Internet-Draft ARC March 2025
Yun & Wood Expires 4 September 2025 [Page]
Workgroup:
Network Working Group
Internet-Draft:
draft-yun-cfrg-arc-latest
Published:
Intended Status:
Informational
Expires:
Authors:
C. Yun
Apple, Inc.
C. A. Wood
Apple, Inc.

Anonymous Rate-Limited Credentials

Abstract

This document specifies the Anonymous Rate-Limited Credential (ARC) protocol, a specialization of keyed-verification anonymous credentials with support for rate limiting. ARC credentials can be presented from client to server up to some fixed number of times, where each presentation is cryptographically bound to client secrets and application-specific public information, such that each presentation is unlinkable from the others as well as the original credential creation. ARC is useful in applications where a server needs to throttle or rate-limit access from anonymous clients.

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://chris-wood.github.io/draft-arc/draft-yun-cfrg-arc.html. Status information for this document may be found at https://datatracker.ietf.org/doc/draft-yun-cfrg-arc/.

Discussion of this document takes place on the Crypto Forum 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/chris-wood/draft-arc.

Status of This Memo

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

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This Internet-Draft will expire on 4 September 2025.

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Table of Contents

1. Introduction

This document specifies the Anonymous Rate-Limited Credential (ARC) protocol, a specialization of keyed-verification anonymous credentials with support for rate limiting.

ARC is privately verifiable (keyed-verification), yet differs from similar token-based protocols in that each credential can be presented multiple times without violating unlinkability of different presentations. Servers issue credentials to clients that are cryptographically bound to client secrets and some public information. Afterwards, clients can present this credential to the server up to some fixed number of times, where each presentation provides proof that it was derived from a valid (previously issued) credential and bound to some public information. Each presentation is pairwise unlinkable, meaning the server cannot link any two presentations to the same client credential, nor can the server link a presentation to the preceding credential issuance flow. Notably, the maximum number of presentations from a credential is fixed by the application.

ARC is useful in settings where applications require a fixed number of zero-knowledge proofs about client secrets that can also be cryptographically bound to some public information. This capability lets servers use credentials in applications that need throttled or rate-limited access from anonymous clients.

2. Conventions and Definitions

2.1. Notation and Terminology

The following functions and notation are used throughout the document.

  • concat(x0, ..., xN): Concatenation of byte strings. For example, concat(0x01, 0x0203, 0x040506) = 0x010203040506.

  • bytes_to_int and int_to_bytes: Convert a byte string to and from a non-negative integer. bytes_to_int and int_to_bytes are implemented as OS2IP and I2OSP as described in [RFC8017], respectively. Note that these functions operate on byte strings in big-endian byte order.

  • random_integer_uniform(M, N): Generate a random, uniformly distributed integer R between M inclusive and N exclusive, i.e., M <= R < N.

  • random_integer_uniform_excluding_set(M, N, S): Generate a random, uniformly distributed integer R between M inclusive and N exclusive, i.e., M <= R < N, such that R does not exist in the set of integers S.

All algorithms and procedures described in this document are laid out in a Python-like pseudocode. Each function takes a set of inputs and parameters and produces a set of output values. Parameters become constant values once the protocol variant and the ciphersuite are fixed.

The notation T U[N] refers to an array called U containing N items of type T. The type opaque means one single byte of uninterpreted data. Items of the array are zero-indexed and referred as U[j] such that 0 <= j < N. The notation {T} refers to a set consisting of elements of type T. For any object x, we write len(x) to denote its length in bytes.

String values such as "CredentialRequest", "CredentialResponse", "Presentation", and "Tag" are ASCII string literals.

The following terms are used throughout this document.

  • Client: Protocol initiator. Creates a credential request, and uses the corresponding server response to make a credential. The client can make multiple presentations of this credential.

  • Server: Computes a response to a credential request, with its server private keys. Later the server can verify the client's presentations with its private keys. Learns nothing about the client's secret attributes, and cannot link a client's request/response and presentation steps.

3. Preliminaries

The construction in this document has one primary dependency:

3.1. Prime-Order Group

In this document, we assume the construction of an additive, prime-order group Group for performing all mathematical operations. In prime-order groups, any element (other than the identity) can generate the other elements of the group. Usually, one element is fixed and defined as the group generator. In the ARC setting, there are two fixed generator elements (generatorG, generatorH). Such groups are uniquely determined by the choice of the prime p that defines the order of the group. (There may, however, exist different representations of the group for a single p. Section 6 lists specific groups which indicate both order and representation.)

The fundamental group operation is addition + with identity element I. For any elements A and B of the group, A + B = B + A is also a member of the group. Also, for any A in the group, there exists an element -A such that A + (-A) = (-A) + A = I. Scalar multiplication by r is equivalent to the repeated application of the group operation on an element A with itself r-1 times, this is denoted as r*A = A + ... + A. For any element A, p*A=I. The case when the scalar multiplication is performed on the group generator is denoted as ScalarMultGen(r). Given two elements A and B, the discrete logarithm problem is to find an integer k such that B = k*A. Thus, k is the discrete logarithm of B with respect to the base A. The set of scalars corresponds to GF(p), a prime field of order p, and are represented as the set of integers defined by {0, 1, ..., p-1}. This document uses types Element and Scalar to denote elements of the group and its set of scalars, respectively.

