



Internet Research Task Force                                     W. Ladd
Internet-Draft                                                  Sealance
Intended status: Informational                             B. Kaduk, Ed.
Expires: 12 August 2022                                           Akamai
                                                           February 2022


                             SPAKE2, a PAKE
                       draft-irtf-cfrg-spake2-26

Abstract

   This document describes SPAKE2 which is a protocol for two parties
   that share a password to derive a strong shared key without
   disclosing the password.  This method is compatible with any group,
   is computationally efficient, and SPAKE2 has a security proof.  This
   document predated the CFRG PAKE competition and it was not selected,
   however, given existing use of variants in Kerberos and other
   applications it was felt publication was beneficial.  Applications
   that need a symmetric PAKE (password authenticated key exchange) and
   where hashing onto an elliptic curve at execution time is not
   possible can use SPAKE2.  This document is a product of the Crypto
   Forum Research Group (CFRG) in the IRTF.

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."

   This Internet-Draft will expire on 5 August 2022.

Copyright Notice

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






Ladd & Kaduk             Expires 12 August 2022                 [Page 1]

Internet-Draft               SPAKE2, a PAKE                February 2022


   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.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
   2.  Requirements Notation . . . . . . . . . . . . . . . . . . . .   3
   3.  Definition of SPAKE2  . . . . . . . . . . . . . . . . . . . .   3
   4.  Key Schedule and Key Confirmation . . . . . . . . . . . . . .   6
   5.  Per-User M and N and M=N  . . . . . . . . . . . . . . . . . .   7
   6.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . . . .   7
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  10
   8.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  11
   9.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .  11
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  11
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  11
     10.2.  Informative References . . . . . . . . . . . . . . . . .  12
   Appendix A.  Algorithm used for Point Generation  . . . . . . . .  13
   Appendix B.  Test Vectors . . . . . . . . . . . . . . . . . . . .  14
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  17

1.  Introduction

   This document describes SPAKE2, a means for two parties that share a
   password to derive a strong shared key without disclosing the
   password.  This password-based key exchange protocol is compatible
   with any group (requiring only a scheme to map a random input of
   fixed length per group to a random group element), is computationally
   efficient, and has a security proof.  Predetermined parameters for a
   selection of commonly used groups are also provided for use by other
   protocols.

   SPAKE2 was not selected as the result of the CFRG PAKE selection
   competition.  However, given existing use of variants in Kerberos and
   other applications it was felt publication was beneficial.  This RFC
   represents the individual opinion(s) of one or more members of the
   Crypto Forum Research Group of the Internet Research Task Force
   (IRTF).

   Many of these applications predated methods to hash to elliptic
   curves being available or predated the publication of the PAKEs that
   were chosen as an outcome of the PAKE selection competition.  In
   cases where a symmetric PAKE is needed, and hashing onto an elliptic
   curve at protocol execution time is not available, SPAKE2 is useful.




Ladd & Kaduk             Expires 12 August 2022                 [Page 2]

Internet-Draft               SPAKE2, a PAKE                February 2022


2.  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.

3.  Definition of SPAKE2

3.1.  Protocol Flow

   SPAKE2 is a two round protocol, wherein the first round establishes a
   shared secret between A and B, and the second round serves as key
   confirmation.  Prior to invocation, A and B are provisioned with
   information such as the input password needed to run the protocol.
   We assume that the roles of A and B are agreed upon by both sides: A
   goes first and uses M, and B goes second and uses N.  If this
   assignment of roles is not possible a symmetric variant MUST be used
   as described later Section 5.  For instance A may be the client when
   using TCP or TLS as an underlying protocol and B the server.  Most
   protocols have such a distinction.  During the first round, A sends a
   public value pA to B, and B responds with its own public value pB.
   Both A and B then derive a shared secret used to produce encryption
   and authentication keys.  The latter are used during the second round
   for key confirmation.  (Section 4 details the key derivation and
   confirmation steps.)  In particular, A sends a key confirmation
   message cA to B, and B responds with its own key confirmation message
   cB.  A MUST NOT consider the protocol complete until it receives and
   verifies cB.  Likewise, B MUST NOT consider the protocol complete
   until it receives and verifies cA.

   This sample flow is shown below.


















