC_VAR: "RO"

   For example, if expand_message is expand_message_xof using SHAKE256,
   the REQUIRED identifier is:

   decaf448_XOF:SHAKE256_D448MAP_RO_

Appendix D.  Rational maps

   This section gives rational maps that can be used when hashing to
   twisted Edwards or Montgomery curves.





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 69]

Internet-Draft                hash-to-curve                    June 2022


   Given a twisted Edwards curve, Appendix D.1 shows how to derive a
   corresponding Montgomery curve and how to map from that curve to the
   twisted Edwards curve.  This mapping may be used when hashing to
   twisted Edwards curves as described in Section 6.8.

   Given a Montgomery curve, Appendix D.2 shows how to derive a
   corresponding Weierstrass curve and how to map from that curve to the
   Montgomery curve.  This mapping can be used to hash to Montgomery or
   twisted Edwards curves via the Shallue-van de Woestijne
   (Section 6.6.1) or Simplified SWU (Section 6.6.2) method, as follows:

   *  For Montgomery curves, first map to the Weierstrass curve, then
      convert to Montgomery coordinates via the mapping.

   *  For twisted Edwards curves, compose the Weierstrass to Montgomery
      mapping with the Montgomery to twisted Edwards mapping
      (Appendix D.1) to obtain a Weierstrass curve and a mapping to the
      target twisted Edwards curve.  Map to this Weierstrass curve, then
      convert to Edwards coordinates via the mapping.

D.1.  Generic Montgomery to twisted Edwards map

   This section gives a generic birational map between twisted Edwards
   and Montgomery curves.

   The map in this section is a simplified version of the map given in
   [BBJLP08], Theorem 3.2.  Specifically, this section's map handles
   exceptional cases in a simplified way that is geared towards hashing
   to a twisted Edwards curve's prime-order subgroup.

   The twisted Edwards curve

       a * v^2 + w^2 = 1 + d * v^2 * w^2

   is birationally equivalent to the Montgomery curve

       K * t^2 = s^3 + J * s^2 + s

   which has the form required by the Elligator 2 mapping of
   Section 6.7.1.  The coefficients of the Montgomery curve are

   *  J = 2 * (a + d) / (a - d)

   *  K = 4 / (a - d)

   The rational map from the point (s, t) on the above Montgomery curve
   to the point (v, w) on the twisted Edwards curve is given by




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 70]

Internet-Draft                hash-to-curve                    June 2022


   *  v = s / t

   *  w = (s - 1) / (s + 1)

   This mapping is undefined when t == 0 or s == -1, i.e., when the
   denominator of either of the above rational functions is zero.
   Implementations MUST detect exceptional cases and return the value
   (v, w) = (0, 1), which is the identity point on all twisted Edwards
   curves.

   The following straight-line implementation of the above rational map
   handles the exceptional cases.

   monty_to_edw_generic(s, t)

   Input: (s, t), a point on the curve K * t^2 = s^3 + J * s^2 + s.
   Output: (v, w), a point on an equivalent twisted Edwards curve.

   1. tv1 = s + 1
   2. tv2 = tv1 * t        # (s + 1) * t
   3. tv2 = inv0(tv2)      # 1 / ((s + 1) * t)
   4.   v = tv2 * tv1      # 1 / t
   5.   v = v * s          # s / t
   6.   w = tv2 * t        # 1 / (s + 1)
   7. tv1 = s - 1
   8.   w = w * tv1        # (s - 1) / (s + 1)
   9.   e = tv2 == 0
   10.  w = CMOV(w, 1, e)  # handle exceptional case
   11. return (v, w)

   For completeness, we also give the inverse relations.  (Note that
   this map is not required when hashing to twisted Edwards curves.)
   The coefficients of the twisted Edwards curve corresponding to the
   above Montgomery curve are

   *  a = (J + 2) / K

   *  d = (J - 2) / K

   The rational map from the point (v, w) on the twisted Edwards curve
   to the point (s, t) on the Montgomery curve is given by

   *  s = (1 + w) / (1 - w)

   *  t = (1 + w) / (v * (1 - w))






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 71]

Internet-Draft                hash-to-curve                    June 2022


   The mapping is undefined when v == 0 or w == 1.  When the goal is to
   map into the prime-order subgroup of the Montgomery curve, it
   suffices to return the identity point on the Montgomery curve in the
   exceptional cases.

D.2.  Weierstrass to Montgomery map

   The rational map from the point (s, t) on the Montgomery curve

       K * t^2 = s^3 + J * s^2 + s

   to the point (x, y) on the equivalent Weierstrass curve

       y^2 = x^3 + A * x + B

   is given by:

   *  A = (3 - J^2) / (3 * K^2)

   *  B = (2 * J^3 - 9 * J) / (27 * K^3)

   *  x = (3 * s + J) / (3 * K)

   *  y = t / K

   The inverse map, from the point (x, y) to the point (s, t), is given
   by

   *  s = (3 * K * x - J) / 3

   *  t = y * K

   This mapping can be used to apply the Shallue-van de Woestijne
   (Section 6.6.1) or Simplified SWU (Section 6.6.2) method to
   Montgomery curves.

Appendix E.  Isogeny maps for suites

   This section specifies the isogeny maps for the secp256k1 and
   BLS12-381 suites listed in Section 8.

   These maps are given in terms of affine coordinates.  Wahby and Boneh
   ([WB19], Section 4.3) show how to evaluate these maps in a projective
   coordinate system (Appendix G.1), which avoids modular inversions.

   Refer to the draft repository [hash2curve-repo] for a Sage [SAGE]
   script that constructs these isogenies.




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 72]

Internet-Draft                hash-to-curve                    June 2022


E.1.  3-isogeny map for secp256k1

   This section specifies the isogeny map for the secp256k1 suite listed
   in Section 8.7.

   The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the
   following rational functions:

   *  x = x_num / x_den, where

      -  x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' +
         k_(1,0)

      -  x_den = x'^2 + k_(2,1) * x' + k_(2,0)

   *  y = y' * y_num / y_den, where

      -  y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' +
         k_(3,0)

      -  y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0)

   The constants used to compute x_num are as follows:

   *  k_(1,0) =
      0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa8c7

   *  k_(1,1) =
      0x7d3d4c80bc321d5b9f315cea7fd44c5d595d2fc0bf63b92dfff1044f17c6581

   *  k_(1,2) =
      0x534c328d23f234e6e2a413deca25caece4506144037c40314ecbd0b53d9dd262

   *  k_(1,3) =
      0x8e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38e38daaaaa88c

   The constants used to compute x_den are as follows:

   *  k_(2,0) =
      0xd35771193d94918a9ca34ccbb7b640dd86cd409542f8487d9fe6b745781eb49b

   *  k_(2,1) =
      0xedadc6f64383dc1df7c4b2d51b54225406d36b641f5e41bbc52a56612a8c6d14

   The constants used to compute y_num are as follows:

   *  k_(3,0) =
      0x4bda12f684bda12f684bda12f684bda12f684bda12f684bda12f684b8e38e23c



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 73]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(3,1) =
      0xc75e0c32d5cb7c0fa9d0a54b12a0a6d5647ab046d686da6fdffc90fc201d71a3

   *  k_(3,2) =
      0x29a6194691f91a73715209ef6512e576722830a201be2018a765e85a9ecee931

   *  k_(3,3) =
      0x2f684bda12f684bda12f684bda12f684bda12f684bda12f684bda12f38e38d84

   The constants used to compute y_den are as follows:

   *  k_(4,0) =
      0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffff93b

   *  k_(4,1) =
      0x7a06534bb8bdb49fd5e9e6632722c2989467c1bfc8e8d978dfb425d2685c2573

   *  k_(4,2) =
      0x6484aa716545ca2cf3a70c3fa8fe337e0a3d21162f0d6299a7bf8192bfd2a76f

E.2.  11-isogeny map for BLS12-381 G1

   The 11-isogeny map from (x', y') on E' to (x, y) on E is given by the
   following rational functions:

   *  x = x_num / x_den, where

      -  x_num = k_(1,11) * x'^11 + k_(1,10) * x'^10 + k_(1,9) * x'^9 +
         ... + k_(1,0)

      -  x_den = x'^10 + k_(2,9) * x'^9 + k_(2,8) * x'^8 + ... + k_(2,0)

   *  y = y' * y_num / y_den, where

      -  y_num = k_(3,15) * x'^15 + k_(3,14) * x'^14 + k_(3,13) * x'^13
         + ... + k_(3,0)

      -  y_den = x'^15 + k_(4,14) * x'^14 + k_(4,13) * x'^13 + ... +
         k_(4,0)

   The constants used to compute x_num are as follows:

   *  k_(1,0) = 0x11a05f2b1e833340b809101dd99815856b303e88a2d7005ff2627b
      56cdb4e2c85610c2d5f2e62d6eaeac1662734649b7

   *  k_(1,1) = 0x17294ed3e943ab2f0588bab22147a81c7c17e75b2f6a8417f565e3
      3c70d1e86b4838f2a6f318c356e834eef1b3cb83bb




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 74]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(1,2) = 0xd54005db97678ec1d1048c5d10a9a1bce032473295983e56878e50
      1ec68e25c958c3e3d2a09729fe0179f9dac9edcb0

   *  k_(1,3) = 0x1778e7166fcc6db74e0609d307e55412d7f5e4656a8dbf25f1b332
      89f1b330835336e25ce3107193c5b388641d9b6861

   *  k_(1,4) = 0xe99726a3199f4436642b4b3e4118e5499db995a1257fb3f086eeb6
      5982fac18985a286f301e77c451154ce9ac8895d9

   *  k_(1,5) = 0x1630c3250d7313ff01d1201bf7a74ab5db3cb17dd952799b9ed3ab
      9097e68f90a0870d2dcae73d19cd13c1c66f652983

   *  k_(1,6) = 0xd6ed6553fe44d296a3726c38ae652bfb11586264f0f8ce19008e21
      8f9c86b2a8da25128c1052ecaddd7f225a139ed84

   *  k_(1,7) = 0x17b81e7701abdbe2e8743884d1117e53356de5ab275b4db1a682c6
      2ef0f2753339b7c8f8c8f475af9ccb5618e3f0c88e

   *  k_(1,8) = 0x80d3cf1f9a78fc47b90b33563be990dc43b756ce79f5574a2c596c
      928c5d1de4fa295f296b74e956d71986a8497e317

   *  k_(1,9) = 0x169b1f8e1bcfa7c42e0c37515d138f22dd2ecb803a0c5c99676314
      baf4bb1b7fa3190b2edc0327797f241067be390c9e

   *  k_(1,10) = 0x10321da079ce07e272d8ec09d2565b0dfa7dccdde6787f96d50af
      36003b14866f69b771f8c285decca67df3f1605fb7b

   *  k_(1,11) = 0x6e08c248e260e70bd1e962381edee3d31d79d7e22c837bc23c0bf
      1bc24c6b68c24b1b80b64d391fa9c8ba2e8ba2d229

   The constants used to compute x_den are as follows:

   *  k_(2,0) = 0x8ca8d548cff19ae18b2e62f4bd3fa6f01d5ef4ba35b48ba9c95886
      17fc8ac62b558d681be343df8993cf9fa40d21b1c

   *  k_(2,1) = 0x12561a5deb559c4348b4711298e536367041e8ca0cf0800c0126c2
      588c48bf5713daa8846cb026e9e5c8276ec82b3bff

   *  k_(2,2) = 0xb2962fe57a3225e8137e629bff2991f6f89416f5a718cd1fca64e0
      0b11aceacd6a3d0967c94fedcfcc239ba5cb83e19

   *  k_(2,3) = 0x3425581a58ae2fec83aafef7c40eb545b08243f16b1655154cca8a
      bc28d6fd04976d5243eecf5c4130de8938dc62cd8

   *  k_(2,4) = 0x13a8e162022914a80a6f1d5f43e7a07dffdfc759a12062bb8d6b44
      e833b306da9bd29ba81f35781d539d395b3532a21e





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 75]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(2,5) = 0xe7355f8e4e667b955390f7f0506c6e9395735e9ce9cad4d0a43bce
      f24b8982f7400d24bc4228f11c02df9a29f6304a5

   *  k_(2,6) = 0x772caacf16936190f3e0c63e0596721570f5799af53a1894e2e073
      062aede9cea73b3538f0de06cec2574496ee84a3a

   *  k_(2,7) = 0x14a7ac2a9d64a8b230b3f5b074cf01996e7f63c21bca68a81996e1
      cdf9822c580fa5b9489d11e2d311f7d99bbdcc5a5e

   *  k_(2,8) = 0xa10ecf6ada54f825e920b3dafc7a3cce07f8d1d7161366b74100da
      67f39883503826692abba43704776ec3a79a1d641

   *  k_(2,9) = 0x95fc13ab9e92ad4476d6e3eb3a56680f682b4ee96f7d03776df533
      978f31c1593174e4b4b7865002d6384d168ecdd0a

   The constants used to compute y_num are as follows:

   *  k_(3,0) = 0x90d97c81ba24ee0259d1f094980dcfa11ad138e48a869522b52af6
      c956543d3cd0c7aee9b3ba3c2be9845719707bb33

   *  k_(3,1) = 0x134996a104ee5811d51036d776fb46831223e96c254f383d0f9063
      43eb67ad34d6c56711962fa8bfe097e75a2e41c696

   *  k_(3,2) = 0xcc786baa966e66f4a384c86a3b49942552e2d658a31ce2c344be4b
      91400da7d26d521628b00523b8dfe240c72de1f6

   *  k_(3,3) = 0x1f86376e8981c217898751ad8746757d42aa7b90eeb791c09e4a3e
      c03251cf9de405aba9ec61deca6355c77b0e5f4cb

   *  k_(3,4) = 0x8cc03fdefe0ff135caf4fe2a21529c4195536fbe3ce50b879833fd
      221351adc2ee7f8dc099040a841b6daecf2e8fedb

   *  k_(3,5) = 0x16603fca40634b6a2211e11db8f0a6a074a7d0d4afadb7bd76505c
      3d3ad5544e203f6326c95a807299b23ab13633a5f0

   *  k_(3,6) = 0x4ab0b9bcfac1bbcb2c977d027796b3ce75bb8ca2be184cb5231413
      c4d634f3747a87ac2460f415ec961f8855fe9d6f2

   *  k_(3,7) = 0x987c8d5333ab86fde9926bd2ca6c674170a05bfe3bdd81ffd038da
      6c26c842642f64550fedfe935a15e4ca31870fb29

   *  k_(3,8) = 0x9fc4018bd96684be88c9e221e4da1bb8f3abd16679dc26c1e8b6e6
      a1f20cabe69d65201c78607a360370e577bdba587

   *  k_(3,9) = 0xe1bba7a1186bdb5223abde7ada14a23c42a0ca7915af6fe06985e7
      ed1e4d43b9b3f7055dd4eba6f2bafaaebca731c30





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 76]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(3,10) = 0x19713e47937cd1be0dfd0b8f1d43fb93cd2fcbcb6caf493fd1183
      e416389e61031bf3a5cce3fbafce813711ad011c132

   *  k_(3,11) = 0x18b46a908f36f6deb918c143fed2edcc523559b8aaf0c2462e6bf
      e7f911f643249d9cdf41b44d606ce07c8a4d0074d8e

   *  k_(3,12) = 0xb182cac101b9399d155096004f53f447aa7b12a3426b08ec02710
      e807b4633f06c851c1919211f20d4c04f00b971ef8

   *  k_(3,13) = 0x245a394ad1eca9b72fc00ae7be315dc757b3b080d4c158013e663
      2d3c40659cc6cf90ad1c232a6442d9d3f5db980133

   *  k_(3,14) = 0x5c129645e44cf1102a159f748c4a3fc5e673d81d7e86568d9ab0f
      5d396a7ce46ba1049b6579afb7866b1e715475224b

   *  k_(3,15) = 0x15e6be4e990f03ce4ea50b3b42df2eb5cb181d8f84965a3957add
      4fa95af01b2b665027efec01c7704b456be69c8b604

   The constants used to compute y_den are as follows:

   *  k_(4,0) = 0x16112c4c3a9c98b252181140fad0eae9601a6de578980be6eec323
      2b5be72e7a07f3688ef60c206d01479253b03663c1

   *  k_(4,1) = 0x1962d75c2381201e1a0cbd6c43c348b885c84ff731c4d59ca4a103
      56f453e01f78a4260763529e3532f6102c2e49a03d

   *  k_(4,2) = 0x58df3306640da276faaae7d6e8eb15778c4855551ae7f310c35a5d
      d279cd2eca6757cd636f96f891e2538b53dbf67f2

   *  k_(4,3) = 0x16b7d288798e5395f20d23bf89edb4d1d115c5dbddbcd30e123da4
      89e726af41727364f2c28297ada8d26d98445f5416

   *  k_(4,4) = 0xbe0e079545f43e4b00cc912f8228ddcc6d19c9f0f69bbb0542eda0
      fc9dec916a20b15dc0fd2ededda39142311a5001d

   *  k_(4,5) = 0x8d9e5297186db2d9fb266eaac783182b70152c65550d881c5ecd87
      b6f0f5a6449f38db9dfa9cce202c6477faaf9b7ac

   *  k_(4,6) = 0x166007c08a99db2fc3ba8734ace9824b5eecfdfa8d0cf8ef5dd365
      bc400a0051d5fa9c01a58b1fb93d1a1399126a775c

   *  k_(4,7) = 0x16a3ef08be3ea7ea03bcddfabba6ff6ee5a4375efa1f4fd7feb34f
      d206357132b920f5b00801dee460ee415a15812ed9

   *  k_(4,8) = 0x1866c8ed336c61231a1be54fd1d74cc4f9fb0ce4c6af5920abc575
      0c4bf39b4852cfe2f7bb9248836b233d9d55535d4a





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 77]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(4,9) = 0x167a55cda70a6e1cea820597d94a84903216f763e13d87bb530859
      2e7ea7d4fbc7385ea3d529b35e346ef48bb8913f55

   *  k_(4,10) = 0x4d2f259eea405bd48f010a01ad2911d9c6dd039bb61a6290e591b
      36e636a5c871a5c29f4f83060400f8b49cba8f6aa8

   *  k_(4,11) = 0xaccbb67481d033ff5852c1e48c50c477f94ff8aefce42d28c0f9a
      88cea7913516f968986f7ebbea9684b529e2561092

   *  k_(4,12) = 0xad6b9514c767fe3c3613144b45f1496543346d98adf02267d5cee
      f9a00d9b8693000763e3b90ac11e99b138573345cc

   *  k_(4,13) = 0x2660400eb2e4f3b628bdd0d53cd76f2bf565b94e72927c1cb748d
      f27942480e420517bd8714cc80d1fadc1326ed06f7

   *  k_(4,14) = 0xe0fa1d816ddc03e6b24255e0d7819c171c40f65e273b853324efc
      d6356caa205ca2f570f13497804415473a1d634b8f

E.3.  3-isogeny map for BLS12-381 G2

   The 3-isogeny map from (x', y') on E' to (x, y) on E is given by the
   following rational functions:

   *  x = x_num / x_den, where

      -  x_num = k_(1,3) * x'^3 + k_(1,2) * x'^2 + k_(1,1) * x' +
         k_(1,0)

      -  x_den = x'^2 + k_(2,1) * x' + k_(2,0)

   *  y = y' * y_num / y_den, where

      -  y_num = k_(3,3) * x'^3 + k_(3,2) * x'^2 + k_(3,1) * x' +
         k_(3,0)

      -  y_den = x'^3 + k_(4,2) * x'^2 + k_(4,1) * x' + k_(4,0)

   The constants used to compute x_num are as follows:

   *  k_(1,0) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842
      3c50ae15d5c2638e343d9c71c6238aaaaaaaa97d6 + 0x5c759507e8e333ebb5b7
      a9a47d7ed8532c52d39fd3a042a88b58423c50ae15d5c2638e343d9c71c6238aaa
      aaaaa97d6 * I

   *  k_(1,1) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
      6b4f20a4181472aaa9cb8d555526a9ffffffffc71a * I





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 78]

Internet-Draft                hash-to-curve                    June 2022


   *  k_(1,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
      6b4f20a4181472aaa9cb8d555526a9ffffffffc71e + 0x8ab05f8bdd54cde1909
      37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff
      ffffffe38d * I

   *  k_(1,3) = 0x171d6541fa38ccfaed6dea691f5fb614cb14b4e7f4e810aa22d610
      8f142b85757098e38d0f671c7188e2aaaaaaaa5ed1

   The constants used to compute x_den are as follows:

   *  k_(2,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
      a0f6b0f6241eabfffeb153ffffb9feffffffffaa63 * I

   *  k_(2,1) = 0xc + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf
      6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa9f * I

   The constants used to compute y_num are as follows:

   *  k_(3,0) = 0x1530477c7ab4113b59a4c18b076d11930f7da5d4a07f649bf54439
      d87d27e500fc8c25ebf8c92f6812cfc71c71c6d706 + 0x1530477c7ab4113b59a
      4c18b076d11930f7da5d4a07f649bf54439d87d27e500fc8c25ebf8c92f6812cfc
      71c71c6d706 * I

   *  k_(3,1) = 0x5c759507e8e333ebb5b7a9a47d7ed8532c52d39fd3a042a88b5842
      3c50ae15d5c2638e343d9c71c6238aaaaaaaa97be * I

   *  k_(3,2) = 0x11560bf17baa99bc32126fced787c88f984f87adf7ae0c7f9a208c
      6b4f20a4181472aaa9cb8d555526a9ffffffffc71c + 0x8ab05f8bdd54cde1909
      37e76bc3e447cc27c3d6fbd7063fcd104635a790520c0a395554e5c6aaaa9354ff
      ffffffe38f * I

   *  k_(3,3) = 0x124c9ad43b6cf79bfbf7043de3811ad0761b0f37a1e26286b0e977
      c69aa274524e79097a56dc4bd9e1b371c71c718b10

   The constants used to compute y_den are as follows:

   *  k_(4,0) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
      a0f6b0f6241eabfffeb153ffffb9feffffffffa8fb + 0x1a0111ea397fe69a4b1
      ba7b6434bacd764774b84f38512bf6730d2a0f6b0f6241eabfffeb153ffffb9fef
      fffffffa8fb * I

   *  k_(4,1) = 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2
      a0f6b0f6241eabfffeb153ffffb9feffffffffa9d3 * I

   *  k_(4,2) = 0x12 + 0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512b
      f6730d2a0f6b0f6241eabfffeb153ffffb9feffffffffaa99 * I





Faz-Hernandez, et al.   Expires 17 December 2022               [Page 79]

Internet-Draft                hash-to-curve                    June 2022


Appendix F.  Straight-line implementations of deterministic mappings

   This section gives straight-line implementations of the mappings of
   Section 6.  These implementations are generic, i.e., they are defined
   for any curve and field.  Appendix G gives further implementations
   that are optimized for specific classes of curves and fields.

F.1.  Shallue-van de Woestijne method

   This section gives a straight-line implementation of the Shallue and
   van de Woestijne method for any Weierstrass curve of the form given
   in Section 6.6.  See Section 6.6.1 for information on the constants
   used in this mapping.

   Note that the constant c3 below MUST be chosen such that sgn0(c3) =
   0.  In other words, if the square-root computation returns a value cx
   such that sgn0(cx) = 1, set c3 = -cx; otherwise, set c3 = cx.

   map_to_curve_svdw(u)

   Input: u, an element of F.
   Output: (x, y), a point on E.

   Constants:
   1. c1 = g(Z)
   2. c2 = -Z / 2
   3. c3 = sqrt(-g(Z) * (3 * Z^2 + 4 * A))     # sgn0(c3) MUST equal 0
   4. c4 = -4 * g(Z) / (3 * Z^2 + 4 * A)

   Steps:
   1.  tv1 = u^2
   2.  tv1 = tv1 * c1
   3.  tv2 = 1 + tv1
   4.  tv1 = 1 - tv1
   5.  tv3 = tv1 * tv2
   6.  tv3 = inv0(tv3)
   7.  tv4 = u * tv1
   8.  tv4 = tv4 * tv3
   9.  tv4 = tv4 * c3
   10.  x1 = c2 - tv4
   11. gx1 = x1^2
   12. gx1 = gx1 + A
   13. gx1 = gx1 * x1
   14. gx1 = gx1 + B
   15.  e1 = is_square(gx1)
   16.  x2 = c2 + tv4
   17. gx2 = x2^2
   18. gx2 = gx2 + A



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 80]

Internet-Draft                hash-to-curve                    June 2022


   19. gx2 = gx2 * x2
   20. gx2 = gx2 + B
   21.  e2 = is_square(gx2) AND NOT e1   # Avoid short-circuit logic ops
   22.  x3 = tv2^2
   23.  x3 = x3 * tv3
   24.  x3 = x3^2
   25.  x3 = x3 * c4
   26.  x3 = x3 + Z
   27.   x = CMOV(x3, x1, e1)   # x = x1 if gx1 is square, else x = x3
   28.   x = CMOV(x, x2, e2)    # x = x2 if gx2 is square and gx1 is not
   29.  gx = x^2
   30.  gx = gx + A
   31.  gx = gx * x
   32.  gx = gx + B
   33.   y = sqrt(gx)
   34.  e3 = sgn0(u) == sgn0(y)
   35.   y = CMOV(-y, y, e3)       # Select correct sign of y
   36. return (x, y)

F.2.  Simplified SWU method

   This section gives a straight-line implementation of the simplified
   SWU method for any Weierstrass curve of the form given in
   Section 6.6.  See Section 6.6.2 for information on the constants used
   in this mapping.

   This optimized, straight-line procedure applies to any base field.
   The sqrt_ratio subroutine is defined in Appendix F.2.1.























Faz-Hernandez, et al.   Expires 17 December 2022               [Page 81]

Internet-Draft                hash-to-curve                    June 2022


   map_to_curve_simple_swu(u)

   Input: u, an element of F.
   Output: (x, y), a point on E.

   Steps:
   1.  tv1 = u^2
   2.  tv1 = Z * tv1
   3.  tv2 = tv1^2
   4.  tv2 = tv2 + tv1
   5.  tv3 = tv2 + 1
   6.  tv3 = B * tv3
   7.  tv4 = CMOV(Z, -tv2, tv2 != 0)
   8.  tv4 = A * tv4
   9.  tv2 = tv3^2
   10. tv6 = tv4^2
   11. tv5 = A * tv6
   12. tv2 = tv2 + tv5
   13. tv2 = tv2 * tv3
   14. tv6 = tv6 * tv4
   15. tv5 = B * tv6
   16. tv2 = tv2 + tv5
   17.   x = tv1 * tv3
   18. (is_gx1_square, y1) = sqrt_ratio(tv2, tv6)
   19.   y = tv1 * u
   20.   y = y * y1
   21.   x = CMOV(x, tv3, is_gx1_square)
   22.   y = CMOV(y, y1, is_gx1_square)
   23.  e1 = sgn0(u) == sgn0(y)
   24.   y = CMOV(-y, y, e1)
   25.   x = x / tv4
   26. return (x, y)

F.2.1.  sqrt_ratio subroutines

   This section defines three variants of the sqrt_ratio subroutine used
   by the above procedure.  The first variant can be used with any
   field; the others are optimized versions for specific fields.

   The routines given in this section depend on the constant Z from the
   simplified SWU map.  For correctness, sqrt_ratio and
   map_to_curve_simple_swu MUST use the same value for Z.