We now detail a number of member functions that can be invoked on a prime-order group.

  • Order(): Outputs the order of the group (i.e. p).

  • Identity(): Outputs the identity element of the group (i.e. I).

  • Generator(): Outputs the fixed generator of the group.

  • HashToGroup(x, info): Deterministically maps an array of bytes x with domain separation value info to an element of Group. The map must ensure that, for any adversary receiving R = HashToGroup(x, info), it is computationally difficult to reverse the mapping. Security properties of this function are described in [I-D.irtf-cfrg-hash-to-curve].

  • HashToScalar(x, info): Deterministically maps an array of bytes x with domain separation value info to an element in GF(p). Security properties of this function are described in [I-D.irtf-cfrg-hash-to-curve], Section 10.5.

  • RandomScalar(): Chooses at random a non-zero element in GF(p).

  • ScalarInverse(s): Returns the inverse of input Scalar s on GF(p).

  • SerializeElement(A): Maps an Element A to a canonical byte array buf of fixed length Ne.

  • DeserializeElement(buf): Attempts to map a byte array buf to an Element A, and fails if the input is not the valid canonical byte representation of an element of the group. This function can raise a DeserializeError if deserialization fails or A is the identity element of the group; see Section 6 for group-specific input validation steps.

  • SerializeScalar(s): Maps a Scalar s to a canonical byte array buf of fixed length Ns.

  • DeserializeScalar(buf): Attempts to map a byte array buf to a Scalar s. This function can raise a DeserializeError if deserialization fails; see Section 6 for group-specific input validation steps.

For each group, there exists two distinct generators, generatorG and generatorH, generatorG = G.Generator() and generatorH = G.HashToGroup(G.SerializeElement(generatorG), "generatorH"). The group member functions GeneratorG() and GeneratorH() are shorthand for returning generatorG and generatorH, respectively.

Section 6 contains details for the implementation of this interface for different prime-order groups instantiated over elliptic curves.

4. ARC Protocol

The ARC protocol is a two-party protocol run between client and server consisting of three distinct phases:

  1. Key generation. In this phase, the server generates its private and public keys to be used for the remaining phases. This phase is described in Section 4.1.

  2. Credential issuance. In this phase, the client and server interact to issue the client a credential that is cryptographically bound to client secrets. This phase is described in Section 4.2.

  3. Presentation. In this phase, the client uses the credential to create a "presentation" to the server, where the server learns nothing more than whether or not the presentation is valid and corresponds to some previously issued credential, without learning which credential it corresponds to. This phase is described in Section 4.3.

This protocol bears resemblance to anonymous token protocols, such as those built on Blind RSA [BLIND-RSA] and Oblivious Pseudorandom Functions [OPRFS] with one critical distinction: unlike anonymous tokens, an anonymous credential can be used multiple times to create unlinkable presentations (up to the fixed presentation limit). This means that a single issuance invocation can drive multiple presentation invocations, whereas with anonymous tokens, each presentation invocation requires exactly one issuance invocation. As a result, credentials are generally longer lived than tokens. Applications configure the credential presentation limit after the credential is issued such that client and server agree on the limit during presentation. Servers are responsible for ensuring this limit is not exceeded. Clients that exceed the agreed-upon presentation limit break the unlinkability guarantees provided by the protocol.

The rest of this section describes the three phases of the ARC protocol.

4.1. Key Generation

In the key generation phase, the server generates its private and public keys, denoted ServerPrivateKey and ServerPublicKey, as follows.

Input: None
Output:
- ServerPrivateKey:
  - x0: Scalar
  - x1: Scalar
  - x2: Scalar
  - x0Blinding: Scalar
- ServerPublicKey:
  - X0: Element
  - X1: Element
  - X2: Element

Parameters
- Group G

def SetupServer():
  x0 = G.RandomScalar()
  x1 = G.RandomScalar()
  x2 = G.RandomScalar()
  x0Blinding = G.RandomScalar()
  X0 = x0 * G.GeneratorG() + x0Blinding * G.GeneratorH()
  X1 = x1 * G.GeneratorH()
  X2 = x2 * G.GeneratorH()
  return ServerPrivateKey(x0, x1, x2, x0Blinding), ServerPublicKey(X0, X1, X2)

The server public keys can be serialized as follows:

struct {
  uint8 X0[Ne]; // G.SerializeElement(X0)
  uint8 X1[Ne]; // G.SerializeElement(X1)
  uint8 X2[Ne]; // G.SerializeElement(X2)
} ServerPublicKey;

The length of this encoded response structure is NserverPublicKey = 3*Ne.