Ladd & Kaduk             Expires 12 August 2022                 [Page 3]

Internet-Draft               SPAKE2, a PAKE                February 2022


                   A                  B
                   | (setup protocol) |
                   |                  |
     (compute pA)  |        pA        |
                   |----------------->|
                   |        pB        | (compute pB)
                   |<-----------------|
                   |                  |
                   | (derive secrets) |
                   |                  |
     (compute cA)  |        cA        |
                   |----------------->|
                   |        cB        | (compute cB)
                   |                  | (check cA)
                   |<-----------------|
     (check cB)    |                  |

3.2.  Setup

   Let G be a group in which the gap Diffie-Hellman (GDH) problem is
   hard.  Suppose G has order p*h where p is a large prime; h will be
   called the cofactor.  Let I be the unit element in G, e.g., the point
   at infinity if G is an elliptic curve group.  We denote the
   operations in the group additively.  We assume there is a
   representation of elements of G as byte strings: common choices would
   be SEC1 [SEC1] uncompressed or compressed for elliptic curve groups
   or big endian integers of a fixed (per-group) length for prime field
   DH.  Applications MUST specify this encoding, typically by referring
   to the document defining the group.  We fix two elements M and N in
   the prime-order subgroup of G as defined in the table in this
   document for common groups, as well as a generator P of the (large)
   prime-order subgroup of G.  In the case of a composite order group we
   will work in the quotient group.  For common groups used in this
   document, P is specified in the document defining the group, and so
   we do not repeat it here.

   For elliptic curves other than the ones in this document the methods
   of [I-D.irtf-cfrg-hash-to-curve] SHOULD be used to generate M and N,
   e.g. via M = hash_to_curve("M SPAKE2 seed OID x") and N =
   hash_to_curve("N SPAKE2 seed OID x"), where x is an OID for the
   curve.  Applications MAY include a DST tag in this step, as specified
   in [I-D.irtf-cfrg-hash-to-curve], though this is not required.

   || denotes concatenation of byte strings.  We also let len(S) denote
   the length of a string in bytes, represented as an eight-byte little-
   endian number.  Finally, let nil represent an empty string, i.e.,
   len(nil) = 0.  Text strings in double quotes are treated as their
   ASCII encodings throughout this document.



Ladd & Kaduk             Expires 12 August 2022                 [Page 4]

Internet-Draft               SPAKE2, a PAKE                February 2022


   KDF(ikm, salt, info, L) is a key-derivation function that takes as
   input a salt, intermediate keying material (IKM), info string, and
   derived key length L to derive a cryptographic key of length L.
   MAC(key, message) is a Message Authentication Code algorithm that
   takes a secret key and message as input to produce an output.  Let
   Hash be a hash function from arbitrary strings to bit strings of a
   fixed length, at least 256 bits long.  Common choices for Hash are
   SHA256 or SHA512 [RFC6234].  Let MHF be a memory-hard hash function
   designed to slow down brute-force attackers.  Scrypt [RFC7914] is a
   common example of this function.  The output length of MHF matches
   that of Hash.  Parameter selection for MHF is out of scope for this
   document.  Section 6 specifies variants of KDF, MAC, and Hash
   suitable for use with the protocols contained herein.

   Let A and B be two parties.  A and B may also have digital
   representations of the parties' identities such as Media Access
   Control addresses or other names (hostnames, usernames, etc).  A and
   B may share Additional Authenticated Data (AAD) of length at most
   2^16 - 128 bits that is separate from their identities which they may
   want to include in the protocol execution.  One example of AAD is a
   list of supported protocol versions if SPAKE2 were used in a higher-
   level protocol which negotiates use of a particular PAKE.  Including
   this list would ensure that both parties agree upon the same set of
   supported protocols and therefore prevent downgrade attacks.  We also
   assume A and B share an integer w; typically w = MHF(pw) mod p, for a
   user-supplied password pw.  Standards such as NIST.SP.800-56Ar3
   suggest taking mod p of a hash value that is 64 bits longer than that
   needed to represent p to remove statistical bias introduced by the
   modulation.  Protocols using this specification MUST define the
   method used to compute w.  In some cases, it may be necessary to
   carry out various forms of normalization of the password before
   hashing [RFC8265].  The hashing algorithm SHOULD be a MHF so as to
   slow down brute-force attackers.

3.3.  SPAKE2

   To begin, A picks x randomly and uniformly from the integers in
   [0,p), and calculates X=x*P and pA=w*M+X, then transmits pA to B.

   B selects y randomly and uniformly from the integers in [0,p), and
   calculates Y=y*P, pB=w*N+Y, then transmits pB to A.

   Both A and B calculate a group element K.  A calculates it as
   h*x*(pB-w*N), while B calculates it as h*y*(pA-w*M).  A knows pB
   because it has received it, and likewise B knows pA.  The
   multiplication by h prevents small subgroup confinement attacks by
   computing a unique value in the quotient group.




