Internet-Draft ACT February 2026
Schlesinger & Katz Expires 17 August 2026 [Page]
Workgroup:
Network Working Group
Internet-Draft:
draft-schlesinger-cfrg-act-01
Published:
Intended Status:
Informational
Expires:
Authors:
S. Schlesinger
Google
J. Katz
Google

Anonymous Credit Tokens

Abstract

This document specifies Anonymous Credit Tokens (ACT), a privacy-preserving authentication protocol that enables numerical credit systems without tracking individual clients. Based on keyed-verification anonymous credentials and privately verifiable BBS-style signatures, the protocol allows issuers to grant tokens containing credits that clients can later spend anonymously with that issuer.

The protocol's key features include: (1) unlinkable transactions - the issuer cannot correlate credit issuance with spending, or link multiple spends by the same client, (2) partial spending - clients can spend a portion of their credits and receive anonymous change, and (3) double-spend prevention through cryptographic nullifiers that preserve privacy while ensuring each token is used only once.

Anonymous Credit Tokens are designed for modern web services requiring rate limiting, usage-based billing, or resource allocation while respecting user privacy. Example applications include rate limiting and API credits.

This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.

About This Document

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

The latest revision of this draft can be found at https://SamuelSchlesinger.github.io/draft-act/draft-schlesinger-cfrg-act.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draft-schlesinger-cfrg-act/.

Discussion of this document takes place on the Crypto Forum Research Group mailing list (mailto:cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/browse/cfrg. Subscribe at https://www.ietf.org/mailman/listinfo/cfrg/.

Source for this draft and an issue tracker can be found at https://github.com/SamuelSchlesinger/draft-act.

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 17 August 2026.

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Table of Contents

1. Introduction

Modern web services face a fundamental tension between operational needs and user privacy. Services need to implement rate limiting to prevent abuse, charge for API usage to sustain operations, and allocate computational resources fairly. However, traditional approaches require tracking client identities and creating detailed logs of client behavior, raising significant privacy concerns in an era of increasing data protection awareness and regulation.

Anonymous Credit Tokens (ACT) help to resolve this tension by providing a cryptographic protocol that enables credit-based systems without client tracking. Built on keyed-verification anonymous credentials [KVAC] and privately verifiable BBS-style signatures [BBS], the protocol allows services to issue, track, and spend credits while maintaining client privacy.

1.1. Key Properties

The protocol provides four essential properties that make it suitable for privacy-preserving credit systems:

  1. Unlinkability: The issuer cannot link credit issuance to spending, or connect multiple transactions by the same client. This property is information-theoretic, not merely computational.

  2. Partial Spending: Clients can spend any amount up to their balance and receive anonymous change without revealing their previous or current balance, enabling flexible spending.

  3. Double-Spend Prevention: Cryptographic nullifiers ensure each token is used only once, without linking it to issuance.

  4. Balance Privacy: During spending, only the amount being spent is revealed, not the total balance in the token, protecting clients from balance-based profiling.

1.2. Use Cases

Anonymous Credit Tokens can be applied to various scenarios:

  • Rate Limiting: Services can issue daily credit allowances that clients spend anonymously for API calls or resource access.

  • API Credits: API providers can sell credit packages that developers use to pay for API requests without creating a detailed usage history linked to their identity. This enables:

    • Pre-paid API access without requiring credit cards for each transaction

    • Anonymous API usage for privacy-sensitive applications

    • Usage-based billing without tracking individual request patterns

    • Protection against competitive analysis through usage monitoring

1.3. Protocol Overview

The protocol involves two parties: an issuer (typically a service provider) and clients (typically users of the service). The interaction follows three main phases:

  1. Setup: The issuer generates a key pair and publishes the public key.

  2. Issuance: A client requests credits from the issuer. The issuer creates a blind signature on the credit value and a client-chosen nullifier, producing a credit token.

  3. Spending: To spend credits, the client reveals a nullifier and proves possession of a valid token associated with that nullifier having sufficient balance. The issuer verifies the proof, checks the nullifier hasn't been used before, and issues a new token (which remains hidden from the issuer) for any remaining balance.

1.4. Design Goals

The protocol is designed with the following goals:

  • Privacy: Unlinkability between issuance and spending; see the Security Properties section for the formal definition.

  • Security: Clients cannot spend more credits than they possess or use the same credits multiple times.

  • Efficiency: All operations should be computationally efficient, with performance characteristics suitable for high-volume web services and a large number of applications.

  • Simplicity: The protocol should be straightforward to implement and integrate into existing systems relative to other comparable solutions.

1.5. Relation to Existing Work

This protocol builds upon several cryptographic primitives:

  • BBS Signatures [BBS]: The core signature scheme that enables efficient proofs of possession. We use a variant that is privately verifiable, which avoids the need for pairings and makes our protocol more efficient.

  • Sigma Protocols [ORRU-SIGMA]: The zero-knowledge proof framework used for spending proofs.

  • Fiat-Shamir Transform [ORRU-FS]: The technique to make the interactive proofs non-interactive.

The protocol can be viewed as a specialized instantiation of keyed-verification anonymous credentials [KVAC] optimized for numerical values and partial spending.

2. Conventions and Definitions

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.

2.1. Notation

This document uses the following notation:

  • ||: Concatenation of byte strings

  • x <- S: Sampling x uniformly from the set S

  • x := y: Assignment of the value y to the variable x

  • [n]: The set of integers {0, 1, ..., n-1}

  • |x|: The length of byte string x

  • 0x prefix: Hexadecimal values

  • We use additive notation for group operations, so group elements are added together like a + b and scalar multiplication of a group element by a scalar is written as a * n, with group element a and scalar n.

2.2. Data Types

The protocol uses the following data types:

  • Scalar: An integer modulo the group order q

  • Element: An element of the Ristretto255 group

  • ByteString: A sequence of bytes

2.3. Cryptographic Parameters

The protocol uses the Ristretto group [RFC9496], which provides a prime-order group abstraction over Curve25519. It would be easy to adapt this approach to using any other prime order group based on the contents of this document. The key parameters are:

  • q: The prime order of the group (2^252 + 27742317777372353535851937790883648493)

  • G: The standard generator of the Ristretto group

  • L: The bit length for credit values

3. Protocol Specification

3.1. System Parameters

The protocol requires the following system parameters:

Parameters:
  - G: Generator of the Ristretto group
  - H1, H2, H3, H4: Additional generators for commitments
  - L: Bit length for credit values (configurable, must satisfy 1 <= L <= 128)

Implementations MUST enforce 1 <= L <= 128. See the Parameter Selection section for the rationale behind this constraint.

The generators H1, H2, H3, and H4 MUST be generated deterministically from a nothing-up-my-sleeve value to ensure they are independent of each other and of G. This prevents attacks whereby malicious parameters could compromise security. Note that these generators are independent of the choice of L.

GenerateParameters(domain_separator):
  Input:
    - domain_separator: ByteString identifying the deployment
  Output:
    - params: System parameters (H1, H2, H3, H4)

  Steps:
    1. seed = BLAKE3(LengthPrefixed(domain_separator))
    2. counter = 0
    3. H1 = HashToRistretto255(seed, counter++)
    4. H2 = HashToRistretto255(seed, counter++)
    5. H3 = HashToRistretto255(seed, counter++)
    6. H4 = HashToRistretto255(seed, counter++)
    7. return (H1, H2, H3, H4)

HashToRistretto255(seed, counter):
  Input:
    - seed: 32-byte seed value
    - counter: Integer counter for domain separation
  Output:
    - P: A valid Ristretto255 point

  Steps:
    1. hasher = BLAKE3.new()
    2. hasher.update(LengthPrefixed(domain_separator))
    3. hasher.update(LengthPrefixed(seed))
    4. hasher.update(LengthPrefixed(counter.to_le_bytes(4)))
    5. uniform_bytes = hasher.finalize_xof(64)
    6. P = OneWayMap(uniform_bytes)
    7. return P

The domain_separator MUST be unique for each deployment to ensure cryptographic isolation between different services. The domain separator SHOULD follow this structured format:

domain_separator = "ACT-v1:" || organization || ":" || service || ":" || deployment_id || ":" || version

Each component (organization, service, deployment_id, version) MUST NOT contain the colon character ':'.