F.2.1.1.  sqrt_ratio for any field







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 82]

Internet-Draft                hash-to-curve                    June 2022


   sqrt_ratio(u, v)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.
   - Z, the constant from the simplified SWU map.

   Input: u and v, elements of F, where v != 0.
   Output: (b, y), where
     b = True and y = sqrt(u / v) if (u / v) is square in F, and
     b = False and y = sqrt(Z * (u / v)) otherwise.

   Constants:
   1. c1, the largest integer such that 2^c1 divides q - 1.
   2. c2 = (q - 1) / (2^c1)        # Integer arithmetic
   3. c3 = (c2 - 1) / 2            # Integer arithmetic
   4. c4 = 2^c1 - 1                # Integer arithmetic
   5. c5 = 2^(c1 - 1)              # Integer arithmetic
   6. c6 = Z^c2
   7. c7 = Z^((c2 + 1) / 2)

   Procedure:
   1. tv1 = c6
   2. tv2 = v^c4
   3. tv3 = tv2^2
   4. tv3 = tv3 * v
   5. tv5 = u * tv3
   6. tv5 = tv5^c3
   7. tv5 = tv5 * tv2
   8. tv2 = tv5 * v
   9. tv3 = tv5 * u
   10. tv4 = tv3 * tv2
   11. tv5 = tv4^c5
   12. isQR = tv5 == 1
   13. tv2 = tv3 * c7
   14. tv5 = tv4 * tv1
   15. tv3 = CMOV(tv2, tv3, isQR)
   16. tv4 = CMOV(tv5, tv4, isQR)
   17. for i in (c1, c1 - 1, ..., 2):
   18.    tv5 = i - 2
   19.    tv5 = 2^tv5
   20.    tv5 = tv4^tv5
   21.    e1 = tv5 == 1
   22.    tv2 = tv3 * tv1
   23.    tv1 = tv1 * tv1
   24.    tv5 = tv4 * tv1
   25.    tv3 = CMOV(tv2, tv3, e1)
   26.    tv4 = CMOV(tv5, tv4, e1)
   27. return (isQR, tv3)



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 83]

Internet-Draft                hash-to-curve                    June 2022


F.2.1.2.  optimized sqrt_ratio for q = 3 mod 4

   sqrt_ratio_3mod4(u, v)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m,
     where q = 3 mod 4.
   - Z, the constant from the simplified SWU map.

   Input: u and v, elements of F, where v != 0.
   Output: (b, y), where
     b = True and y = sqrt(u / v) if (u / v) is square in F, and
     b = False and y = sqrt(Z * (u / v)) otherwise.

   Constants:
   1. c1 = (q - 3) / 4     # Integer arithmetic
   2. c2 = sqrt(-Z)

   Procedure:
   1. tv1 = v^2
   2. tv2 = u * v
   3. tv1 = tv1 * tv2
   4. y1 = tv1^c1
   5. y1 = y1 * tv2
   6. y2 = y1 * c2
   7. tv3 = y1^2
   8. tv3 = tv3 * v
   9. isQR = tv3 == u
   10. y = CMOV(y2, y1, isQR)
   11. return (isQR, y)

F.2.1.3.  optimized sqrt_ratio for q = 5 mod 8



















Faz-Hernandez, et al.   Expires 17 December 2022               [Page 84]

Internet-Draft                hash-to-curve                    June 2022


   sqrt_ratio_5mod8(u, v)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m,
     where q = 5 mod 8.
   - Z, the constant from the simplified SWU map.

   Input: u and v, elements of F, where v != 0.
   Output: (b, y), where
     b = True and y = sqrt(u / v) if (u / v) is square in F, and
     b = False and y = sqrt(Z * (u / v)) otherwise.

   Constants:
   1. c1 = (q - 5) / 8
   2. c2 = sqrt(-1)
   3. c3 = sqrt(Z / c2)

   Steps:
   1. tv1 = v^2
   2. tv2 = tv1 * v
   3. tv1 = tv1^2
   4. tv2 = tv2 * u
   5. tv1 = tv1 * tv2
   6. y1 = tv1^c1
   7. y1 = y1 * tv2
   8. tv1 = y1 * c2
   9. tv2 = tv1^2
   10. tv2 = tv2 * v
   11. e1 = tv2 == u
   12. y1 = CMOV(y1, tv1, e1)
   13. tv2 = y1^2
   14. tv2 = tv2 * v
   15. isQR = tv2 == u
   16. y2 = y1 * c3
   17. tv1 = y2 * c2
   18. tv2 = tv1^2
   19. tv2 = tv2 * v
   20. tv3 = Z * u
   21. e2 = tv2 == tv3
   22. y2 = CMOV(y2, tv1, e2)
   23. y = CMOV(y2, y1, isQR)
   24. return (isQR, y)









Faz-Hernandez, et al.   Expires 17 December 2022               [Page 85]

Internet-Draft                hash-to-curve                    June 2022


F.3.  Elligator 2 method

   This section gives a straight-line implementation of the Elligator 2
   method for any Montgomery curve of the form given in Section 6.7.
   See Section 6.7.1 for information on the constants used in this
   mapping.

   Appendix G.2 gives optimized straight-line procedures that apply to
   specific classes of curves and base fields, including curve25519 and
   curve448 [RFC7748].

   map_to_curve_elligator2(u)

   Input: u, an element of F.
   Output: (s, t), a point on M.

   Constants:
   1.   c1 = J / K
   2.   c2 = 1 / K^2

   Steps:
   1.  tv1 = u^2
   2.  tv1 = Z * tv1            # Z * u^2
   3.   e1 = tv1 == -1          # exceptional case: Z * u^2 == -1
   4.  tv1 = CMOV(tv1, 0, e1)   # if tv1 == -1, set tv1 = 0
   5.   x1 = tv1 + 1
   6.   x1 = inv0(x1)
   7.   x1 = -c1 * x1           # x1 = -(J / K) / (1 + Z * u^2)
   8.  gx1 = x1 + c1
   9.  gx1 = gx1 * x1
   10. gx1 = gx1 + c2
   11. gx1 = gx1 * x1           # gx1 = x1^3 + (J / K) * x1^2 + x1 / K^2
   12.  x2 = -x1 - c1
   13. gx2 = tv1 * gx1
   14.  e2 = is_square(gx1)     # If is_square(gx1)
   15.   x = CMOV(x2, x1, e2)   #   then  x = x1,  else  x = x2
   16.  y2 = CMOV(gx2, gx1, e2) #   then y2 = gx1, else y2 = gx2
   17.   y = sqrt(y2)
   18.  e3 = sgn0(y) == 1
   19.   y = CMOV(y, -y, e2 XOR e3)    # fix sign of y
   20.   s = x * K
   21.   t = y * K
   22. return (s, t)








Faz-Hernandez, et al.   Expires 17 December 2022               [Page 86]

Internet-Draft                hash-to-curve                    June 2022


Appendix G.  Curve-specific optimized sample code

   This section gives sample implementations optimized for some of the
   elliptic curves listed in Section 8.  Sample Sage [SAGE] code for
   each algorithm can also be found in the draft repository
   [hash2curve-repo].

G.1.  Interface and projective coordinate systems

   The sample code in this section uses a different interface than the
   mappings of Section 6.  Specifically, each mapping function in this
   section has the following signature:

       (xn, xd, yn, yd) = map_to_curve(u)

   The resulting affine point (x, y) is given by (xn / xd, yn / yd).

   The reason for this modified interface is that it enables further
   optimizations when working with points in a projective coordinate
   system.  This is desirable, for example, when the resulting point
   will be immediately multiplied by a scalar, since most scalar
   multiplication algorithms operate on projective points.

   Projective coordinates are also useful when implementing random
   oracle encodings (Section 3).  One reason is that, in general, point
   addition is faster using projective coordinates.  Another reason is
   that, for Weierstrass curves, projective coordinates allow using
   complete addition formulas [RCB16].  This is especially convenient
   when implementing a constant-time encoding, because it eliminates the
   need for a special case when Q0 == Q1, which incomplete addition
   formulas usually do not handle.

   The following are two commonly used projective coordinate systems and
   the corresponding conversions:

   *  A point (X, Y, Z) in homogeneous projective coordinates
      corresponds to the affine point (x, y) = (X / Z, Y / Z); the
      inverse conversion is given by (X, Y, Z) = (x, y, 1).  To convert
      (xn, xd, yn, yd) to homogeneous projective coordinates, compute
      (X, Y, Z) = (xn * yd, yn * xd, xd * yd).

   *  A point (X', Y', Z') in Jacobian projective coordinates
      corresponds to the affine point (x, y) = (X' / Z'^2, Y' / Z'^3);
      the inverse conversion is given by (X', Y', Z') = (x, y, 1).  To
      convert (xn, xd, yn, yd) to Jacobian projective coordinates,
      compute (X', Y', Z') = (xn * xd * yd^2, yn * yd^2 * xd^3, xd *
      yd).




Faz-Hernandez, et al.   Expires 17 December 2022               [Page 87]

Internet-Draft                hash-to-curve                    June 2022


G.2.  Elligator 2

G.2.1.  curve25519 (q = 5 (mod 8), K = 1)

   The following is a straight-line implementation of Elligator 2 for
   curve25519 [RFC7748] as specified in Section 8.5.

   This implementation can also be used for any Montgomery curve with K
   = 1 over GF(q) where q = 5 (mod 8).

   map_to_curve_elligator2_curve25519(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on curve25519.

   Constants:
   1. c1 = (q + 3) / 8       # Integer arithmetic
   2. c2 = 2^c1
   3. c3 = sqrt(-1)
   4. c4 = (q - 5) / 8       # Integer arithmetic

   Steps:
   1.  tv1 = u^2
   2.  tv1 = 2 * tv1
   3.   xd = tv1 + 1         # Nonzero: -1 is square (mod p), tv1 is not
   4.  x1n = -J              # x1 = x1n / xd = -J / (1 + 2 * u^2)
   5.  tv2 = xd^2
   6.  gxd = tv2 * xd        # gxd = xd^3
   7.  gx1 = J * tv1         # x1n + J * xd
   8.  gx1 = gx1 * x1n       # x1n^2 + J * x1n * xd
   9.  gx1 = gx1 + tv2       # x1n^2 + J * x1n * xd + xd^2
   10. gx1 = gx1 * x1n       # x1n^3 + J * x1n^2 * xd + x1n * xd^2
   11. tv3 = gxd^2
   12. tv2 = tv3^2           # gxd^4
   13. tv3 = tv3 * gxd       # gxd^3
   14. tv3 = tv3 * gx1       # gx1 * gxd^3
   15. tv2 = tv2 * tv3       # gx1 * gxd^7
   16. y11 = tv2^c4          # (gx1 * gxd^7)^((p - 5) / 8)
   17. y11 = y11 * tv3       # gx1 * gxd^3 * (gx1 * gxd^7)^((p - 5) / 8)
   18. y12 = y11 * c3
   19. tv2 = y11^2
   20. tv2 = tv2 * gxd
   21.  e1 = tv2 == gx1
   22.  y1 = CMOV(y12, y11, e1)  # If g(x1) is square, this is its sqrt
   23. x2n = x1n * tv1           # x2 = x2n / xd = 2 * u^2 * x1n / xd
   24. y21 = y11 * u
   25. y21 = y21 * c2



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 88]

Internet-Draft                hash-to-curve                    June 2022


   26. y22 = y21 * c3
   27. gx2 = gx1 * tv1           # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
   28. tv2 = y21^2
   29. tv2 = tv2 * gxd
   30.  e2 = tv2 == gx2
   31.  y2 = CMOV(y22, y21, e2)  # If g(x2) is square, this is its sqrt
   32. tv2 = y1^2
   33. tv2 = tv2 * gxd
   34.  e3 = tv2 == gx1
   35.  xn = CMOV(x2n, x1n, e3)  # If e3, x = x1, else x = x2
   36.   y = CMOV(y2, y1, e3)    # If e3, y = y1, else y = y2
   37.  e4 = sgn0(y) == 1        # Fix sign of y
   38.   y = CMOV(y, -y, e3 XOR e4)
   39. return (xn, xd, y, 1)

G.2.2.  edwards25519

   The following is a straight-line implementation of Elligator 2 for
   edwards25519 [RFC7748] as specified in Section 8.5.  The subroutine
   map_to_curve_elligator2_curve25519 is defined in Appendix G.2.1.

   Note that the sign of the constant c1 below is chosen as specified in
   Section 6.8.1, i.e., applying the rational map to the edwards25519
   base point yields the curve25519 base point (see erratum [EID4730]).

   map_to_curve_elligator2_edwards25519(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on edwards25519.

   Constants:
   1. c1 = sqrt(-486664) # sgn0(c1) MUST equal 0

   Steps:
   1.  (xMn, xMd, yMn, yMd) = map_to_curve_elligator2_curve25519(u)
   2.  xn = xMn * yMd
   3.  xn = xn * c1
   4.  xd = xMd * yMn    # xn / xd = c1 * xM / yM
   5.  yn = xMn - xMd
   6.  yd = xMn + xMd    # (n / d - 1) / (n / d + 1) = (n - d) / (n + d)
   7. tv1 = xd * yd
   8.   e = tv1 == 0
   9.  xn = CMOV(xn, 0, e)
   10. xd = CMOV(xd, 1, e)
   11. yn = CMOV(yn, 1, e)
   12. yd = CMOV(yd, 1, e)
   13. return (xn, xd, yn, yd)



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 89]

Internet-Draft                hash-to-curve                    June 2022


G.2.3.  curve448 (q = 3 (mod 4), K = 1)

   The following is a straight-line implementation of Elligator 2 for
   curve448 [RFC7748] as specified in Section 8.6.

   This implementation can also be used for any Montgomery curve with K
   = 1 over GF(q) where q = 3 (mod 4).

   map_to_curve_elligator2_curve448(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on curve448.

   Constants:
   1. c1 = (q - 3) / 4         # Integer arithmetic

   Steps:
   1.  tv1 = u^2
   2.   e1 = tv1 == 1
   3.  tv1 = CMOV(tv1, 0, e1)  # If Z * u^2 == -1, set tv1 = 0
   4.   xd = 1 - tv1
   5.  x1n = -J
   6.  tv2 = xd^2
   7.  gxd = tv2 * xd          # gxd = xd^3
   8.  gx1 = -J * tv1          # x1n + J * xd
   9.  gx1 = gx1 * x1n         # x1n^2 + J * x1n * xd
   10. gx1 = gx1 + tv2         # x1n^2 + J * x1n * xd + xd^2
   11. gx1 = gx1 * x1n         # x1n^3 + J * x1n^2 * xd + x1n * xd^2
   12. tv3 = gxd^2
   13. tv2 = gx1 * gxd         # gx1 * gxd
   14. tv3 = tv3 * tv2         # gx1 * gxd^3
   15.  y1 = tv3^c1            # (gx1 * gxd^3)^((p - 3) / 4)
   16.  y1 = y1 * tv2          # gx1 * gxd * (gx1 * gxd^3)^((p - 3) / 4)
   17. x2n = -tv1 * x1n        # x2 = x2n / xd = -1 * u^2 * x1n / xd
   18.  y2 = y1 * u
   19.  y2 = CMOV(y2, 0, e1)
   20. tv2 = y1^2
   21. tv2 = tv2 * gxd
   22.  e2 = tv2 == gx1
   23.  xn = CMOV(x2n, x1n, e2)  # If e2, x = x1, else x = x2
   24.   y = CMOV(y2, y1, e2)    # If e2, y = y1, else y = y2
   25.  e3 = sgn0(y) == 1        # Fix sign of y
   26.   y = CMOV(y, -y, e2 XOR e3)
   27. return (xn, xd, y, 1)






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 90]

Internet-Draft                hash-to-curve                    June 2022


G.2.4.  edwards448

   The following is a straight-line implementation of Elligator 2 for
   edwards448 [RFC7748] as specified in Section 8.6.  The subroutine
   map_to_curve_elligator2_curve448 is defined in Appendix G.2.3.














































Faz-Hernandez, et al.   Expires 17 December 2022               [Page 91]

Internet-Draft                hash-to-curve                    June 2022


   map_to_curve_elligator2_edwards448(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on edwards448.

   Steps:
   1. (xn, xd, yn, yd) = map_to_curve_elligator2_curve448(u)
   2.  xn2 = xn^2
   3.  xd2 = xd^2
   4.  xd4 = xd2^2
   5.  yn2 = yn^2
   6.  yd2 = yd^2
   7.  xEn = xn2 - xd2
   8.  tv2 = xEn - xd2
   9.  xEn = xEn * xd2
   10. xEn = xEn * yd
   11. xEn = xEn * yn
   12. xEn = xEn * 4
   13. tv2 = tv2 * xn2
   14. tv2 = tv2 * yd2
   15. tv3 = 4 * yn2
   16. tv1 = tv3 + yd2
   17. tv1 = tv1 * xd4
   18. xEd = tv1 + tv2
   19. tv2 = tv2 * xn
   20. tv4 = xn * xd4
   21. yEn = tv3 - yd2
   22. yEn = yEn * tv4
   23. yEn = yEn - tv2
   24. tv1 = xn2 + xd2
   25. tv1 = tv1 * xd2
   26. tv1 = tv1 * xd
   27. tv1 = tv1 * yn2
   28. tv1 = -2 * tv1
   29. yEd = tv2 + tv1
   30. tv4 = tv4 * yd2
   31. yEd = yEd + tv4
   32. tv1 = xEd * yEd
   33.   e = tv1 == 0
   34. xEn = CMOV(xEn, 0, e)
   35. xEd = CMOV(xEd, 1, e)
   36. yEn = CMOV(yEn, 1, e)
   37. yEd = CMOV(yEd, 1, e)
   38. return (xEn, xEd, yEn, yEd)






Faz-Hernandez, et al.   Expires 17 December 2022               [Page 92]

Internet-Draft                hash-to-curve                    June 2022


G.2.5.  Montgomery curves with q = 3 (mod 4)

   The following is a straight-line implementation of Elligator 2 that
   applies to any Montgomery curve defined over GF(q) where q = 3 (mod
   4).

   For curves where K = 1, the implementation given in Appendix G.2.3
   gives identical results with slightly reduced cost.











































Faz-Hernandez, et al.   Expires 17 December 2022               [Page 93]

Internet-Draft                hash-to-curve                    June 2022


   map_to_curve_elligator2_3mod4(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on the target curve.

   Constants:
   1. c1 = (q - 3) / 4        # Integer arithmetic
   2. c2 = K^2

   Steps:
   1.  tv1 = u^2
   2.   e1 = tv1 == 1
   3.  tv1 = CMOV(tv1, 0, e1) # If Z * u^2 == -1, set tv1 = 0
   4.   xd = 1 - tv1
   5.   xd = xd * K
   6.  x1n = -J          # x1 = x1n / xd = -J / (K * (1 + 2 * u^2))
   7.  tv2 = xd^2
   8.  gxd = tv2 * xd
   9.  gxd = gxd * c2    # gxd = xd^3 * K^2
   10. gx1 = x1n * K
   11. tv3 = xd * J
   12. tv3 = gx1 + tv3   # x1n * K + xd * J
   13. gx1 = gx1 * tv3   # K^2 * x1n^2 + J * K * x1n * xd
   14. gx1 = gx1 + tv2   # K^2 * x1n^2 + J * K * x1n * xd + xd^2
   15. gx1 = gx1 * x1n   # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2
   16. tv3 = gxd^2
   17. tv2 = gx1 * gxd   # gx1 * gxd
   18. tv3 = tv3 * tv2   # gx1 * gxd^3
   19.  y1 = tv3^c1      # (gx1 * gxd^3)^((q - 3) / 4)
   20.  y1 = y1 * tv2    # gx1 * gxd * (gx1 * gxd^3)^((q - 3) / 4)
   21. x2n = -tv1 * x1n  # x2 = x2n / xd = -1 * u^2 * x1n / xd
   22.  y2 = y1 * u
   23.  y2 = CMOV(y2, 0, e1)
   24. tv2 = y1^2
   25. tv2 = tv2 * gxd
   26.  e2 = tv2 == gx1
   27.  xn = CMOV(x2n, x1n, e2)  # If e2, x = x1, else x = x2
   28.  xn = xn * K
   29.   y = CMOV(y2, y1, e2)    # If e2, y = y1, else y = y2
   30.  e3 = sgn0(y) == 1        # Fix sign of y
   31.   y = CMOV(y, -y, e2 XOR e3)
   32.   y = y * K
   33. return (xn, xd, y, 1)







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 94]

Internet-Draft                hash-to-curve                    June 2022


G.2.6.  Montgomery curves with q = 5 (mod 8)

   The following is a straight-line implementation of Elligator 2 that
   applies to any Montgomery curve defined over GF(q) where q = 5 (mod
   8).

   For curves where K = 1, the implementation given in Appendix G.2.1
   gives identical results with slightly reduced cost.

   map_to_curve_elligator2_5mod8(u)

   Input: u, an element of F.
   Output: (xn, xd, yn, yd) such that (xn / xd, yn / yd) is a
           point on the target curve.

   Constants:
   1. c1 = (q + 3) / 8           # Integer arithmetic
   2. c2 = 2^c1
   3. c3 = sqrt(-1)
   4. c4 = (q - 5) / 8           # Integer arithmetic
   5. c5 = K^2

   Steps:
   1.  tv1 = u^2
   2.  tv1 = 2 * tv1
   3.   xd = tv1 + 1     # Nonzero: -1 is square (mod p), tv1 is not
   4.   xd = xd * K
   5.  x1n = -J          # x1 = x1n / xd = -J / (K * (1 + 2 * u^2))
   6.  tv2 = xd^2
   7.  gxd = tv2 * xd
   8.  gxd = gxd * c5    # gxd = xd^3 * K^2
   9.  gx1 = x1n * K
   10. tv3 = xd * J
   11. tv3 = gx1 + tv3   # x1n * K + xd * J
   12. gx1 = gx1 * tv3   # K^2 * x1n^2 + J * K * x1n * xd
   13. gx1 = gx1 + tv2   # K^2 * x1n^2 + J * K * x1n * xd + xd^2
   14. gx1 = gx1 * x1n   # K^2 * x1n^3 + J * K * x1n^2 * xd + x1n * xd^2
   15. tv3 = gxd^2
   16. tv2 = tv3^2       # gxd^4
   17. tv3 = tv3 * gxd   # gxd^3
   18. tv3 = tv3 * gx1   # gx1 * gxd^3
   19. tv2 = tv2 * tv3   # gx1 * gxd^7
   20. y11 = tv2^c4      # (gx1 * gxd^7)^((q - 5) / 8)
   21. y11 = y11 * tv3   # gx1 * gxd^3 * (gx1 * gxd^7)^((q - 5) / 8)
   22. y12 = y11 * c3
   23. tv2 = y11^2
   24. tv2 = tv2 * gxd
   25.  e1 = tv2 == gx1



Faz-Hernandez, et al.   Expires 17 December 2022               [Page 95]

Internet-Draft                hash-to-curve                    June 2022


   26.  y1 = CMOV(y12, y11, e1)  # If g(x1) is square, this is its sqrt
   27. x2n = x1n * tv1           # x2 = x2n / xd = 2 * u^2 * x1n / xd
   28. y21 = y11 * u
   29. y21 = y21 * c2
   30. y22 = y21 * c3
   31. gx2 = gx1 * tv1           # g(x2) = gx2 / gxd = 2 * u^2 * g(x1)
   32. tv2 = y21^2
   33. tv2 = tv2 * gxd
   34.  e2 = tv2 == gx2
   35.  y2 = CMOV(y22, y21, e2)  # If g(x2) is square, this is its sqrt
   36. tv2 = y1^2
   37. tv2 = tv2 * gxd
   38.  e3 = tv2 == gx1
   39.  xn = CMOV(x2n, x1n, e3)  # If e3, x = x1, else x = x2
   40.  xn = xn * K
   41.   y = CMOV(y2, y1, e3)    # If e3, y = y1, else y = y2
   42.  e4 = sgn0(y) == 1        # Fix sign of y
   43.   y = CMOV(y, -y, e3 XOR e4)
   44.   y = y * K
   45. return (xn, xd, y, 1)

G.3.  Cofactor clearing for BLS12-381 G2

   The curve BLS12-381, whose parameters are defined in Section 8.8.2,
   admits an efficiently-computable endomorphism psi that can be used to
   speed up cofactor clearing for G2 [SBCDK09] [FKR11] [BP17] (see also
   Section 7).  This section implements the endomorphism psi and a fast
   cofactor clearing method described by Budroni and Pintore [BP17].

   The functions in this section operate on points whose coordinates are
   represented as ratios, i.e., (xn, xd, yn, yd) corresponds to the
   point (xn / xd, yn / yd); see Appendix G.1 for further discussion of
   projective coordinates.  When points are represented in affine
   coordinates, one can simply ignore the denominators (xd == 1 and yd
   == 1).

   The following function computes the Frobenius endomorphism for an
   element of F = GF(p^2) with basis (1, I), where I^2 + 1 == 0 in F.
   (This is the base field of the elliptic curve E defined in
   Section 8.8.2.)











Faz-Hernandez, et al.   Expires 17 December 2022               [Page 96]

Internet-Draft                hash-to-curve                    June 2022


   frobenius(x)

   Input: x, an element of GF(p^2).
   Output: a, an element of GF(p^2).

   Notation: x = x0 + I * x1, where x0 and x1 are elements of GF(p).

   Steps:
   1. a = x0 - I * x1
   2. return a

   The following function computes the endomorphism psi for points on
   the elliptic curve E defined in Section 8.8.2.

   psi(xn, xd, yn, yd)

   Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
   Output: Q, a point on the same curve.

   Constants:
   1. c1 = 1 / (1 + I)^((p - 1) / 3)           # in GF(p^2)
   2. c2 = 1 / (1 + I)^((p - 1) / 2)           # in GF(p^2)

   Steps:
   1. qxn = c1 * frobenius(xn)
   2. qxd = frobenius(xd)
   3. qyn = c2 * frobenius(yn)
   4. qyd = frobenius(yd)
   5. return (qxn, qxd, qyn, qyd)

   The following function efficiently computes psi(psi(P)).

   psi2(xn, xd, yn, yd)

   Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
   Output: Q, a point on the same curve.

   Constants:
   1. c1 = 1 / 2^((p - 1) / 3)                 # in GF(p^2)

   Steps:
   1. qxn = c1 * xn
   2. qyn = -yn
   3. return (qxn, xd, qyn, yd)







Faz-Hernandez, et al.   Expires 17 December 2022               [Page 97]

Internet-Draft                hash-to-curve                    June 2022


   The following function maps any point on the elliptic curve E
   (Section 8.8.2) into the prime-order subgroup G2.  This function
   returns a point equal to h_eff * P, where h_eff is the parameter
   given in Section 8.8.2.

   clear_cofactor_bls12381_g2(P)

   Input: P, a point (xn / xd, yn / yd) on the curve E (see above).
   Output: Q, a point in the subgroup G2 of BLS12-381.

   Constants:
   1. c1 = -15132376222941642752       # the BLS parameter for BLS12-381
                                       # i.e., -0xd201000000010000

   Notation: in this procedure, + and - represent elliptic curve point
   addition and subtraction, respectively, and * represents scalar
   multiplication.

   Steps:
   1.  t1 = c1 * P
   2.  t2 = psi(P)
   3.  t3 = 2 * P
   4.  t3 = psi2(t3)
   5.  t3 = t3 - t2
   6.  t2 = t1 + t2
   7.  t2 = c1 * t2
   8.  t3 = t3 + t2
   9.  t3 = t3 - t1
   10.  Q = t3 - P
   11. return Q

Appendix H.  Scripts for parameter generation

   This section gives Sage [SAGE] scripts used to generate parameters
   for the mappings of Section 6.