Ladd & Kaduk             Expires 12 August 2022                 [Page 5]

Internet-Draft               SPAKE2, a PAKE                February 2022


   K is a shared value, though it MUST NOT be used or output as a shared
   secret from the protocol.  Both A and B must derive two additional
   shared secrets from the protocol transcript, which includes K.  This
   prevents man-in-the-middle attackers from inserting themselves into
   the exchange.  The transcript TT is encoded as follows:

           TT = len(A)  || A
             || len(B)  || B
             || len(pA) || pA
             || len(pB) || pB
             || len(K)  || K
             || len(w)  || w

   Here w is encoded as a big endian number padded to the length of p.
   This representation prevents timing attacks that otherwise would
   reveal the length of w. len(w) is thus a constant.  We include it for
   consistency.

   If an identity is absent, it is encoded as a zero-length string.
   This MUST only be done for applications in which identities are
   implicit.  Otherwise, the protocol risks unknown key share attacks,
   where both sides of a connection disagree over who is authenticated.

   Upon completion of this protocol, A and B compute shared secrets Ke,
   KcA, and KcB as specified in Section 4.  A MUST send B a key
   confirmation message so both parties agree upon these shared secrets.
   This confirmation message cA is computed as a MAC over the protocol
   transcript TT using KcA, as follows: cA = MAC(KcA, TT).  Similarly, B
   MUST send A a confirmation message using a MAC computed equivalently
   except with the use of KcB.  Key confirmation verification requires
   computing cB and checking for equality against that which was
   received.

4.  Key Schedule and Key Confirmation

   The protocol transcript TT, as defined in Section 3.3, is unique and
   secret to A and B.  Both parties use TT to derive shared symmetric
   secrets Ke and Ka as Ke || Ka = Hash(TT), with |Ke| = |Ka|.  The
   length of each key is equal to half of the digest output, e.g., 128
   bits for SHA-256.  Keys MUST be at least 128 bits in length.











Ladd & Kaduk             Expires 12 August 2022                 [Page 6]

Internet-Draft               SPAKE2, a PAKE                February 2022


   Both endpoints use Ka to derive subsequent MAC keys for key
   confirmation messages.  Specifically, let KcA and KcB be the MAC keys
   used by A and B, respectively.  A and B compute them as KcA || KcB =
   KDF(Ka, nil, "ConfirmationKeys" || AAD, L), where AAD is the
   associated data each given to each endpoint, or nil if none was
   provided.  The length of each of KcA and KcB is equal to half of the
   underlying hash output length, e.g., |KcA| = |KcB| = 128 bits for
   HKDF(SHA256), with L=256 bits.

   The resulting key schedule for this protocol, given transcript TT and
   additional associated data AAD, is as follows.

       TT  -> Hash(TT) = Ke || Ka
       AAD -> KDF(Ka, nil, "ConfirmationKeys" || AAD) = KcA || KcB

   A and B output Ke as the shared secret from the protocol.  Ka and its
   derived keys are not used for anything except key confirmation.

5.  Per-User M and N and M=N

   To avoid concerns that an attacker needs to solve a single ECDH
   instance to break the authentication of SPAKE2, it is possible to
   vary M and N using [I-D.irtf-cfrg-hash-to-curve] as follows:

       M = hash_to_curve(Hash("M SPAKE2" || len(A) || A || len(B) || B))
       N = hash_to_curve(Hash("N SPAKE2" || len(A) || A || len(B) || B))

   There is also a symmetric variant where M=N.  For this variant we set

       M = hash_to_curve(Hash("M AND N SPAKE2"))

   This variant MUST be used when it is not possible to determine which
   of A and B should use M or N, due to asymmetries in the protocol
   flows or the desire to use only a single shared secret with nil
   identities for authentication.  The security of these variants is
   examined in [MNVAR].  The variant with per-user M and N may not be
   suitable for protocols that require the initial messages to be
   generated by each party at the same time and do not know the exact
   identity of the parties before the flow begins.

6.  Ciphersuites

   This section documents SPAKE2 ciphersuite configurations.  A
   ciphersuite indicates a group, cryptographic hash function, and pair
   of KDF and MAC functions, e.g., SPAKE2-P256-SHA256-HKDF-HMAC.  This
   ciphersuite indicates a SPAKE2 protocol instance over P-256 that uses
   SHA256 along with HKDF [RFC5869] and HMAC [RFC2104] for G, Hash, KDF,
   and MAC functions, respectively.  For Ed25519 the compressed encoding



Ladd & Kaduk             Expires 12 August 2022                 [Page 7]

Internet-Draft               SPAKE2, a PAKE                February 2022


   is used [RFC8032], all others use the uncompressed SEC1 encoding.