Where:

  • organization: A unique identifier for the organization (e.g., "example-corp", "acme-inc")

  • service: The specific service or application name (e.g., "payment-api", "rate-limiter")

  • deployment_id: The deployment environment (e.g., "production", "staging", "us-west-1")

  • version: An ISO 8601 date (YYYY-MM-DD) indicating when parameters were generated

Example: "ACT-v1:example-corp:payment-api:production:2024-01-15"

This structured format ensures: 1. Protocol identification through the "ACT-v1:" prefix 2. Organizational namespace isolation 3. Service-level separation within organizations 4. Environment isolation (production vs staging) 5. Version tracking for parameter updates

Using generic or unstructured domain separators creates security risks through parameter collision and MUST NOT be used. When parameters need to be updated (e.g., for security reasons or protocol upgrades), a new version date MUST be used, creating entirely new parameters.

The OneWayMap function is defined in [RFC9496] Section 4.3.4, which provides a cryptographically secure mapping from uniformly random byte strings to valid Ristretto255 points.

3.2. Key Generation

The issuer generates a key pair as follows:

KeyGen():
  Input: None
  Output:
    - sk: Private key (Scalar)
    - pk: Public key (Group Element)

  Steps:
    1. x <- Zq
    2. W = G * x
    3. sk = x
    4. pk = W
    5. return (sk, pk)

3.3. Token Issuance

The issuance protocol is an interactive protocol between a client and the issuer:

3.3.1. Client: Issuance Request 

IssueRequest():
  Output:
    - request: Issuance request
    - state: Client state for later verification

  Steps:
    1. k <- Zq  // Nullifier (will prevent double-spending)
    2. r <- Zq  // Blinding factor
    3. K = H2 * k + H3 * r
    4. // Generate proof of knowledge of k, r
    5. k' <- Zq
    6. r' <- Zq
    7. K1 = H2 * k' + H3 * r'
    8. transcript = CreateTranscript("request")
    9. AddToTranscript(transcript, K)
    10. AddToTranscript(transcript, K1)
    11. gamma = GetChallenge(transcript)
    12. k_bar = k' + gamma * k
    13. r_bar = r' + gamma * r
    14. request = (K, gamma, k_bar, r_bar)
    15. state = (k, r)
    16. return (request, state)

3.3.2. Issuer: Issuance Response 

IssueResponse(sk, request, c, ctx):
  Input:
    - sk: Issuer's private key
    - request: Client's issuance request
    - c: Credit amount to issue (c > 0)
    - ctx: Request context (Scalar)
  Output:
    - response: Issuance response or INVALID
  Exceptions:
    - InvalidIssuanceRequestProof, raised when the client proof verification fails

  Steps:
    1. Parse request as (K, gamma, k_bar, r_bar)
    2. // Verify proof of knowledge
    3. K1 = H2 * k_bar + H3 * r_bar - K * gamma
    4. transcript = CreateTranscript("request")
    5. AddToTranscript(transcript, K)
    6. AddToTranscript(transcript, K1)
    7. if GetChallenge(transcript) != gamma:
    8.     raise InvalidIssuanceRequestProof
    9. // Create BBS signature on (c, ctx, k, r)
    10. e <- Zq
    11. A = (G + H1 * c + H4 * ctx + K) * (1/(e + sk))  // K = H2 * k + H3 * r
    12. // Generate proof of correct computation
    13. alpha <- Zq
    14. Y_A = A * alpha
    15. Y_G = G * alpha
    16. X_A = G + H1 * c + H4 * ctx + K
    17. X_G = G * e + pk
    18. transcript_resp = CreateTranscript("respond")
    19. AddToTranscript(transcript_resp, c)
    20. AddToTranscript(transcript_resp, ctx)
    21. AddToTranscript(transcript_resp, e)
    22. AddToTranscript(transcript_resp, A)
    23. AddToTranscript(transcript_resp, X_A)
    24. AddToTranscript(transcript_resp, X_G)
    25. AddToTranscript(transcript_resp, Y_A)
    26. AddToTranscript(transcript_resp, Y_G)
    27. gamma_resp = GetChallenge(transcript_resp)
    28. z = gamma_resp * (sk + e) + alpha
    29. response = (A, e, gamma_resp, z, c, ctx)
    30. return response

3.3.3. Client: Token Verification 

VerifyIssuance(pk, request, response, state):
  Input:
    - pk: Issuer's public key
    - request: The issuance request sent
    - response: Issuer's response
    - state: Client state from request generation
  Output:
    - token: Credit token
  Exceptions:
    - InvalidIssuanceResponseProof, raised when the server proof verification fails

  Steps:
    1. Parse request as (K, gamma, k_bar, r_bar)
    2. Parse response as (A, e, gamma_resp, z, c, ctx)
    3. Parse state as (k, r)
    4. // Verify proof
    5. X_A = G + H1 * c + H4 * ctx + K
    6. X_G = G * e + pk
    7. Y_A = A * z - X_A * gamma_resp
    8. Y_G = G * z - X_G * gamma_resp
    9. transcript_resp = CreateTranscript("respond")
    10. AddToTranscript(transcript_resp, c)
    11. AddToTranscript(transcript_resp, ctx)
    12. AddToTranscript(transcript_resp, e)
    13. AddToTranscript(transcript_resp, A)
    14. AddToTranscript(transcript_resp, X_A)
    15. AddToTranscript(transcript_resp, X_G)
    16. AddToTranscript(transcript_resp, Y_A)
    17. AddToTranscript(transcript_resp, Y_G)
    18. if GetChallenge(transcript_resp) != gamma_resp:
    19.     raise InvalidIssuanceResponseProof
    20. token = (A, e, k, r, c, ctx)
    21. return token

3.4. Token Spending

The spending protocol allows a client to spend s credits from a token containing c credits (where 0 <= s <= c).

Note: Spending s = 0 is permitted and produces a new token with the same balance but a fresh nullifier. This "re-anonymization" operation is useful for securely transferring a token to another party: after a zero-spend, the original holder can no longer use the old nullifier, and the recipient obtains a token that is cryptographically unlinkable to the original.

3.4.1. Client: Spend Proof Generation 

ProveSpend(token, s):
  Input:
    - token: Credit token (A, e, k, r, c, ctx)
    - s: Amount to spend (0 <= s <= c)
  Output:
    - proof: Spend proof
    - state: Client state for receiving change
  Exceptions:
    - InvalidAmount: raised when s > c or s >= 2^L or c >= 2^L

  Steps:
    1. // Validate inputs
    2. if s >= 2^L:
    3.     raise InvalidAmount
    4. if c >= 2^L:
    5.     raise InvalidAmount
    6. if s > c:
    7.     raise InvalidAmount

    8. // Randomize the signature
    9. r1, r2 <- Zq
    10. B = G + H1 * c + H2 * k + H3 * r + H4 * ctx
    11. A' = A * (r1 * r2)
    12. B_bar = B * r1
    13. r3 = 1/r1

    14. // Generate initial proof components
    15. c' <- Zq
    16. r' <- Zq
    17. e' <- Zq
    18. r2' <- Zq
    19. r3' <- Zq

    20. // Compute first round messages
    21. A1 = A' * e' + B_bar * r2'
    22. A2 = B_bar * r3' + H1 * c' + H3 * r'

    23. // Decompose c - s into bits
    24. m = c - s
    25. (i[0], ..., i[L-1]) = BitDecompose(m)  // See Section 3.7

    26. // Create commitments for each bit
    27. k* <- Zq
    28. s[0] <- Zq
    29. Com[0] = H1 * i[0] + H2 * k* + H3 * s[0]
    30. For j = 1 to L-1:
    31.     s[j] <- Zq
    32.     Com[j] = H1 * i[j] + H3 * s[j]