H.1.  Finding Z for the Shallue-van de Woestijne map

   The below function outputs an appropriate Z for the Shallue and van
   de Woestijne map (Section 6.6.1).











Faz-Hernandez, et al.   Expires 17 December 2022               [Page 98]

Internet-Draft                hash-to-curve                    June 2022


   # Arguments:
   # - F, a field object, e.g., F = GF(2^521 - 1)
   # - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
   def find_z_svdw(F, A, B, init_ctr=1):
       g = lambda x: F(x)^3 + F(A) * F(x) + F(B)
       h = lambda Z: -(F(3) * Z^2 + F(4) * A) / (F(4) * g(Z))
       # NOTE: if init_ctr=1 fails to find Z, try setting it to F.gen()
       ctr = init_ctr
       while True:
           for Z_cand in (F(ctr), F(-ctr)):
               # Criterion 1:
               #   g(Z) != 0 in F.
               if g(Z_cand) == F(0):
                   continue
               # Criterion 2:
               #   -(3 * Z^2 + 4 * A) / (4 * g(Z)) != 0 in F.
               if h(Z_cand) == F(0):
                   continue
               # Criterion 3:
               #   -(3 * Z^2 + 4 * A) / (4 * g(Z)) is square in F.
               if not is_square(h(Z_cand)):
                   continue
               # Criterion 4:
               #   At least one of g(Z) and g(-Z / 2) is square in F.
               if is_square(g(Z_cand)) or is_square(g(-Z_cand / F(2))):
                   return Z_cand
           ctr += 1

H.2.  Finding Z for Simplified SWU

   The below function outputs an appropriate Z for the Simplified SWU
   map (Section 6.6.2).



















Faz-Hernandez, et al.   Expires 17 December 2022               [Page 99]

Internet-Draft                hash-to-curve                    June 2022


   # Arguments:
   # - F, a field object, e.g., F = GF(2^521 - 1)
   # - A and B, the coefficients of the curve y^2 = x^3 + A * x + B
   def find_z_sswu(F, A, B):
       R.<xx> = F[]                       # Polynomial ring over F
       g = xx^3 + F(A) * xx + F(B)        # y^2 = g(x) = x^3 + A * x + B
       ctr = F.gen()
       while True:
           for Z_cand in (F(ctr), F(-ctr)):
               # Criterion 1: Z is non-square in F.
               if is_square(Z_cand):
                   continue
               # Criterion 2: Z != -1 in F.
               if Z_cand == F(-1):
                   continue
               # Criterion 3: g(x) - Z is irreducible over F.
               if not (g - Z_cand).is_irreducible():
                   continue
               # Criterion 4: g(B / (Z * A)) is square in F.
               if is_square(g(B / (Z_cand * A))):
                   return Z_cand
           ctr += 1

H.3.  Finding Z for Elligator 2

   The below function outputs an appropriate Z for the Elligator 2 map
   (Section 6.7.1).

   # Argument:
   # - F, a field object, e.g., F = GF(2^255 - 19)
   def find_z_ell2(F):
       ctr = F.gen()
       while True:
           for Z_cand in (F(ctr), F(-ctr)):
               # Z must be a non-square in F.
               if is_square(Z_cand):
                   continue
               return Z_cand
           ctr += 1

Appendix I.  sqrt and is_square functions

   This section defines special-purpose sqrt functions for the three
   most common cases, q = 3 (mod 4), q = 5 (mod 8), and q = 9 (mod 16),
   plus a generic constant-time algorithm that works for any prime
   modulus.

   In addition, it gives an optimized is_square method for GF(p^2).



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 100]

Internet-Draft                hash-to-curve                    June 2022


I.1.  sqrt for q = 3 (mod 4)

   sqrt_3mod4(x)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.

   Input: x, an element of F.
   Output: z, an element of F such that (z^2) == x, if x is square in F.

   Constants:
   1. c1 = (q + 1) / 4     # Integer arithmetic

   Procedure:
   1. return x^c1

I.2.  sqrt for q = 5 (mod 8)

   sqrt_5mod8(x)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.

   Input: x, an element of F.
   Output: z, an element of F such that (z^2) == x, if x is square in F.

   Constants:
   1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
   2. c2 = (q + 3) / 8     # Integer arithmetic

   Procedure:
   1. tv1 = x^c2
   2. tv2 = tv1 * c1
   3.   e = (tv1^2) == x
   4.   z = CMOV(tv2, tv1, e)
   5. return z

I.3.  sqrt for q = 9 (mod 16)













Faz-Hernandez, et al.   Expires 17 December 2022              [Page 101]

Internet-Draft                hash-to-curve                    June 2022


   sqrt_9mod16(x)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.

   Input: x, an element of F.
   Output: z, an element of F such that (z^2) == x, if x is square in F.

   Constants:
   1. c1 = sqrt(-1) in F, i.e., (c1^2) == -1 in F
   2. c2 = sqrt(c1) in F, i.e., (c2^2) == c1 in F
   3. c3 = sqrt(-c1) in F, i.e., (c3^2) == -c1 in F
   4. c4 = (q + 7) / 16         # Integer arithmetic

   Procedure:
   1. tv1 = x^c4
   2. tv2 = c1 * tv1
   3. tv3 = c2 * tv1
   4. tv4 = c3 * tv1
   5.  e1 = (tv2^2) == x
   6.  e2 = (tv3^2) == x
   7. tv1 = CMOV(tv1, tv2, e1)  # Select tv2 if (tv2^2) == x
   8. tv2 = CMOV(tv4, tv3, e2)  # Select tv3 if (tv3^2) == x
   9.  e3 = (tv2^2) == x
   10.  z = CMOV(tv1, tv2, e3)  # Select the sqrt from tv1 and tv2
   11. return z

I.4.  Constant-time Tonelli-Shanks algorithm

   This algorithm is a constant-time version of the classic Tonelli-
   Shanks algorithm ([C93], Algorithm 1.5.1) due to Sean Bowe, Jack
   Grigg, and Eirik Ogilvie-Wigley [jubjub-fq], adapted and optimized by
   Michael Scott.

   This algorithm applies to GF(p) for any p.  Note, however, that the
   special-purpose algorithms given in the prior sections are faster,
   when they apply.














Faz-Hernandez, et al.   Expires 17 December 2022              [Page 102]

Internet-Draft                hash-to-curve                    June 2022


   sqrt_ts_ct(x)

   Parameters:
   - F, a finite field of characteristic p and order q = p^m.

   Input x, an element of F.
   Output: z, an element of F such that z^2 == x, if x is square in F.

   Constants:
   1. c1, the largest integer such that 2^c1 divides q - 1.
   2. c2 = (q - 1) / (2^c1)        # Integer arithmetic
   3. c3 = (c2 - 1) / 2            # Integer arithmetic
   4. c4, a non-square value in F
   5. c5 = c4^c2 in F

   Procedure:
   1.  z = x^c3
   2.  t = z * z
   3.  t = t * x
   4.  z = z * x
   5.  b = t
   6.  c = c5
   7.  for i in (c1, c1 - 1, ..., 2):
   8.    for j in (1, 2, ..., i - 2):
   9.      b = b * b
   10.   e = b == 1
   11.   zt = z * c
   12.   z = CMOV(zt, z, e)
   13.   c = c * c
   14.   tt = t * c
   15.   t = CMOV(tt, t, e)
   16.   b = t
   17. return z

I.5.  is_square for F = GF(p^2)

   The following is_square method applies to any field F = GF(p^2) with
   basis (1, I) represented as described in Section 2.1, i.e., an
   element x = (x_1, x_2) = x_1 + x_2 * I.

   Other optimizations of this type are possible in other extension
   fields; see, e.g., [AR13] for more information.









Faz-Hernandez, et al.   Expires 17 December 2022              [Page 103]

Internet-Draft                hash-to-curve                    June 2022


   is_square(x)

   Parameters:
   - F, an extension field of characteristic p and order q = p^2
     with basis (1, I).

   Input: x, an element of F.
   Output: True if x is square in F, and False otherwise.

   Constants:
   1. c1 = (p - 1) / 2         # Integer arithmetic

   Procedure:
   1. tv1 = x_1^2
   2. tv2 = I * x_2
   3. tv2 = tv2^2
   4. tv1 = tv1 - tv2
   5. tv1 = tv1^c1
   6.  e1 = tv1 != -1          # Note: -1 in F
   7. return e1

Appendix J.  Suite test vectors

   This section gives test vectors for each suite defined in Section 8.
   The test vectors in this section were generated using code that is
   available from [hash2curve-repo].

   Each test vector in this section lists values computed by the
   appropriate encoding function, with variable names defined as in
   Section 3.  For example, for a suite whose encoding type is random
   oracle, the test vector gives the value for msg, u, Q0, Q1, and the
   output point P.

J.1.  NIST P-256

J.1.1.  P256_XMD:SHA-256_SSWU_RO_

   suite   = P256_XMD:SHA-256_SSWU_RO_
   dst     = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_RO_

   msg     =
   P.x     = 2c15230b26dbc6fc9a37051158c95b79656e17a1a920b11394ca91
             c44247d3e4
   P.y     = 8a7a74985cc5c776cdfe4b1f19884970453912e9d31528c060be9a
             b5c43e8415
   u[0]    = ad5342c66a6dd0ff080df1da0ea1c04b96e0330dd89406465eeba1
             1582515009
   u[1]    = 8c0f1d43204bd6f6ea70ae8013070a1518b43873bcd850aafa0a9e



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 104]

Internet-Draft                hash-to-curve                    June 2022


             220e2eea5a
   Q0.x    = ab640a12220d3ff283510ff3f4b1953d09fad35795140b1c5d64f3
             13967934d5
   Q0.y    = dccb558863804a881d4fff3455716c836cef230e5209594ddd33d8
             5c565b19b1
   Q1.x    = 51cce63c50d972a6e51c61334f0f4875c9ac1cd2d3238412f84e31
             da7d980ef5
   Q1.y    = b45d1a36d00ad90e5ec7840a60a4de411917fbe7c82c3949a6e699
             e5a1b66aac

   msg     = abc
   P.x     = 0bb8b87485551aa43ed54f009230450b492fead5f1cc91658775da
             c4a3388a0f
   P.y     = 5c41b3d0731a27a7b14bc0bf0ccded2d8751f83493404c84a88e71
             ffd424212e
   u[0]    = afe47f2ea2b10465cc26ac403194dfb68b7f5ee865cda61e9f3e07
             a537220af1
   u[1]    = 379a27833b0bfe6f7bdca08e1e83c760bf9a338ab335542704edcd
             69ce9e46e0
   Q0.x    = 5219ad0ddef3cc49b714145e91b2f7de6ce0a7a7dc7406c7726c7e
             373c58cb48
   Q0.y    = 7950144e52d30acbec7b624c203b1996c99617d0b61c2442354301
             b191d93ecf
   Q1.x    = 019b7cb4efcfeaf39f738fe638e31d375ad6837f58a852d032ff60
             c69ee3875f
   Q1.y    = 589a62d2b22357fed5449bc38065b760095ebe6aeac84b01156ee4
             252715446e

   msg     = abcdef0123456789
   P.x     = 65038ac8f2b1def042a5df0b33b1f4eca6bff7cb0f9c6c15268118
             64e544ed80
   P.y     = cad44d40a656e7aff4002a8de287abc8ae0482b5ae825822bb870d
             6df9b56ca3
   u[0]    = 0fad9d125a9477d55cf9357105b0eb3a5c4259809bf87180aa01d6
             51f53d312c
   u[1]    = b68597377392cd3419d8fcc7d7660948c8403b19ea78bbca4b133c
             9d2196c0fb
   Q0.x    = a17bdf2965eb88074bc01157e644ed409dac97cfcf0c61c998ed0f
             a45e79e4a2
   Q0.y    = 4f1bc80c70d411a3cc1d67aeae6e726f0f311639fee560c7f5a664
             554e3c9c2e
   Q1.x    = 7da48bb67225c1a17d452c983798113f47e438e4202219dd0715f8
             419b274d66
   Q1.y    = b765696b2913e36db3016c47edb99e24b1da30e761a8a3215dc0ec
             4d8f96e6f9

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 105]

Internet-Draft                hash-to-curve                    June 2022


             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 4be61ee205094282ba8a2042bcb48d88dfbb609301c49aa8b07853
             3dc65a0b5d
   P.y     = 98f8df449a072c4721d241a3b1236d3caccba603f916ca680f4539
             d2bfb3c29e
   u[0]    = 3bbc30446f39a7befad080f4d5f32ed116b9534626993d2cc5033f
             6f8d805919
   u[1]    = 76bb02db019ca9d3c1e02f0c17f8baf617bbdae5c393a81d9ce11e
             3be1bf1d33
   Q0.x    = c76aaa823aeadeb3f356909cb08f97eee46ecb157c1f56699b5efe
             bddf0e6398
   Q0.y    = 776a6f45f528a0e8d289a4be12c4fab80762386ec644abf2bffb9b
             627e4352b1
   Q1.x    = 418ac3d85a5ccc4ea8dec14f750a3a9ec8b85176c95a7022f39182
             6794eb5a75
   Q1.y    = fd6604f69e9d9d2b74b072d14ea13050db72c932815523305cb9e8
             07cc900aff

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 457ae2981f70ca85d8e24c308b14db22f3e3862c5ea0f652ca38b5
             e49cd64bc5
   P.y     = ecb9f0eadc9aeed232dabc53235368c1394c78de05dd96893eefa6
             2b0f4757dc
   u[0]    = 4ebc95a6e839b1ae3c63b847798e85cb3c12d3817ec6ebc10af6ee
             51adb29fec
   u[1]    = 4e21af88e22ea80156aff790750121035b3eefaa96b425a8716e0d
             20b4e269ee
   Q0.x    = d88b989ee9d1295df413d4456c5c850b8b2fb0f5402cc5c4c7e815
             412e926db8
   Q0.y    = bb4a1edeff506cf16def96afff41b16fc74f6dbd55c2210e5b8f01
             1ba32f4f40
   Q1.x    = a281e34e628f3a4d2a53fa87ff973537d68ad4fbc28d3be5e8d9f6
             a2571c5a4b
   Q1.y    = f6ed88a7aab56a488100e6f1174fa9810b47db13e86be999644922
             961206e184

J.1.2.  P256_XMD:SHA-256_SSWU_NU_





Faz-Hernandez, et al.   Expires 17 December 2022              [Page 106]

Internet-Draft                hash-to-curve                    June 2022


   suite   = P256_XMD:SHA-256_SSWU_NU_
   dst     = QUUX-V01-CS02-with-P256_XMD:SHA-256_SSWU_NU_

   msg     =
   P.x     = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a
             ce8ddd14d1
   P.y     = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4
             f011ddc99b
   u[0]    = b22d487045f80e9edcb0ecc8d4bf77833e2bf1f3a54004d7df1d57
             f4802d311f
   Q.x     = f871caad25ea3b59c16cf87c1894902f7e7b2c822c3d3f73596c5a
             ce8ddd14d1
   Q.y     = 87b9ae23335bee057b99bac1e68588b18b5691af476234b8971bc4
             f011ddc99b

   msg     = abc
   P.x     = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92
             b5237befa4
   P.y     = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac
             7b47157866
   u[0]    = c7f96eadac763e176629b09ed0c11992225b3a5ae99479760601cb
             d69c221e58
   Q.x     = fc3f5d734e8dce41ddac49f47dd2b8a57257522a865c124ed02b92
             b5237befa4
   Q.y     = fe4d197ecf5a62645b9690599e1d80e82c500b22ac705a0b421fac
             7b47157866

   msg     = abcdef0123456789
   P.x     = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17
             7e5d8faa84
   P.y     = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012
             b6fcee7f97
   u[0]    = 314e8585fa92068b3ea2c3bab452d4257b38be1c097d58a2189045
             6c2929614d
   Q.x     = f164c6674a02207e414c257ce759d35eddc7f55be6d7f415e2cc17
             7e5d8faa84
   Q.y     = 3aa274881d30db70485368c0467e97da0e73c18c1d00f34775d012
             b6fcee7f97

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 324532006312be4f162614076460315f7a54a6f85544da773dc659
             aca0311853
   P.y     = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4
             b1af6ef883
   u[0]    = 752d8eaa38cd785a799a31d63d99c2ae4261823b4a367b133b2c66
             27f48858ab



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 107]

Internet-Draft                hash-to-curve                    June 2022


   Q.x     = 324532006312be4f162614076460315f7a54a6f85544da773dc659
             aca0311853
   Q.y     = 8d8197374bcd52de2acfefc8a54fe2c8d8bebd2a39f16be9b710e4
             b1af6ef883

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b
             0691bea5d9
   P.y     = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd
             cb16f7ef7b
   u[0]    = 0e1527840b9df2dfbef966678ff167140f2b27c4dccd884c25014d
             ce0e41dfa3
   Q.x     = 5c4bad52f81f39c8e8de1260e9a06d72b8b00a0829a8ea004a610b
             0691bea5d9
   Q.y     = c801e7c0782af1f74f24fc385a8555da0582032a3ce038de637ccd
             cb16f7ef7b

J.2.  NIST P-384

J.2.1.  P384_XMD:SHA-384_SSWU_RO_

   suite   = P384_XMD:SHA-384_SSWU_RO_
   dst     = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_RO_

   msg     =
   P.x     = eb9fe1b4f4e14e7140803c1d99d0a93cd823d2b024040f9c067a8e
             ca1f5a2eeac9ad604973527a356f3fa3aeff0e4d83
   P.y     = 0c21708cff382b7f4643c07b105c2eaec2cead93a917d825601e63
             c8f21f6abd9abc22c93c2bed6f235954b25048bb1a
   u[0]    = 25c8d7dc1acd4ee617766693f7f8829396065d1b447eedb155871f
             effd9c6653279ac7e5c46edb7010a0e4ff64c9f3b4
   u[1]    = 59428be4ed69131df59a0c6a8e188d2d4ece3f1b2a3a02602962b4
             7efa4d7905945b1e2cc80b36aa35c99451073521ac
   Q0.x    = e4717e29eef38d862bee4902a7d21b44efb58c464e3e1f0d03894d
             94de310f8ffc6de86786dd3e15a1541b18d4eb2846
   Q0.y    = 6b95a6e639822312298a47526bb77d9cd7bcf76244c991c8cd7007
             5e2ee6e8b9a135c4a37e3c0768c7ca871c0ceb53d4
   Q1.x    = 509527cfc0750eedc53147e6d5f78596c8a3b7360e0608e2fab056
             3a1670d58d8ae107c9f04bcf90e89489ace5650efd



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 108]

Internet-Draft                hash-to-curve                    June 2022


   Q1.y    = 33337b13cb35e173fdea4cb9e8cce915d836ff57803dbbeb7998aa
             49d17df2ff09b67031773039d09fbd9305a1566bc4

   msg     = abc
   P.x     = e02fc1a5f44a7519419dd314e29863f30df55a514da2d655775a81
             d413003c4d4e7fd59af0826dfaad4200ac6f60abe1
   P.y     = 01f638d04d98677d65bef99aef1a12a70a4cbb9270ec55248c0453
             0d8bc1f8f90f8a6a859a7c1f1ddccedf8f96d675f6
   u[0]    = 53350214cb6bef0b51abb791b1c4209a2b4c16a0c67e1ab1401017
             fad774cd3b3f9a8bcdf7f6229dd8dd5a075cb149a0
   u[1]    = c0473083898f63e03f26f14877a2407bd60c75ad491e7d26cbc6cc
             5ce815654075ec6b6898c7a41d74ceaf720a10c02e
   Q0.x    = fc853b69437aee9a19d5acf96a4ee4c5e04cf7b53406dfaa2afbdd
             7ad2351b7f554e4bbc6f5db4177d4d44f933a8f6ee
   Q0.y    = 7e042547e01834c9043b10f3a8221c4a879cb156f04f72bfccab0c
             047a304e30f2aa8b2e260d34c4592c0c33dd0c6482
   Q1.x    = 57912293709b3556b43a2dfb137a315d256d573b82ded120ef8c78
             2d607c05d930d958e50cb6dc1cc480b9afc38c45f1
   Q1.y    = de9387dab0eef0bda219c6f168a92645a84665c4f2137c14270fb4
             24b7532ff84843c3da383ceea24c47fa343c227bb8

   msg     = abcdef0123456789
   P.x     = bdecc1c1d870624965f19505be50459d363c71a699a496ab672f9a
             5d6b78676400926fbceee6fcd1780fe86e62b2aa89
   P.y     = 57cf1f99b5ee00f3c201139b3bfe4dd30a653193778d89a0accc5e
             0f47e46e4e4b85a0595da29c9494c1814acafe183c
   u[0]    = aab7fb87238cf6b2ab56cdcca7e028959bb2ea599d34f68484139d
             de85ec6548a6e48771d17956421bdb7790598ea52e
   u[1]    = 26e8d833552d7844d167833ca5a87c35bcfaa5a0d86023479fb28e
             5cd6075c18b168bf1f5d2a0ea146d057971336d8d1
   Q0.x    = 0ceece45b73f89844671df962ad2932122e878ad2259e650626924
             e4e7f132589341dec1480ebcbbbe3509d11fb570b7
   Q0.y    = fafd71a3115298f6be4ae5c6dfc96c400cfb55760f185b7b03f3fa
             45f3f91eb65d27628b3c705cafd0466fafa54883ce
   Q1.x    = dea1be8d3f9be4cbf4fab9d71d549dde76875b5d9b876832313a08
             3ec81e528cbc2a0a1d0596b3bcb0ba77866b129776
   Q1.y    = eb15fe71662214fb03b65541f40d3eb0f4cf5c3b559f647da138c9
             f9b7484c48a08760e02c16f1992762cb7298fa52cf

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 03c3a9f401b78c6c36a52f07eeee0ec1289f178adf78448f43a385
             0e0456f5dd7f7633dd31676d990eda32882ab486c0
   P.y     = cc183d0d7bdfd0a3af05f50e16a3f2de4abbc523215bf57c848d5e
             a662482b8c1f43dc453a93b94a8026db58f3f5d878
   u[0]    = 04c00051b0de6e726d228c85bf243bf5f4789efb512b22b498cde3
             821db9da667199b74bd5a09a79583c6d353a3bb41c



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 109]

Internet-Draft                hash-to-curve                    June 2022


   u[1]    = 97580f218255f899f9204db64cd15e6a312cb4d8182375d1e5157c
             8f80f41d6a1a4b77fb1ded9dce56c32058b8d5202b
   Q0.x    = 051a22105e0817a35d66196338c8d85bd52690d79bba373ead8a86
             dd9899411513bb9f75273f6483395a7847fb21edb4
   Q0.y    = f168295c1bbcff5f8b01248e9dbc885335d6d6a04aea960f7384f7
             46ba6502ce477e624151cc1d1392b00df0f5400c06
   Q1.x    = 6ad7bc8ed8b841efd8ad0765c8a23d0b968ec9aa360a558ff33500
             f164faa02bee6c704f5f91507c4c5aad2b0dc5b943
   Q1.y    = 47313cc0a873ade774048338fc34ca5313f96bbf6ae22ac6ef475d
             85f03d24792dc6afba8d0b4a70170c1b4f0f716629

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 7b18d210b1f090ac701f65f606f6ca18fb8d081e3bc6cbd937c560
             4325f1cdea4c15c10a54ef303aabf2ea58bd9947a4
   P.y     = ea857285a33abb516732915c353c75c576bf82ccc96adb63c094dd
             e580021eddeafd91f8c0bfee6f636528f3d0c47fd2
   u[0]    = 480cb3ac2c389db7f9dac9c396d2647ae946db844598971c26d1af
             d53912a1491199c0a5902811e4b809c26fcd37a014
   u[1]    = d28435eb34680e148bf3908536e42231cba9e1f73ae2c6902a222a
             89db5c49c97db2f8fa4d4cd6e424b17ac60bdb9bb6
   Q0.x    = 42e6666f505e854187186bad3011598d9278b9d6e3e4d2503c3d23
             6381a56748dec5d139c223129b324df53fa147c4df
   Q0.y    = 8ee51dbda46413bf621838cc935d18d617881c6f33f3838a79c767
             a1e5618e34b22f79142df708d2432f75c7366c8512
   Q1.x    = 4ff01ceeba60484fa1bc0d825fe1e5e383d8f79f1e5bb78e5fb26b
             7a7ef758153e31e78b9d60ce75c5e32e43869d4e12
   Q1.y    = 0f84b978fac8ceda7304b47e229d6037d32062e597dc7a9b95bcd9
             af441f3c56c619a901d21635f9ec6ab4710b9fcd0e

J.2.2.  P384_XMD:SHA-384_SSWU_NU_

   suite   = P384_XMD:SHA-384_SSWU_NU_
   dst     = QUUX-V01-CS02-with-P384_XMD:SHA-384_SSWU_NU_

   msg     =
   P.x     = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725
             6b033ef61058029a3bfb13c1c7ececd6641881ae20
   P.y     = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379
             3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 110]

Internet-Draft                hash-to-curve                    June 2022


   u[0]    = bc7dc1b2cdc5d588a66de3276b0f24310d4aca4977efda7d6272e1
             be25187b001493d267dc53b56183c9e28282368e60
   Q.x     = de5a893c83061b2d7ce6a0d8b049f0326f2ada4b966dc7e7292725
             6b033ef61058029a3bfb13c1c7ececd6641881ae20
   Q.y     = 63f46da6139785674da315c1947e06e9a0867f5608cf24724eb379
             3a1f5b3809ee28eb21a0c64be3be169afc6cdb38ca

   msg     = abc
   P.x     = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f
             b4bd5194583f40d052f32df66afe5249c9915d139b
   P.y     = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88
             24bc5118613f0ac8de04a1041d90ff8a5ad555f96c
   u[0]    = 9de6cf41e6e41c03e4a7784ac5c885b4d1e49d6de390b3cdd5a1ac
             5dd8c40afb3dfd7bb2686923bab644134483fc1926
   Q.x     = 1f08108b87e703c86c872ab3eb198a19f2b708237ac4be53d7929f
             b4bd5194583f40d052f32df66afe5249c9915d139b
   Q.y     = 1369dc8d5bf038032336b989994874a2270adadb67a7fcc32f0f88
             24bc5118613f0ac8de04a1041d90ff8a5ad555f96c

   msg     = abcdef0123456789
   P.x     = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1
             d813c2d7963c7be81aa6db39e78ff315a10115c0d0
   P.y     = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b
             13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f
   u[0]    = 84e2d430a5e2543573e58e368af41821ca3ccc97baba7e9aab51a8
             4543d5a0298638a22ceee6090d9d642921112af5b7
   Q.x     = 4dac31ec8a82ee3c02ba2d7c9fa431f1e59ffe65bf977b948c59e1
             d813c2d7963c7be81aa6db39e78ff315a10115c0d0
   Q.y     = 845333cdb5702ad5c525e603f302904d6fc84879f0ef2ee2014a6b
             13edd39131bfd66f7bd7cdc2d9ccf778f0c8892c3f