4.2. Issuance

The purpose of the issuance phase is for the client and server to cooperatively compute a credential that is cryptographically bound to the client's secrets. Clients do not choose these secrets; they are computed by the protocol.

The issuance phase of the protocol requires clients to know the server public key a priori, as well as an arbitrary, application-specific request context. It requires no other input. It consists of three distinct steps:

  1. The client generates and sends a credential request to the server. This credential request contains a proof that the request is valid with respect to the client's secrets and request context. See Section 4.2.1 for details about this step.

  2. The server validates the credential request. If valid, it computes a credential response with the server private keys. The response includes a proof that the credential response is valid with respect to the server keys. The server sends the response to the client. See Section 4.2.2 for details about this step.

  3. The client finalizes the credential by processing the server response. If valid, this step yields a credential that can then be used in the presentation phase of the protocol. See Section 4.2.3 for details about this step.

Each of these steps are described in the following subsections.

4.2.1. Credential Request

Given a request context, the process for creating a credential request is as follows:

(clientSecrets, request) = CredentialRequest(requestContext)

Inputs:
- requestContext: Data, context for the credential request

Outputs:
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.
- clientSecrets:
  - m1: Scalar, first secret.
  - m2: Scalar, second secret.
  - r1: Scalar, blinding factor for first secret.
  - r2: Scalar, blinding factor for second secret.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

def CredentialRequest(requestContext):
  m1 = G.RandomScalar()
  m2 = G.HashToScalar(requestContext, "requestContext")
  r1 = G.RandomScalar()
  r2 = G.RandomScalar()
  m1Enc = m1 * generatorG + r1 * generatorH
  m2Enc = m2 * generatorG + r2 * generatorH
  requestProof = MakeCredentialRequestProof(m1, m2, r1, r2, m1Enc, m2Enc)
  request = (m1Enc, m2Enc, requestProof)
  clientSecrets = (m1, m2, r1, r2)
  return (clientSecrets, request)

See Section 5.2 for more details on the generation of the credential request proof.

The resulting request can be serialized as follows.

struct {
  uint8 m1Enc[Ne];
  uint8 m2Enc[Ne];
  uint8 challenge[Ns];
  uint8 response0[Ns];
  uint8 response1[Ns];
  uint8 response2[Ns];
  uint8 response3[Ns];
} CredentialRequest;

The length of this encoded request structure is Nrequest = 2*Ne + 5*Ns.

4.2.2. Credential Response

Given a credential request and server public and private keys, the process for creating a credential response is as follows.

response = CredentialResponse(serverPrivateKey, serverPublicKey, request)

Inputs:
- serverPrivateKey:
  - x0: Scalar (private), server private key 0.
  - x1: Scalar (private), server private key 1.
  - x2: Scalar (private), server private key 2.
  - x0Blinding: Scalar (private), blinding value for x0.
- serverPublicKey:
  - X0: Element, server public key 0.
  - X1: Element, server public key 1.
  - X2: Element, server public key 2.
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.

Outputs:
- U: Element, a randomized generator for the response, `b*G`.
- encUPrime: Element, encrypted UPrime.
- X0Aux: Element, auxiliary point for X0.
- X1Aux: Element, auxiliary point for X1.
- X2Aux: Element, auxiliary point for X2.
- HAux: Element, auxiliary point for generatorH.
- responseProof: ZKProof, a proof of correct generation of
  U, encUPrime, server public keys, and auxiliary points.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

Exceptions:
- VerifyError, raised when response verification fails

def CredentialResponse(serverPrivateKeys, serverPublicKey, request):
  if VerifyCredentialRequestProof(request) == false:
    raise VerifyError

  b = G.RandomScalar()
  U = b * generatorG
  encUPrime = b * (serverPublicKey.X0 +
        serverPrivateKeys.x1 * request.m1Enc +
        serverPrivateKeys.x2 * request.m2Enc)
  X0Aux = b * serverPrivateKeys.x0Blinding * generatorH
  X1Aux = b * serverPublicKey.X1
  X2Aux = b * serverPublicKey.X2
  HAux = b * generatorH

  responseProof = MakeCredentialResponseProof(serverPrivateKey,
    serverPublicKey, request, b, U, encUPrime, X0Aux, X1Aux, X2Aux, HAux)
  return (U, encUPrime, X0Aux, X1Aux, X2Aux, HAux, responseProof)

The resulting response can be serialized as follows. See Section 5.3 for more details on the generation of the credential response proof.

struct {
  uint8 U[Ne];
  uint8 encUPrime[Ne];
  uint8 X0Aux[Ne];
  uint8 X1Aux[Ne];
  uint8 X2Aux[Ne];
  uint8 HAux[Ne];
  uint8 challenge[Ns];
  uint8 response0[Ns];
  uint8 response1[Ns];
  uint8 response2[Ns];
  uint8 response3[Ns];
  uint8 response4[Ns];
  uint8 response5[Ns];
  uint8 response6[Ns];
} CredentialResponse

The length of this encoded response structure is Nresponse = 6*Ne + 8*Ns.