    33. // Initialize range proof arrays
    34. C = array[L][2]
    35. C' = array[L][2]
    36. gamma0 = array[L]
    37. z = array[L][2]

    38. // Process bit 0 (with k* component)
    39. C[0][0] = Com[0]
    40. C[0][1] = Com[0] - H1
    41. k0' <- Zq
    42. s_prime = array[L]
    43. s_prime[0] <- Zq
    44. gamma0[0] <- Zq
    45. w0 <- Zq
    46. z[0] <- Zq

    47. if i[0] == 0:
    48.     C'[0][0] = H2 * k0' + H3 * s_prime[0]
    49.     C'[0][1] = H2 * w0 + H3 * z[0] - C[0][1] * gamma0[0]
    50. else:
    51.     C'[0][0] = H2 * w0 + H3 * z[0] - C[0][0] * gamma0[0]
    52.     C'[0][1] = H2 * k0' + H3 * s_prime[0]

    53. // Process remaining bits
    54. For j = 1 to L-1:
    55.     C[j][0] = Com[j]
    56.     C[j][1] = Com[j] - H1
    57.     s_prime[j] <- Zq
    58.     gamma0[j] <- Zq
    59.     z[j] <- Zq
    60.
    61.     if i[j] == 0:
    62.         C'[j][0] = H3 * s_prime[j]
    63.         C'[j][1] = H3 * z[j] - C[j][1] * gamma0[j]
    64.     else:
    65.         C'[j][0] = H3 * z[j] - C[j][0] * gamma0[j]
    66.         C'[j][1] = H3 * s_prime[j]

    67. // Compute K' commitment
    68. K' = Sum(Com[j] * 2^j for j in [L])
    69. r* = Sum(s[j] * 2^j for j in [L])
    70. k' <- Zq
    71. s' <- Zq
    72. C_final = H1 * (-c') + H2 * k' + H3 * s'

    73. // Generate challenge using transcript
    74. transcript = CreateTranscript("spend")
    75. AddToTranscript(transcript, k)
    76. AddToTranscript(transcript, ctx)
    77. AddToTranscript(transcript, A')
    78. AddToTranscript(transcript, B_bar)
    79. AddToTranscript(transcript, A1)
    80. AddToTranscript(transcript, A2)
    81. For j = 0 to L-1:
    82.     AddToTranscript(transcript, Com[j])
    83. For j = 0 to L-1:
    84.     AddToTranscript(transcript, C'[j][0])
    85.     AddToTranscript(transcript, C'[j][1])
    86. AddToTranscript(transcript, C_final)
    87. gamma = GetChallenge(transcript)

    88. // Compute responses
    89. e_bar = -gamma * e + e'
    90. r2_bar = gamma * r2 + r2'
    91. r3_bar = gamma * r3 + r3'
    92. c_bar = -gamma * c + c'
    93. r_bar = -gamma * r + r'

    94. // Complete range proof responses
    95. z_final = array[L][2]
    96. gamma0_final = array[L]
    97.
    98. // For bit 0
    99. if i[0] == 0:
    100.    gamma0_final[0] = gamma - gamma0[0]
    101.    w00 = gamma0_final[0] * k* + k0'
    102.    w01 = w0
    103.    z_final[0][0] = gamma0_final[0] * s[0] + s_prime[0]
    104.    z_final[0][1] = z[0]
    105. else:
    106.    gamma0_final[0] = gamma0[0]
    107.    w00 = w0
    108.    w01 = (gamma - gamma0_final[0]) * k* + k0'
    109.    z_final[0][0] = z[0]
    110.    z_final[0][1] = (gamma - gamma0_final[0]) * s[0] + s_prime[0]

    111. // For remaining bits
    112. For j = 1 to L-1:
    113.     if i[j] == 0:
    114.         gamma0_final[j] = gamma - gamma0[j]
    115.         z_final[j][0] = gamma0_final[j] * s[j] + s_prime[j]
    116.         z_final[j][1] = z[j]
    117.     else:
    118.         gamma0_final[j] = gamma0[j]
    119.         z_final[j][0] = z[j]
    120.         z_final[j][1] = (gamma - gamma0_final[j]) * s[j] + s_prime[j]

    121. k_bar = gamma * k* + k'
    122. s_bar = gamma * r* + s'

    123. // Construct proof
    124. proof = (k, s, ctx, A', B_bar, Com, gamma, e_bar,
    125.          r2_bar, r3_bar, c_bar, r_bar,
    126.          w00, w01, gamma0_final, z_final,
    127.          k_bar, s_bar)
    128. state = (k*, r*, m, ctx)
    129. return (proof, state)

3.4.2. Issuer: Spend Verification and Refund 

VerifyAndRefund(sk, proof, t):
  Input:
    - sk: Issuer's private key
    - proof: Client's spend proof
    - t: Credits to return to the client (0 <= t <= s, t < 2^L)
  Output:
    - refund: Refund for remaining credits
  Exceptions:
    - DoubleSpendError: raised when the nullifier has been used before
    - InvalidSpendProof: raised when the spend proof verification fails

  Steps:
    1. Parse proof and extract nullifier k
    2. // Check nullifier hasn't been used
    3. if k in used_nullifiers:
    4.     raise DoubleSpendError
    5. // Verify the proof (see Section 3.5.2)
    6. if not VerifySpendProof(sk, proof):
    7.     raise InvalidSpendProof
    8. // Record nullifier
    9. used_nullifiers.add(k)
    10. // Issue refund for remaining balance
    11. K' = Sum(Com[j] * 2^j for j in [L])
    12. refund = IssueRefund(sk, K', proof.ctx, proof.s, t)
    13. return refund

3.4.3. Refund Issuance

After verifying a spend proof, the issuer creates a refund token for the remaining balance. The issuer may optionally return t credits (where 0 <= t <= s) back to the client via a partial credit return. This enables pre-authorization patterns where the client holds s credits but only t are returned unused. The resulting token will have c - s + t credits. Use t = 0 to consume the full spend amount:

IssueRefund(sk, K', ctx, s, t):
  Input:
    - sk: Issuer's private key
    - K': Commitment to remaining balance and new nullifier
    - ctx: Request context from the spend proof
    - s: The spend amount from the proof
    - t: Credits to return to the client (0 <= t <= s, t < 2^L)
  Output:
    - refund: Refund response
  Exceptions:
    - InvalidAmount: raised when t > s or t >= 2^L

  Steps:
    1. // Validate partial return amount
    2. if t >= 2^L:
    3.     raise InvalidAmount
    4. if t > s:
    5.     raise InvalidAmount

    6. // Create new BBS signature on remaining balance + partial return
    7. e* <- Zq
    8. X_A* = G + K' + H1 * t + H4 * ctx
    9. A* = X_A* * (1/(e* + sk))

    10. // Generate proof of correct computation
    11. alpha <- Zq
    12. Y_A = A* * alpha
    13. Y_G = G * alpha
    14. X_G = G * e* + pk

    15. // Create challenge using transcript
    16. transcript = CreateTranscript("refund")
    17. AddToTranscript(transcript, e*)
    18. AddToTranscript(transcript, t)
    19. AddToTranscript(transcript, ctx)
    20. AddToTranscript(transcript, A*)
    21. AddToTranscript(transcript, X_A*)
    22. AddToTranscript(transcript, X_G)
    23. AddToTranscript(transcript, Y_A)
    24. AddToTranscript(transcript, Y_G)
    25. gamma = GetChallenge(transcript)

    26. // Compute response
    27. z = gamma * (sk + e*) + alpha

    28. refund = (A*, e*, gamma, z, t)
    29. return refund

3.4.4. Client: Refund Token Construction

The client verifies the refund and constructs a new credit token:

ConstructRefundToken(pk, spend_proof, refund, state):
  Input:
    - pk: Issuer's public key
    - spend_proof: The spend proof sent to issuer
    - refund: Issuer's refund response
    - state: Client state (k*, r*, m, ctx)
  Output:
    - token: New credit token or INVALID
  Exceptions:
    - InvalidRefundProof: When the refund proof verification fails