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9
             ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d
   P.y     = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801
             579820420b5546b579008df2145fd770f584a1a54c
   u[0]    = 504e4d5a529333b9205acaa283107bd1bffde753898f7744161f7d
             d19ba57fbb6a64214a2e00ddd2613d76cd508ddb30
   Q.x     = 13c1f8c52a492183f7c28e379b0475486718a7e3ac1dfef39283b9
             ce5fb02b73f70c6c1f3dfe0c286b03e2af1af12d1d
   Q.y     = 57e101887e73e40eab8963324ed16c177d55eb89f804ec9df06801
             579820420b5546b579008df2145fd770f584a1a54c

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 111]

Internet-Draft                hash-to-curve                    June 2022


             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9
             bf1204537bdde73c7cefb752a6ed5ebcd44e183302
   P.y     = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b
             5327943eca95d90b23b009ba45f58b72906f2a99e2
   u[0]    = 7b01ce9b8c5a60d9fbc202d6dde92822e46915d8c17e03fcb92ece
             1ed6074d01e149fc9236def40d673de903c1d4c166
   Q.x     = af129727a4207a8cb9e9dce656d88f79fce25edbcea350499d65e9
             bf1204537bdde73c7cefb752a6ed5ebcd44e183302
   Q.y     = ce68a3d5e161b2e6a968e4ddaa9e51504ad1516ec170c7eef3ca6b
             5327943eca95d90b23b009ba45f58b72906f2a99e2

J.3.  NIST P-521

J.3.1.  P521_XMD:SHA-512_SSWU_RO_

   suite   = P521_XMD:SHA-512_SSWU_RO_
   dst     = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_RO_

   msg     =
   P.x     = 00fd767cebb2452030358d0e9cf907f525f50920c8f607889a6a35
             680727f64f4d66b161fafeb2654bea0d35086bec0a10b30b14adef
             3556ed9f7f1bc23cecc9c088
   P.y     = 0169ba78d8d851e930680322596e39c78f4fe31b97e57629ef6460
             ddd68f8763fd7bd767a4e94a80d3d21a3c2ee98347e024fc73ee1c
             27166dc3fe5eeef782be411d
   u[0]    = 01e5f09974e5724f25286763f00ce76238c7a6e03dc396600350ee
             2c4135fb17dc555be99a4a4bae0fd303d4f66d984ed7b6a3ba3860
             93752a855d26d559d69e7e9e
   u[1]    = 00ae593b42ca2ef93ac488e9e09a5fe5a2f6fb330d18913734ff60
             2f2a761fcaaf5f596e790bcc572c9140ec03f6cccc38f767f1c197
             5a0b4d70b392d95a0c7278aa
   Q0.x    = 00b70ae99b6339fffac19cb9bfde2098b84f75e50ac1e80d6acb95
             4e4534af5f0e9c4a5b8a9c10317b8e6421574bae2b133b4f2b8c6c
             e4b3063da1d91d34fa2b3a3c
   Q0.y    = 007f368d98a4ddbf381fb354de40e44b19e43bb11a1278759f4ea7
             b485e1b6db33e750507c071250e3e443c1aaed61f2c28541bb54b1
             b456843eda1eb15ec2a9b36e
   Q1.x    = 01143d0e9cddcdacd6a9aafe1bcf8d218c0afc45d4451239e821f5
             d2a56df92be942660b532b2aa59a9c635ae6b30e803c45a6ac8714
             32452e685d661cd41cf67214
   Q1.y    = 00ff75515df265e996d702a5380defffab1a6d2bc232234c7bcffa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 112]

Internet-Draft                hash-to-curve                    June 2022


             433cd8aa791fbc8dcf667f08818bffa739ae25773b32073213cae9
             a0f2a917a0b1301a242dda0c

   msg     = abc
   P.x     = 002f89a1677b28054b50d15e1f81ed6669b5a2158211118ebdef8a
             6efc77f8ccaa528f698214e4340155abc1fa08f8f613ef14a04371
             7503d57e267d57155cf784a4
   P.y     = 010e0be5dc8e753da8ce51091908b72396d3deed14ae166f66d8eb
             f0a4e7059ead169ea4bead0232e9b700dd380b316e9361cfdba55a
             08c73545563a80966ecbb86d
   u[0]    = 003d00c37e95f19f358adeeaa47288ec39998039c3256e13c2a4c0
             0a7cb61a34c8969472960150a27276f2390eb5e53e47ab193351c2
             d2d9f164a85c6a5696d94fe8
   u[1]    = 01f3cbd3df3893a45a2f1fecdac4d525eb16f345b03e2820d69bc5
             80f5cbe9cb89196fdf720ef933c4c0361fcfe29940fd0db0a5da6b
             afb0bee8876b589c41365f15
   Q0.x    = 01b254e1c99c835836f0aceebba7d77750c48366ecb07fb658e4f5
             b76e229ae6ca5d271bb0006ffcc42324e15a6d3daae587f9049de2
             dbb0494378ffb60279406f56
   Q0.y    = 01845f4af72fc2b1a5a2fe966f6a97298614288b456cfc385a425b
             686048b25c952fbb5674057e1eb055d04568c0679a8e2dda3158dc
             16ac598dbb1d006f5ad915b0
   Q1.x    = 007f08e813c620e527c961b717ffc74aac7afccb9158cebc347d57
             15d5c2214f952c97e194f11d114d80d3481ed766ac0a3dba3eb73f
             6ff9ccb9304ad10bbd7b4a36
   Q1.y    = 0022468f92041f9970a7cc025d71d5b647f822784d29ca7b3bc3b0
             829d6bb8581e745f8d0cc9dc6279d0450e779ac2275c4c3608064a
             d6779108a7828ebd9954caeb

   msg     = abcdef0123456789
   P.x     = 006e200e276a4a81760099677814d7f8794a4a5f3658442de63c18
             d2244dcc957c645e94cb0754f95fcf103b2aeaf94411847c24187b
             89fb7462ad3679066337cbc4
   P.y     = 001dd8dfa9775b60b1614f6f169089d8140d4b3e4012949b52f98d
             b2deff3e1d97bf73a1fa4d437d1dcdf39b6360cc518d8ebcc0f899
             018206fded7617b654f6b168
   u[0]    = 00183ee1a9bbdc37181b09ec336bcaa34095f91ef14b66b1485c16
             6720523dfb81d5c470d44afcb52a87b704dbc5c9bc9d0ef524dec2
             9884a4795f55c1359945baf3
   u[1]    = 00504064fd137f06c81a7cf0f84aa7e92b6b3d56c2368f0a08f447
             76aa8930480da1582d01d7f52df31dca35ee0a7876500ece3d8fe0
             293cd285f790c9881c998d5e
   Q0.x    = 0021482e8622aac14da60e656043f79a6a110cbae5012268a62dd6
             a152c41594549f373910ebed170ade892dd5a19f5d687fae7095a4
             61d583f8c4295f7aaf8cd7da
   Q0.y    = 0177e2d8c6356b7de06e0b5712d8387d529b848748e54a8bc0ef5f
             1475aa569f8f492fa85c3ad1c5edc51faf7911f11359bfa2a12d2e
             f0bd73df9cb5abd1b101c8b1



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 113]

Internet-Draft                hash-to-curve                    June 2022


   Q1.x    = 00abeafb16fdbb5eb95095678d5a65c1f293291dfd20a3751dbe05
             d0a9bfe2d2eef19449fe59ec32cdd4a4adc3411177c0f2dffd0159
             438706159a1bbd0567d9b3d0
   Q1.y    = 007cc657f847db9db651d91c801741060d63dab4056d0a1d3524e2
             eb0e819954d8f677aa353bd056244a88f00017e00c3ce8beeedb43
             82d83d74418bd48930c6c182

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 01b264a630bd6555be537b000b99a06761a9325c53322b65bdc41b
             f196711f9708d58d34b3b90faf12640c27b91c70a507998e559406
             48caa8e71098bf2bc8d24664
   P.y     = 01ea9f445bee198b3ee4c812dcf7b0f91e0881f0251aab272a1220
             1fd89b1a95733fd2a699c162b639e9acdcc54fdc2f6536129b6beb
             0432be01aa8da02df5e59aaa
   u[0]    = 0159871e222689aad7694dc4c3480a49807b1eedd9c8cb4ae1b219
             d5ba51655ea5b38e2e4f56b36bf3e3da44a7b139849d28f598c816
             fe1bc7ed15893b22f63363c3
   u[1]    = 004ef0cffd475152f3858c0a8ccbdf7902d8261da92744e98df9b7
             fadb0a5502f29c5086e76e2cf498f47321434a40b1504911552ce4
             4ad7356a04e08729ad9411f5
   Q0.x    = 0005eac7b0b81e38727efcab1e375f6779aea949c3e409b53a1d37
             aa2acbac87a7e6ad24aafbf3c52f82f7f0e21b872e88c55e17b7fa
             21ce08a94ea2121c42c2eb73
   Q0.y    = 00a173b6a53a7420dbd61d4a21a7c0a52de7a5c6ce05f31403bef7
             47d16cc8604a039a73bdd6e114340e55dacd6bea8e217ffbadfb8c
             292afa3e1b2afc839a6ce7bb
   Q1.x    = 01881e3c193a69e4d88d8180a6879b74782a0bc7e529233e9f84bf
             7f17d2f319c36920ffba26f9e57a1e045cc7822c834c239593b6e1
             42a694aa00c757b0db79e5e8
   Q1.y    = 01558b16d396d866e476e001f2dd0758927655450b84e12f154032
             c7c2a6db837942cd9f44b814f79b4d729996ced61eec61d85c6751
             39cbffe3fbf071d2c21cfecb

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 00c12bc3e28db07b6b4d2a2b1167ab9e26fc2fa85c7b0498a17b03
             47edf52392856d7e28b8fa7a2dd004611159505835b687ecf1a764
             857e27e9745848c436ef3925



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 114]

Internet-Draft                hash-to-curve                    June 2022


   P.y     = 01cd287df9a50c22a9231beb452346720bb163344a41c5f5a24e83
             35b6ccc595fd436aea89737b1281aecb411eb835f0b939073fdd1d
             d4d5a2492e91ef4a3c55bcbd
   u[0]    = 0033d06d17bc3b9a3efc081a05d65805a14a3050a0dd4dfb488461
             8eb5c73980a59c5a246b18f58ad022dd3630faa22889fbb8ba1593
             466515e6ab4aeb7381c26334
   u[1]    = 0092290ab99c3fea1a5b8fb2ca49f859994a04faee3301cefab312
             d34227f6a2d0c3322cf76861c6a3683bdaa2dd2a6daa5d6906c663
             e065338b2344d20e313f1114
   Q0.x    = 00041f6eb92af8777260718e4c22328a7d74203350c6c8f5794d99
             d5789766698f459b83d5068276716f01429934e40af3d1111a2278
             0b1e07e72238d2207e5386be
   Q0.y    = 001c712f0182813942b87cab8e72337db017126f52ed797dd23458
             4ac9ae7e80dfe7abea11db02cf1855312eae1447dbaecc9d7e8c88
             0a5e76a39f6258074e1bc2e0
   Q1.x    = 0125c0b69bcf55eab49280b14f707883405028e05c927cd7625d4e
             04115bd0e0e6323b12f5d43d0d6d2eff16dbcf244542f84ec05891
             1260dc3bb6512ab5db285fbd
   Q1.y    = 008bddfb803b3f4c761458eb5f8a0aee3e1f7f68e9d7424405fa69
             172919899317fb6ac1d6903a432d967d14e0f80af63e7035aaae0c
             123e56862ce969456f99f102

J.3.2.  P521_XMD:SHA-512_SSWU_NU_

   suite   = P521_XMD:SHA-512_SSWU_NU_
   dst     = QUUX-V01-CS02-with-P521_XMD:SHA-512_SSWU_NU_

   msg     =
   P.x     = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97
             be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c
             ee2a3672d4f1987d13974705
   P.y     = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd
             12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da
             3286eddf19be47e9c4cf0e91
   u[0]    = 01e4947fe62a4e47792cee2798912f672fff820b2556282d9843b4
             b465940d7683a986f93ccb0e9a191fbc09a6e770a564490d2a4ae5
             1b287ca39f69c3d910ba6a4f
   Q.x     = 01ec604b4e1e3e4c7449b7a41e366e876655538acf51fd40d08b97
             be066f7d020634e906b1b6942f9174b417027c953d75fb6ec64b8c
             ee2a3672d4f1987d13974705
   Q.y     = 00944fc439b4aad2463e5c9cfa0b0707af3c9a42e37c5a57bb4ecd
             12fef9fb21508568aedcdd8d2490472df4bbafd79081c81e99f4da
             3286eddf19be47e9c4cf0e91

   msg     = abc
   P.x     = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e
             b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b
             998311fcf46924ce66a2be9a



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 115]

Internet-Draft                hash-to-curve                    June 2022


   P.y     = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5
             bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9
             45e13d7421ad8121dbc72b1d
   u[0]    = 0019b85ef78596efc84783d42799e80d787591fe7432dee1d9fa2b
             7651891321be732ddf653fa8fefa34d86fb728db569d36b5b6ed39
             83945854b2fc2dc6a75aa25b
   Q.x     = 00c720ab56aa5a7a4c07a7732a0a4e1b909e32d063ae1b58db5f0e
             b5e09f08a9884bff55a2bef4668f715788e692c18c1915cd034a6b
             998311fcf46924ce66a2be9a
   Q.y     = 003570e87f91a4f3c7a56be2cb2a078ffc153862a53d5e03e5dad5
             bccc6c529b8bab0b7dbb157499e1949e4edab21cf5d10b782bc1e9
             45e13d7421ad8121dbc72b1d

   msg     = abcdef0123456789
   P.x     = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838
             7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585
             8f45ee0bf370aba25fc990cc
   P.y     = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e
             25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5
             0c1bae25044fe1dd77744bbd
   u[0]    = 01dba0d7fa26a562ee8a9014ebc2cca4d66fd9de036176aca8fc11
             ef254cd1bc208847ab7701dbca7af328b3f601b11a1737a899575a
             5c14f4dca5aaca45e9935e07
   Q.x     = 00bcaf32a968ff7971b3bbd9ce8edfbee1309e2019d7ff373c3838
             7a782b005dce6ceffccfeda5c6511c8f7f312f343f3a891029c585
             8f45ee0bf370aba25fc990cc
   Q.y     = 00923517e767532d82cb8a0b59705eec2b7779ce05f9181c7d5d5e
             25694ef8ebd4696343f0bc27006834d2517215ecf79482a84111f5
             0c1bae25044fe1dd77744bbd

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe
             9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83
             187da02bf3db967d867ce748
   P.y     = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a
             051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952
             4cf89ad70b5d6e0b4a3b7341
   u[0]    = 00844da980675e1244cb209dcf3ea0aabec23bd54b2cda69fff86e
             b3acc318bf3d01bae96e9cd6f4c5ceb5539df9a7ad7fcc5e9d5469
             6081ba9782f3a0f6d14987e3
   Q.x     = 001ac69014869b6c4ad7aa8c443c255439d36b0e48a0f57b03d6fe
             9c40a66b4e2eaed2a93390679a5cc44b3a91862b34b673f0e92c83
             187da02bf3db967d867ce748
   Q.y     = 00d5603d530e4d62b30fccfa1d90c2206654d74291c1db1c25b86a
             051ee3fffc294e5d56f2e776853406bd09206c63d40f37ad882952
             4cf89ad70b5d6e0b4a3b7341



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 116]

Internet-Draft                hash-to-curve                    June 2022


   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d
             721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68
             7668f15ec178a17e3d27349b
   P.y     = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a
             3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2
             938959a83a1f7dd4a6fd395b
   u[0]    = 01aab1fb7e5cd44ba4d9f32353a383cb1bb9eb763ed40b32bdd5f6
             66988970205998c0e44af6e2b5f6f8e48e969b3f649cae3c6ab463
             e1b274d968d91c02f00cce91
   Q.x     = 01801de044c517a80443d2bd4f503a9e6866750d2f94a22970f62d
             721f96e4310e4a828206d9cdeaa8f2d476705cc3bbc490a6165c68
             7668f15ec178a17e3d27349b
   Q.y     = 0068889ea2e1442245fe42bfda9e58266828c0263119f35a61631a
             3358330f3bb84443fcb54fcd53a1d097fccbe310489b74ee143fc2
             938959a83a1f7dd4a6fd395b

J.4.  curve25519

J.4.1.  curve25519_XMD:SHA-512_ELL2_RO_

   suite   = curve25519_XMD:SHA-512_ELL2_RO_
   dst     = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_RO_

   msg     =
   P.x     = 2de3780abb67e861289f5749d16d3e217ffa722192d16bbd9d1bfb
             9d112b98c0
   P.y     = 3b5dc2a498941a1033d176567d457845637554a2fe7a3507d21abd
             1c1bd6e878
   u[0]    = 005fe8a7b8fef0a16c105e6cadf5a6740b3365e18692a9c05bfbb4
             d97f645a6a
   u[1]    = 1347edbec6a2b5d8c02e058819819bee177077c9d10a4ce165aab0
             fd0252261a
   Q0.x    = 36b4df0c864c64707cbf6cf36e9ee2c09a6cb93b28313c169be295
             61bb904f98
   Q0.y    = 6cd59d664fb58c66c892883cd0eb792e52055284dac3907dd756b4
             5d15c3983d
   Q1.x    = 3fa114783a505c0b2b2fbeef0102853c0b494e7757f2a089d0daae
             7ed9a0db2b



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 117]

Internet-Draft                hash-to-curve                    June 2022


   Q1.y    = 76c0fe7fec932aaafb8eefb42d9cbb32eb931158f469ff3050af15
             cfdbbeff94

   msg     = abc
   P.x     = 2b4419f1f2d48f5872de692b0aca72cc7b0a60915dd70bde432e82
             6b6abc526d
   P.y     = 1b8235f255a268f0a6fa8763e97eb3d22d149343d495da1160eff9
             703f2d07dd
   u[0]    = 49bed021c7a3748f09fa8cdfcac044089f7829d3531066ac9e74e0
             994e05bc7d
   u[1]    = 5c36525b663e63389d886105cee7ed712325d5a97e60e140aba7e2
             ce5ae851b6
   Q0.x    = 16b3d86e056b7970fa00165f6f48d90b619ad618791661b7b5e1ec
             78be10eac1
   Q0.y    = 4ab256422d84c5120b278cbdfc4e1facc5baadffeccecf8ee9bf39
             46106d50ca
   Q1.x    = 7ec29ddbf34539c40adfa98fcb39ec36368f47f30e8f888cc7e86f
             4d46e0c264
   Q1.y    = 10d1abc1cae2d34c06e247f2141ba897657fb39f1080d54f09ce0a
             f128067c74

   msg     = abcdef0123456789
   P.x     = 68ca1ea5a6acf4e9956daa101709b1eee6c1bb0df1de3b90d46023
             82a104c036
   P.y     = 2a375b656207123d10766e68b938b1812a4a6625ff83cb8d5e86f5
             8a4be08353
   u[0]    = 6412b7485ba26d3d1b6c290a8e1435b2959f03721874939b21782d
             f17323d160
   u[1]    = 24c7b46c1c6d9a21d32f5707be1380ab82db1054fde82865d5c9e3
             d968f287b2
   Q0.x    = 71de3dadfe268872326c35ac512164850860567aea0e7325e6b91a
             98f86533ad
   Q0.y    = 26a08b6e9a18084c56f2147bf515414b9b63f1522e1b6c5649f7d4
             b0324296ec
   Q1.x    = 5704069021f61e41779e2ba6b932268316d6d2a6f064f997a22fef
             16d1eaeaca
   Q1.y    = 50483c7540f64fb4497619c050f2c7fe55454ec0f0e79870bb4430
             2e34232210

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 096e9c8bae6c06b554c1ee69383bb0e82267e064236b3a30608d4e
             d20b73ac5a
   P.y     = 1eb5a62612cafb32b16c3329794645b5b948d9f8ffe501d4e26b07
             3fef6de355
   u[0]    = 5e123990f11bbb5586613ffabdb58d47f64bb5f2fa115f8ea8df01
             88e0c9e1b5



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 118]

Internet-Draft                hash-to-curve                    June 2022


   u[1]    = 5e8553eb00438a0bb1e7faa59dec6d8087f9c8011e5fb8ed9df31c
             b6c0d4ac19
   Q0.x    = 7a94d45a198fb5daa381f45f2619ab279744efdd8bd8ed587fc5b6
             5d6cea1df0
   Q0.y    = 67d44f85d376e64bb7d713585230cdbfafc8e2676f7568e0b6ee59
             361116a6e1
   Q1.x    = 30506fb7a32136694abd61b6113770270debe593027a968a01f271
             e146e60c18
   Q1.y    = 7eeee0e706b40c6b5174e551426a67f975ad5a977ee2f01e8e20a6
             d612458c3b

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 1bc61845a138e912f047b5e70ba9606ba2a447a4dade024c8ef3dd
             42b7bbc5fe
   P.y     = 623d05e47b70e25f7f1d51dda6d7c23c9a18ce015fe3548df596ea
             9e38c69bf1
   u[0]    = 20f481e85da7a3bf60ac0fb11ed1d0558fc6f941b3ac5469aa8b56
             ec883d6d7d
   u[1]    = 017d57fd257e9a78913999a23b52ca988157a81b09c5442501d07f
             ed20869465
   Q0.x    = 02d606e2699b918ee36f2818f2bc5013e437e673c9f9b9cdc15fd0
             c5ee913970
   Q0.y    = 29e9dc92297231ef211245db9e31767996c5625dfbf92e1c8107ef
             887365de1e
   Q1.x    = 38920e9b988d1ab7449c0fa9a6058192c0c797bb3d42ac34572434
             1a1aa98745
   Q1.y    = 24dcc1be7c4d591d307e89049fd2ed30aae8911245a9d8554bf603
             2e5aa40d3d

J.4.2.  curve25519_XMD:SHA-512_ELL2_NU_

   suite   = curve25519_XMD:SHA-512_ELL2_NU_
   dst     = QUUX-V01-CS02-with-curve25519_XMD:SHA-512_ELL2_NU_

   msg     =
   P.x     = 1bb913f0c9daefa0b3375378ffa534bda5526c97391952a7789eb9
             76edfe4d08
   P.y     = 4548368f4f983243e747b62a600840ae7c1dab5c723991f85d3a97
             68479f3ec4



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 119]

Internet-Draft                hash-to-curve                    June 2022


   u[0]    = 608d892b641f0328523802a6603427c26e55e6f27e71a91a478148
             d45b5093cd
   Q.x     = 51125222da5e763d97f3c10fcc92ea6860b9ccbbd2eb1285728f56
             6721c1e65b
   Q.y     = 343d2204f812d3dfc5304a5808c6c0d81a903a5d228b342442aa3c
             9ba5520a3d

   msg     = abc
   P.x     = 7c22950b7d900fa866334262fcaea47a441a578df43b894b4625c9
             b450f9a026
   P.y     = 5547bc00e4c09685dcbc6cb6765288b386d8bdcb595fa5a6e3969e
             08097f0541
   u[0]    = 46f5b22494bfeaa7f232cc8d054be68561af50230234d7d1d63d1d
             9abeca8da5
   Q.x     = 7d56d1e08cb0ccb92baf069c18c49bb5a0dcd927eff8dcf75ca921
             ef7f3e6eeb
   Q.y     = 404d9a7dc25c9c05c44ab9a94590e7c3fe2dcec74533a0b24b188a
             5d5dacf429

   msg     = abcdef0123456789
   P.x     = 31ad08a8b0deeb2a4d8b0206ca25f567ab4e042746f792f4b7973f
             3ae2096c52
   P.y     = 405070c28e78b4fa269427c82827261991b9718bd6c6e95d627d70
             1a53c30db1
   u[0]    = 235fe40c443766ce7e18111c33862d66c3b33267efa50d50f9e8e5
             d252a40aaa
   Q.x     = 3fbe66b9c9883d79e8407150e7c2a1c8680bee496c62fabe4619a7
             2b3cabe90f
   Q.y     = 08ec476147c9a0a3ff312d303dbbd076abb7551e5fce82b48ab14b
             433f8d0a7b

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 027877759d155b1997d0d84683a313eb78bdb493271d935b622900
             459d52ceaa
   P.y     = 54d691731a53baa30707f4a87121d5169fb5d587d70fb0292b5830
             dedbec4c18
   u[0]    = 001e92a544463bda9bd04ddbe3d6eed248f82de32f522669efc5dd
             ce95f46f5b
   Q.x     = 227e0bb89de700385d19ec40e857db6e6a3e634b1c32962f370d26
             f84ff19683
   Q.y     = 5f86ff3851d262727326a32c1bf7655a03665830fa7f1b8b1e5a09
             d85bc66e4a

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 120]

Internet-Draft                hash-to-curve                    June 2022


             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 5fd892c0958d1a75f54c3182a18d286efab784e774d1e017ba2fb2
             52998b5dc1
   P.y     = 750af3c66101737423a4519ac792fb93337bd74ee751f19da4cf1e
             94f4d6d0b8
   u[0]    = 1a68a1af9f663592291af987203393f707305c7bac9c8d63d6a729
             bdc553dc19
   Q.x     = 3bcd651ee54d5f7b6013898aab251ee8ecc0688166fce6e9548d38
             472f6bd196
   Q.y     = 1bb36ad9197299f111b4ef21271c41f4b7ecf5543db8bb5931307e
             bdb2eaa465

J.5.  edwards25519

J.5.1.  edwards25519_XMD:SHA-512_ELL2_RO_

   suite   = edwards25519_XMD:SHA-512_ELL2_RO_
   dst     = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_RO_

   msg     =
   P.x     = 3c3da6925a3c3c268448dcabb47ccde5439559d9599646a8260e47
             b1e4822fc6
   P.y     = 09a6c8561a0b22bef63124c588ce4c62ea83a3c899763af26d7953
             02e115dc21
   u[0]    = 03fef4813c8cb5f98c6eef88fae174e6e7d5380de2b007799ac7ee
             712d203f3a
   u[1]    = 780bdddd137290c8f589dc687795aafae35f6b674668d92bf92ae7
             93e6a60c75
   Q0.x    = 6549118f65bb617b9e8b438decedc73c496eaed496806d3b2eb9ee
             60b88e09a7
   Q0.y    = 7315bcc8cf47ed68048d22bad602c6680b3382a08c7c5d3f439a97
             3fb4cf9feb
   Q1.x    = 31dcfc5c58aa1bee6e760bf78cbe71c2bead8cebb2e397ece0f37a
             3da19c9ed2
   Q1.y    = 7876d81474828d8a5928b50c82420b2bd0898d819e9550c5c82c39
             fc9bafa196

   msg     = abc
   P.x     = 608040b42285cc0d72cbb3985c6b04c935370c7361f4b7fbdb1ae7
             f8c1a8ecad
   P.y     = 1a8395b88338f22e435bbd301183e7f20a5f9de643f11882fb237f
             88268a5531



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 121]