4.2.3. Finalize Credential

Given a credential request and response, server public keys, and the client secrets produced when creating a credential request, the process for finalizing the issuance flow and creating a credential is as follows.

credential = FinalizeCredential(clientSecrets, serverPublicKey, request, response)

Inputs:
- clientSecrets:
  - m1: Scalar, first secret.
  - m2: Scalar, second secret.
  - r1: Scalar, blinding factor for first secret.
  - r2: Scalar, blinding factor for second secret.
- serverPublicKey: ServerPublicKey, shared with the client out-of-band
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.
- response:
  - U: Element, a randomized generator for the response. `b*G`.
  - encUPrime: Element, encrypted UPrime.
  - X0Aux: Element, auxiliary point for X0.
  - X1Aux: Element, auxiliary point for X1.
  - X2Aux: Element, auxiliary point for X2.
  - HAux: Element, auxiliary point for generatorH.
  - responseProof: ZKProof, a proof of correct generation of U, encUPrime, server public keys, and auxiliary points.

Outputs:
- credential:
  - m1: Scalar, client's first secret.
  - U: Element, a randomized generator for the response. `b*G`.
  - UPrime: Element, the MAC over the server's private keys and the client's secret secrets.
  - X1: Element, server public key 1.

Exceptions:
- VerifyError, raised when response verification fails

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

def FinalizeCredential(clientSecrets, serverPublicKey, request, response):
  if VerifyCredentialResponseProof(serverPublicKey, response, request) == false:
    raise VerifyError
  UPrime = response.encUPrime - response.X0Aux - clientSecrets.r1 * response.X1Aux - clientSecrets.r2 * response.X2Aux
  return (clientSecrets.m1, response.U, UPrime, serverPublicKey.X1)

4.3. Presentation

The purpose of the presentation phase is for the client to create a "presentation" to the server which can be verified using the server private key. This phase is non-interactive, i.e., there is no state stored between client and server in order to produce and then verify a presentation. Client and server agree upon a fixed limit of presentations in order to create and verify presentations; presentations will not verify correctly if the client and server use different limits.

This phase consists of three steps:

  1. The client creates a presentation state for a given presentation context and presentation limit. This state is used to produce a fixed amount of presentations.

  2. The client creates a presentation from the presentation state and sends it to the server. The presentation is cryptographically bound to the state's presentation context, and contains proof that the presentation is valid with respect to the presentation context. Moreover, the presentation contains proof that the nonce (an integer) associated with this presentation is within the presentation limit.

  3. The server verifies the presentation with respect to the presentation context and presentation limit.

Details for each each of these steps are in the following subsections.

4.3.1. Presentation State

Presentation state is used to track the number of presentations for a given credential. This state is important for ARC's unlinkability goals: reuse of state can break unlinkability properties of credential presentations. State is initialized with a credential, presentation context, and presentation limit. It is then mutated after each presentation construction (as described in Section 4.3.2).

state = MakePresentationState(credential, presentationContext, presentationLimit)

Inputs:
- credential:
  - m1: Scalar, client's first secret.
  - U: Element, a randomized generator for the response `b*G`.
  - UPrime: Element, the MAC over the server's private keys and the client's secrets.
  - X1: Element, server public key 1.
- presentationContext: Data (public), used for presentation tag computation.
- presentationLimit: Integer, the fixed presentation limit.

Outputs:
- credential
- presentationContext: Data (public), used for presentation tag computation.
- presentationNonceSet: {Integer}, the set of nonces that have been used for this presentation
- presentationLimit: Integer, the fixed presentation limit.

def MakePresentationState(credential, presentationContext, presentationLimit):
  nonce = random_integer_uniform(0, presentationLimit)
  return PresentationState(credential, presentationContext, [nonce], presentationLimit)

4.3.2. Presentation Construction

Creating a presentation requires a credential, presentation context, and presentation limit. This process is necessarily stateful on the client since the number of times a credential is used for a given presentation context cannot exceed the presentation limit; doing so would break presentation unlinkability, as two presentations created with the same nonce can be directly compared for equality (via the "tag"). As a result, the process for creating a presentation accepts as input a presentation state and then outputs an updated presentation state.

newState, nonce, presentation = Present(state)

Inputs:
state: input PresentationState
  - credential
  - presentationContext: Data (public), used for presentation tag computation.
  - presentationNonceSet: {Integer}, the set of nonces that have been used for this presentation
  - presentationLimit: Integer, the fixed presentation limit.