   +==============+==================+================+================+
   |      G       |       Hash       |      KDF       |      MAC       |
   +==============+==================+================+================+
   |    P-256     | SHA256 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |              |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |    P-256     | SHA512 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |              |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |    P-384     | SHA256 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |              |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |    P-384     | SHA512 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |              |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |    P-521     | SHA512 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |              |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   | edwards25519 | SHA256 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |  [RFC8032]   |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |  edwards448  | SHA512 [RFC6234] | HKDF [RFC5869] |      HMAC      |
   |  [RFC8032]   |                  |                |   [RFC2104]    |
   +--------------+------------------+----------------+----------------+
   |    P-256     | SHA256 [RFC6234] | HKDF [RFC5869] |  CMAC-AES-128  |
   |              |                  |                |   [RFC4493]    |
   +--------------+------------------+----------------+----------------+
   |    P-256     | SHA512 [RFC6234] | HKDF [RFC5869] |  CMAC-AES-128  |
   |              |                  |                |   [RFC4493]    |
   +--------------+------------------+----------------+----------------+

                        Table 1: SPAKE2 Ciphersuites

   The following points represent permissible point generation seeds for
   the groups listed in the Table Table 1, using the algorithm presented
   in Appendix A.  These bytestrings are compressed points as in [SEC1]
   for curves from [SEC1].

   For P256:










Ladd & Kaduk             Expires 12 August 2022                 [Page 8]

Internet-Draft               SPAKE2, a PAKE                February 2022


   M =
   02886e2f97ace46e55ba9dd7242579f2993b64e16ef3dcab95afd497333d8fa12f
   seed: 1.2.840.10045.3.1.7 point generation seed (M)

   N =
   03d8bbd6c639c62937b04d997f38c3770719c629d7014d49a24b4f98baa1292b49
   seed: 1.2.840.10045.3.1.7 point generation seed (N)

   For P384:

   M =
   030ff0895ae5ebf6187080a82d82b42e2765e3b2f8749c7e05eba366434b363d3dc
   36f15314739074d2eb8613fceec2853
   seed: 1.3.132.0.34 point generation seed (M)

   N =
   02c72cf2e390853a1c1c4ad816a62fd15824f56078918f43f922ca21518f9c543bb
   252c5490214cf9aa3f0baab4b665c10
   seed: 1.3.132.0.34 point generation seed (N)

   For P521:

   M =
   02003f06f38131b2ba2600791e82488e8d20ab889af753a41806c5db18d37d85608
   cfae06b82e4a72cd744c719193562a653ea1f119eef9356907edc9b56979962d7aa
   seed: 1.3.132.0.35 point generation seed (M)

   N =
   0200c7924b9ec017f3094562894336a53c50167ba8c5963876880542bc669e494b25
   32d76c5b53dfb349fdf69154b9e0048c58a42e8ed04cef052a3bc349d95575cd25
   seed: 1.3.132.0.35 point generation seed (N)

   For edwards25519:

   M =
   d048032c6ea0b6d697ddc2e86bda85a33adac920f1bf18e1b0c6d166a5cecdaf
   seed: edwards25519 point generation seed (M)

   N =
   d3bfb518f44f3430f29d0c92af503865a1ed3281dc69b35dd868ba85f886c4ab
   seed: edwards25519 point generation seed (N)

   For edwards448:








Ladd & Kaduk             Expires 12 August 2022                 [Page 9]

Internet-Draft               SPAKE2, a PAKE                February 2022


   M =
   b6221038a775ecd007a4e4dde39fd76ae91d3cf0cc92be8f0c2fa6d6b66f9a12
   942f5a92646109152292464f3e63d354701c7848d9fc3b8880
   seed: edwards448 point generation seed (M)

   N =
   6034c65b66e4cd7a49b0edec3e3c9ccc4588afd8cf324e29f0a84a072531c4db
   f97ff9af195ed714a689251f08f8e06e2d1f24a0ffc0146600
   seed: edwards448 point generation seed (N)

7.  Security Considerations

   A security proof of SPAKE2 for prime order groups is found in [REF],
   reducing the security of SPAKE2 to the gap Diffie-Hellman assumption.
   Note that the choice of M and N is critical for the security proof.
   The generation methods specified in this document are designed to
   eliminate concerns related to knowing discrete logs of M and N.

   Elements received from a peer MUST be checked for group membership:
   failure to properly deserialize and validate group elements can lead
   to attacks.  An endpoint MUST abort the protocol if any received
   public value is not a member of G.