Internet-Draft                hash-to-curve                    June 2022


   u[0]    = 5081955c4141e4e7d02ec0e36becffaa1934df4d7a270f70679c78
             f9bd57c227
   u[1]    = 005bdc17a9b378b6272573a31b04361f21c371b256252ae5463119
             aa0b925b76
   Q0.x    = 5c1525bd5d4b4e034512949d187c39d48e8cd84242aa4758956e4a
             dc7d445573
   Q0.y    = 2bf426cf7122d1a90abc7f2d108befc2ef415ce8c2d09695a74072
             40faa01f29
   Q1.x    = 37b03bba828860c6b459ddad476c83e0f9285787a269df2156219b
             7e5c86210c
   Q1.y    = 285ebf5412f84d0ad7bb4e136729a9ffd2195d5b8e73c0dc85110c
             e06958f432

   msg     = abcdef0123456789
   P.x     = 6d7fabf47a2dc03fe7d47f7dddd21082c5fb8f86743cd020f3fb14
             7d57161472
   P.y     = 53060a3d140e7fbcda641ed3cf42c88a75411e648a1add71217f70
             ea8ec561a6
   u[0]    = 285ebaa3be701b79871bcb6e225ecc9b0b32dff2d60424b4c50642
             636a78d5b3
   u[1]    = 2e253e6a0ef658fedb8e4bd6a62d1544fd6547922acb3598ec6b36
             9760b81b31
   Q0.x    = 3ac463dd7fddb773b069c5b2b01c0f6b340638f54ee3bd92d452fc
             ec3015b52d
   Q0.y    = 7b03ba1e8db9ec0b390d5c90168a6a0b7107156c994c674b61fe69
             6cbeb46baf
   Q1.x    = 0757e7e904f5e86d2d2f4acf7e01c63827fde2d363985aa7432106
             f1b3a444ec
   Q1.y    = 50026c96930a24961e9d86aa91ea1465398ff8e42015e2ec1fa397
             d416f6a1c0

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 5fb0b92acedd16f3bcb0ef83f5c7b7a9466b5f1e0d8d217421878e
             a3686f8524
   P.y     = 2eca15e355fcfa39d2982f67ddb0eea138e2994f5956ed37b7f72e
             ea5e89d2f7
   u[0]    = 4fedd25431c41f2a606952e2945ef5e3ac905a42cf64b8b4d4a83c
             533bf321af
   u[1]    = 02f20716a5801b843987097a8276b6d869295b2e11253751ca72c1
             09d37485a9
   Q0.x    = 703e69787ea7524541933edf41f94010a201cc841c1cce60205ec3
             8513458872
   Q0.y    = 32bb192c4f89106466f0874f5fd56a0d6b6f101cb714777983336c
             159a9bec75
   Q1.x    = 0c9077c5c31720ed9413abe59bf49ce768506128d810cb882435aa
             90f713ef6b



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 122]

Internet-Draft                hash-to-curve                    June 2022


   Q1.y    = 7d5aec5210db638c53f050597964b74d6dda4be5b54fa73041bf90
             9ccb3826cb

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 0efcfde5898a839b00997fbe40d2ebe950bc81181afbd5cd6b9618
             aa336c1e8c
   P.y     = 6dc2fc04f266c5c27f236a80b14f92ccd051ef1ff027f26a07f8c0
             f327d8f995
   u[0]    = 6e34e04a5106e9bd59f64aba49601bf09d23b27f7b594e56d5de06
             df4a4ea33b
   u[1]    = 1c1c2cb59fc053f44b86c5d5eb8c1954b64976d0302d3729ff66e8
             4068f5fd96
   Q0.x    = 21091b2e3f9258c7dfa075e7ae513325a94a3d8a28e1b1cb3b5b6f
             5d65675592
   Q0.y    = 41a33d324c89f570e0682cdf7bdb78852295daf8084c669f2cc969
             2896ab5026
   Q1.x    = 4c07ec48c373e39a23bd7954f9e9b66eeab9e5ee1279b867b3d531
             5aa815454f
   Q1.y    = 67ccac7c3cb8d1381242d8d6585c57eabaddbb5dca5243a68a8aeb
             5477d94b3a

J.5.2.  edwards25519_XMD:SHA-512_ELL2_NU_

   suite   = edwards25519_XMD:SHA-512_ELL2_NU_
   dst     = QUUX-V01-CS02-with-edwards25519_XMD:SHA-512_ELL2_NU_

   msg     =
   P.x     = 1ff2b70ecf862799e11b7ae744e3489aa058ce805dd323a936375a
             84695e76da
   P.y     = 222e314d04a4d5725e9f2aff9fb2a6b69ef375a1214eb19021ceab
             2d687f0f9b
   u[0]    = 7f3e7fb9428103ad7f52db32f9df32505d7b427d894c5093f7a0f0
             374a30641d
   Q.x     = 42836f691d05211ebc65ef8fcf01e0fb6328ec9c4737c26050471e
             50803022eb
   Q.y     = 22cb4aaa555e23bd460262d2130d6a3c9207aa8bbb85060928beb2
             63d6d42a95

   msg     = abc



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 123]

Internet-Draft                hash-to-curve                    June 2022


   P.x     = 5f13cc69c891d86927eb37bd4afc6672360007c63f68a33ab423a3
             aa040fd2a8
   P.y     = 67732d50f9a26f73111dd1ed5dba225614e538599db58ba30aaea1
             f5c827fa42
   u[0]    = 09cfa30ad79bd59456594a0f5d3a76f6b71c6787b04de98be5cd20
             1a556e253b
   Q.x     = 333e41b61c6dd43af220c1ac34a3663e1cf537f996bab50ab66e33
             c4bd8e4e19
   Q.y     = 51b6f178eb08c4a782c820e306b82c6e273ab22e258d972cd0c511
             787b2a3443

   msg     = abcdef0123456789
   P.x     = 1dd2fefce934ecfd7aae6ec998de088d7dd03316aa1847198aecf6
             99ba6613f1
   P.y     = 2f8a6c24dd1adde73909cada6a4a137577b0f179d336685c4a955a
             0a8e1a86fb
   u[0]    = 475ccff99225ef90d78cc9338e9f6a6bb7b17607c0c4428937de75
             d33edba941
   Q.x     = 55186c242c78e7d0ec5b6c9553f04c6aeef64e69ec2e824472394d
             a32647cfc6
   Q.y     = 5b9ea3c265ee42256a8f724f616307ef38496ef7eba391c08f99f3
             bea6fa88f0

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 35fbdc5143e8a97afd3096f2b843e07df72e15bfca2eaf6879bf97
             c5d3362f73
   P.y     = 2af6ff6ef5ebba128b0774f4296cb4c2279a074658b083b8dcca91
             f57a603450
   u[0]    = 049a1c8bd51bcb2aec339f387d1ff51428b88d0763a91bcdf69298
             14ac95d03d
   Q.x     = 024b6e1621606dca8071aa97b43dce4040ca78284f2a527dcf5d0f
             bfac2b07e7
   Q.y     = 5102353883d739bdc9f8a3af650342b171217167dcce34f8db5720
             8ec1dfdbf2

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 6e5e1f37e99345887fc12111575fc1c3e36df4b289b8759d23af14



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 124]

Internet-Draft                hash-to-curve                    June 2022


             d774b66bff
   P.y     = 2c90c3d39eb18ff291d33441b35f3262cdd307162cc97c31bfcc7a
             4245891a37
   u[0]    = 3cb0178a8137cefa5b79a3a57c858d7eeeaa787b2781be4a362a2f
             0750d24fa0
   Q.x     = 3e6368cff6e88a58e250c54bd27d2c989ae9b3acb6067f2651ad28
             2ab8c21cd9
   Q.y     = 38fb39f1566ca118ae6c7af42810c0bb9767ae5960abb5a8ca7925
             30bfb9447d

J.6.  curve448

J.6.1.  curve448_XOF:SHAKE256_ELL2_RO_

   suite   = curve448_XOF:SHAKE256_ELL2_RO_
   dst     = QUUX-V01-CS02-with-curve448_XOF:SHAKE256_ELL2_RO_

   msg     =
   P.x     = 5ea5ff623d27c75e73717514134e73e419f831a875ca9e82915fdf
             c7069d0a9f8b532cfb32b1d8dd04ddeedbe3fa1d0d681c01e825d6
             a9ea
   P.y     = afadd8de789f8f8e3516efbbe313a7eba364c939ecba00dabf4ced
             5c563b18e70a284c17d8f46b564c4e6ce11784a3825d9411166221
             28c1
   u[0]    = c704c7b3d3b36614cf3eedd0324fe6fe7d1402c50efd16cff89ff6
             3f50938506280d3843478c08e24f7842f4e3ef45f6e3c4897f9d97
             6148
   u[1]    = c25427dc97fff7a5ad0a78654e2c6c27b1c1127b5b53c7950cd1fd
             6edd2703646b25f341e73deedfebf022d1d3cecd02b93b4d585ead
             3ed7
   Q0.x    = 3ba318806f89c19cc019f51e33eb6b8c038dab892e858ce7c7f2c2
             ac58618d06146a5fef31e49af49588d4d3db1bcf02bd4e4a733e37
             065d
   Q0.y    = b30b4cfc2fd14d9d4b70456c0f5c6f6070be551788893d570e7955
             675a20f6c286d01d6e90d2fb500d2efb8f4e18db7f8268bb9b7fbc
             5975
   Q1.x    = f03a48cf003f63be61ca055fec87c750434da07a15f8aa6210389f
             f85943b5166484339c8bea1af9fc571313d35ed2fbb779408b760c
             4cbd
   Q1.y    = 23943a33b2954dc54b76a8222faf5b7e18405a41f5ecc61bf1b8df
             1f9cbfad057307ed0c7b721f19c0390b8ee3a2dec223671f9ff905
             fda7

   msg     = abc
   P.x     = 9b2f7ce34878d7cebf34c582db14958308ea09366d1ec71f646411
             d3de0ae564d082b06f40cd30dfc08d9fb7cb21df390cf207806ad9
             d0e4
   P.y     = 138a0eef0a4993ea696152ed7db61f7ddb4e8100573591e7466d61



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 125]

Internet-Draft                hash-to-curve                    June 2022


             c0c568ecaec939e36a84d276f34c402526d8989a96e99760c4869e
             d633
   u[0]    = 2dd95593dfee26fe0d218d3d9a0a23d9e1a262fd1d0b602483d084
             15213e75e2db3c69b0a5bc89e71bcefc8c723d2b6a0cf263f02ad2
             aa70
   u[1]    = 272e4c79a1290cc6d2bc4f4f9d31bf7fbe956ca303c04518f117d7
             7c0e9d850796fc3e1e2bcb9c75e8eaaded5e150333cae993186804
             7c9d
   Q0.x    = 26714783887ec444fbade9ae350dc13e8d5a64150679232560726a
             73d36e28bd56766d7d0b0899d79c8d1c889ae333f601c57532ff3c
             4f09
   Q0.y    = 080e486f8f5740dbbe82305160cab9fac247b0b22a54d961de6750
             37c3036fa68464c8756478c322ae0aeb9ba386fe626cebb0bcca46
             840c
   Q1.x    = 0d9741d10421691a8ebc7778b5f623260fdf8b28ae28d776efcb8e
             0d5fbb65139a2f828617835f527cb2ca24a8f5fc8e84378343c43d
             096d
   Q1.y    = 54f4c499bf3d5b154511913f9615bd914969b65cfb74508d7ae5a1
             69e9595b7cbcab9a1485e07b2ce426e4fbed052f03842c4313b7db
             e39a

   msg     = abcdef0123456789
   P.x     = f54ecd14b85a50eeeee0618452df3a75be7bfba11da5118774ae4e
             a55ac204e153f77285d780c4acee6c96abe3577a0c0b00be6e790c
             f194
   P.y     = 935247a64bf78c107069943c7e3ecc52acb27ce4a3230407c83573
             41685ea2152e8c3da93f8cd77da1bddb5bb759c6e7ae7d516dced4
             2850
   u[0]    = 6aab71a38391639f27e49eae8b1cb6b7172a1f478190ece293957e
             7cdb2391e7cc1c4261970d9c1bbf9c3915438f74fbd7eb5cd4d4d1
             7ace
   u[1]    = c80b8380ca47a3bcbf76caa75cef0e09f3d270d5ee8f676cde11ae
             df41aaca6741bd81a86232bd336ccb42efad39f06542bc06a67b65
             909e
   Q0.x    = 946d91bd50c90ef70743e0dd194bddd68bb630f4e67e5b93e15a9b
             94e62cb85134467993501759525c1f4fdbf06f10ddaf817847d735
             e062
   Q0.y    = 185cf511262ec1e9b3c3cbdc015ab93df4e71cbe87766917d81c9f
             3419d480407c1462385122c84982d4dae60c3ae4acce0089e37ad6
             5934
   Q1.x    = 01778f4797b717cd6f83c193b2dfb92a1606a36ede941b0f6ab0ac
             71ad0eac756d17604bf054398887da907e41065d3595f178ae802f
             2087
   Q1.y    = b4ca727d0bda895e0eee7eb3cbc28710fa2e90a73b568cae26bd7c
             2e73b70a9fa0affe1096f0810198890ed65d8935886b6e60dc4c56
             9dc6

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 126]

Internet-Draft                hash-to-curve                    June 2022


             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 5bd67c4f88adf6beb10f7e0d0054659776a55c97b809ec8b310172
             9e104fd0f684e103792f267fd87cc4afc25a073956ef4f268fb028
             24d5
   P.y     = da1f5cb16a352719e4cb064cf47ba72aeba7752d03e8ca2c56229f
             419b4ef378785a5af1a53dd7ab4d467c1f92f7b139b3752faf29c9
             6432
   u[0]    = cb5c27e51f9c18ee8ffdb6be230f4eb4f2c2481963b2293484f08d
             a2241c1ff59f80978e6defe9d70e34abba2fcbe12dc3a1eb2c5d3d
             2e4a
   u[1]    = c895e8afecec5466e126fa70fc4aa784b8009063afb10e3ee06a9b
             22318256aa8693b0c85b955cf2d6540b8ed71e729af1b8d5ca3b11
             6cd7
   Q0.x    = c2d275826d6ad55e41a22318f6b6240f1f862a2e231120ff41eadb
             ec319756032e8cef2a7ac6c10214fa0608c17fcaf61ec2694a8a2b
             358b
   Q0.y    = 93d2e092762b135509840e609d413200df800d99da91d8b8284066
             6cac30e7a3520adbaa4b089bfdc86132e42729f651d022f4782502
             f12c
   Q1.x    = 3c0880ece7244036e9a45944a85599f9809d772f770cc237ac41b2
             1aa71615e4f3bb08f64fca618896e4f6cf5bd92e16b89d2cf6e195
             6bfb
   Q1.y    = 45cce4beb96505cac5976b3d2673641e9bcd18d3462bbb453d293e
             5282740a6389cfeae610adc7bd425c728541ceec83fcc999164af4
             3fb5

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = ea441c10b3636ecedd5c0dfcae96384cc40de8390a0ab648765b45
             08da12c586d55dc981275776507ebca0e4d1bcaa302bb69dcfa31b
             3451
   P.y     = fee0192d49bcc0c28d954763c2cbe739b9265c4bebe3883803c649
             71220cfda60b9ac99ad986cd908c0534b260b5cfca46f6c2b0f3f2
             1bda
   u[0]    = 8cba93a007bb2c801b1769e026b1fa1640b14a34cf3029db3c7fd6
             392745d6fec0f7870b5071d6da4402cedbbde28ae4e50ab30e1049
             a238
   u[1]    = 4223746145069e4b8a981acc3404259d1a2c3ecfed5d864798a89d
             45f81a2c59e2d40eb1d5f0fe11478cbb2bb30246dd388cb932ad7b



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 127]

Internet-Draft                hash-to-curve                    June 2022


             b330
   Q0.x    = 4321ab02a9849128691e9b80a5c5576793a218de14885fddccb91f
             17ceb1646ea00a28b69ad211e1f14f17739612dbde3782319bdf00
             9689
   Q0.y    = 1b8a7b539519eec0ea9f7a46a43822e16cba39a439733d6847ac44
             a806b8adb3e1a75ea48a1228b8937ba85c6cb6ee01046e10cad895
             3b1e
   Q1.x    = 126d744da6a14fddec0f78a9cee4571c1320ac7645b600187812e4
             d7021f98fc4703732c54daec787206e1f34d9dbbf4b292c68160b8
             bfbd
   Q1.y    = 136eebe6020f2389d448923899a1a38a4c8ad74254e0686e91c4f9
             3c1f8f8e1bd619ffb7c1281467882a9c957d22d50f65c5b72b2aee
             11af

J.6.2.  curve448_XOF:SHAKE256_ELL2_NU_

   suite   = curve448_XOF:SHAKE256_ELL2_NU_
   dst     = QUUX-V01-CS02-with-curve448_XOF:SHAKE256_ELL2_NU_

   msg     =
   P.x     = b65e8dbb279fd656f926f68d463b13ca7a982b32f5da9c7cc58afc
             f6199e4729863fb75ca9ae3c95c6887d95a5102637a1c5c40ff0aa
             fadc
   P.y     = ea1ea211cf29eca11c057fe8248181591a19f6ac51d45843a65d4b
             b8b71bc83a64c771ed7686218a278ef1c5d620f3d26b5316218864
             5453
   u[0]    = 242c70f74eac8184116c71630d284cf8a742fc463e710545847ff6
             4d8e9161cb9f599728a18a32dbd8b67c3bec5d64c9b1d2f2cde7b5
             888d
   Q.x     = e6304424de5af3f556d3e645600530c53ad949891c3e60ba041dd5
             f68a93901beff8440164477d348c13d28e27bfcd360c44c80b4c7d
             4cea
   Q.y     = 4160a8f2043a347185406a6a7e50973b98b82edbdfa3209b0e1c90
             118e10eeb45045b0990d4b2b0708a30eca17df40ad53c9100f20c1
             0b44

   msg     = abc
   P.x     = 51aceca4fa95854bbaba58d8a5e17a86c07acadef32e1188cafda2
             6232131800002cc2f27c7aec454e5e0c615bddffb7df6a5f7f0f14
             793f
   P.y     = c590c9246eb28b08dee816d608ef233ea5d76e305dc458774a1e1b
             d880387e6734219e2018e4aa50a49486dce0ba8740065da37e6cf5
             212c
   u[0]    = ef6dcb75b696d325fb36d66b104700df1480c4c17ea9190d447eee
             1e7e4c9b7f36bbfb8ba7ba7c4cb6b07fed16531c1ac7a26a3618b4
             0b34
   Q.x     = de0dc93df9ce7953452f20e270699c1e7dacd5d571c226d77f53b7
             e3053d16f8a81b1601efb362054e973c8e733b663af93f00cb81ba



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 128]

Internet-Draft                hash-to-curve                    June 2022


             f130
   Q.y     = 8c5bdec6fa6690905f6eff966b0f98f5a8161493bd04976684d4ec
             1f4512fa8743d86860b2ff2c5d67e9c145fd906f2cb89ff812c6b9
             883f

   msg     = abcdef0123456789
   P.x     = c6d65987f146b8d0cb5d2c44e1872ac3af1f458f6a8bd8c232ffe8
             b9d09496229a5a27f350eb7d97305bcc4e0f38328718352e8e3129
             ed71
   P.y     = 4d2f901bf333fdc4135b954f20d59207e9f6a4ecf88ce5af11c892
             b44f79766ec4ecc9f60d669b95ca8940f39b1b7044140ac2040c1b
             f659
   u[0]    = 3012ba5d9b3bb648e4613833a26ecaeadb3e8c8bba07fc90ac3da0
             375769289c44d3dc87474b23df7f45f9a4030892cda689e343aeee
             a6ad
   Q.x     = dc29532761f03c24d57f530da4c24acc4c676d185becaa89fcc083
             266541fb7f10ecec91dac64a34cd988274633ae25c4d784aee52de
             47a8
   Q.y     = a5f6da11259c69f2e07fce6a7b6afec4c25bd2df83426765f9c070
             4111da24c6a0550d5c7aac7d648d55f7640d50be99c926195e852a
             daac

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 9b8d008863beb4a02fb9e4efefd2eba867307fb1c7ce01746115d3
             2e1db551bb254e8e3e4532d5c74a83949a69a60519ecc9178083cb
             e943
   P.y     = 346a1fca454d1e67c628437c270ec0f0c4256bb774fe6c0e49de70
             04ff6d9199e2cd99d8f7575a96aafc4dc8db1811ba0a44317581f4
             1371
   u[0]    = fe952ac0149f92436bba12ea2e542aa226f4fc074d79ff462c41b3
             27968a649a495a8a93b6c3044af2273456abb5e166ce4fb8c9b10c
             8c2e
   Q.x     = 512803d89f59c57376e6570cd54c4e901643e089cd9456f549daa4
             372b8b52679860b68aa8bedfaa88970f15ab6098d5f252083ac98a
             58c9
   Q.y     = 3d9b6593c7941a20d76161c9a171f1e507495a08f03dfcae33a2ac
             3602698e46a74d1039b583c984036f590eaa43d20ba5aada3ffb55
             2f77

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 129]

Internet-Draft                hash-to-curve                    June 2022


             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 8746dc34799112d1f20acda9d7f722c9abb29b1fb6b7e9e5669838
             43c20bd7c9bfad21b45c5166b808d2f5d44e188f1fdaf29cdee8a7
             2e4c
   P.y     = 7c1293484c9287c298a1a0600c64347eee8530acf563cd8705e057
             28274d8cd8101835f8003b6f3b78b5beb28f5be188a3d7bce1ec5a
             36b1
   u[0]    = afd3d7ad9d819be7561706e050d4f30b634b203387ab682739365f
             62cd7393ca2cf18cd07a3d3af8dd163f043ac7457c2eb145b4a561
             70a9
   Q.x     = 08aed6480793218034fd3b3b0867943d7e0bd1b6f76b4929e0885b
             d082b84d4449341da6038bb08229ad9eb7d518dff2c7ea50148e70
             a4db
   Q.y     = e00d32244561ebd4b5f4ef70fcac75a06416be0a1c1b304e7bd361
             a6a6586915bb902a323eaf73cf7738e70d34282f61485395ab2833
             d2c1

J.7.  edwards448

J.7.1.  edwards448_XOF:SHAKE256_ELL2_RO_

   suite   = edwards448_XOF:SHAKE256_ELL2_RO_
   dst     = QUUX-V01-CS02-with-edwards448_XOF:SHAKE256_ELL2_RO_

   msg     =
   P.x     = 73036d4a88949c032f01507005c133884e2f0d81f9a950826245dd
             a9e844fc78186c39daaa7147ead3e462cff60e9c6340b58134480b
             4d17
   P.y     = 94c1d61b43728e5d784ef4fcb1f38e1075f3aef5e99866911de5a2
             34f1aafdc26b554344742e6ba0420b71b298671bbeb2b773661863
             4610
   u[0]    = 0847c5ebf957d3370b1f98fde499fb3e659996d9fc9b5707176ade
             785ba72cd84b8a5597c12b1024be5f510fa5ba99642c4cec7f3f69
             d3e7
   u[1]    = f8cbd8a7ae8c8deed071f3ac4b93e7cfcb8f1eac1645d699fd6d38
             81cb295a5d3006d9449ed7cad412a77a1fe61e84a9e41d59ef384d
             6f9a
   Q0.x    = c08177330869db17fb81a5e6e53b36d29086d806269760f2e4caba
             a4015f5dbadb7ca2ba594d96a89d0ca4f0944489e1ef393d53db85
             096f
   Q0.y    = 02e894598c050eeb7195f5791f1a5f65da3776b7534be37640bcbf
             95d4b915bd22333c50387583507169708fbd7bea0d7aa385dcc614
             be9c
   Q1.x    = 770877fd3b6c5503398157b68a9d3609f585f40e1ebebdd69bb0e4
             d3d9aa811995ce75333fdadfa50db886a35959cc59cffd5c9710da
             ca25



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 130]

Internet-Draft                hash-to-curve                    June 2022


   Q1.y    = b27fef77aa6231fbbc27538fa90eaca8abd03eb1e62fdae4ec5e82
             8117c3b8b3ff8c34d0a6e6d79fff16d339b94ae8ede33331d5b464
             c792

   msg     = abc
   P.x     = 4e0158acacffa545adb818a6ed8e0b870e6abc24dfc1dc45cf9a05
             2e98469275d9ff0c168d6a5ac7ec05b742412ee090581f12aa398f
             9f8c
   P.y     = 894d3fa437b2d2e28cdc3bfaade035430f350ec5239b6b406b5501
             da6f6d6210ff26719cad83b63e97ab26a12df6dec851d6bf38e294
             af9a
   u[0]    = 04d975cd938ab49be3e81703d6a57cca84ed80d2ff6d4756d3f229
             47fb5b70ab0231f0087cbfb4b7cae73b41b0c9396b356a4831d9a1
             4322
   u[1]    = 2547ca887ac3db7b5fad3a098aa476e90078afe1358af6c63d677d
             6edfd2100bc004e0f5db94dd2560fc5b308e223241d00488c9ca6b
             0ef2
   Q0.x    = 7544612a97f4419c94ab0f621a1ee8ccf46c6657b8e0778ec9718b
             f4b41bc774487ad87d9b1e617aa49d3a4dd35a3cf57cd390ebf042
             9952
   Q0.y    = d3ab703e60267d796b485bb58a28f934bd0133a6d1bbdfeda5277f
             a293310be262d7f653a5adffa608c37ed45c0e6008e54a16e1a342
             e4df
   Q1.x    = 6262f18d064bc131ade1b8bbcf1cbdf984f4f88153fcc9f94c888a
             f35d5e41aae84c12f169a55d8abf06e6de6c5b23079e587a58cf73
             303e
   Q1.y    = 6d57589e901abe7d947c93ab02c307ad9093ed9a83eb0b6e829fb7
             318d590381ca25f3cc628a36a924a9ddfcf3cbedf94edf3b338ea7
             7403

   msg     = abcdef0123456789
   P.x     = 2c25b4503fadc94b27391933b557abdecc601c13ed51c5de683894
             84f93dbd6c22e5f962d9babf7a39f39f994312f8ca23344847e1fb
             f176
   P.y     = d5e6f5350f430e53a110f5ac7fcc82a96cb865aeca982029522d32
             601e41c042a9dfbdfbefa2b0bdcdc3bc58cca8a7cd546803083d3a
             8548
   u[0]    = 10659ce25588db4e4be6f7c791a79eb21a7f24aaaca76a6ca3b83b
             80aaf95aa328fe7d569a1ac99f9cd216edf3915d72632f1a8b990e
             250c
   u[1]    = 9243e5b6c480683fd533e81f4a778349a309ce00bd163a29eb9fa8
             dbc8f549242bef33e030db21cffacd408d2c4264b93e476c6a8590
             e7aa
   Q0.x    = 1457b60c12e00e47ceb3ce64b57e7c3c61636475443d704a8e2b2a
             b0a5ac7e4b3909435416784e16e19929c653b1bdcd9478a8e5331c
             a9ae
   Q0.y    = 935d9f75f7a0babbc39c0a1c3b412518ed8a24bc2c4886722fb4b7
             d4a747af98e4e2528c75221e2dffd3424abb436e10539a74caaafa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 131]

Internet-Draft                hash-to-curve                    June 2022


             3ea3
   Q1.x    = b44d9e34211b4028f24117e856585ed81448f3c8b934987a1c5939
             c86048737a08d85934fec6b3c2ef9f09cbd365cf22744f2e4ce697
             62a4
   Q1.y    = dc996c1736f4319868f897d9a27c45b02dd3bc6b7ca356a039606e
             5406e131a0bbe8238208b327b00853e8af84b58b13443e70542556
             3323