Outputs:
- newState: updated PresentationState
- nonce: Integer, the nonce associated with this presentation.
- presentation:
  - U: Element, re-randomized from the U in the response.
  - UPrimeCommit: Element, a public key to the issued UPrime.
  - m1Commit: Element, a public key to the client secret (m1).
  - tag: Element, the tag element used for enforcing the presentation limit.
  - presentationProof: ZKProof, a proof of correct generation of the presentation.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

Exceptions:
- LimitExceededError, raised when the presentation count meets or exceeds the presentation limit for the given presentation context

def Present(state):
  if len(state.presentationNonceSet) >= state.presentationLimit:
    raise LimitExceededError

  a = G.RandomScalar()
  r = G.RandomScalar()
  z = G.RandomScalar()

  U = a * state.credential.U
  UPrime = a * state.credential.UPrime
  UPrimeCommit = UPrime + r * generatorG
  m1Commit = state.credential.m1 * U + z * generatorH

  # This step mutates the state by keeping track of
  # what nonces have already been spent.
  nonce = random_integer_uniform_excluding_set(0,
    state.presentationLimit, state.presentationNonceSet)
  state.presentationNonceSet.add(nonce)

  generatorT = G.HashToGroup(presentationContext, "Tag")
  tag = (credential.m1 + nonce)^(-1) * generatorT
  V = z * credential.X1 - r * generatorG
  m1Tag = state.credential.m1 * tag

  presentationProof = MakePresentationProof(U, UPrimeCommit, m1Commit, tag, generatorT, credential, V, r, z, nonce, m1Tag)

  presentation = (U, UPrimeCommit, m1Commit, tag, presentationProof)

  return state, nonce, presentation

[[OPEN ISSUE: should the tag also fold in the presentation limit?]]

The resulting presentation can be serialized as follows. See Section 5.4 for more details on the generation of the presentation proof.

struct {
  uint8 U[Ne];
  uint8 UPrimeCommit[Ne];
  uint8 m1Commit[Ne];
  uint8 tag[Ne];
  uint8 challenge[Ns];
  uint8 response0[Ns];
  uint8 response1[Ns];
  uint8 response2[Ns];
  uint8 response3[Ns];
} Presentation

The length of this structure is Npresentation = 4*Ne + 5*Ns.

4.3.3. Presentation Verification

The server processes the presentation by verifying the presentation proof against server-computed values, and performing a check that the presentation conforms to the presentation limit.

validity = VerifyPresentation(serverPrivateKey, serverPublicKey, requestContext, presentationContext, nonce, presentation, presentationLimit)

Inputs:
- serverPrivateKey:
  - x0: Scalar (private), server private key 0.
  - x1: Scalar (private), server private key 1.
  - x2: Scalar (private), server private key 2.
  - x0Blinding: Scalar (private), blinding value for x0.
- serverPublicKey:
  - X0: Element, server public key 0.
  - X1: Element, server public key 1.
  - X2: Element, server public key 2.
- requestContext: Data, context for the credential request.
- presentationContext: Data (public), used for presentation tag computation.
- nonce: Integer, the nonce associated with this presentation.
- presentation:
  - U: Element, re-randomized from the U in the response.
  - UPrimeCommit: Element, a public key to the issued UPrime.
  - m1Commit: Element, a public key to the client secret (m1).
  - tag: Element, the tag element used for enforcing the presentation limit.
  - presentationProof: ZKProof, a proof of correct generation of the presentation.
- presentationLimit: Integer, the fixed presentation limit.

Outputs:
- validity: Boolean, True if the presentation is valid, False otherwise.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

Exceptions:
- InvalidNonceError, raised when the nonce associated with the presentation is invalid

def VerifyPresentation(serverPrivateKey, serverPublicKey, requestContext, presentationContext, nonce, presentation, presentationLimit):
  if nonce < 0 or nonce > presentationLimit:
    raise InvalidNonceError

  generatorT = G.HashToGroup(presentationContext, "Tag")
  m1Tag = generatorT - (nonce * presentation.tag)

  validity = VerifyPresentationProof(serverPrivateKey, serverPublicKey, requestContext, presentationContext, presentation, m1Tag)
  # Implementation-specific step: perform double-spending check on tag.
  # Implementation-specific step: store tag for future double-spending check.
  return validity

Implementation-specific steps: the server must perform a check that the tag (presentation.tag) has not previously been seen, to prevent double spending. It then stores the tag for use in future double spending checks. To reduce the overhead of performing double spend checks, the server can store and look up the tags corresponding to the associated requestContext and presentationContext values.

5. Zero-Knowledge Proofs

This section describes a Schnorr proof compiler that is used for the construction of other proofs needed throughout the ARC protocol. Section 5.1 describes the compiler, and the remaining sections describe how it is used for the purposes of producing ARC proofs.

5.1. Schnorr Compiler

The compiler specified in this section automates the Fiat-Shamir transform that is often used to transform interactive zero-knowledge proofs into non-interactive proofs such that they can be used to non-interactively prove various statements of importance in higher-level protocols, such as ARC. The compiler consists of a prover and verifier role. The prover constructs a transcript for the proof and then applies the Fiat-Shamir heuristic to generate the resulting challenge and response values. The verifier reconstructs the same transcript to verify the proof.

The prover and verifier roles are specified below in Section 5.1.1 and Section 5.1.2, respectively.

5.1.1. Prover

The prover role consists of four functions:

  • AppendScalar: This function adds a scalar representation to the transcript.