   The choices of random numbers MUST BE uniform.  Randomly generated
   values, e.g., x and y, MUST NOT be reused; such reuse violates the
   security assumptions of the protocol and results in significant
   insecurity.  It is RECOMMENDED to generate these uniform numbers
   using rejection sampling.

   Some implementations of elliptic curve multiplication may leak
   information about the length of the scalar.  These MUST NOT be used.
   All operations on elliptic curve points must take time independent of
   the inputs.  Hashing of the transcript may take time depending only
   on the length of the transcript, but not the contents.

   SPAKE2 does not support augmentation.  As a result, the server has to
   store a password equivalent.  This is considered a significant
   drawback in some use cases.  Applications that need augmented PAKEs
   should use [I-D.irtf-cfrg-opaque].

   The HMAC keys in this document are shorter than recommended in
   [RFC8032].  This is appropriate as the difficulty of the discrete
   logarithm problem is comparable with the difficulty of brute forcing
   the keys.







Ladd & Kaduk             Expires 12 August 2022                [Page 10]

Internet-Draft               SPAKE2, a PAKE                February 2022


8.  IANA Considerations

   No IANA action is required.

9.  Acknowledgments

   Special thanks to Nathaniel McCallum and Greg Hudson for generation
   of M and N, and Chris Wood for test vectors.  Thanks to Mike Hamburg
   for advice on how to deal with cofactors.  Greg Hudson also suggested
   the addition of warnings on the reuse of x and y.  Thanks to Fedor
   Brunner, Adam Langley, Liliya Akhmetzyanova, and the members of the
   CFRG for comments and advice.  Thanks to Scott Fluhrer and those
   Crypto Panel experts involved in the PAKE selection process
   (https://github.com/cfrg/pake-selection) who have provided valuable
   comments.  Chris Wood contributed substantial text and reformatting
   to address the excellent review comments from Kenny Paterson.

10.  References

10.1.  Normative References

   [I-D.irtf-cfrg-hash-to-curve]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and
              C. Wood, "Hashing to Elliptic Curves", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-hash-to-curve-05, 2
              November 2019, <http://www.ietf.org/internet-drafts/draft-
              irtf-cfrg-hash-to-curve-05.txt>.

   [RFC2104]  Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed-
              Hashing for Message Authentication", RFC 2104,
              DOI 10.17487/RFC2104, February 1997,
              <https://www.rfc-editor.org/info/rfc2104>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [RFC4493]  Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
              AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
              2006, <https://www.rfc-editor.org/info/rfc4493>.

   [RFC5480]  Turner, S., Brown, D., Yiu, K., Housley, R., and T. Polk,
              "Elliptic Curve Cryptography Subject Public Key
              Information", RFC 5480, DOI 10.17487/RFC5480, March 2009,
              <https://www.rfc-editor.org/info/rfc5480>.





Ladd & Kaduk             Expires 12 August 2022                [Page 11]

Internet-Draft               SPAKE2, a PAKE                February 2022


   [RFC5869]  Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand
              Key Derivation Function (HKDF)", RFC 5869,
              DOI 10.17487/RFC5869, May 2010,
              <https://www.rfc-editor.org/info/rfc5869>.

   [RFC6234]  Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
              (SHA and SHA-based HMAC and HKDF)", RFC 6234,
              DOI 10.17487/RFC6234, May 2011,
              <https://www.rfc-editor.org/info/rfc6234>.

   [RFC7914]  Percival, C. and S. Josefsson, "The scrypt Password-Based
              Key Derivation Function", RFC 7914, DOI 10.17487/RFC7914,
              August 2016, <https://www.rfc-editor.org/info/rfc7914>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/info/rfc8032>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

10.2.  Informative References

   [SEC1]     Standards for Efficient Cryptography Group, "SEC 1:
              Elliptic Curve Cryptography", May 2009.

   [MNVAR]    Abdalla, M., Barbosa, M., Bradley, T., Jarecki, S., Katz,
              J., and J. Xu, "Universally Composable Relaxed Password
              Authentication", August 2020.  Appears in Micciancio D.,
              Ristenpart T. (eds) Advances in Cryptology -CRYPTO 2020.
              Crypto 2020.  Lecture notes in Computer Science volume
              12170.  Springer.

   [REF]      Abdalla, M. and D. Pointcheval, "Simple Password-Based
              Encrypted Key Exchange Protocols.", February 2005.
              Appears in A.  Menezes, editor.  Topics in Cryptography-
              CT-RSA 2005, Volume 3376 of Lecture Notes in Computer
              Science, pages 191-208, San Francisco, CA, US.  Springer-
              Verlag, Berlin, Germany.