  Steps:
    1. Parse refund as (A*, e*, gamma, z, t)
    2. Parse state as (k*, r*, m, ctx)

    3. // Reconstruct commitment with partial return
    4. K' = Sum(spend_proof.Com[j] * 2^j for j in [L])
    5. X_A* = G + K' + H1 * t + H4 * ctx
    6. X_G = G * e* + pk

    7. // Verify proof
    8. Y_A = A* * z + X_A* * (-gamma)
    9. Y_G = G * z + X_G * (-gamma)

    10. // Check challenge using transcript
    11. transcript = CreateTranscript("refund")
    12. AddToTranscript(transcript, e*)
    13. AddToTranscript(transcript, t)
    14. AddToTranscript(transcript, ctx)
    15. AddToTranscript(transcript, A*)
    16. AddToTranscript(transcript, X_A*)
    17. AddToTranscript(transcript, X_G)
    18. AddToTranscript(transcript, Y_A)
    19. AddToTranscript(transcript, Y_G)
    20. if GetChallenge(transcript) != gamma:
    21.     raise InvalidRefundProof

    22. // Construct new token with remaining balance + partial return
    23. token = (A*, e*, k*, r*, m + t, ctx)
    24. return token

3.4.5. Spend Proof Verification

The issuer verifies a spend proof as follows:

VerifySpendProof(sk, proof):
  Input:
    - sk: Issuer's private key
    - proof: Spend proof from client
  Output:
    - valid: Boolean indicating if proof is valid
  Exceptions:
    - IdentityPointError: raised when A' is the identity
    - InvalidClientSpendProof: raised when the challenge does not match the reconstruction

  Steps:
    1. Parse proof as (k, s, ctx, A', B_bar, Com, gamma, e_bar,
                      r2_bar, r3_bar, c_bar, r_bar, w00, w01,
                      gamma0, z, k_bar, s_bar)

    2. // Check A' is not identity
    3. if A' == Identity:
    4.     raise IdentityPointError

    5. // Compute issuer's view of signature
    6. A_bar = A' * sk
    7. H1_prime = G + H2 * k + H4 * ctx

    8. // Verify sigma protocol
    9. A1 = A' * e_bar + B_bar * r2_bar - A_bar * gamma
    10. A2 = B_bar * r3_bar + H1 * c_bar + H3 * r_bar - H1_prime * gamma

    11. // Initialize arrays for range proof verification
    12. gamma1 = array[L]
    13. C = array[L][2]
    14. C' = array[L][2]

    15. // Process bit 0 (with k* component)
    16. gamma1[0] = gamma - gamma0[0]
    17. C[0][0] = Com[0]
    18. C[0][1] = Com[0] - H1
    19. C'[0][0] = H2 * w00 + H3 * z[0][0] - C[0][0] * gamma0[0]
    20. C'[0][1] = H2 * w01 + H3 * z[0][1] - C[0][1] * gamma1[0]

    21. // Verify remaining bits
    22. For j = 1 to L-1:
    23.     gamma1[j] = gamma - gamma0[j]
    24.     C[j][0] = Com[j]
    25.     C[j][1] = Com[j] - H1
    26.     C'[j][0] = H3 * z[j][0] - C[j][0] * gamma0[j]
    27.     C'[j][1] = H3 * z[j][1] - C[j][1] * gamma1[j]

    28. // Verify final commitment
    29. K' = Sum(Com[j] * 2^j for j in [L])
    30. Com_total = H1 * s + K'
    31. C_final = H1 * (-c_bar) + H2 * k_bar + H3 * s_bar - Com_total * gamma

    32. // Recompute challenge using transcript
    33. transcript = CreateTranscript("spend")
    34. AddToTranscript(transcript, k)
    35. AddToTranscript(transcript, ctx)
    36. AddToTranscript(transcript, A')
    37. AddToTranscript(transcript, B_bar)
    38. AddToTranscript(transcript, A1)
    39. AddToTranscript(transcript, A2)
    40. For j = 0 to L-1:
    41.     AddToTranscript(transcript, Com[j])
    42. For j = 0 to L-1:
    43.     AddToTranscript(transcript, C'[j][0])
    44.     AddToTranscript(transcript, C'[j][1])
    45. AddToTranscript(transcript, C_final)
    46. gamma_check = GetChallenge(transcript)

    47. // Verify challenge matches
    48. if gamma != gamma_check:
    49.     raise InvalidVerifySpendProof

    50. return true

3.5. Cryptographic Primitives

3.5.1. Protocol Version

The protocol version string for domain separation is:

PROTOCOL_VERSION = "curve25519-ristretto anonymous-credits v1.0"

This version string MUST be used consistently across all implementations for interoperability. The curve specification is included to prevent cross-curve attacks and ensure implementations using different curves cannot accidentally interact.

3.5.2. Hash Function and Fiat-Shamir Transform

The protocol uses BLAKE3 [BLAKE3] as the underlying hash function for the Fiat-Shamir transform [ORRU-FS]. Following the sigma protocol framework [ORRU-SIGMA], challenges are generated using a transcript that accumulates all protocol messages:

CreateTranscript(label):
  Input:
    - label: ASCII string identifying the proof type
  Output:
    - transcript: A new transcript object

  Steps:
    1. hasher = BLAKE3.new()
    2. hasher.update(LengthPrefixed(PROTOCOL_VERSION))
    3. hasher.update(LengthPrefixed(Encode(H1)))
    4. hasher.update(LengthPrefixed(Encode(H2)))
    5. hasher.update(LengthPrefixed(Encode(H3)))
    6. hasher.update(LengthPrefixed(Encode(H4)))
    7. hasher.update(LengthPrefixed(label))
    8. return transcript with hasher

AddToTranscript(transcript, value):
  Input:
    - transcript: Existing transcript
    - value: Element or Scalar to add

  Steps:
    1. encoded = Encode(value)
    2. transcript.hasher.update(LengthPrefixed(encoded))

GetChallenge(transcript):
  Input:
    - transcript: Completed transcript
  Output:
    - challenge: Scalar challenge value

  Steps:
    1. hash = transcript.hasher.output(64)  // 64 bytes of output
    2. challenge = from_little_endian_bytes(hash) mod q
    3. return challenge

This approach ensures:

  • Domain separation through the label and protocol version

  • Inclusion of all public parameters to prevent parameter substitution attacks

  • Proper ordering with length prefixes to prevent ambiguity

  • Deterministic challenge generation from the complete transcript

3.5.3. Encoding Functions

Elements and scalars are encoded as follows:

Encode(value):
  Input:
    - value: Element or Scalar
  Output:
    - encoding: ByteString

  Steps:
    1. If value is an Element:
    2.     return value.compress()  // 32 bytes, compressed Ristretto point
    3. If value is a Scalar:
    4.     return value.to_bytes_le()  // 32 bytes, little-endian

The following function provides consistent length-prefixing for hash inputs:

LengthPrefixed(data):
  Input:
    - data: ByteString to be length-prefixed
  Output:
    - prefixed: ByteString with length prefix

  Steps:
    1. length = len(data)
    2. return length.to_be_bytes(8) || data  // 8-byte big-endian length prefix

Note: Implementations MAY use standard serialization formats (e.g. CBOR) for complex structures, but MUST ensure deterministic encoding for hash inputs.

3.5.4. Binary Decomposition

To decompose a scalar into its binary representation:

BitDecompose(s):
  Input:
    - s: Scalar value
  Output:
    - bits: Array of L scalars (each 0 or 1)

  Steps:
    1. bytes = s.to_bytes_le()  // 32 bytes, little-endian
    2. For i = 0 to L-1:
    3.     byte_index = i / 8
    4.     bit_position = i % 8
    5.     bit = (bytes[byte_index] >> bit_position) & 1
    6.     bits[i] = Scalar(bit)
    7. return bits

Note: This algorithm produces bits in LSB-first order (i.e., bits[0] is the least significant bit). See Section 3.1 for constraints on L.