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = a1861a9464ae31249a0e60bf38791f3663049a3f5378998499a832
             92e159a2fecff838eb9bc6939e5c6ae76eb074ad4aae39b55b72ca
             0b9a
   P.y     = 580a2798c5b904f8adfec5bd29fb49b4633cd9f8c2935eb4a0f12e
             5dfa0285680880296bb729c6405337525fb5ed3dff930c137314f6
             0401
   u[0]    = c80390020e578f009ead417029eff6cd0926110922db63ab98395e
             3bdfdd5d8a65b1a2b8d495dc8c5e59b7f3518731f7dfc0f93ace5d
             ee4b
   u[1]    = 1c4dc6653a445bbef2add81d8e90a6c8591a788deb91d0d3f1519a
             2e4a460313041b77c1b0817f2e80b388e5c3e49f37d787dc1f85e4
             324a
   Q0.x    = 9d355251e245e4b13ed4ea3e5a3c55bf9b7211f1704771f2e1d8f1
             a65610c468b1cf70c6c2ce30dcaad54ad9e5439471ec554b862ec8
             875a
   Q0.y    = 6689ba36a242af69ac2aadb955d15e982d9b04f5d77f7609ebf742
             9587feb7e5ce27490b9c72114509f89565122074e46a614d7fd7c8
             00bd
   Q1.x    = c4b3d3ad4d2d62739a62989532992c1081e9474a201085b4616da5
             706cab824693b9fb428a201bcd1639a4588cc43b9eb841dbca7421
             9b1f
   Q1.y    = 265286f5dee8f3d894b5649da8565b58e96b4cfd44b462a2883ea6
             4dbcda21a00706ea3fea53fc2d769084b0b74589e91d0384d71189
             09fb

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 987c5ac19dd4b47835466a50b2d9feba7c8491b8885a04edf577e1
             5a9f2c98b203ec2cd3e5390b3d20bba0fa6fc3eecefb5029a31723



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 132]

Internet-Draft                hash-to-curve                    June 2022


             4401
   P.y     = 5e273fcfff6b007bb6771e90509275a71ff1480c459ded26fc7b10
             664db0a68aaa98bc7ecb07e49cf05b80ae5ac653fbdd14276bbd35
             ccbc
   u[0]    = 163c79ab0210a4b5e4f44fb19437ea965bf5431ab233ef16606f0b
             03c5f16a3feb7d46a5a675ce8f606e9c2bf74ee5336c54a1e54919
             f13f
   u[1]    = f99666bde4995c4088333d6c2734687e815f80a99c6da02c47df4b
             51f6c9d9ed466b4fecf7d9884990a8e0d0be6907fa437e0b1a27f4
             9265
   Q0.x    = d1a5eba4a332514b69760948af09ceaeddbbb9fd4cb1f19b78349c
             2ee4cf9ee86dbcf9064659a4a0566fe9c34d90aec86f0801edc131
             ad9b
   Q0.y    = 5d0a75a3014c3269c33b1b5da80706a4f097893461df286353484d
             8031cd607c98edc2a846c77a841f057c7251eb45077853c7b20595
             7e52
   Q1.x    = 69583b00dc6b2aced6ffa44630cc8c8cd0dd0649f57588dd0fb1da
             ad2ce132e281d01e3f25ccd3f405be759975c6484268bfe8f5e5f2
             3c30
   Q1.y    = 8418484035f60bdccf48cb488634c2dfb40272123435f7e654fb6f
             254c6c42e7e38f1fa79a637a168a28de6c275232b704f9ded0ff76
             dd94

J.7.2.  edwards448_XOF:SHAKE256_ELL2_NU_

   suite   = edwards448_XOF:SHAKE256_ELL2_NU_
   dst     = QUUX-V01-CS02-with-edwards448_XOF:SHAKE256_ELL2_NU_

   msg     =
   P.x     = eb5a1fc376fd73230af2de0f3374087cc7f279f0460114cf0a6c12
             d6d044c16de34ec2350c34b26bf110377655ab77936869d085406a
             f71e
   P.y     = df5dcea6d42e8f494b279a500d09e895d26ac703d75ca6d118e8ca
             58bf6f608a2a383f292fce1563ff995dce75aede1fdc8e7c0c737a
             e9ad
   u[0]    = 1368aefc0416867ea2cfc515416bcbeecc9ec81c4ecbd52ccdb91e
             06996b3f359bc930eef6743c7a2dd7adb785bc7093ed044efed950
             86d7
   Q.x     = 4b2abf8c0fca49d027c2a81bf73bb5990e05f3e76c7ba137cc0b89
             415ccd55ce7f191cc0c11b0560c1cdc2a8085dd56996079e05a3cd
             8dde
   Q.y     = 82532f5b0cb3bfb8542d3228d055bfe61129dbeae8bace80cf61f1
             7725e8ec8226a24f0e687f78f01da88e3b2715194a03dca7c0a96b
             bf04

   msg     = abc
   P.x     = 4623a64bceaba3202df76cd8b6e3daf70164f3fcbda6d6e340f7fa
             b5cdf89140d955f722524f5fe4d968fef6ba2853ff4ea086c2f67d



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 133]

Internet-Draft                hash-to-curve                    June 2022


             8110
   P.y     = abaac321a169761a8802ab5b5d10061fec1a83c670ac6bc9595470
             0317ee5f82870120e0e2c5a21b12a0c7ad17ebd343363604c4bcec
             afd1
   u[0]    = cda3b0ecfe054c4077007d7300969ec24f4c741300b630ec9188eb
             ab31a5ae0065612ee22d9f793733179ffc2e10c53ca5b539057aaf
             dc2f
   Q.x     = b1ca5bef2f157673a210f56c9b0039db8399e4749585abac64f831
             f74ed1ec5f591928976c687c06d57686bacb98440e77af878349cd
             f2d2
   Q.y     = 5bbfd6a3730d517b03c3cd9e2eed94af12891334ec090e0495c2ed
             c588e9e10b6f63b03a62076808cbcd6da95adfb5af76c136b2d42e
             0dac

   msg     = abcdef0123456789
   P.x     = e9eb562e76db093baa43a31b7edd04ec4aadcef3389a7b9c58a19c
             f87f8ae3d154e134b6b3ed45847a741e33df51903da681629a4b8b
             cc2e
   P.y     = 0cf6606927ad7eb15dbc193993bc7e4dda744b311a8ec4274c8f73
             8f74f605934582474c79260f60280fe35bd37d4347e59184cbfa12
             cbc4
   u[0]    = d36bae98351512c382c7a3e1eba22497574f11fef9867901b1a270
             0b39fa2cd0d38ed4380387a99162b7ba0240c743f0532ef60d577c
             413d
   Q.x     = 958a51e2f02e0dfd3930709010d5d16f869adb9d8a8f7c01139911
             d206c20cdb7bfb40ee33ba30536a99f49362fa7633d0f417fc3914
             fe21
   Q.y     = f4307a36ab6612fa97501497f01afa109733ce85875935551c3ca9
             0f0fa7e0097a8640bb7e5dbcc38ab32b23b748790f2261f2c44c3b
             f3ba

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 122a3234d34b26c69749f23356452bf9501efa2d94859d5ef741fe
             f024156d9d191a03a2ad24c38186f93e02d05572575968b083d8a3
             9738
   P.y     = ddf55e74eb4414c2c1fa4aa6bc37c4ab470a3fed6bb5af1e435703
             09b162fb61879bb15f9ea49c712efd42d0a71666430f9f0d4a2050
             5050
   u[0]    = 5945744d27122f89da3daf76ab4db9616053df64e25d30ec9a0066
             7ee6710240579c1db8f8ef3386f3f4f413cfb325ac14094d582026
             a971
   Q.x     = e7e1f2d13548ac2c8fcd346e4c63606545bf93652011721e83ac3b
             64226f77a8823d3881e164bc6ca45505b236e8e3721c028052fcc9
             ade5
   Q.y     = 7e0f340501bf25f018b9d374c2acbdd43c07261d85a6ef3c855113
             d4e023634db59a87b8fab9efe04ed1fee302c8a4994e83bdda32bd



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 134]

Internet-Draft                hash-to-curve                    June 2022


             9c0b

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 221704949b1ce1ab8dd174dc9b8c56fcffa27179569ce9219c0c2f
             e183d3d23343a4c42a0e2e9d6b9d0feb1df3883ec489b6671d1fa6
             4089
   P.y     = ebdecfdc87142d1a919034bf22ecfad934c9a85effff14b594ae2c
             00943ca62a39d6ee3be9df0bb504ce8a9e1669bc6959c42ad6a1d3
             b686
   u[0]    = 1192e378043f01cedc7ea0209321519213b0184ea0d8575816bcd9
             182a367823e1eecc2faf1df8f79b24027a4b9bfa208cd320e79bef
             06ea
   Q.x     = 0fd3bb833c1d7a5b319d1d4117406a23b9aece976186ecb18a11a6
             35e6fbdb920d47e04762b1f2a8c59d2f8435d0fdefe501f544cda2
             3dbf
   Q.y     = f13b0dad4d5eeb120f2443ac4392f8096a1396f5014ec2a3506a34
             7fef8076a7282035cf619599b1919cf29df5ce87711c11688aab77
             00a6

J.8.  secp256k1

J.8.1.  secp256k1_XMD:SHA-256_SSWU_RO_

   suite   = secp256k1_XMD:SHA-256_SSWU_RO_
   dst     = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_RO_

   msg     =
   P.x     = c1cae290e291aee617ebaef1be6d73861479c48b841eaba9b7b585
             2ddfeb1346
   P.y     = 64fa678e07ae116126f08b022a94af6de15985c996c3a91b64c406
             a960e51067
   u[0]    = 6b0f9910dd2ba71c78f2ee9f04d73b5f4c5f7fc773a701abea1e57
             3cab002fb3
   u[1]    = 1ae6c212e08fe1a5937f6202f929a2cc8ef4ee5b9782db68b0d579
             9fd8f09e16
   Q0.x    = 74519ef88b32b425a095e4ebcc84d81b64e9e2c2675340a720bb1a
             1857b99f1e
   Q0.y    = c174fa322ab7c192e11748beed45b508e9fdb1ce046dee9c2cd3a2
             a86b410936



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 135]

Internet-Draft                hash-to-curve                    June 2022


   Q1.x    = 44548adb1b399263ded3510554d28b4bead34b8cf9a37b4bd0bd2b
             a4db87ae63
   Q1.y    = 96eb8e2faf05e368efe5957c6167001760233e6dd2487516b46ae7
             25c4cce0c6

   msg     = abc
   P.x     = 3377e01eab42db296b512293120c6cee72b6ecf9f9205760bd9ff1
             1fb3cb2c4b
   P.y     = 7f95890f33efebd1044d382a01b1bee0900fb6116f94688d487c6c
             7b9c8371f6
   u[0]    = 128aab5d3679a1f7601e3bdf94ced1f43e491f544767e18a4873f3
             97b08a2b61
   u[1]    = 5897b65da3b595a813d0fdcc75c895dc531be76a03518b044daaa0
             f2e4689e00
   Q0.x    = 07dd9432d426845fb19857d1b3a91722436604ccbbbadad8523b8f
             c38a5322d7
   Q0.y    = 604588ef5138cffe3277bbd590b8550bcbe0e523bbaf1bed4014a4
             67122eb33f
   Q1.x    = e9ef9794d15d4e77dde751e06c182782046b8dac05f8491eb88764
             fc65321f78
   Q1.y    = cb07ce53670d5314bf236ee2c871455c562dd76314aa41f012919f
             e8e7f717b3

   msg     = abcdef0123456789
   P.x     = bac54083f293f1fe08e4a70137260aa90783a5cb84d3f35848b324
             d0674b0e3a
   P.y     = 4436476085d4c3c4508b60fcf4389c40176adce756b398bdee27bc
             a19758d828
   u[0]    = ea67a7c02f2cd5d8b87715c169d055a22520f74daeb080e6180958
             380e2f98b9
   u[1]    = 7434d0d1a500d38380d1f9615c021857ac8d546925f5f2355319d8
             23a478da18
   Q0.x    = 576d43ab0260275adf11af990d130a5752704f7947862876172080
             8862544b5d
   Q0.y    = 643c4a7fb68ae6cff55edd66b809087434bbaff0c07f3f9ec4d49b
             b3c16623c3
   Q1.x    = f89d6d261a5e00fe5cf45e827b507643e67c2a947a20fd9ad71039
             f8b0e29ff8
   Q1.y    = b33855e0cc34a9176ead91c6c3acb1aacb1ce936d563bc1cee1dcf
             fc806caf57

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = e2167bc785333a37aa562f021f1e881defb853839babf52a7f72b1
             02e41890e9
   P.y     = f2401dd95cc35867ffed4f367cd564763719fbc6a53e969fb8496a
             1e6685d873



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 136]

Internet-Draft                hash-to-curve                    June 2022


   u[0]    = eda89a5024fac0a8207a87e8cc4e85aa3bce10745d501a30deb873
             41b05bcdf5
   u[1]    = dfe78cd116818fc2c16f3837fedbe2639fab012c407eac9dfe9245
             bf650ac51d
   Q0.x    = 9c91513ccfe9520c9c645588dff5f9b4e92eaf6ad4ab6f1cd720d1
             92eb58247a
   Q0.y    = c7371dcd0134412f221e386f8d68f49e7fa36f9037676e163d4a06
             3fbf8a1fb8
   Q1.x    = 10fee3284d7be6bd5912503b972fc52bf4761f47141a0015f1c6ae
             36848d869b
   Q1.y    = 0b163d9b4bf21887364332be3eff3c870fa053cf508732900fc69a
             6eb0e1b672

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = e3c8d35aaaf0b9b647e88a0a0a7ee5d5bed5ad38238152e4e6fd8c
             1f8cb7c998
   P.y     = 8446eeb6181bf12f56a9d24e262221cc2f0c4725c7e3803024b588
             8ee5823aa6
   u[0]    = 8d862e7e7e23d7843fe16d811d46d7e6480127a6b78838c277bca1
             7df6900e9f
   u[1]    = 68071d2530f040f081ba818d3c7188a94c900586761e9115efa47a
             e9bd847938
   Q0.x    = b32b0ab55977b936f1e93fdc68cec775e13245e161dbfe556bbb1f
             72799b4181
   Q0.y    = 2f5317098360b722f132d7156a94822641b615c91f8663be691698
             70a12af9e8
   Q1.x    = 148f98780f19388b9fa93e7dc567b5a673e5fca7079cd9cdafd719
             82ec4c5e12
   Q1.y    = 3989645d83a433bc0c001f3dac29af861f33a6fd1e04f4b36873f5
             bff497298a

J.8.2.  secp256k1_XMD:SHA-256_SSWU_NU_

   suite   = secp256k1_XMD:SHA-256_SSWU_NU_
   dst     = QUUX-V01-CS02-with-secp256k1_XMD:SHA-256_SSWU_NU_

   msg     =
   P.x     = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf
             65c4c55c5b



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 137]

Internet-Draft                hash-to-curve                    June 2022


   P.y     = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f
             31639ff3a7
   u[0]    = 0137fcd23bc3da962e8808f97474d097a6c8aa2881fceef4514173
             635872cf3b
   Q.x     = a4792346075feae77ac3b30026f99c1441b4ecf666ded19b7522cf
             65c4c55c5b
   Q.y     = 62c59e2a6aeed1b23be5883e833912b08ba06be7f57c0e9cdc663f
             31639ff3a7

   msg     = abc
   P.x     = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1
             d7ee426b9d
   P.y     = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493
             c4f5d876a5
   u[0]    = e03f894b4d7caf1a50d6aa45cac27412c8867a25489e32c5ddeb50
             3229f63a2e
   Q.x     = 3f3b5842033fff837d504bb4ce2a372bfeadbdbd84a1d2b678b6e1
             d7ee426b9d
   Q.y     = 902910d1fef15d8ae2006fc84f2a5a7bda0e0407dc913062c3a493
             c4f5d876a5

   msg     = abcdef0123456789
   P.x     = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a
             a50c59bfdf
   P.y     = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626
             b506c21e7b
   u[0]    = e7a6525ae7069ff43498f7f508b41c57f80563c1fe4283510b3224
             46f32af41b
   Q.x     = 07644fa6281c694709f53bdd21bed94dab995671e4a8cd1904ec4a
             a50c59bfdf
   Q.y     = c79f8d1dad79b6540426922f7fbc9579c3018dafeffcd4552b1626
             b506c21e7b

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4
             83aa9acc33
   P.y     = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801
             f2d0b1d6ee
   u[0]    = d97cf3d176a2f26b9614a704d7d434739d194226a706c886c5c3c3
             9806bc323c
   Q.x     = b734f05e9b9709ab631d960fa26d669c4aeaea64ae62004b9d34f4
             83aa9acc33
   Q.y     = 03fc8a4a5a78632e2eb4d8460d69ff33c1d72574b79a35e402e801
             f2d0b1d6ee

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 138]

Internet-Draft                hash-to-curve                    June 2022


             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136
             070fd9fa6c
   P.y     = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289
             ecc2a51718
   u[0]    = a9ffbeee1d6e41ac33c248fb3364612ff591b502386c1bf6ac4aaf
             1ea51f8c3b
   Q.x     = 17d22b867658977b5002dbe8d0ee70a8cfddec3eec50fb93f36136
             070fd9fa6c
   Q.y     = e9178ff02f4dab73480f8dd590328aea99856a7b6cc8e5a6cdf289
             ecc2a51718

J.9.  BLS12-381 G1

J.9.1.  BLS12381G1_XMD:SHA-256_SSWU_RO_

   suite   = BLS12381G1_XMD:SHA-256_SSWU_RO_
   dst     = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_RO_

   msg     =
   P.x     = 052926add2207b76ca4fa57a8734416c8dc95e24501772c8142787
             00eed6d1e4e8cf62d9c09db0fac349612b759e79a1
   P.y     = 08ba738453bfed09cb546dbb0783dbb3a5f1f566ed67bb6be0e8c6
             7e2e81a4cc68ee29813bb7994998f3eae0c9c6a265
   u[0]    = 0ba14bd907ad64a016293ee7c2d276b8eae71f25a4b941eece7b0d
             89f17f75cb3ae5438a614fb61d6835ad59f29c564f
   u[1]    = 019b9bd7979f12657976de2884c7cce192b82c177c80e0ec604436
             a7f538d231552f0d96d9f7babe5fa3b19b3ff25ac9
   Q0.x    = 11a3cce7e1d90975990066b2f2643b9540fa40d6137780df4e753a
             8054d07580db3b7f1f03396333d4a359d1fe3766fe
   Q0.y    = 0eeaf6d794e479e270da10fdaf768db4c96b650a74518fc67b04b0
             3927754bac66f3ac720404f339ecdcc028afa091b7
   Q1.x    = 160003aaf1632b13396dbad518effa00fff532f604de1a7fc2082f
             f4cb0afa2d63b2c32da1bef2bf6c5ca62dc6b72f9c
   Q1.y    = 0d8bb2d14e20cf9f6036152ed386d79189415b6d015a20133acb4e
             019139b94e9c146aaad5817f866c95d609a361735e

   msg     = abc
   P.x     = 03567bc5ef9c690c2ab2ecdf6a96ef1c139cc0b2f284dca0a9a794
             3388a49a3aee664ba5379a7655d3c68900be2f6903



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 139]

Internet-Draft                hash-to-curve                    June 2022


   P.y     = 0b9c15f3fe6e5cf4211f346271d7b01c8f3b28be689c8429c85b67
             af215533311f0b8dfaaa154fa6b88176c229f2885d
   u[0]    = 0d921c33f2bad966478a03ca35d05719bdf92d347557ea166e5bba
             579eea9b83e9afa5c088573c2281410369fbd32951
   u[1]    = 003574a00b109ada2f26a37a91f9d1e740dffd8d69ec0c35e1e9f4
             652c7dba61123e9dd2e76c655d956e2b3462611139
   Q0.x    = 125435adce8e1cbd1c803e7123f45392dc6e326d292499c2c45c58
             65985fd74fe8f042ecdeeec5ecac80680d04317d80
   Q0.y    = 0e8828948c989126595ee30e4f7c931cbd6f4570735624fd25aef2
             fa41d3f79cfb4b4ee7b7e55a8ce013af2a5ba20bf2
   Q1.x    = 11def93719829ecda3b46aa8c31fc3ac9c34b428982b898369608e
             4f042babee6c77ab9218aad5c87ba785481eff8ae4
   Q1.y    = 0007c9cef122ccf2efd233d6eb9bfc680aa276652b0661f4f820a6
             53cec1db7ff69899f8e52b8e92b025a12c822a6ce6

   msg     = abcdef0123456789
   P.x     = 11e0b079dea29a68f0383ee94fed1b940995272407e3bb916bbf26
             8c263ddd57a6a27200a784cbc248e84f357ce82d98
   P.y     = 03a87ae2caf14e8ee52e51fa2ed8eefe80f02457004ba4d486d6aa
             1f517c0889501dc7413753f9599b099ebcbbd2d709
   u[0]    = 062d1865eb80ebfa73dcfc45db1ad4266b9f3a93219976a3790ab8
             d52d3e5f1e62f3b01795e36834b17b70e7b76246d4
   u[1]    = 0cdc3e2f271f29c4ff75020857ce6c5d36008c9b48385ea2f2bf6f
             96f428a3deb798aa033cd482d1cdc8b30178b08e3a
   Q0.x    = 08834484878c217682f6d09a4b51444802fdba3d7f2df9903a0dda
             db92130ebbfa807fffa0eabf257d7b48272410afff
   Q0.y    = 0b318f7ecf77f45a0f038e62d7098221d2dbbca2a394164e2e3fe9
             53dc714ac2cde412d8f2d7f0c03b259e6795a2508e
   Q1.x    = 158418ed6b27e2549f05531a8281b5822b31c3bf3144277fbb977f
             8d6e2694fedceb7011b3c2b192f23e2a44b2bd106e
   Q1.y    = 1879074f344471fac5f839e2b4920789643c075792bec5af4282c7
             3f7941cda5aa77b00085eb10e206171b9787c4169f

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 15f68eaa693b95ccb85215dc65fa81038d69629f70aeee0d0f677c
             f22285e7bf58d7cb86eefe8f2e9bc3f8cb84fac488
   P.y     = 1807a1d50c29f430b8cafc4f8638dfeeadf51211e1602a5f184443
             076715f91bb90a48ba1e370edce6ae1062f5e6dd38
   u[0]    = 010476f6a060453c0b1ad0b628f3e57c23039ee16eea5e71bb87c3
             b5419b1255dc0e5883322e563b84a29543823c0e86
   u[1]    = 0b1a912064fb0554b180e07af7e787f1f883a0470759c03c1b6509
             eb8ce980d1670305ae7b928226bb58fdc0a419f46e
   Q0.x    = 0cbd7f84ad2c99643fea7a7ac8f52d63d66cefa06d9a56148e58b9
             84b3dd25e1f41ff47154543343949c64f88d48a710
   Q0.y    = 052c00e4ed52d000d94881a5638ae9274d3efc8bc77bc0e5c650de
             04a000b2c334a9e80b85282a00f3148dfdface0865



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 140]

Internet-Draft                hash-to-curve                    June 2022


   Q1.x    = 06493fb68f0d513af08be0372f849436a787e7b701ae31cb964d96
             8021d6ba6bd7d26a38aaa5a68e8c21a6b17dc8b579
   Q1.y    = 02e98f2ccf5802b05ffaac7c20018bc0c0b2fd580216c4aa2275d2
             909dc0c92d0d0bdc979226adeb57a29933536b6bb4

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 082aabae8b7dedb0e78aeb619ad3bfd9277a2f77ba7fad20ef6aab
             dc6c31d19ba5a6d12283553294c1825c4b3ca2dcfe
   P.y     = 05b84ae5a942248eea39e1d91030458c40153f3b654ab7872d779a
             d1e942856a20c438e8d99bc8abfbf74729ce1f7ac8
   u[0]    = 0a8ffa7447f6be1c5a2ea4b959c9454b431e29ccc0802bc052413a
             9c5b4f9aac67a93431bd480d15be1e057c8a08e8c6
   u[1]    = 05d487032f602c90fa7625dbafe0f4a49ef4a6b0b33d7bb349ff4c
             f5410d297fd6241876e3e77b651cfc8191e40a68b7
   Q0.x    = 0cf97e6dbd0947857f3e578231d07b309c622ade08f2c08b32ff37
             2bd90db19467b2563cc997d4407968d4ac80e154f8
   Q0.y    = 127f0cddf2613058101a5701f4cb9d0861fd6c2a1b8e0afe194fcc
             f586a3201a53874a2761a9ab6d7220c68661a35ab3
   Q1.x    = 092f1acfa62b05f95884c6791fba989bbe58044ee6355d100973bf
             9553ade52b47929264e6ae770fb264582d8dce512a
   Q1.y    = 028e6d0169a72cfedb737be45db6c401d3adfb12c58c619c82b93a
             5dfcccef12290de530b0480575ddc8397cda0bbebf

J.9.2.  BLS12381G1_XMD:SHA-256_SSWU_NU_

   suite   = BLS12381G1_XMD:SHA-256_SSWU_NU_
   dst     = QUUX-V01-CS02-with-BLS12381G1_XMD:SHA-256_SSWU_NU_

   msg     =
   P.x     = 184bb665c37ff561a89ec2122dd343f20e0f4cbcaec84e3c3052ea
             81d1834e192c426074b02ed3dca4e7676ce4ce48ba
   P.y     = 04407b8d35af4dacc809927071fc0405218f1401a6d15af775810e
             4e460064bcc9468beeba82fdc751be70476c888bf3
   u[0]    = 156c8a6a2c184569d69a76be144b5cdc5141d2d2ca4fe341f011e2
             5e3969c55ad9e9b9ce2eb833c81a908e5fa4ac5f03
   Q.x     = 11398d3b324810a1b093f8e35aa8571cced95858207e7f49c4fd74
             656096d61d8a2f9a23cdb18a4dd11cd1d66f41f709
   Q.y     = 19316b6fb2ba7717355d5d66a361899057e1e84a6823039efc7bec
             cefe09d023fb2713b1c415fcf278eb0c39a89b4f72



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 141]

Internet-Draft                hash-to-curve                    June 2022


   msg     = abc
   P.x     = 009769f3ab59bfd551d53a5f846b9984c59b97d6842b20a2c565ba
             a167945e3d026a3755b6345df8ec7e6acb6868ae6d
   P.y     = 1532c00cf61aa3d0ce3e5aa20c3b531a2abd2c770a790a26138183
             03c6b830ffc0ecf6c357af3317b9575c567f11cd2c
   u[0]    = 147e1ed29f06e4c5079b9d14fc89d2820d32419b990c1c7bb7dbea
             2a36a045124b31ffbde7c99329c05c559af1c6cc82
   Q.x     = 1998321bc27ff6d71df3051b5aec12ff47363d81a5e9d2dff55f44
             4f6ca7e7d6af45c56fd029c58237c266ef5cda5254
   Q.y     = 034d274476c6307ae584f951c82e7ea85b84f72d28f4d647173235
             6121af8d62a49bc263e8eb913a6cf6f125995514ee

   msg     = abcdef0123456789
   P.x     = 1974dbb8e6b5d20b84df7e625e2fbfecb2cdb5f77d5eae5fb2955e
             5ce7313cae8364bc2fff520a6c25619739c6bdcb6a
   P.y     = 15f9897e11c6441eaa676de141c8d83c37aab8667173cbe1dfd6de
             74d11861b961dccebcd9d289ac633455dfcc7013a3
   u[0]    = 04090815ad598a06897dd89bcda860f25837d54e897298ce31e694
             7378134d3761dc59a572154963e8c954919ecfa82d
   Q.x     = 17d502fa43bd6a4cad2859049a0c3ecefd60240d129be65da271a4
             c03a9c38fa78163b9d2a919d2beb57df7d609b4919
   Q.y     = 109019902ae93a8732abecf2ff7fecd2e4e305eb91f41c9c3267f1
             6b6c19de138c7272947f25512745da6c466cdfd1ac