   [TDH]      Cash, D., Kiltz, E., and V. Shoup, "The Twin-Diffie
              Hellman Problem and Applications", 2008.  EUROCRYPT 2008.
              Volume 4965 of Lecture notes in Computer Science, pages
              127-145.  Springer-Verlag, Berlin, Germany.





Ladd & Kaduk             Expires 12 August 2022                [Page 12]

Internet-Draft               SPAKE2, a PAKE                February 2022


   [RFC8265]  Saint-Andre, P. and A. Melnikov, "Preparation,
              Enforcement, and Comparison of Internationalized Strings
              Representing Usernames and Passwords", RFC 8265,
              DOI 10.17487/RFC8265, October 2017,
              <https://www.rfc-editor.org/info/rfc8265>.

   [I-D.irtf-cfrg-opaque]
              Krawczyk, H., Bourdrez, D., Lewi, K., and C. A. Wood, "The
              OPAQUE Asymmetric PAKE Protocol", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-opaque-06, 12 July 2021,
              <https://www.ietf.org/archive/id/draft-irtf-cfrg-opaque-
              06.txt>.

Appendix A.  Algorithm used for Point Generation

   This section describes the algorithm that was used to generate the
   points M and N in the table in Section 6.

   For each curve in the table below, we construct a string using the
   curve OID from [RFC5480] (as an ASCII string) or its name, combined
   with the needed constant, e.g., "1.3.132.0.35 point generation seed
   (M)" for P-521.  This string is turned into a series of blocks by
   hashing with SHA256, and hashing that output again to generate the
   next 32 bytes, and so on.  This pattern is repeated for each group
   and value, with the string modified appropriately.

   A byte string of length equal to that of an encoded group element is
   constructed by concatenating as many blocks as are required, starting
   from the first block, and truncating to the desired length.  The byte
   string is then formatted as required for the group.  In the case of
   Weierstrass curves, we take the desired length as the length for
   representing a compressed point (section 2.3.4 of [SEC1]), and use
   the low-order bit of the first byte as the sign bit.  In order to
   obtain the correct format, the value of the first byte is set to 0x02
   or 0x03 (clearing the first six bits and setting the seventh bit),
   leaving the sign bit as it was in the byte string constructed by
   concatenating hash blocks.  For the [RFC8032] curves a different
   procedure is used.  For edwards448 the 57-byte input has the least-
   significant 7 bits of the last byte set to zero, and for edwards25519
   the 32-byte input is not modified.  For both the [RFC8032] curves the
   (modified) input is then interpreted as the representation of the
   group element.  If this interpretation yields a valid group element
   with the correct order (p), the (modified) byte string is the output.
   Otherwise, the initial hash block is discarded and a new byte string
   constructed from the remaining hash blocks.  The procedure of
   constructing a byte string of the appropriate length, formatting it
   as required for the curve, and checking if it is a valid point of the
   correct order, is repeated until a valid element is found.



Ladd & Kaduk             Expires 12 August 2022                [Page 13]

Internet-Draft               SPAKE2, a PAKE                February 2022


   The following python snippet generates the above points, assuming an
   elliptic curve implementation following the interface of
   Edwards25519Point.stdbase() and Edwards448Point.stdbase() in
   Appendix A of [RFC8032]:

  def iterated_hash(seed, n):
      h = seed
      for i in range(n):
          h = hashlib.sha256(h).digest()
      return h

  def bighash(seed, start, sz):
      n = -(-sz // 32)
      hashes = [iterated_hash(seed, i) for i in range(start, start + n)]
      return b''.join(hashes)[:sz]

  def canon_pointstr(ecname, s):
      if ecname == 'edwards25519':
          return s
      elif ecname == 'edwards448':
          return s[:-1] + bytes([s[-1] & 0x80])
      else:
          return bytes([(s[0] & 1) | 2]) + s[1:]

  def gen_point(seed, ecname, ec):
      for i in range(1, 1000):
          hval = bighash(seed, i, len(ec.encode()))
          pointstr = canon_pointstr(ecname, hval)
          try:
              p = ec.decode(pointstr)
              if p != ec.zero_elem() and p * p.l() == ec.zero_elem():
                  return pointstr, i
          except Exception:
              pass

Appendix B.  Test Vectors

   This section contains test vectors for SPAKE2 using the P256-SHA256-
   HKDF-HMAC ciphersuite.  (Choice of MHF is omitted and values for w,
   x, and y are provided directly.)  All points are encoded using the
   uncompressed format, i.e., with a 0x04 octet prefix, specified in
   [SEC1] A and B identity strings are provided in the protocol
   invocation.