  • AppendElement: This function adds an element representation to the transcript.

  • Constrain: This function applies an explicit constraint to the proof, where the constraint is expressed as equality between some element and a linear combination of scalar and element representations. An example constraint might be Z = aX + bY, for scalars a, b, and elements X, Y, Z.

  • Prove: This function applies the Fiat-Shamir heuristic to the protocol transcript and set of constraints to produce a zero-knowledge proof that can be verified.

These functions are defined in the following sub-sections.

In addition, the prover role consists of the following state:

  • label: Data, a value representing the context in which the proof will be used

  • scalars: [Integer], An ordered set of representation of scalar variables to use in the proof. Each scalar has a label associated with it, stored in a list called scalar_labels.

  • elements: [Integer], An ordered set of representation of element variables to use in the proof. Each element has a label associated with it, stored in a list called element_labels.

  • constraints: a set of constraints, where each constraint consists of a constraint element and a linear combination of variables.

5.1.1.1. AppendScalar 
AppendScalar(label, assignment)

Inputs:
- label: Data, Scalar variable label
- assignment: Scalar variable

Outputs:
- Integer: Integer representation of the new scalar variable

def AppendScalar(label, assignment):
  state.scalars.append(assignment)
  state.scalar_labels.append(label)
  return len(state.scalars) - 1
5.1.1.2. AppendElement 
AppendElement(label, assignment)

Inputs:
- label: Data, Element variable label
- assignment: Element variable

Outputs:
- Integer: Integer representation of the new element variable

def AppendElement(label, assignment):
  state.elements.append(assignment)
  state.element_labels.append(label)
  return len(state.elements) - 1
5.1.1.3. Constrain 
Constrain(result, linearCombination)

Inputs:
- result: Integer, representation of constraint element
- assignment: linear combination of scalar and element variable (representations)

def Constrain(label, linearCombination):
  state.constraints.append((result, linearCombination))
5.1.1.4. Prove

The Prove function is defined below.

Prove()

Outputs:
- ZKProof, a proof consisting of a challenge Scalar and then fixed number of response Scalar values

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

Exceptions:
- InvalidVariableAllocationError, raised when the prover was incorrectly configured

def Prove():
  blindings = [G.RandomScalar() for i in range(len(state.scalars))]

  blinded_elements = []
  for (constraint_point, linear_combination) in state.constraints:
    if constraint_point.index > len(state.elements):
      raise InvalidVariableAllocationError

    for (scalar_var, element_var) in linear_combination:
      if scalar_var.index > len(state.scalars):
        raise InvalidVariableAllocationError
      if element_var.index > len(state.elements):
        raise InvalidVariableAllocationError

    scalar_index = linear_combination[0][0]
    element_index = linear_combination[0][1]
    blinded_element = blindings[scalar_index] * state.elements[element_index]

    for i, pair in enumerate(linear_combination):
      if i > 0:
        scalar_index = pair[0]
        element_index = pair[1]
        blinded_element += blindings[scalar_index] * state.elements[element_index]

        blinded_elements.append(blinded_element)

  # Obtain a scalar challenge
  challenge = ComposeChallenge(state.label, state.elements, blinded_elements)

  # Compute response scalars from the challenge, scalars, and blindings.
  responses = []
  for (index, scalar) in enumerate(state.scalars):
    blinding = blindings[index]
    responses.append(blinding - challenge * scalar)

  return ZKProof(challenge, responses)

The function ComposeChallenge is defined below.

ComposeChallenge(label, elements, blinded_elements)

Inputs:
- label: Data, the proof label
- elements: [Element], ordered list of elements
- blinded_elements: [Element], ordered list of blinded elements

Outputs:
- challenge, Scalar

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

def ComposeChallenge(label, elements, blinded_elements):
  challenge_input = Data() # Empty Data

  for element in elements:
    serialized_element = G.SerializeElement(element)
    challenge_input += I2OSP(len(serialized_element), 2) + serialized_element

  for blinded_element in blinded_elements:
    serialized_blinded_element = G.SerializeElement(blinded_element)
    challenge_input += I2OSP(len(serialized_blinded_element), 2) + serialized_blinded_element

  return G.HashToScalar(challenge_input, label)

5.1.2. Verifier

The verifier role consists of four functions:

  • AppendScalar: This function adds a scalar representation to the transcript.

  • AppendElement: This function adds an element representation to the transcript.

  • Constrain: This function applies an explicit constraint to the proof, where the constraint is expressed as equality between some element and a linear combination of scalar and element representations. An example constraint might be Z = aX + bY, for scalars a, b, and elements X, Y, Z.

  • Verify: This function applies the Fiat-Shamir heuristic to verify the zero-knowledge proof.

AppendScalar and Verify are defined in the following sub-sections. AppendElement and Constrain matches the functionality used in the prover role.