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 0a7a047c4a8397b3446450642c2ac64d7239b61872c9ae7a59707a
             8f4f950f101e766afe58223b3bff3a19a7f754027c
   P.y     = 1383aebba1e4327ccff7cf9912bda0dbc77de048b71ef8c8a81111
             d71dc33c5e3aa6edee9cf6f5fe525d50cc50b77cc9
   u[0]    = 08dccd088ca55b8bfbc96fb50bb25c592faa867a8bb78d4e94a8cc
             2c92306190244532e91feba2b7fed977e3c3bb5a1f
   Q.x     = 112eb92dd2b3aa9cd38b08de4bef603f2f9fb0ca226030626a9a2e
             47ad1e9847fe0a5ed13766c339e38f514bba143b21
   Q.y     = 17542ce2f8d0a54f2c5ba8c4b14e10b22d5bcd7bae2af3c965c8c8
             72b571058c720eac448276c99967ded2bf124490e1

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 142]

Internet-Draft                hash-to-curve                    June 2022


   P.x     = 0e7a16a975904f131682edbb03d9560d3e48214c9986bd50417a77
             108d13dc957500edf96462a3d01e62dc6cd468ef11
   P.y     = 0ae89e677711d05c30a48d6d75e76ca9fb70fe06c6dd6ff988683d
             89ccde29ac7d46c53bb97a59b1901abf1db66052db
   u[0]    = 0dd824886d2123a96447f6c56e3a3fa992fbfefdba17b6673f9f63
             0ff19e4d326529db37e1c1be43f905bf9202e0278d
   Q.x     = 1775d400a1bacc1c39c355da7e96d2d1c97baa9430c4a3476881f8
             521c09a01f921f592607961efc99c4cd46bd78ca19
   Q.y     = 1109b5d59f65964315de65a7a143e86eabc053104ed289cf480949
             317a5685fad7254ff8e7fe6d24d3104e5d55ad6370

J.10.  BLS12-381 G2

J.10.1.  BLS12381G2_XMD:SHA-256_SSWU_RO_

   suite   = BLS12381G2_XMD:SHA-256_SSWU_RO_
   dst     = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_RO_

   msg     =
   P.x     = 0141ebfbdca40eb85b87142e130ab689c673cf60f1a3e98d693352
             66f30d9b8d4ac44c1038e9dcdd5393faf5c41fb78a
       + I * 05cb8437535e20ecffaef7752baddf98034139c38452458baeefab
             379ba13dff5bf5dd71b72418717047f5b0f37da03d
   P.y     = 0503921d7f6a12805e72940b963c0cf3471c7b2a524950ca195d11
             062ee75ec076daf2d4bc358c4b190c0c98064fdd92
       + I * 12424ac32561493f3fe3c260708a12b7c620e7be00099a974e259d
             dc7d1f6395c3c811cdd19f1e8dbf3e9ecfdcbab8d6
   u[0]    = 03dbc2cce174e91ba93cbb08f26b917f98194a2ea08d1cce75b2b9
             cc9f21689d80bd79b594a613d0a68eb807dfdc1cf8
       + I * 05a2acec64114845711a54199ea339abd125ba38253b70a92c876d
             f10598bd1986b739cad67961eb94f7076511b3b39a
   u[1]    = 02f99798e8a5acdeed60d7e18e9120521ba1f47ec090984662846b
             c825de191b5b7641148c0dbc237726a334473eee94
       + I * 145a81e418d4010cc027a68f14391b30074e89e60ee7a22f87217b
             2f6eb0c4b94c9115b436e6fa4607e95a98de30a435
   Q0.x    = 019ad3fc9c72425a998d7ab1ea0e646a1f6093444fc6965f1cad5a
             3195a7b1e099c050d57f45e3fa191cc6d75ed7458c
       + I * 171c88b0b0efb5eb2b88913a9e74fe111a4f68867b59db252ce586
             8af4d1254bfab77ebde5d61cd1a86fb2fe4a5a1c1d
   Q0.y    = 0ba10604e62bdd9eeeb4156652066167b72c8d743b050fb4c1016c
             31b505129374f76e03fa127d6a156213576910fef3
       + I * 0eb22c7a543d3d376e9716a49b72e79a89c9bfe9feee8533ed931c
             bb5373dde1fbcd7411d8052e02693654f71e15410a
   Q1.x    = 113d2b9cd4bd98aee53470b27abc658d91b47a78a51584f3d4b950
             677cfb8a3e99c24222c406128c91296ef6b45608be
       + I * 13855912321c5cb793e9d1e88f6f8d342d49c0b0dbac613ee9e17e
             3c0b3c97dfbb5a49cc3fb45102fdbaf65e0efe2632
   Q1.y    = 0fd3def0b7574a1d801be44fde617162aa2e89da47f464317d9bb5



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 143]

Internet-Draft                hash-to-curve                    June 2022


             abc3a7071763ce74180883ad7ad9a723a9afafcdca
       + I * 056f617902b3c0d0f78a9a8cbda43a26b65f602f8786540b9469b0
             60db7b38417915b413ca65f875c130bebfaa59790c

   msg     = abc
   P.x     = 02c2d18e033b960562aae3cab37a27ce00d80ccd5ba4b7fe0e7a21
             0245129dbec7780ccc7954725f4168aff2787776e6
       + I * 139cddbccdc5e91b9623efd38c49f81a6f83f175e80b06fc374de9
             eb4b41dfe4ca3a230ed250fbe3a2acf73a41177fd8
   P.y     = 1787327b68159716a37440985269cf584bcb1e621d3a7202be6ea0
             5c4cfe244aeb197642555a0645fb87bf7466b2ba48
       + I * 00aa65dae3c8d732d10ecd2c50f8a1baf3001578f71c694e03866e
             9f3d49ac1e1ce70dd94a733534f106d4cec0eddd16
   u[0]    = 15f7c0aa8f6b296ab5ff9c2c7581ade64f4ee6f1bf18f55179ff44
             a2cf355fa53dd2a2158c5ecb17d7c52f63e7195771
       + I * 01c8067bf4c0ba709aa8b9abc3d1cef589a4758e09ef53732d670f
             d8739a7274e111ba2fcaa71b3d33df2a3a0c8529dd
   u[1]    = 187111d5e088b6b9acfdfad078c4dacf72dcd17ca17c82be35e79f
             8c372a693f60a033b461d81b025864a0ad051a06e4
       + I * 08b852331c96ed983e497ebc6dee9b75e373d923b729194af8e72a
             051ea586f3538a6ebb1e80881a082fa2b24df9f566
   Q0.x    = 12b2e525281b5f4d2276954e84ac4f42cf4e13b6ac4228624e1776
             0faf94ce5706d53f0ca1952f1c5ef75239aeed55ad
       + I * 05d8a724db78e570e34100c0bc4a5fa84ad5839359b40398151f37
             cff5a51de945c563463c9efbdda569850ee5a53e77
   Q0.y    = 02eacdc556d0bdb5d18d22f23dcb086dd106cad713777c7e640794
             3edbe0b3d1efe391eedf11e977fac55f9b94f2489c
       + I * 04bbe48bfd5814648d0b9e30f0717b34015d45a861425fabc1ee06
             fdfce36384ae2c808185e693ae97dcde118f34de41
   Q1.x    = 19f18cc5ec0c2f055e47c802acc3b0e40c337256a208001dde14b2
             5afced146f37ea3d3ce16834c78175b3ed61f3c537
       + I * 15b0dadc256a258b4c68ea43605dffa6d312eef215c19e6474b3e1
             01d33b661dfee43b51abbf96fee68fc6043ac56a58
   Q1.y    = 05e47c1781286e61c7ade887512bd9c2cb9f640d3be9cf87ea0bad
             24bd0ebfe946497b48a581ab6c7d4ca74b5147287f
       + I * 19f98db2f4a1fcdf56a9ced7b320ea9deecf57c8e59236b0dc21f6
             ee7229aa9705ce9ac7fe7a31c72edca0d92370c096

   msg     = abcdef0123456789
   P.x     = 121982811d2491fde9ba7ed31ef9ca474f0e1501297f68c298e9f4
             c0028add35aea8bb83d53c08cfc007c1e005723cd0
       + I * 190d119345b94fbd15497bcba94ecf7db2cbfd1e1fe7da034d26cb
             ba169fb3968288b3fafb265f9ebd380512a71c3f2c
   P.y     = 05571a0f8d3c08d094576981f4a3b8eda0a8e771fcdcc8ecceaf13
             56a6acf17574518acb506e435b639353c2e14827c8
       + I * 0bb5e7572275c567462d91807de765611490205a941a5a6af3b169
             1bfe596c31225d3aabdf15faff860cb4ef17c7c3be
   u[0]    = 0313d9325081b415bfd4e5364efaef392ecf69b087496973b22930



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 144]

Internet-Draft                hash-to-curve                    June 2022


             3e1816d2080971470f7da112c4eb43053130b785e1
       + I * 062f84cb21ed89406890c051a0e8b9cf6c575cf6e8e18ecf63ba86
             826b0ae02548d83b483b79e48512b82a6c0686df8f
   u[1]    = 1739123845406baa7be5c5dc74492051b6d42504de008c635f3535
             bb831d478a341420e67dcc7b46b2e8cba5379cca97
       + I * 01897665d9cb5db16a27657760bbea7951f67ad68f8d55f7113f24
             ba6ddd82caef240a9bfa627972279974894701d975
   Q0.x    = 0f48f1ea1318ddb713697708f7327781fb39718971d72a9245b973
             1faaca4dbaa7cca433d6c434a820c28b18e20ea208
       + I * 06051467c8f85da5ba2540974758f7a1e0239a5981de441fdd8768
             0a995649c211054869c50edbac1f3a86c561ba3162
   Q0.y    = 168b3d6df80069dbbedb714d41b32961ad064c227355e1ce5fac8e
             105de5e49d77f0c64867f3834848f152497eb76333
       + I * 134e0e8331cee8cb12f9c2d0742714ed9eee78a84d634c9a95f6a7
             391b37125ed48bfc6e90bf3546e99930ff67cc97bc
   Q1.x    = 004fd03968cd1c99a0dd84551f44c206c84dcbdb78076c5bfee24e
             89a92c8508b52b88b68a92258403cbe1ea2da3495f
       + I * 1674338ea298281b636b2eb0fe593008d03171195fd6dcd4531e8a
             1ed1f02a72da238a17a635de307d7d24aa2d969a47
   Q1.y    = 0dc7fa13fff6b12558419e0a1e94bfc3cfaf67238009991c5f24ee
             94b632c3d09e27eca329989aee348a67b50d5e236c
       + I * 169585e164c131103d85324f2d7747b23b91d66ae5d947c449c819
             4a347969fc6bbd967729768da485ba71868df8aed2

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 19a84dd7248a1066f737cc34502ee5555bd3c19f2ecdb3c7d9e24d
             c65d4e25e50d83f0f77105e955d78f4762d33c17da
       + I * 0934aba516a52d8ae479939a91998299c76d39cc0c035cd18813be
             c433f587e2d7a4fef038260eef0cef4d02aae3eb91
   P.y     = 14f81cd421617428bc3b9fe25afbb751d934a00493524bc4e06563
             5b0555084dd54679df1536101b2c979c0152d09192
       + I * 09bcccfa036b4847c9950780733633f13619994394c23ff0b32fa6
             b795844f4a0673e20282d07bc69641cee04f5e5662
   u[0]    = 025820cefc7d06fd38de7d8e370e0da8a52498be9b53cba9927b2e
             f5c6de1e12e12f188bbc7bc923864883c57e49e253
       + I * 034147b77ce337a52e5948f66db0bab47a8d038e712123bb381899
             b6ab5ad20f02805601e6104c29df18c254b8618c7b
   u[1]    = 0930315cae1f9a6017c3f0c8f2314baa130e1cf13f6532bff0a8a1
             790cd70af918088c3db94bda214e896e1543629795
       + I * 10c4df2cacf67ea3cb3108b00d4cbd0b3968031ebc8eac4b1ebcef
             e84d6b715fde66bef0219951ece29d1facc8a520ef
   Q0.x    = 09eccbc53df677f0e5814e3f86e41e146422834854a224bf5a83a5
             0e4cc0a77bfc56718e8166ad180f53526ea9194b57
       + I * 0c3633943f91daee715277bd644fba585168a72f96ded64fc5a384
             cce4ec884a4c3c30f08e09cd2129335dc8f67840ec
   Q0.y    = 0eb6186a0457d5b12d132902d4468bfeb7315d83320b6c32f1c875



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 145]

Internet-Draft                hash-to-curve                    June 2022


             f344efcba979952b4aa418589cb01af712f98cc555
       + I * 119e3cf167e69eb16c1c7830e8df88856d48be12e3ff0a40791a5c
             d2f7221311d4bf13b1847f371f467357b3f3c0b4c7
   Q1.x    = 0eb3aabc1ddfce17ff18455fcc7167d15ce6b60ddc9eb9b59f8d40
             ab49420d35558686293d046fc1e42f864b7f60e381
       + I * 198bdfb19d7441ebcca61e8ff774b29d17da16547d2c10c273227a
             635cacea3f16826322ae85717630f0867539b5ed8b
   Q1.y    = 0aaf1dee3adf3ed4c80e481c09b57ea4c705e1b8d25b897f0ceeec
             3990748716575f92abff22a1c8f4582aff7b872d52
       + I * 0d058d9061ed27d4259848a06c96c5ca68921a5d269b078650c882
             cb3c2bd424a8702b7a6ee4e0ead9982baf6843e924

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 01a6ba2f9a11fa5598b2d8ace0fbe0a0eacb65deceb476fbbcb64f
             d24557c2f4b18ecfc5663e54ae16a84f5ab7f62534
       + I * 11fca2ff525572795a801eed17eb12785887c7b63fb77a42be46ce
             4a34131d71f7a73e95fee3f812aea3de78b4d01569
   P.y     = 0b6798718c8aed24bc19cb27f866f1c9effcdbf92397ad6448b5c9
             db90d2b9da6cbabf48adc1adf59a1a28344e79d57e
       + I * 03a47f8e6d1763ba0cad63d6114c0accbef65707825a511b251a66
             0a9b3994249ae4e63fac38b23da0c398689ee2ab52
   u[0]    = 190b513da3e66fc9a3587b78c76d1d132b1152174d0b83e3c11140
             66392579a45824c5fa17649ab89299ddd4bda54935
       + I * 12ab625b0fe0ebd1367fe9fac57bb1168891846039b4216b9d9400
             7b674de2d79126870e88aeef54b2ec717a887dcf39
   u[1]    = 0e6a42010cf435fb5bacc156a585e1ea3294cc81d0ceb81924d950
             40298380b164f702275892cedd81b62de3aba3f6b5
       + I * 117d9a0defc57a33ed208428cb84e54c85a6840e7648480ae42883
             8989d25d97a0af8e3255be62b25c2a85630d2dddd8
   Q0.x    = 17cadf8d04a1a170f8347d42856526a24cc466cb2ddfd506cff011
             91666b7f944e31244d662c904de5440516a2b09004
       + I * 0d13ba91f2a8b0051cf3279ea0ee63a9f19bc9cb8bfcc7d78b3cbd
             8cc4fc43ba726774b28038213acf2b0095391c523e
   Q0.y    = 17ef19497d6d9246fa94d35575c0f8d06ee02f21a284dbeaa78768
             cb1e25abd564e3381de87bda26acd04f41181610c5
       + I * 12c3c913ba4ed03c24f0721a81a6be7430f2971ffca8fd1729aafe
             496bb725807531b44b34b59b3ae5495e5a2dcbd5c8
   Q1.x    = 16ec57b7fe04c71dfe34fb5ad84dbce5a2dbbd6ee085f1d8cd17f4
             5e8868976fc3c51ad9eeda682c7869024d24579bfd



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 146]

Internet-Draft                hash-to-curve                    June 2022


       + I * 13103f7aace1ae1420d208a537f7d3a9679c287208026e4e3439ab
             8cd534c12856284d95e27f5e1f33eec2ce656533b0
   Q1.y    = 0958b2c4c2c10fcef5a6c59b9e92c4a67b0fae3e2e0f1b6b5edad9
             c940b8f3524ba9ebbc3f2ceb3cfe377655b3163bd7
       + I * 0ccb594ed8bd14ca64ed9cb4e0aba221be540f25dd0d6ba15a4a4b
             e5d67bcf35df7853b2d8dad3ba245f1ea3697f66aa

J.10.2.  BLS12381G2_XMD:SHA-256_SSWU_NU_

   suite   = BLS12381G2_XMD:SHA-256_SSWU_NU_
   dst     = QUUX-V01-CS02-with-BLS12381G2_XMD:SHA-256_SSWU_NU_

   msg     =
   P.x     = 00e7f4568a82b4b7dc1f14c6aaa055edf51502319c723c4dc2688c
             7fe5944c213f510328082396515734b6612c4e7bb7
       + I * 126b855e9e69b1f691f816e48ac6977664d24d99f8724868a18418
             6469ddfd4617367e94527d4b74fc86413483afb35b
   P.y     = 0caead0fd7b6176c01436833c79d305c78be307da5f6af6c133c47
             311def6ff1e0babf57a0fb5539fce7ee12407b0a42
       + I * 1498aadcf7ae2b345243e281ae076df6de84455d766ab6fcdaad71
             fab60abb2e8b980a440043cd305db09d283c895e3d
   u[0]    = 07355d25caf6e7f2f0cb2812ca0e513bd026ed09dda65b177500fa
             31714e09ea0ded3a078b526bed3307f804d4b93b04
       + I * 02829ce3c021339ccb5caf3e187f6370e1e2a311dec9b753631170
             63ab2015603ff52c3d3b98f19c2f65575e99e8b78c
   Q.x     = 18ed3794ad43c781816c523776188deafba67ab773189b8f18c49b
             c7aa841cd81525171f7a5203b2a340579192403bef
       + I * 0727d90785d179e7b5732c8a34b660335fed03b913710b60903cf4
             954b651ed3466dc3728e21855ae822d4a0f1d06587
   Q.y     = 00764a5cf6c5f61c52c838523460eb2168b5a5b43705e19cb612e0
             06f29b717897facfd15dd1c8874c915f6d53d0342d
       + I * 19290bb9797c12c1d275817aa2605ebe42275b66860f0e4d04487e
             bc2e47c50b36edd86c685a60c20a2bd584a82b011a

   msg     = abc
   P.x     = 108ed59fd9fae381abfd1d6bce2fd2fa220990f0f837fa30e0f279
             14ed6e1454db0d1ee957b219f61da6ff8be0d6441f
       + I * 0296238ea82c6d4adb3c838ee3cb2346049c90b96d602d7bb1b469
             b905c9228be25c627bffee872def773d5b2a2eb57d
   P.y     = 033f90f6057aadacae7963b0a0b379dd46750c1c94a6357c99b65f
             63b79e321ff50fe3053330911c56b6ceea08fee656
       + I * 153606c417e59fb331b7ae6bce4fbf7c5190c33ce9402b5ebe2b70
             e44fca614f3f1382a3625ed5493843d0b0a652fc3f
   u[0]    = 138879a9559e24cecee8697b8b4ad32cced053138ab913b9987277
             2dc753a2967ed50aabc907937aefb2439ba06cc50c
       + I * 0a1ae7999ea9bab1dcc9ef8887a6cb6e8f1e22566015428d220b7e
             ec90ffa70ad1f624018a9ad11e78d588bd3617f9f2
   Q.x     = 0f40e1d5025ecef0d850aa0bb7bbeceab21a3d4e85e6bee857805b



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 147]

Internet-Draft                hash-to-curve                    June 2022


             09693051f5b25428c6be343edba5f14317fcc30143
       + I * 02e0d261f2b9fee88b82804ec83db330caa75fbb12719cfa71ccce
             1c532dc4e1e79b0a6a281ed8d3817524286c8bc04c
   Q.y     = 0cf4a4adc5c66da0bca4caddc6a57ecd97c8252d7526a8ff478e0d
             fed816c4d321b5c3039c6683ae9b1e6a3a38c9c0ae
       + I * 11cad1646bb3768c04be2ab2bbe1f80263b7ff6f8f9488f5bc3b68
             50e5a3e97e20acc583613c69cf3d2bfe8489744ebb

   msg     = abcdef0123456789
   P.x     = 038af300ef34c7759a6caaa4e69363cafeed218a1f207e93b2c70d
             91a1263d375d6730bd6b6509dcac3ba5b567e85bf3
       + I * 0da75be60fb6aa0e9e3143e40c42796edf15685cafe0279afd2a67
             c3dff1c82341f17effd402e4f1af240ea90f4b659b
   P.y     = 19b148cbdf163cf0894f29660d2e7bfb2b68e37d54cc83fd4e6e62
             c020eaa48709302ef8e746736c0e19342cc1ce3df4
       + I * 0492f4fed741b073e5a82580f7c663f9b79e036b70ab3e51162359
             cec4e77c78086fe879b65ca7a47d34374c8315ac5e
   u[0]    = 18c16fe362b7dbdfa102e42bdfd3e2f4e6191d479437a59db4eb71
             6986bf08ee1f42634db66bde97d6c16bbfd342b3b8
       + I * 0e37812ce1b146d998d5f92bdd5ada2a31bfd63dfe18311aa91637
             b5f279dd045763166aa1615e46a50d8d8f475f184e
   Q.x     = 13a9d4a738a85c9f917c7be36b240915434b58679980010499b9ae
             8d7a1bf7fbe617a15b3cd6060093f40d18e0f19456
       + I * 16fa88754e7670366a859d6f6899ad765bf5a177abedb2740aacc9
             252c43f90cd0421373fbd5b2b76bb8f5c4886b5d37
   Q.y     = 0a7fa7d82c46797039398253e8765a4194100b330dfed6d7fbb46d
             6fbf01e222088779ac336e3675c7a7a0ee05bbb6e3
       + I * 0c6ee170ab766d11fa9457cef53253f2628010b2cffc102b3b2835
             1eb9df6c281d3cfc78e9934769d661b72a5265338d

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   P.x     = 0c5ae723be00e6c3f0efe184fdc0702b64588fe77dda152ab13099
             a3bacd3876767fa7bbad6d6fd90b3642e902b208f9
       + I * 12c8c05c1d5fc7bfa847f4d7d81e294e66b9a78bc9953990c35894
             5e1f042eedafce608b67fdd3ab0cb2e6e263b9b1ad
   P.y     = 04e77ddb3ede41b5ec4396b7421dd916efc68a358a0d7425bddd25
             3547f2fb4830522358491827265dfc5bcc1928a569
       + I * 11c624c56dbe154d759d021eec60fab3d8b852395a89de497e4850
             4366feedd4662d023af447d66926a28076813dd646
   u[0]    = 08d4a0997b9d52fecf99427abb721f0fa779479963315fe21c6445
             250de7183e3f63bfdf86570da8929489e421d4ee95
       + I * 16cb4ccad91ec95aab070f22043916cd6a59c4ca94097f7f510043
             d48515526dc8eaaea27e586f09151ae613688d5a89
   Q.x     = 0a08b2f639855dfdeaaed972702b109e2241a54de198b2b4cd12ad
             9f88fa419a6086a58d91fc805de812ea29bee427c2
       + I * 04a7442e4cb8b42ef0f41dac9ee74e65ecad3ce0851f0746dc4756



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 148]

Internet-Draft                hash-to-curve                    June 2022


             8b0e7a8134121ed09ba054509232c49148aef62cda
   Q.y     = 05d60b1f04212b2c87607458f71d770f43973511c260f0540eef3a
             565f42c7ce59aa1cea684bb2a7bcab84acd2f36c8c
       + I * 1017aa5747ba15505ece266a86b0ca9c712f41a254b76ca04094ca
             442ce45ecd224bd5544cd16685d0d1b9d156dd0531

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   P.x     = 0ea4e7c33d43e17cc516a72f76437c4bf81d8f4eac69ac355d3bf9
             b71b8138d55dc10fd458be115afa798b55dac34be1
       + I * 1565c2f625032d232f13121d3cfb476f45275c303a037faa255f9d
             a62000c2c864ea881e2bcddd111edc4a3c0da3e88d
   P.y     = 043b6f5fe4e52c839148dc66f2b3751e69a0f6ebb3d056d6465d50
             d4108543ecd956e10fa1640dfd9bc0030cc2558d28
       + I * 0f8991d2a1ad662e7b6f58ab787947f1fa607fce12dde171bc1790
             3b012091b657e15333e11701edcf5b63ba2a561247
   u[0]    = 03f80ce4ff0ca2f576d797a3660e3f65b274285c054feccc3215c8
             79e2c0589d376e83ede13f93c32f05da0f68fd6a10
       + I * 006488a837c5413746d868d1efb7232724da10eca410b07d8b505b
             9363bdccf0a1fc0029bad07d65b15ccfe6dd25e20d
   Q.x     = 19592c812d5a50c5601062faba14c7d670711745311c879de1235a
             0a11c75aab61327bf2d1725db07ec4d6996a682886
       + I * 0eef4fa41ddc17ed47baf447a2c498548f3c72a02381313d13bef9
             16e240b61ce125539090d62d9fbb14a900bf1b8e90
   Q.y     = 1260d6e0987eae96af9ebe551e08de22b37791d53f4db9e0d59da7
             36e66699735793e853e26362531fe4adf99c1883e3
       + I * 0dbace5df0a4ac4ac2f45d8fdf8aee45484576fdd6efc4f98ab9b9
             f4112309e628255e183022d98ea5ed6e47ca00306c

Appendix K.  Expand test vectors

   This section gives test vectors for expand_message variants specified
   in Section 5.3.  The test vectors in this section were generated
   using code that is available from [hash2curve-repo].









Faz-Hernandez, et al.   Expires 17 December 2022              [Page 149]

Internet-Draft                hash-to-curve                    June 2022


   Each test vector in this section lists the expand_message name, hash
   function, and DST, along with a series of tuples of the function
   inputs (msg and len_in_bytes), output (uniform_bytes), and
   intermediate values (dst_prime and msg_prime).  DST and msg are
   represented as ASCII strings.  Intermediate and output values are
   represented as byte strings in hexadecimal.