B.1.  SPAKE2 Test Vectors






Ladd & Kaduk             Expires 12 August 2022                [Page 14]

Internet-Draft               SPAKE2, a PAKE                February 2022


spake2: A='server', B='client'
w = 0x2ee57912099d31560b3a44b1184b9b4866e904c49d12ac5042c97dca461b1a5f
x = 0x43dd0fd7215bdcb482879fca3220c6a968e66d70b1356cac18bb26c84a78d729
pA = 0x04a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751470
f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c
y = 0xdcb60106f276b02606d8ef0a328c02e4b629f84f89786af5befb0bc75b6e66be
pB = 0x0406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf4367a420676
d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a1e0b7
K = 0x0412af7e89717850671913e6b469ace67bd90a4df8ce45c2af19010175e37eed
69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db277b715d3325
TT = 0x06000000000000007365727665720600000000000000636c69656e744100000
00000000004a56fa807caaa53a4d28dbb9853b9815c61a411118a6fe516a8798434751
470f9010153ac33d0d5f2047ffdb1a3e42c9b4e6be662766e1eeb4116988ede5f912c4
1000000000000000406557e482bd03097ad0cbaa5df82115460d951e3451962f1eaf43
67a420676d09857ccbc522686c83d1852abfa8ed6e4a1155cf8f1543ceca528afb591a
1e0b741000000000000000412af7e89717850671913e6b469ace67bd90a4df8ce45c2a
f19010175e37eed69f75897996d539356e2fa6a406d528501f907e04d97515fbe83db2
77b715d332520000000000000002ee57912099d31560b3a44b1184b9b4866e904c49d1
2ac5042c97dca461b1a5f
Hash(TT) = 0x0e0672dc86f8e45565d338b0540abe6915bdf72e2b35b5c9e5663168e960a91b
Ke = 0x0e0672dc86f8e45565d338b0540abe69
Ka = 0x15bdf72e2b35b5c9e5663168e960a91b
KcA = 0x00c12546835755c86d8c0db7851ae86f
KcB = 0xa9fa3406c3b781b93d804485430ca27a
A conf = 0x58ad4aa88e0b60d5061eb6b5dd93e80d9c4f00d127c65b3b35b1b5281fee38f0
B conf = 0xd3e2e547f1ae04f2dbdbf0fc4b79f8ecff2dff314b5d32fe9fcef2fb26dc459b


spake2: A='', B='client'
w = 0x0548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e883e2efae
x = 0x403abbe3b1b4b9ba17e3032849759d723939a27a27b9d921c500edde18ed654b
pA = 0x04a897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f
44fb92ba966578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a
y = 0x903023b6598908936ea7c929bd761af6039577a9c3f9581064187c3049d87065
pB = 0x04e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a6
1b881a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc
K = 0x048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a735e
bda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa6
TT = 0x00000000000000000600000000000000636c69656e74410000000000000004a
897b769e681c62ac1c2357319a3d363f610839c4477720d24cbe32f5fd85f44fb92ba9
66578c1b712be6962498834078262caa5b441ecfa9d4a9485720e918a4100000000000
00004e0f816fd1c35e22065d5556215c097e799390d16661c386e0ecc84593974a61b8
81a8c82327687d0501862970c64565560cb5671f696048050ca66ca5f8cc7fc4100000
000000000048f83ec9f6e4f87cc6f9dc740bdc2769725f923364f01c84148c049a39a7
35ebda82eac03e00112fd6a5710682767cff5361f7e819e53d8d3c3a2922e0d837aa62
0000000000000000548d8729f730589e579b0475a582c1608138ddf7054b73b5381c7e
883e2efae
Hash(TT) = 0x642f05c473c2cd79909f9a841e2f30a70bf89b18180af97353ba198789c2b963



Ladd & Kaduk             Expires 12 August 2022                [Page 15]