3.5.5. Scalar Conversion

Converting between credit amounts and scalars:

CreditToScalar(amount):
  Input:
    - amount: Integer credit amount (0 <= amount < 2^L)
  Output:
    - s: Scalar representation
  Exceptions:
    - AmountTooBigError: raised when the amount exceeds 2^L

  Steps:
    1. if amount >= 2^L:
    2.     return AmountTooBigError
    3. return Scalar(amount)

ScalarToCredit(s):
  Input:
    - s: Scalar value
  Output:
    - amount: Integer credit amount or ERROR
  Exceptions:
    - ScalarOutOfRangeError: raised when the scalar value is >= 2^L

  Steps:
    1. amount = s as integer  // Interpret little-endian scalar bytes as integer
    2. if amount >= 2^L:
    3.     return ScalarOutOfRangeError
    4. return amount

4. Protocol Messages and Wire Format

4.1. Message Encoding

All protocol messages SHOULD be encoded using deterministic CBOR (RFC 8949) for interoperability. Decoders MUST reject messages containing unknown CBOR map keys. The following sections define the structure of each message type.

4.1.1. Issuance Request Message 

IssuanceRequestMsg = {
    1: bstr,  ; K (compressed Ristretto point, 32 bytes)
    2: bstr,  ; gamma (scalar, 32 bytes)
    3: bstr,  ; k_bar (scalar, 32 bytes)
    4: bstr   ; r_bar (scalar, 32 bytes)
}

4.1.2. Issuance Response Message 

IssuanceResponseMsg = {
    1: bstr,  ; A (compressed Ristretto point, 32 bytes)
    2: bstr,  ; e (scalar, 32 bytes)
    3: bstr,  ; gamma_resp (scalar, 32 bytes)
    4: bstr,  ; z (scalar, 32 bytes)
    5: bstr,  ; c (scalar, 32 bytes)
    6: bstr   ; ctx (scalar, 32 bytes)
}

4.1.3. Spend Proof Message 

SpendProofMsg = {
    1: bstr,           ; k (nullifier, 32 bytes)
    2: bstr,           ; s (spend amount, 32 bytes)
    3: bstr,           ; A' (compressed point, 32 bytes)
    4: bstr,           ; B_bar (compressed point, 32 bytes)
    5: [* bstr],       ; Com array (L compressed points)
    6: bstr,           ; gamma (scalar, 32 bytes)
    7: bstr,           ; e_bar (scalar, 32 bytes)
    8: bstr,           ; r2_bar (scalar, 32 bytes)
    9: bstr,           ; r3_bar (scalar, 32 bytes)
    10: bstr,          ; c_bar (scalar, 32 bytes)
    11: bstr,          ; r_bar (scalar, 32 bytes)
    12: bstr,          ; w00 (scalar, 32 bytes)
    13: bstr,          ; w01 (scalar, 32 bytes)
    14: [* bstr],      ; gamma0 array (L scalars)
    15: [* [bstr, bstr]], ; z array (L pairs of scalars)
    16: bstr,          ; k_bar (scalar, 32 bytes)
    17: bstr,          ; s_bar (scalar, 32 bytes)
    18: bstr           ; ctx (scalar, 32 bytes)
}

4.1.4. Refund Message 

RefundMsg = {
    1: bstr,  ; A* (compressed Ristretto point, 32 bytes)
    2: bstr,  ; e* (scalar, 32 bytes)
    3: bstr,  ; gamma (scalar, 32 bytes)
    4: bstr,  ; z (scalar, 32 bytes)
    5: bstr   ; t (partial return, scalar, 32 bytes)
}

4.2. Error Responses

Error responses SHOULD use the following format:

ErrorMsg = {
    1: uint,   ; error_code
    2: tstr    ; error_message (for debugging only)
}

Error codes are defined in Section 5.3.

4.3. Protocol Flow

The complete protocol flow with message types:

Client                                          Issuer
  |                                               |
  |-- IssuanceRequestMsg ------------------------>|
  |                                               |
  |<-- IssuanceResponseMsg -----------------------|
  |                                               |
  | (client creates token)                        |
  |                                               |
  |-- SpendProofMsg ----------------------------->|
  |                                               |
  |<-- RefundMsg or ErrorMsg ---------------------|
  |                                               |

4.3.1. Example Usage Scenario

Consider an API service that sells credits in bundles of 1000:

  1. Purchase: Alice buys 1000 API credits

    • Alice generates a random nullifier k and blinding factor r

    • Alice sends IssuanceRequestMsg to the service

    • Service creates a BBS signature on (1000, k, r) and returns it

    • Alice now has a token worth 1000 credits

  2. First API Call: Alice makes an API call costing 50 credits

    • Alice creates a SpendProofMsg proving she has ≥ 50 credits

    • Alice reveals nullifier k to prevent double-spending

    • Service verifies the proof and records k as used

    • Service issues a RefundMsg for a new token worth 950 credits

    • Alice generates new nullifier k' for the refund token

  3. Subsequent Calls: Alice continues using the API

    • Each call repeats the spend/refund process

    • Each new token has a fresh nullifier

    • The service cannot link Alice's calls together

This example demonstrates how the protocol maintains privacy while preventing double-spending and enabling flexible partial payments.

5. Implementation Considerations

5.1. Nullifier Management

Implementations MUST maintain a persistent database of used nullifiers to prevent double-spending. The nullifier storage requirements grow linearly with the number of spent tokens. Implementations MAY use the following strategies to manage storage:

  1. Expiration: If tokens have expiration dates, old nullifiers can be pruned.

  2. Sharding: Nullifiers can be partitioned across multiple databases.

  3. Bloom Filters: Probabilistic data structures can reduce memory usage with a small false-positive rate. WARNING: false positives cause legitimate spends to be rejected. Bloom filters MUST NOT be the sole nullifier check; a positive result MUST be confirmed against authoritative storage before rejecting a spend.

5.2. Constant-Time Operations

Implementations MUST use constant-time operations for all secret-dependent computations. See the Security Considerations section for detailed requirements and mitigations.

5.3. Randomness Generation

The security of the protocol critically depends on the quality of random number generation. Implementations MUST use cryptographically secure random number generators (CSPRNGs) for:

  • Private key generation

  • Blinding factors (r, k)

  • Proof randomness (nonces)

5.3.1. RNG Requirements

  1. Entropy Source: Use OS-provided entropy (e.g., /dev/urandom on Unix systems)

  2. Fork Safety: Reseed after fork() to prevent nonce reuse

  3. Backtracking Resistance: Use forward-secure PRNGs when possible

5.3.2. Nonce Generation

Following [ORRU-SIGMA], nonces (the randomness used in proofs) MUST be generated with extreme care:

  1. Fresh Randomness: Generate new nonces for every proof

  2. No Reuse: Never reuse nonces across different proofs

  3. Full Entropy: Use the full security parameter (256 bits) of randomness

  4. Zeroization: Clear nonces from memory after use

WARNING: Leakage of even a few bits of a nonce can allow complete recovery of the witness (secret values). Implementations MUST use constant-time operations and secure memory handling for all nonce-related computations.

5.4. Point Validation

All Ristretto points received from external sources MUST be validated:

  1. Deserialization: Verify the point deserializes to a valid Ristretto point

  2. Non-Identity: Verify the point is not the identity element

  3. Subgroup Check: Ristretto guarantees prime-order subgroup membership

Example validation:

ValidatePoint(P):
  1. If P fails to deserialize:
  2.     return INVALID
  3. If P == Identity:
  4.     return INVALID
  5. // Ristretto ensures prime-order subgroup membership
  6. return VALID

All implementations MUST validate points at these locations:

  • When receiving K in issuance request

  • When receiving A in issuance response

  • When receiving A' and B_bar in spend proof

  • When receiving Com[j] commitments in spend proof

  • When receiving A* in refund response

5.5. Error Handling

Implementations SHOULD NOT provide detailed error messages that could leak information about the verification process. A single INVALID response should be returned for all verification failures.