K.1.  expand_message_xmd(SHA-256)

   name    = expand_message_xmd
   DST     = QUUX-V01-CS02-with-expander-SHA256-128
   hash    = SHA256
   k       = 128

   msg     =
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000002000515555582d5630312d43533032
             2d776974682d657870616e6465722d5348413235362d31323826
   uniform_bytes = 68a985b87eb6b46952128911f2a4412bbc302a9d759667f8
             7f7a21d803f07235

   msg     = abc
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000616263002000515555582d5630312d43
             5330322d776974682d657870616e6465722d5348413235362d3132
             3826
   uniform_bytes = d8ccab23b5985ccea865c6c97b6e5b8350e794e603b4b979
             02f53a8a0d605615

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000061626364656630313233343536373839
             002000515555582d5630312d435330322d776974682d657870616e
             6465722d5348413235362d31323826
   uniform_bytes = eff31487c770a893cfb36f912fbfcbff40d5661771ca4b2c
             b4eafe524333f5c1



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 150]

Internet-Draft                hash-to-curve                    June 2022


   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000713132385f7171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171002000515555582d5630312d435330322d77
             6974682d657870616e6465722d5348413235362d31323826
   uniform_bytes = b23a1d2b4d97b2ef7785562a7e8bac7eed54ed6e97e29aa5
             1bfe3f12ddad1ff9

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000613531325f6161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 151]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161002000515555582d5630312d
             435330322d776974682d657870616e6465722d5348413235362d31
             323826
   uniform_bytes = 4623227bcc01293b8c130bf771da8c298dede7383243dc09
             93d2d94823958c4c

   msg     =
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000008000515555582d5630312d43533032
             2d776974682d657870616e6465722d5348413235362d31323826
   uniform_bytes = af84c27ccfd45d41914fdff5df25293e221afc53d8ad2ac0
             6d5e3e29485dadbee0d121587713a3e0dd4d5e69e93eb7cd4f5df4
             cd103e188cf60cb02edc3edf18eda8576c412b18ffb658e3dd6ec8
             49469b979d444cf7b26911a08e63cf31f9dcc541708d3491184472
             c2c29bb749d4286b004ceb5ee6b9a7fa5b646c993f0ced

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000616263008000515555582d5630312d43
             5330322d776974682d657870616e6465722d5348413235362d3132
             3826
   uniform_bytes = abba86a6129e366fc877aab32fc4ffc70120d8996c88aee2
             fe4b32d6c7b6437a647e6c3163d40b76a73cf6a5674ef1d890f95b
             664ee0afa5359a5c4e07985635bbecbac65d747d3d2da7ec2b8221
             b17b0ca9dc8a1ac1c07ea6a1e60583e2cb00058e77b7b72a298425
             cd1b941ad4ec65e8afc50303a22c0f99b0509b4c895f40

   msg     = abcdef0123456789
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000061626364656630313233343536373839
             008000515555582d5630312d435330322d776974682d657870616e
             6465722d5348413235362d31323826



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 152]

Internet-Draft                hash-to-curve                    June 2022


   uniform_bytes = ef904a29bffc4cf9ee82832451c946ac3c8f8058ae97d8d6
             29831a74c6572bd9ebd0df635cd1f208e2038e760c4994984ce73f
             0d55ea9f22af83ba4734569d4bc95e18350f740c07eef653cbb9f8
             7910d833751825f0ebefa1abe5420bb52be14cf489b37fe1a72f7d
             e2d10be453b2c9d9eb20c7e3f6edc5a60629178d9478df

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000713132385f7171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171008000515555582d5630312d435330322d77
             6974682d657870616e6465722d5348413235362d31323826
   uniform_bytes = 80be107d0884f0d881bb460322f0443d38bd222db8bd0b0a
             5312a6fedb49c1bbd88fd75d8b9a09486c60123dfa1d73c1cc3169
             761b17476d3c6b7cbbd727acd0e2c942f4dd96ae3da5de368d26b3
             2286e32de7e5a8cb2949f866a0b80c58116b29fa7fabb3ea7d520e
             e603e0c25bcaf0b9a5e92ec6a1fe4e0391d1cdbce8c68a

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413235362d31323826
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000613531325f6161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 153]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161008000515555582d5630312d
             435330322d776974682d657870616e6465722d5348413235362d31
             323826
   uniform_bytes = 546aff5444b5b79aa6148bd81728704c32decb73a3ba76e9
             e75885cad9def1d06d6792f8a7d12794e90efed817d96920d72889
             6a4510864370c207f99bd4a608ea121700ef01ed879745ee3e4cee
             f777eda6d9e5e38b90c86ea6fb0b36504ba4a45d22e86f6db5dd43
             d98a294bebb9125d5b794e9d2a81181066eb954966a487

K.2.  expand_message_xmd(SHA-256) (long DST)

   name    = expand_message_xmd
   DST     = QUUX-V01-CS02-with-expander-SHA256-128-long-DST-111111
             111111111111111111111111111111111111111111111111111111
             111111111111111111111111111111111111111111111111111111
             111111111111111111111111111111111111111111111111111111
             1111111111111111111111111111111111111111
   hash    = SHA256
   k       = 128

   msg     =
   len_in_bytes = 0x20
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000002000412717974da474d0f8c420f320
             ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = e8dc0c8b686b7ef2074086fbdd2f30e3f8bfbd3bdf177f73
             f04b97ce618a3ed3

   msg     = abc
   len_in_bytes = 0x20
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 154]

Internet-Draft                hash-to-curve                    June 2022


   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000616263002000412717974da474d0f8c4
             20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 52dbf4f36cf560fca57dedec2ad924ee9c266341d8f3d6af
             e5171733b16bbb12

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000061626364656630313233343536373839
             002000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0
             82b4fb4d16c0a23620
   uniform_bytes = 35387dcf22618f3728e6c686490f8b431f76550b0b2c61cb
             c1ce7001536f4521

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000713132385f7171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171002000412717974da474d0f8c420f320ff81
             e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 01b637612bb18e840028be900a833a74414140dde0c4754c
             198532c3a0ba42bc

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 155]

Internet-Draft                hash-to-curve                    June 2022


   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000613531325f6161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161002000412717974da474d0f8
             c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 20cce7033cabc5460743180be6fa8aac5a103f56d481cf36
             9a8accc0c374431b

   msg     =
   len_in_bytes = 0x80
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000008000412717974da474d0f8c420f320
             ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 14604d85432c68b757e485c8894db3117992fc57e0e136f7
             1ad987f789a0abc287c47876978e2388a02af86b1e8d1342e5ce4f
             7aaa07a87321e691f6fba7e0072eecc1218aebb89fb14a0662322d
             5edbd873f0eb35260145cd4e64f748c5dfe60567e126604bcab1a3
             ee2dc0778102ae8a5cfd1429ebc0fa6bf1a53c36f55dfc

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 156]

Internet-Draft                hash-to-curve                    June 2022


             0000000000000000000000616263008000412717974da474d0f8c4
             20f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 1a30a5e36fbdb87077552b9d18b9f0aee16e80181d5b951d
             0471d55b66684914aef87dbb3626eaabf5ded8cd0686567e503853
             e5c84c259ba0efc37f71c839da2129fe81afdaec7fbdc0ccd4c794
             727a17c0d20ff0ea55e1389d6982d1241cb8d165762dbc39fb0cee
             4474d2cbbd468a835ae5b2f20e4f959f56ab24cd6fe267

   msg     = abcdef0123456789
   len_in_bytes = 0x80
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000061626364656630313233343536373839
             008000412717974da474d0f8c420f320ff81e8432adb7c927d9bd0
             82b4fb4d16c0a23620
   uniform_bytes = d2ecef3635d2397f34a9f86438d772db19ffe9924e28a1ca
             f6f1c8f15603d4028f40891044e5c7e39ebb9b31339979ff33a424
             9206f67d4a1e7c765410bcd249ad78d407e303675918f20f26ce6d
             7027ed3774512ef5b00d816e51bfcc96c3539601fa48ef1c07e494
             bdc37054ba96ecb9dbd666417e3de289d4f424f502a982

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x80
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000713132385f7171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171008000412717974da474d0f8c420f320ff81
             e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = ed6e8c036df90111410431431a232d41a32c86e296c05d42
             6e5f44e75b9a50d335b2412bc6c91e0a6dc131de09c43110d9180d
             0a70f0d6289cb4e43b05f7ee5e9b3f42a1fad0f31bac6a625b3b5c
             50e3a83316783b649e5ecc9d3b1d9471cb5024b7ccf40d41d1751a
             04ca0356548bc6e703fca02ab521b505e8e45600508d32

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 157]

Internet-Draft                hash-to-curve                    June 2022


             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = 412717974da474d0f8c420f320ff81e8432adb7c927d9bd082b4
             fb4d16c0a23620
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             0000000000000000000000613531325f6161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161008000412717974da474d0f8
             c420f320ff81e8432adb7c927d9bd082b4fb4d16c0a23620
   uniform_bytes = 78b53f2413f3c688f07732c10e5ced29a17c6a16f717179f
             fbe38d92d6c9ec296502eb9889af83a1928cd162e845b0d3c5424e
             83280fed3d10cffb2f8431f14e7a23f4c68819d40617589e4c4116
             9d0b56e0e3535be1fd71fbb08bb70c5b5ffed953d6c14bf7618b35
             fc1f4c4b30538236b4b08c9fbf90462447a8ada60be495

K.3.  expand_message_xmd(SHA-512)

   name    = expand_message_xmd
   DST     = QUUX-V01-CS02-with-expander-SHA512-256
   hash    = SHA512
   k       = 256

   msg     =
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 158]

Internet-Draft                hash-to-curve                    June 2022


             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000002000515555
             582d5630312d435330322d776974682d657870616e6465722d5348
             413531322d32353626
   uniform_bytes = 6b9a7312411d92f921c6f68ca0b6380730a1a4d982c50721
             1a90964c394179ba

   msg     = abc
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000616263002000
             515555582d5630312d435330322d776974682d657870616e646572
             2d5348413531322d32353626
   uniform_bytes = 0da749f12fbe5483eb066a5f595055679b976e93abe9be6f
             0f6318bce7aca8dc

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000616263646566
             30313233343536373839002000515555582d5630312d435330322d
             776974682d657870616e6465722d5348413531322d32353626
   uniform_bytes = 087e45a86e2939ee8b91100af1583c4938e0f5fc6c9db4b1
             07b83346bc967f58

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 159]

Internet-Draft                hash-to-curve                    June 2022


             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000713132385f71
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             71717171717171717171717171717171717171002000515555582d
             5630312d435330322d776974682d657870616e6465722d53484135
             31322d32353626
   uniform_bytes = 7336234ee9983902440f6bc35b348352013becd88938d2af
             ec44311caf8356b3

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000613531325f61
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 160]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161610020
             00515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   uniform_bytes = 57b5f7e766d5be68a6bfe1768e3c2b7f1228b3e4b3134956
             dd73a59b954c66f4

   msg     =
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000008000515555
             582d5630312d435330322d776974682d657870616e6465722d5348
             413531322d32353626
   uniform_bytes = 41b037d1734a5f8df225dd8c7de38f851efdb45c372887be
             655212d07251b921b052b62eaed99b46f72f2ef4cc96bfaf254ebb
             bec091e1a3b9e4fb5e5b619d2e0c5414800a1d882b62bb5cd1778f
             098b8eb6cb399d5d9d18f5d5842cf5d13d7eb00a7cff859b605da6
             78b318bd0e65ebff70bec88c753b159a805d2c89c55961

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000616263008000
             515555582d5630312d435330322d776974682d657870616e646572
             2d5348413531322d32353626
   uniform_bytes = 7f1dddd13c08b543f2e2037b14cefb255b44c83cc397c178
             6d975653e36a6b11bdd7732d8b38adb4a0edc26a0cef4bb4521713
             5456e58fbca1703cd6032cb1347ee720b87972d63fbf232587043e
             d2901bce7f22610c0419751c065922b488431851041310ad659e4b
             23520e1772ab29dcdeb2002222a363f0c2b1c972b3efe1

   msg     = abcdef0123456789
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 161]

Internet-Draft                hash-to-curve                    June 2022


             000000000000000000000000000000000000000000616263646566
             30313233343536373839008000515555582d5630312d435330322d
             776974682d657870616e6465722d5348413531322d32353626
   uniform_bytes = 3f721f208e6199fe903545abc26c837ce59ac6fa45733f1b
             aaf0222f8b7acb0424814fcb5eecf6c1d38f06e9d0a6ccfbf85ae6
             12ab8735dfdf9ce84c372a77c8f9e1c1e952c3a61b7567dd069301
             6af51d2745822663d0c2367e3f4f0bed827feecc2aaf98c949b5ed
             0d35c3f1023d64ad1407924288d366ea159f46287e61ac

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000713132385f71
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             71717171717171717171717171717171717171008000515555582d
             5630312d435330322d776974682d657870616e6465722d53484135
             31322d32353626
   uniform_bytes = b799b045a58c8d2b4334cf54b78260b45eec544f9f2fb5bd
             12fb603eaee70db7317bf807c406e26373922b7b8920fa29142703
             dd52bdf280084fb7ef69da78afdf80b3586395b433dc66cde048a2
             58e476a561e9deba7060af40adf30c64249ca7ddea79806ee5beb9
             a1422949471d267b21bc88e688e4014087a0b592b695ed

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   msg_prime = 0000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 162]

Internet-Draft                hash-to-curve                    June 2022


             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000000000000000
             000000000000000000000000000000000000000000613531325f61
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161610080
             00515555582d5630312d435330322d776974682d657870616e6465
             722d5348413531322d32353626
   uniform_bytes = 05b0bfef265dcee87654372777b7c44177e2ae4c13a27f10
             3340d9cd11c86cb2426ffcad5bd964080c2aee97f03be1ca18e30a
             1f14e27bc11ebbd650f305269cc9fb1db08bf90bfc79b42a952b46
             daf810359e7bc36452684784a64952c343c52e5124cd1f71d474d5
             197fefc571a92929c9084ffe1112cf5eea5192ebff330b

K.4.  expand_message_xof(SHAKE128)

   name    = expand_message_xof
   DST     = QUUX-V01-CS02-with-expander-SHAKE128
   hash    = SHAKE128
   k       = 128

   msg     =
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 0020515555582d5630312d435330322d776974682d657870616e
             6465722d5348414b4531323824
   uniform_bytes = 86518c9cd86581486e9485aa74ab35ba150d1c75c88e26b7
             043e44e2acd735a2

   msg     = abc
   len_in_bytes = 0x20



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 163]

Internet-Draft                hash-to-curve                    June 2022


   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 6162630020515555582d5630312d435330322d776974682d6578
             70616e6465722d5348414b4531323824
   uniform_bytes = 8696af52a4d862417c0763556073f47bc9b9ba43c99b5053
             05cb1ec04a9ab468

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 616263646566303132333435363738390020515555582d563031
             2d435330322d776974682d657870616e6465722d5348414b453132
             3824
   uniform_bytes = 912c58deac4821c3509dbefa094df54b34b8f5d01a191d1d
             3108a2c89077acca

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             20515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   uniform_bytes = 1adbcc448aef2a0cebc71dac9f756b22e51839d348e031e6
             3b33ebb50faeaf3f

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 164]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             61616161610020515555582d5630312d435330322d776974682d65
             7870616e6465722d5348414b4531323824
   uniform_bytes = df3447cc5f3e9a77da10f819218ddf31342c310778e0e4ef
             72bbaecee786a4fe

   msg     =
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 0080515555582d5630312d435330322d776974682d657870616e
             6465722d5348414b4531323824
   uniform_bytes = 7314ff1a155a2fb99a0171dc71b89ab6e3b2b7d59e38e644
             19b8b6294d03ffee42491f11370261f436220ef787f8f76f5b26bd
             cd850071920ce023f3ac46847744f4612b8714db8f5db83205b2e6
             25d95afd7d7b4d3094d3bdde815f52850bb41ead9822e08f22cf41
             d615a303b0d9dde73263c049a7b9898208003a739a2e57

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 6162630080515555582d5630312d435330322d776974682d6578
             70616e6465722d5348414b4531323824
   uniform_bytes = c952f0c8e529ca8824acc6a4cab0e782fc3648c563ddb00d
             a7399f2ae35654f4860ec671db2356ba7baa55a34a9d7f79197b60
             ddae6e64768a37d699a78323496db3878c8d64d909d0f8a7de4927
             dcab0d3dbbc26cb20a49eceb0530b431cdf47bc8c0fa3e0d88f53b
             318b6739fbed7d7634974f1b5c386d6230c76260d5337a

   msg     = abcdef0123456789
   len_in_bytes = 0x80



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 165]

Internet-Draft                hash-to-curve                    June 2022


   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 616263646566303132333435363738390080515555582d563031
             2d435330322d776974682d657870616e6465722d5348414b453132
             3824
   uniform_bytes = 19b65ee7afec6ac06a144f2d6134f08eeec185f1a890fe34
             e68f0e377b7d0312883c048d9b8a1d6ecc3b541cb4987c26f45e0c
             82691ea299b5e6889bbfe589153016d8131717ba26f07c3c14ffbe
             f1f3eff9752e5b6183f43871a78219a75e7000fbac6a7072e2b83c
             790a3a5aecd9d14be79f9fd4fb180960a3772e08680495

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             80515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   uniform_bytes = ca1b56861482b16eae0f4a26212112362fcc2d76dcc80c93
             c4182ed66c5113fe41733ed68be2942a3487394317f3379856f482
             2a611735e50528a60e7ade8ec8c71670fec6661e2c59a09ed36386
             513221688b35dc47e3c3111ee8c67ff49579089d661caa29db1ef1
             0eb6eace575bf3dc9806e7c4016bd50f3c0e2a6481ee6d

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4531323824
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 166]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             61616161610080515555582d5630312d435330322d776974682d65
             7870616e6465722d5348414b4531323824
   uniform_bytes = 9d763a5ce58f65c91531b4100c7266d479a5d9777ba76169
             3d052acd37d149e7ac91c796a10b919cd74a591a1e38719fb91b72
             03e2af31eac3bff7ead2c195af7d88b8bc0a8adf3d1e90ab9bed6d
             dc2b7f655dd86c730bdeaea884e73741097142c92f0e3fc1811b69
             9ba593c7fbd81da288a29d423df831652e3a01a9374999

K.5.  expand_message_xof(SHAKE128) (long DST)

   name    = expand_message_xof
   DST     = QUUX-V01-CS02-with-expander-SHAKE128-long-DST-11111111
             111111111111111111111111111111111111111111111111111111
             111111111111111111111111111111111111111111111111111111
             111111111111111111111111111111111111111111111111111111
             1111111111111111111111111111111111111111
   hash    = SHAKE128
   k       = 128

   msg     =
   len_in_bytes = 0x20
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 0020acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732
             12950132d035792f20
   uniform_bytes = 827c6216330a122352312bccc0c8d6e7a146c5257a776dbd
             9ad9d75cd880fc53

   msg     = abc
   len_in_bytes = 0x20
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 6162630020acb9736c0867fdfbd6385519b90fc8c034b5af04a9
             58973212950132d035792f20



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 167]

Internet-Draft                hash-to-curve                    June 2022


   uniform_bytes = 690c8d82c7213b4282c6cb41c00e31ea1d3e2005f93ad19b
             bf6da40f15790c5c

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 616263646566303132333435363738390020acb9736c0867fdfb
             d6385519b90fc8c034b5af04a958973212950132d035792f20
   uniform_bytes = 979e3a15064afbbcf99f62cc09fa9c85028afcf3f825eb07
             11894dcfc2f57057

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             20acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   uniform_bytes = c5a9220962d9edc212c063f4f65b609755a1ed96e62f9db5
             d1fd6adb5a8dc52b

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 168]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             61616161610020acb9736c0867fdfbd6385519b90fc8c034b5af04
             a958973212950132d035792f20
   uniform_bytes = f7b96a5901af5d78ce1d071d9c383cac66a1dfadb508300e
             c6aeaea0d62d5d62

   msg     =
   len_in_bytes = 0x80
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 0080acb9736c0867fdfbd6385519b90fc8c034b5af04a9589732
             12950132d035792f20
   uniform_bytes = 3890dbab00a2830be398524b71c2713bbef5f4884ac2e6f0
             70b092effdb19208c7df943dc5dcbaee3094a78c267ef276632ee2
             c8ea0c05363c94b6348500fae4208345dd3475fe0c834c2beac7fa
             7bc181692fb728c0a53d809fc8111495222ce0f38468b11becb15b
             32060218e285c57a60162c2c8bb5b6bded13973cd41819

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 6162630080acb9736c0867fdfbd6385519b90fc8c034b5af04a9
             58973212950132d035792f20
   uniform_bytes = 41b7ffa7a301b5c1441495ebb9774e2a53dbbf4e54b9a1af
             6a20fd41eafd69ef7b9418599c5545b1ee422f363642b01d4a5344
             9313f68da3e49dddb9cd25b97465170537d45dcbdf92391b5bdff3
             44db4bd06311a05bca7dcd360b6caec849c299133e5c9194f4e15e
             3e23cfaab4003fab776f6ac0bfae9144c6e2e1c62e7d57

   msg     = abcdef0123456789
   len_in_bytes = 0x80
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 616263646566303132333435363738390080acb9736c0867fdfb
             d6385519b90fc8c034b5af04a958973212950132d035792f20
   uniform_bytes = 55317e4a21318472cd2290c3082957e1242241d9e0d04f47



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 169]

Internet-Draft                hash-to-curve                    June 2022


             026f03401643131401071f01aa03038b2783e795bdfa8a3541c194
             ad5de7cb9c225133e24af6c86e748deb52e560569bd54ef4dac034
             65111a3a44b0ea490fb36777ff8ea9f1a8a3e8e0de3cf0880b4b2f
             8dd37d3a85a8b82375aee4fa0e909f9763319b55778e71

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x80
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             80acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   uniform_bytes = 19fdd2639f082e31c77717ac9bb032a22ff0958382b2dbb3
             9020cdc78f0da43305414806abf9a561cb2d0067eb2f7bc544482f
             75623438ed4b4e39dd9e6e2909dd858bd8f1d57cd0fce2d3150d90
             aa67b4498bdf2df98c0100dd1a173436ba5d0df6be1defb0b2ce55
             ccd2f4fc05eb7cb2c019c35d5398b85adc676da4238bc7

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = acb9736c0867fdfbd6385519b90fc8c034b5af04a95897321295
             0132d035792f20
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 170]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             61616161610080acb9736c0867fdfbd6385519b90fc8c034b5af04
             a958973212950132d035792f20
   uniform_bytes = 945373f0b3431a103333ba6a0a34f1efab2702efde41754c
             4cb1d5216d5b0a92a67458d968562bde7fa6310a83f53dda138368
             0a276a283438d58ceebfa7ab7ba72499d4a3eddc860595f63c93b1
             c5e823ea41fc490d938398a26db28f61857698553e93f0574eb8c5
             017bfed6249491f9976aaa8d23d9485339cc85ca329308

K.6.  expand_message_xof(SHAKE256)

   name    = expand_message_xof
   DST     = QUUX-V01-CS02-with-expander-SHAKE256
   hash    = SHAKE256
   k       = 256

   msg     =
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 0020515555582d5630312d435330322d776974682d657870616e
             6465722d5348414b4532353624
   uniform_bytes = 2ffc05c48ed32b95d72e807f6eab9f7530dd1c2f013914c8
             fed38c5ccc15ad76

   msg     = abc
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 6162630020515555582d5630312d435330322d776974682d6578
             70616e6465722d5348414b4532353624
   uniform_bytes = b39e493867e2767216792abce1f2676c197c0692aed06156
             0ead251821808e07

   msg     = abcdef0123456789
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 616263646566303132333435363738390020515555582d563031
             2d435330322d776974682d657870616e6465722d5348414b453235
             3624



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 171]

Internet-Draft                hash-to-curve                    June 2022


   uniform_bytes = 245389cf44a13f0e70af8665fe5337ec2dcd138890bb7901
             c4ad9cfceb054b65

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             20515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   uniform_bytes = 719b3911821e6428a5ed9b8e600f2866bcf23c8f0515e52d
             6c6c019a03f16f0e

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x20
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 172]

Internet-Draft                hash-to-curve                    June 2022


             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             61616161610020515555582d5630312d435330322d776974682d65
             7870616e6465722d5348414b4532353624
   uniform_bytes = 9181ead5220b1963f1b5951f35547a5ea86a820562287d6c
             a4723633d17ccbbc

   msg     =
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 0080515555582d5630312d435330322d776974682d657870616e
             6465722d5348414b4532353624
   uniform_bytes = 7a1361d2d7d82d79e035b8880c5a3c86c5afa719478c007d
             96e6c88737a3f631dd74a2c88df79a4cb5e5d9f7504957c70d669e
             c6bfedc31e01e2bacc4ff3fdf9b6a00b17cc18d9d72ace7d6b81c2
             e481b4f73f34f9a7505dccbe8f5485f3d20c5409b0310093d5d649
             2dea4e18aa6979c23c8ea5de01582e9689612afbb353df

   msg     = abc
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 6162630080515555582d5630312d435330322d776974682d6578
             70616e6465722d5348414b4532353624
   uniform_bytes = a54303e6b172909783353ab05ef08dd435a558c3197db0c1
             32134649708e0b9b4e34fb99b92a9e9e28fc1f1d8860d85897a8e0
             21e6382f3eea10577f968ff6df6c45fe624ce65ca25932f679a42a
             404bc3681efe03fcd45ef73bb3a8f79ba784f80f55ea8a3c367408
             f30381299617f50c8cf8fbb21d0f1e1d70b0131a7b6fbe

   msg     = abcdef0123456789
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 616263646566303132333435363738390080515555582d563031
             2d435330322d776974682d657870616e6465722d5348414b453235
             3624
   uniform_bytes = e42e4d9538a189316e3154b821c1bafb390f78b2f010ea40
             4e6ac063deb8c0852fcd412e098e231e43427bd2be1330bb47b403
             9ad57b30ae1fc94e34993b162ff4d695e42d59d9777ea18d3848d9
             d336c25d2acb93adcad009bcfb9cde12286df267ada283063de0bb
             1505565b2eb6c90e31c48798ecdc71a71756a9110ff373

   msg     = q128_qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq
             qqqqqqqqqqqqqqqqqqqqqqqqq



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 173]

Internet-Draft                hash-to-curve                    June 2022


   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 713132385f717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717171
             717171717171717171717171717171717171717171717171717100
             80515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   uniform_bytes = 4ac054dda0a38a65d0ecf7afd3c2812300027c8789655e47
             aecf1ecc1a2426b17444c7482c99e5907afd9c25b991990490bb9c
             686f43e79b4471a23a703d4b02f23c669737a886a7ec28bddb92c3
             a98de63ebf878aa363a501a60055c048bea11840c4717beae7eee2
             8c3cfa42857b3d130188571943a7bd747de831bd6444e0

   msg     = a512_aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
             aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa
   len_in_bytes = 0x80
   DST_prime = 515555582d5630312d435330322d776974682d657870616e6465
             722d5348414b4532353624
   msg_prime = 613531325f616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161
             616161616161616161616161616161616161616161616161616161



Faz-Hernandez, et al.   Expires 17 December 2022              [Page 174]

Internet-Draft                hash-to-curve                    June 2022


             61616161610080515555582d5630312d435330322d776974682d65
             7870616e6465722d5348414b4532353624
   uniform_bytes = 09afc76d51c2cccbc129c2315df66c2be7295a231203b8ab
             2dd7f95c2772c68e500bc72e20c602abc9964663b7a03a389be128
             c56971ce81001a0b875e7fd17822db9d69792ddf6a23a151bf4700
             79c518279aef3e75611f8f828994a9988f4a8a256ddb8bae161e65
             8d5a2a09bcfe839c6396dc06ee5c8ff3c22d3b1f9deb7e

Authors' Addresses

   Armando Faz-Hernandez
   Cloudflare, Inc.
   101 Townsend St
   San Francisco,
   United States of America
   Email: armfazh@cloudflare.com


   Sam Scott
   Cornell Tech
   2 West Loop Rd
   New York, New York 10044,
   United States of America
   Email: sam.scott@cornell.edu


   Nick Sullivan
   Cloudflare, Inc.
   101 Townsend St
   San Francisco,
   United States of America
   Email: nick@cloudflare.com


   Riad S. Wahby
   Stanford University
   Email: rsw@cs.stanford.edu


   Christopher A. Wood
   Cloudflare, Inc.
   101 Townsend St
   San Francisco,
   United States of America
   Email: caw@heapingbits.net






Faz-Hernandez, et al.   Expires 17 December 2022              [Page 175]