Internet-Draft               SPAKE2, a PAKE                February 2022


Ke = 0x642f05c473c2cd79909f9a841e2f30a7
Ka = 0x0bf89b18180af97353ba198789c2b963
KcA = 0xc6be376fc7cd1301fd0a13adf3e7bffd
KcB = 0xb7243f4ae60440a49b3f8cab3c1fba07
A conf = 0x47d29e6666af1b7dd450d571233085d7a9866e4d49d2645e2df975489521232b
B conf = 0x3313c5cefc361d27fb16847a91c2a73b766ffa90a4839122a9b70a2f6bd1d6df

spake2: A='server', B=''
w = 0x626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b14eae8c254
x = 0x07adb3db6bc623d3399726bfdbfd3d15a58ea776ab8a308b00392621291f9633
pA = 0x04f88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2
e15a873b298503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b0
y = 0xb6a4fc8dbb629d4ba51d6f91ed1532cf87adec98f25dd153a75accafafedec16
pB = 0x040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101
e7cca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db
K = 0x0445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986afa
f3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe
TT = 0x06000000000000007365727665720000000000000000410000000000000004f
88fb71c99bfffaea370966b7eb99cd4be0ff1a7d335caac4211c4afd855e2e15a873b2
98503ad8ba1d9cbb9a392d2ba309b48bfd7879aefd0f2cea6009763b04100000000000
000040c269d6be017dccb15182ac6bfcd9e2a14de019dd587eaf4bdfd353f031101e7c
ca177f8eb362a6e83e7d5e729c0732e1b528879c086f39ba0f31a9661bd34db4100000
0000000000445ee233b8ecb51ebd6e7da3f307e88a1616bae2166121221fdc0dadb986
afaf3ec8a988dc9c626fa3b99f58a7ca7c9b844bb3e8dd9554aafc5b53813504c1cbe2
000000000000000626e0cdc7b14c9db3e52a0b1b3a768c98e37852d5db30febe0497b1
4eae8c254
Hash(TT) = 0x005184ff460da2ce59062c87733c299c3521297d736598fc0a1127600efa1afb
Ke = 0x005184ff460da2ce59062c87733c299c
Ka = 0x3521297d736598fc0a1127600efa1afb
KcA = 0xf3da53604f0aeecea5a33be7bddf6edf
KcB = 0x9e3f86848736f159bd92b6e107ec6799
A conf = 0xbc9f9bbe99f26d0b2260e6456e05a86196a3307ec6663a18bf6ac825736533b2
B conf = 0xc2370e1bf813b086dff0d834e74425a06e6390f48f5411900276dcccc5a297ec

spake2: A='', B=''
w = 0x7bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
x = 0x8cef65df64bb2d0f83540c53632de911b5b24b3eab6cc74a97609fd659e95473
pA = 0x04a65b367a3f613cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f4
6ca1ef6a676598889fc28de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66
y = 0xd7a66f64074a84652d8d623a92e20c9675c61cb5b4f6a0063e4648a2fdc02d53
pB = 0x04589f13218822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494
662eea8ba6b64692dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d
K = 0x041a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40
a994b4477f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e135543772
TT = 0x00000000000000000000000000000000410000000000000004a65b367a3f613
cf9f0654b1b28a1e3a8a40387956c8ba6063e8658563890f46ca1ef6a676598889fc28
de2950ab8120b79a5ef1ea4c9f44bc98f585634b46d66410000000000000004589f132
18822710d98d8b2123a079041052d9941b9cf88c6617ddb2fcc0494662eea8ba6b6469



Ladd & Kaduk             Expires 12 August 2022                [Page 16]

Internet-Draft               SPAKE2, a PAKE                February 2022


2dc318250030c6af045cb738bc81ba35b043c3dcb46adf6f58d4100000000000000041
a3c03d51b452537ca2a1fea6110353c6d5ed483c4f0f86f4492ca3f378d40a994b4477
f93c64d928edbbcd3e85a7c709b7ea73ee97986ce3d1438e1355437722000000000000
0007bf46c454b4c1b25799527d896508afd5fc62ef4ec59db1efb49113063d70cca
Hash(TT) = 0xfc6374762ba5cf11f4b2caa08b2cd1b9907ae0e26e8d6234318d91583cd74c86
Ke = 0xfc6374762ba5cf11f4b2caa08b2cd1b9
Ka = 0x907ae0e26e8d6234318d91583cd74c86
KcA = 0x5dbd2f477166b7fb6d61febbd77a5563
KcB = 0x7689b4654407a5faeffdc8f18359d8a3
A conf = 0xdfb4db8d48ae5a675963ea5e6c19d98d4ea028d8e898dad96ea19a80ade95dca
B conf = 0xd0f0609d1613138d354f7e95f19fb556bf52d751947241e8c7118df5ef0ae175


Authors' Addresses

   Watson Ladd
   Sealance

   Email: watsonbladd@gmail.com


   Benjamin Kaduk (editor)
   Akamai Technologies

   Email: kaduk@mit.edu


























Ladd & Kaduk             Expires 12 August 2022                [Page 17]