5.5.1. Error Codes

While detailed error messages should not be exposed to untrusted parties, implementations MAY use the following internal error codes:

  • INVALID_PROOF: Proof verification failed

  • NULLIFIER_REUSE: Double-spend attempt detected

  • MALFORMED_REQUEST: Request format is invalid

  • INVALID_AMOUNT: Credit amount is invalid (exceeds 2^L - 1, spend exceeds balance, or partial return exceeds spend)

5.6. Parameter Selection

Implementations MUST choose L based on their maximum credit requirements and performance constraints. See Section 3.1 for constraints on L.

The bit length L is configurable and determines the range of credit values (0 to 2^L - 1). The choice of L involves several trade-offs:

  1. Range: Larger L supports higher credit values

  2. Performance: Proof size and verification time scale linearly with L

5.6.1. Performance Characteristics

The protocol has the following computational complexity:

Notation for Operations:

  • Group Operations: Point additions in the Ristretto255 group (e.g., P + Q)

  • Group Exponentiations: Scalar multiplication of group elements (e.g., P * s)

  • Scalar Additions/Multiplications: Arithmetic operations modulo the group order q

  • Issuance:

Table 1
Operation Group Operations Group Exponentiations Scalar Additions Scalar Multiplications Hashes
Client Request 2 4 2 1 1
Issuer Response 5 8 3 1 2
Client Credit Token Construction 5 5 0 0 1

  • Spending:

Table 2
Operation Group Operations Group Exponentiations Scalar Additions Scalar Multiplications Hashes
Client Request 17 + 4L 27 + 8L 13 + 5L 12 + 3L 1
Issuer Response 16 + 4L 24 + 5L 4 + L 1 1
Client Credit Token Construction 3 5 L L 1

Note: L is the configurable bit length for credit values.

  • Storage:

Table 3
Component Size
Token size 192 bytes (6 × 32 bytes)
Spend proof size 32 × (14 + 4L) bytes
Nullifier database entry 32 bytes per spent token

Note: Token size is independent of L.

6. Security Considerations

6.1. Security Model and Definitions

6.1.1. Threat Model

We consider a setting with:

  • Multiple issuers who can operate independently, though malicious issuers may collude with each other

  • Potentially malicious clients who may attempt to spend more credits than they should (whether by forging tokens, spending more credits than a token has, or double-spending a token)

6.1.2. Security Properties

The protocol provides the following security guarantees:

  1. Unforgeability: For an honest isser I, no probabilistic polynomial-time (PPT) adversary controlling a set of malicious clients and other malicious issuers can spend more credits than have been issued by I.

  2. Anonymity/Unlinkability: For an honest client C, no adversary controlling a set of malicious issuers and other malicious clients can link a token issuance/refund to C with a token spend by C. This property is information-theoretic in nature.

6.2. Cryptographic Assumptions

Security relies on:

  1. The q-SDH Assumption in the Ristretto255 group. We refer to [TZ23] for the formal definition.

  2. Random Oracle Model: The BLAKE3 hash function H is modeled as a random oracle.

6.3. Privacy Properties

The protocol provides the following privacy guarantees:

  1. Unlinkability: The issuer cannot link a token issuance/refund to a later spend of that token.

However, the protocol does NOT provide:

  1. Network-Level Privacy: IP addresses and network metadata can still link transactions.

  2. Amount Privacy: The spent amount s is revealed to the issuer.

  3. Timing Privacy: Transaction timing patterns could potentially be used for correlation.

  4. Context Privacy: The request context (ctx) is revealed in the clear during spending. If the issuer assigns distinct ctx values per issuance, the resulting token chain (issuance, spend, refund, subsequent spends) becomes linkable through the shared ctx value. This is by design for application-level context binding, but deployments that require full unlinkability MUST use a shared ctx across all clients within the same context (e.g., per-service or per-epoch), not per-client values. The ctx value persists across refunds: a token produced by a refund inherits the ctx of the original token.

6.4. Implementation Vulnerabilities and Mitigations

6.4.1. Critical Security Requirements

  1. RNG Failures: Weak randomness can completely break the protocol's security.

    Attack Vector: Predictable or repeated nonces in proofs can allow complete recovery of secret values including private keys and token contents.

    Mitigations:

    • MUST use cryptographically secure RNGs (e.g., OS-provided entropy sources)

    • MUST reseed after fork() operations to prevent nonce reuse

    • MUST implement forward-secure RNG state management

    • SHOULD use separate RNG instances for different protocol components

    • MUST zeroize RNG state on process termination

  2. Timing Attacks: Variable-time operations can leak information about secret values.

    Attack Vector: Timing variations in scalar arithmetic or bit operations can reveal secret bit patterns, potentially exposing credit balances or allowing token forgery.

    Mitigations:

    • MUST use constant-time scalar arithmetic libraries

    • MUST use constant-time conditional selection for range proof conditionals

    • MUST avoid early-exit conditions based on secret values

    • Critical constant-time operations include:

      • Scalar multiplication and addition

      • Binary decomposition in range proofs

      • Conditional assignments based on secret bits

      • Challenge verification comparisons

  3. Nullifier Database Attacks: Corruption or manipulation of the nullifier database enables double-spending.

    Attack Vectors:

    • Database corruption allowing nullifier deletion

    • Race conditions in concurrent nullifier checks

    Mitigations:

    • MUST use ACID-compliant database transactions

    • MUST check nullifier uniqueness within the same transaction as insertion

    • SHOULD implement append-only audit logs for nullifier operations

    • MUST implement proper database backup and recovery procedures

  4. Eavesdropping/Message Modification Attacks: A network-level adversary can copy spend proofs or modify messages sent between an honest client and issuer.

    Attack Vectors:

    • Eavesdropping and copying of proofs

    • Message modifications causing protocol failure

    Mitigations:

    • Client and issuer MUST use TLS 1.3 or above when communicating.

  5. State Management Vulnerabilities: Improper state handling can lead to security breaches.

    Attack Vectors:

    • State confusion between protocol sessions

    • Memory disclosure of sensitive state

    • Incomplete state cleanup

    Mitigations:

    • MUST use separate state objects for each protocol session

    • MUST zeroize all sensitive data (keys, nonces, intermediate values) after use

    • SHOULD use memory protection mechanisms (e.g., mlock) for sensitive data

    • MUST implement proper error handling that doesn't leak state information

    • SHOULD use explicit state machines for protocol flow

  6. Concurrency and Race Conditions: Parallel operations can introduce vulnerabilities.

    Attack Vectors:

    • TOCTOU (Time-of-check to time-of-use) vulnerabilities in nullifier checking

    • Race conditions in balance updates

    • Concurrent modification of shared state

    Mitigations:

    • MUST use appropriate locking for all shared resources

    • MUST perform nullifier check and insertion atomically

    • SHOULD document thread-safety guarantees

    • MUST ensure atomic read-modify-write for all critical operations

6.5. Known Attack Scenarios

6.5.1. 1. Parallel Spend Attack

Scenario: A malicious client attempts to spend the same token multiple times by initiating parallel spend operations before any nullifier is recorded.

Prevention: The issuer MUST ensure atomic nullifier checking and recording within a single database transaction. Network-level rate limiting can provide additional protection.

6.5.2. 2. Balance Inflation Attack

Scenario: An attacker attempts to create a proof claiming to have more credits than actually issued by manipulating the range proof.

Prevention: The cryptographic soundness of the range proof prevents this attack.

6.5.3. 3. Token Linking Attack

Scenario: An issuer attempts to link transactions by analyzing patterns in nullifiers, amounts, or timing.

Prevention: Nullifiers are cryptographically random and unlinkable. However, implementations MAY add random delays and amount obfuscation where possible.

6.6. Protocol Composition and State Management

6.6.1. State Management Requirements

Before they make a spend request or an issue request, the client MUST store their private state (the nullifier, the blinding factor, and the new balance) durably.

For the issuer, the spend and refund operations MUST be treated as an atomic transaction. However, even more is required. If a nullifier associated with a given spend is persisted to the database, clients MUST be able to access the associated refund. If they cannot access this, then they can lose access to the rest of their credits. For performance reasons, an issuer SHOULD automatically clean these up after some expiry, but if they do so, they MUST inform the client of this policy so the client can ensure they can retry to retrieve the rest of their credits in time. Issuers MAY implement functionality to notify the issuer that the refund request was processed, so they can delete the refund record. It is not clear that this is worth the cost relative to just cleaning them up in bulk at some specified expiration date, however if you are memory constrained this could be useful.

6.6.2. Session Management

Each protocol session (issuance or spend/refund) MUST use fresh randomness. See the Randomness Generation section for detailed RNG requirements.

6.6.3. Version Negotiation

To support protocol evolution, implementations MAY include version negotiation in the initial handshake. All parties MUST agree on the protocol version before proceeding.

6.7. Quantum Resistance

This protocol is NOT quantum-resistant. The discrete logarithm problem can be solved efficiently by quantum computers using Shor's algorithm. Organizations requiring long-term security should consider post-quantum alternatives. However, user privacy is preserved even in the presence of a cryptographically relevant quantum computer.

7. IANA Considerations

This document has no IANA actions.

8. References

8.1. Normative References

[BLAKE3]
"BLAKE3: One Function, Fast Everywhere", , <https://github.com/BLAKE3-team/BLAKE3-specs/blob/master/blake3.pdf>.
[RFC2119]
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfc-editor.org/rfc/rfc2119>.
[RFC8174]
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfc-editor.org/rfc/rfc8174>.
[RFC8949]
Bormann, C. and P. Hoffman, "Concise Binary Object Representation (CBOR)", STD 94, RFC 8949, DOI 10.17487/RFC8949, , <https://www.rfc-editor.org/rfc/rfc8949>.
[RFC9380]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, , <https://www.rfc-editor.org/rfc/rfc9380>.
[RFC9496]
de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I., Tankersley, G., and F. Valsorda, "The ristretto255 and decaf448 Groups", RFC 9496, DOI 10.17487/RFC9496, , <https://www.rfc-editor.org/rfc/rfc9496>.

8.2. Informative References

[BBS]
"Short Group Signatures", , <https://crypto.stanford.edu/~dabo/pubs/papers/groupsigs.pdf>.
[KVAC]
"Keyed-Verification Anonymous Credentials", , <https://eprint.iacr.org/2013/516.pdf>.
[ORRU-FS]
"The Fiat-Shamir Transform", , <https://mmaker.github.io/draft-zkproof-sigma-protocols/draft-orru-zkproof-fiat-shamir.html>.
[ORRU-SIGMA]
"Sigma Protocols", , <https://www.ietf.org/archive/id/draft-orru-zkproof-sigma-protocols-00.txt>.
[RFC9474]
Denis, F., Jacobs, F., and C. A. Wood, "RSA Blind Signatures", RFC 9474, DOI 10.17487/RFC9474, , <https://www.rfc-editor.org/rfc/rfc9474>.
[TZ23]
"Revisiting BBS Signatures", , <https://eprint.iacr.org/2023/275>.

Appendix A. Test Vectors

This appendix provides test vectors for implementers to verify their implementations. All values are encoded in hexadecimal.

The following test vector was generated deterministically using a ChaCha20 RNG seeded with the bytes 00 01 02 ... 1e 1f and L=8. The domain separator is "ACT-v1:test:vectors:v0:2025-01-01", credit amount c=100, spend amount s=30, partial return t=10, and ctx=0. Values labelled *_cbor are the CBOR wire-format encodings (Section 4) of each protocol message, displayed in hexadecimal.

Implementations SHOULD verify they can deserialize these CBOR messages and that a full protocol run with the same deterministic RNG produces identical output.

A.1. Parameters 

domain_separator: "ACT-v1:test:vectors:v0:2025-01-01"
L: 8
c: 100
s: 30
t: 10
ctx: 0000000000000000000000000000000000000000000000000000000000000000

A.2. Key Generation 

sk_cbor:
  a201582036e5b43419551a92c809a995a3d2c817a86ce8f5dd973b06fe9cb5a3
  f012870b0258204aceeb1d507e50957db46b6bcd374614b8ea080cbbc77ad060
  666bf5788c8121

pk_cbor:
  58204aceeb1d507e50957db46b6bcd374614b8ea080cbbc77ad060666bf5788c
  8121

A.3. Issuance 

preissuance_cbor:
  a20158206102398efee33b886f4bb7042b897d83db59b71a05aff76e9b633b87
  cade7d0002582069e5d557cb6094acfa586118e602e90aa6fe6cbabd4571eeb0
  d2f63b8c8a8f07

issuance_request_cbor:
  a4015820aa9315999f76c89406fe743dc7ff12e8fab85871f8c36987c6ec25ee
  ca2cd84e025820811880b9160decfbb41006af6c39056c9b0c139f7acf647fb5
  b0b22486039504035820c319066c466ef36d08809279c02dac8430c119fae886
  7f0c235cd6f6e4514c0f0458206af5dcb3e7138eb2a0f5b523054e05137b558f
  1bb6711b10b689e3565c241506

issuance_response_cbor:
  a60158201eeda51d75404be1bdd06c31aa72bbd38470a4717e732ccf372b91bc
  77161a0e0258204778cf09b14bf78e89e5ef5bcb523863d4e70f9d84ae1fbe75
  778e60a92c290e035820e94bd324b71702e9f29a239b3a064caa4c46d85d9693
  0f75bc39eca6211d2c0f04582029002c4fa8a9f71b8c015fb7869ad64a0fc4e0
  50c7c4955ef6ebff27f722890e05582064000000000000000000000000000000
  0000000000000000000000000000000006582000000000000000000000000000
  00000000000000000000000000000000000000

credit_token_cbor:
  a60158201eeda51d75404be1bdd06c31aa72bbd38470a4717e732ccf372b91bc
  77161a0e0258204778cf09b14bf78e89e5ef5bcb523863d4e70f9d84ae1fbe75
  778e60a92c290e03582069e5d557cb6094acfa586118e602e90aa6fe6cbabd45
  71eeb0d2f63b8c8a8f070458206102398efee33b886f4bb7042b897d83db59b7
  1a05aff76e9b633b87cade7d0005582064000000000000000000000000000000
  0000000000000000000000000000000006582000000000000000000000000000
  00000000000000000000000000000000000000

A.4. Spending 

nullifier:
  69e5d557cb6094acfa586118e602e90aa6fe6cbabd4571eeb0d2f63b8c8a8f07

context:
  0000000000000000000000000000000000000000000000000000000000000000

charge:
  1e00000000000000000000000000000000000000000000000000000000000000

spend_proof_cbor:
  b201582069e5d557cb6094acfa586118e602e90aa6fe6cbabd4571eeb0d2f63b
  8c8a8f070258201e000000000000000000000000000000000000000000000000
  00000000000000035820b221966bf32eafaac154b44f6037083d01314eee2e46
  e11b5003bcd07b6bb074045820562528491b9c19a10cefd74b3e45a89fcfdd62
  440997758ae8fa34d1a8d8583d05885820581721b2e63035cfe346849bd92c2b
  937b8e9404dabe1e5ca29e77de3321852a5820448491c7d81ecc20a9705b1bdd
  4df0c71570df3ad5fd9334758f8df9f56ede32582096431dd3f6ea1d7cb43165
  8d6f0fe14795e569b731c3cbe79bf293d78b6674255820cc718e6924e761e85b
  abf7a2b2336e892bb722c36d5aad88ee962b2d8d6b003a5820349e11dd9143d4
  8cf90e62386ebcada9490ab0ebb87739c49863ddd8a984a43a5820cc635618f8
  1e27bb91252afb2d66ea443cc17e48d4f08c0041f1071e6ead92145820e6e7cd
  cde60cf3fcef195aa9c511636eb31ee52ed1a328a821f9600bc997d25d582092
  857b094af803da09df1a95db12761e860425b53cb7cd967d3c227e535b991d06
  5820c68d2efa21dce1ada7a23b30c34edce8db9d6bc71ce2e8c75921fb179f6c
  d10607582003918610c7af601b6e22c22d0861e781252a24c6f759c4cda08b8f
  1fe7a09003085820c273eff86e662b4d44bcbdb73b0e4eec7714662dd1c0db61
  5ad53260a2b7650b0958207dbd42848d4dc3cb0a5461f26a08c3761f5decf00c
  d97976302a12a1ec46dc0e0a5820e97e15f04be7008f2f1c0bc69d2d4bebfbd1
  8bcca213f1ccf6187e933166e9050b5820a19468507a55c3e47628a435e3210e
  8024dc8259a06fdf7efb8c0e70d4a07e0d0c5820786081f2c4f6061d3e313f67
  cc7f73500283ffefa13a0f829e7ce85c8b19640b0d5820b8b352e7cb2066cef5
  cbbb79ae17852b0e44c82dc29b4b9a76d3422037e14c0c0e8858209411a6afca
  a687cba25c7a3ba2e3f3e12917bce920615a6073b7dceebad92b0b58205d5690
  3268f420e1ccc725685a7be6edd935f9a1ddc0e11508d648268e198b09582046
  040cd466c38375c1811153671aeed9530a3f052ed30c660d33141252255f0458
  209aa8b4018a9a0d53701b704bcd5636b1870749f1fd45dda322764282784707
  0a5820e3c62f417f88e5b7ec2182560e0d852e50e37a1d7ac00a8f5e090a7205
  f41f0d58205f22918a53fb88546521a44238ce5608c13c6ebf0f11f6784c835a
  c531b8720a58209d04a1b04cc9f59818dbaf7d04da70d2446145f9534886bce9
  5ed2b70df7cd095820346599fabc841424884cd029f3d63ae63158c6b4c975af
  e0d1147715d462aa020f88825820677b5e013537036d2e0c11ec8117799424bc
  ee744e7b1bd9571b7c464c7b72085820ba017329a9e86ccbe5ed1c76a1362f39
  9724cc7211b77940b6c513155bb1b30e825820ab6c718a173c91b3d30670fa50
  862021895ca40c818bc9d15e4110f7c0a4640958203c36da850f85cf51fdc896
  54b0a853677308974b9ea38e6e7e49ce94560ca8018258201b7d378abce009fc
  e50133f2d77380f3492794f72b209407e155f8ed8341bd05582057aa6d5d46a6
  d3b63c1d45af5c8d66ebc78494ee94df100b6cefc775fa6db8098258205612d9
  c1afb3c81bc81860c99b18466af0797eec6ebb3c6b8207b0521ffdf20c58209f
  5184c3bd8bc8d7a3f33941a3e8126850fbc795d4809dfeb5f363a30ff2ee0482
  58204c7c14d9881aa291a136ab4979d832cf5c80b75c8fe00ca976f65b3d4c9d
  150c5820150b6c20f4b894b4d491a252b4956c72e880639b7880f25f6ad0cba1
  6dbd020f8258207b2bc0d260f1f3ac32dfa6643a721a845faf9fe8e5de52751f
  5d824fa34e770f58209fe39299e57caa120937957a8957cd6c25f14958ea8e7b
  6333a2ac501bc144068258201f19d99a4e9955cc464fe48912f51399358bc678
  4a22312a95233e592dd1070558204c666f3adfaa3806b290247faa6950bb74e2
  70c9155f5004d407d21f223c0802825820d0b1d956f1b1a4ffb3c1a447bcb22a
  0e5679b70ca98357ce0dd5b6670bdc480e582017636a9a14caef700e53785bcb
  bd98380ac45d53af06107e99911fbf4ccf24001058201c7ad54f635ebd976c1e
  4fc275d93ce2e0981bbcaacb3745deda0be61cfe9f07115820c8451b222fdf76
  abc8a76dd90d1b5526de7e510809750825eea0cce6a4fe1c0212582000000000
  00000000000000000000000000000000000000000000000000000000

prerefund_cbor:
  a40158200f9288d8ef1360d8ef4967e041bf09a716c093956464370d30dfe283
  2be71b06025820ebada4fb4050db92729a58f0ae585f76154103a2ef2166c401
  12638f006d280b03582046000000000000000000000000000000000000000000
  0000000000000000000004582000000000000000000000000000000000000000
  00000000000000000000000000

A.5. Refund 

refund_cbor:
  a5015820880974b47fd0d4d06333e2f047abc4420992bd903ed44dae86199a54
  361f9c540258208a0977b088e9d17a637f71a013c67774648f0da03b141404ae
  678a0e5e090b04035820fdcd645c0d6e13905fff07e56d63465e4cc585f3c247
  8500c96cd361a4ad01070458202c9f3110e53540738100e7e636949ce7ac08bf
  b4ac6867fb72ac6ec847a2f90e0558200a000000000000000000000000000000
  00000000000000000000000000000000

A.6. Refund Token 

refund_token_cbor:
  a6015820880974b47fd0d4d06333e2f047abc4420992bd903ed44dae86199a54
  361f9c540258208a0977b088e9d17a637f71a013c67774648f0da03b141404ae
  678a0e5e090b04035820ebada4fb4050db92729a58f0ae585f76154103a2ef21
  66c40112638f006d280b0458200f9288d8ef1360d8ef4967e041bf09a716c093
  956464370d30dfe2832be71b0605582050000000000000000000000000000000
  0000000000000000000000000000000006582000000000000000000000000000
  00000000000000000000000000000000000000

refund_token_credits:
  5000000000000000000000000000000000000000000000000000000000000000

refund_token_nullifier:
  ebada4fb4050db92729a58f0ae585f76154103a2ef2166c40112638f006d280b

remaining_balance: 80

Appendix B. Implementation Status

This section records the status of known implementations of the protocol defined by this specification at the time of posting of this Internet-Draft, and is based on a proposal described in RFC 7942.

B.1. anonymous-credit-tokens

Organization: Google

Description: Reference implementation in Rust

Maturity: Beta

Coverage: Complete protocol implementation

License: Apache 2.0

Contact: sgschlesinger@gmail.com

URL: https://github.com/SamuelSchlesinger/anonymous-credit-tokens

Appendix C. Terminology Glossary

This glossary provides quick definitions of key terms used throughout this document:

ACT (Anonymous Credit Tokens): The privacy-preserving authentication protocol specified in this document.

Blind Signature: A cryptographic signature where the signer signs a message without seeing its content.

Refund: The refund issued for the remaining balance after a partial spend.

Credit: A numerical unit of authorization that can be spent by clients.

Domain Separator: A unique string used to ensure cryptographic isolation between different deployments.

Element: A point in the Ristretto255 elliptic curve group.

Issuer: The entity that creates and signs credit tokens.

Nullifier: A unique value revealed during spending that prevents double-spending of the same token.

Partial Spending: The ability to spend less than the full value of a token and receive change.

Scalar: An integer modulo the group order q, used in cryptographic operations.

Sigma Protocol: An interactive zero-knowledge proof protocol following a commit-challenge-response pattern.

Token: A cryptographic credential containing a BBS signature and associated data (A, e, k, r, c, ctx).

Unlinkability: The property that transactions cannot be correlated with each other or with token issuance.

Appendix D. Acknowledgments

The authors would like to thank the Crypto Forum Research Group for their valuable feedback and suggestions. Special thanks to the contributors who provided implementation guidance and security analysis.

This work builds upon the foundational research in anonymous credentials and zero-knowledge proofs by numerous researchers in the cryptographic community, particularly the work on BBS signatures by Boneh, Boyen, and Shacham, and keyed-verification anonymous credentials by Chase, Meiklejohn, and Zaverucha.

Authors' Addresses

Samuel Schlesinger
Google
Jonathan Katz
Google