5.1.2.1. AppendScalar 
AppendScalar(label)

Inputs:
- label: Data, Scalar variable label

Outputs:
- Integer: Integer representation of the new scalar variable

def AppendScalar(label):
  state.scalar_labels.append(label)
  return len(state.scalar_labels) - 1
5.1.2.2. Verify 
Verify(proof)

Inputs:
- ZKProof, a proof consisting of a challenge Scalar and then fixed number of response Scalar values

Outputs:
- Boolean, True if the proof is valid, False otherwise.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

Exceptions:
- InvalidVariableAllocationError, raised when the prover was incorrectly configured

def Verify(proof):
  if len(state.elements) != len(state.element_labels):
    raise InvalidVariableAllocationError

  blinded_elements = []
  for (constraint_element, linear_combination) in state.constraints:
    if constraint_element > len(state.elements):
      raise InvalidVariableAllocationError
    for (_, element_var) in linear_combination:
      if element_var > len(state.elements):
        raise InvalidVariableAllocationError

    challenge_element = proof.challenge * state.elements[constraint_element]
    for i, pair in enumerate(linear_combination):
      challenge_element += proof.responses[pair[0]] * state.elements[pair[1]]

    blinded_elements.append(challenge_element)

  challenge = ComposeChallenge(state.label, self.elements, blinded_elements)
  return challenge == proof.challenge

5.2. CredentialRequest Proof

The request proof is a proof of knowledge of (m1, m2, r1, r2) used to generate the encrypted request. Statements to prove:

1. m1Enc = m1 * generatorG + r1 * generatorH
2. m2Enc = m2 * generatorG + r2 * generatorH

5.2.1. CredentialRequest Proof Creation 

requestProof = MakeCredentialRequestProof(m1, m2, r1, r2, m1Enc, m2Enc)

Inputs:
- m1: Scalar, first secret.
- m2: Scalar, second secret.
- r1: Scalar, blinding factor for first secret.
- r2: Scalar, blinding factor for second secret.
- m1Enc: Element, first encrypted secret.
- m2Enc: Element, second encrypted secret.

Outputs:
- proof: ZKProof
  - challenge: Scalar, the challenge used in the proof of valid encryption.
  - response0: Scalar, the response corresponding to m1.
  - response1: Scalar, the response corresponding to m2.
  - response2: Scalar, the response corresponding to r1.
  - response3: Scalar, the response corresponding to r2.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()
- contextString: public input

def MakeCredentialRequestProof(m1, m2, r1, r2, m1Enc, m2Enc):
  prover = Prover(contextString + "CredentialRequest")

  m1Var = prover.AppendScalar("m1", m1)
  m2Var = prover.AppendScalar("m2", m2)
  r1Var = prover.AppendScalar("r1", r1)
  r2Var = prover.AppendScalar("r2", r2)

  genGVar = prover.AppendElement("genG", generatorG)
  genHVar = prover.AppendElement("genH", generatorH)
  m1EncVar = prover.AppendElement("m1Enc", m1Enc)
  m2EncVar = prover.AppendElement("m2Enc", m2Enc)

  # 1. m1Enc = m1 * generatorG + r1 * generatorH
  prover.Constrain(m1EncVar, [(m1Var, genGVar), (r1Var, genHVar)])

  # 2. m2Enc = m2 * generatorG + r2 * generatorH
  prover.Constrain(m2EncVar, [(m2Var, genGVar), (r2Var, genHVar)])

  return prover.Prove()

5.2.2. CredentialRequest Proof Verification 

validity = VerifyCredentialRequestProof(request)

Inputs:
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.
    - challenge: Scalar, the challenge used in the proof of valid encryption.
    - response0: Scalar, the response corresponding to m1.
    - response1: Scalar, the response corresponding to m2.
    - response2: Scalar, the response corresponding to r1.
    - response3: Scalar, the response corresponding to r2.

Outputs:
- validity: Boolean, True if the proof verifies correctly, False otherwise.

Parameters:
- G: group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()
- contextString: public input

def VerifyCredentialRequestProof(request):
  verifier = Verifier(contextString + "CredentialRequest")

  m1Var = verifier.AppendScalar("m1")
  m2Var = verifier.AppendScalar("m2")
  r1Var = verifier.AppendScalar("r1")
  r2Var = verifier.AppendScalar("r2")

  genGVar = verifier.AppendElement("genG", generatorG)
  genHVar = verifier.AppendElement("genH", generatorH)
  m1EncVar = verifier.AppendElement("m1Enc", request.m1Enc)
  m2EncVar = verifier.AppendElement("m2Enc", request.m2Enc)

  # 1. m1Enc = m1 * generatorG + r1 * generatorH
  verifier.Constrain(m1EncVar, [(m1Var, genGVar), (r1Var, genHVar)])

  # 2. m2Enc = m2 * generatorG + r2 * generatorH
  verifier.Constrain(m2EncVar, [(m2Var, genGVar), (r2Var, genHVar)])

  return verifier.Verify(request.proof)

5.3. CredentialResponse Proof

The response proof is a proof of knowledge of (x0, x1, x2, x0Blinding, b) used in the server's CredentialResponse for the client's CredentialRequest. Statements to prove:

1. X0 = x0 * generatorG + x0Blinding * generatorH
2. X1 = x1 * generatorH
3. X2 = x2 * generatorH
4. X0Aux = b * x0Blinding * generatorH
  4a. HAux = b * generatorH
  4b: X0Aux = x0Blinding * HAux (= b * x0Blinding * generatorH)
5. X1Aux = b * x1 * generatorH
  5a. X1Aux = t1 * generatorH (t1 = b * x1)
  5b. X1Aux = b * X1 (X1 = x1 * generatorH)
6. X2Aux = b * x2 * generatorH
  6a. X2Aux = b * X2 (X2 = x2 * generatorH)
  6b. X2Aux = t2 * generatorH (t2 = b * x2)
7. U = b * generatorG
8. encUPrime = b * (X0 + x1 * Enc(m1) + x2 * Enc(m2))

5.3.1. CredentialResponse Proof Creation 

responseProof = MakeCredentialResponseProof(serverPrivateKey, serverPublicKey, request, b, U, encUPrime, X0Aux, X1Aux, X2Aux, HAux)

Inputs:
- serverPrivateKey:
  - x0: Scalar (private), server private key 0.
  - x1: Scalar (private), server private key 1.
  - x2: Scalar (private), server private key 2.
  - x0Blinding: Scalar (private), blinding value for x0.
- serverPublicKey:
  - X0: Element, server public key 0.
  - X1: Element, server public key 1.
  - X2: Element, server public key 2.
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.
- encUPrime: Element, encrypted UPrime.
- X0Aux: Element, auxiliary point for X0.
- X1Aux: Element, auxiliary point for X1.
- X2Aux: Element, auxiliary point for X2.
- HAux: Element, auxiliary point for generatorH.

Outputs:
- proof: ZKProof
  - challenge: Scalar, the challenge used in the proof of valid response.
  - response0: Scalar, the response corresponding to x0.
  - response1: Scalar, the response corresponding to x1.
  - response2: Scalar, the response corresponding to x2.
  - response3: Scalar, the response corresponding to x0Blinding.
  - response4: Scalar, the response corresponding to b.
  - response5: Scalar, the response corresponding to t1.
  - response6: Scalar, the response corresponding to t2.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()
- contextString: public input

def MakeCredentialResponseProof(serverPrivateKey, serverPublicKey, request, b, U, encUPrime, X0Aux, X1Aux, X2Aux, HAux):
  prover = Prover(contextString + "CredentialResponse")

  x0Var = prover.AppendScalar("x0", serverPrivateKey.x0)
  x1Var = prover.AppendScalar("x1", serverPrivateKey.x1)
  x2Var = prover.AppendScalar("x2", serverPrivateKey.x2)
  x0BlindingVar = prover.AppendScalar("x0Blinding", serverPrivateKey.x0Blinding)
  bVar = prover.AppendScalar("b", b)
  t1Var = prover.AppendScalar("t1", b * serverPrivateKey.x1)
  t2Var = prover.AppendScalar("t2", b * serverPrivateKey.x2)

  genGVar = prover.AppendElement("genG", generatorG)
  genHVar = prover.AppendElement("genH", generatorH)
  m1EncVar = prover.AppendElement("m1Enc", request.m1Enc)
  m2EncVar = prover.AppendElement("m2Enc", request.m2Enc)
  UVar = prover.AppendElement("U", U)
  encUPrimeVar = prover.AppendElement("encUPrime", encUPrime)
  X0Var = prover.AppendElement("X0", serverPublicKey.X0)
  X1Var = prover.AppendElement("X1", serverPublicKey.X1)
  X2Var = prover.AppendElement("X2", serverPublicKey.X2)
  X0AuxVar = prover.AppendElement("X0Aux", X0Aux)
  X1AuxVar = prover.AppendElement("X1Aux", X1Aux)
  X2AuxVar = prover.AppendElement("X2Aux", X2Aux)
  HAuxVar = prover.AppendElement("HAux", HAux)

  # 1. X0 = x0 * generatorG + x0Blinding * generatorH
  prover.Constrain(X0Var, [(x0Var, genGVar), (x0BlindingVar, genHVar)])
  # 2. X1 = x1 * generatorH
  prover.Constrain(X1Var, [(x1Var, genHVar)])
  # 3. X2 = x2 * generatorH
  prover.Constrain(X2Var, [(x2Var, genHVar)])

  # 4. X0Aux = b * x0Blinding * generatorH
  # 4a. HAux = b * generatorH
  prover.Constrain(HAuxVar, [(bVar, genHVar)])
  # 4b: X0Aux = x0Blinding * HAux (= b * x0Blinding * generatorH)
  prover.Constrain(X0AuxVar, [(x0BlindingVar, HAuxVar)])

  # 5. X1Aux = b * x1 * generatorH
  # 5a. X1Aux = t1 * generatorH (t1 = b * x1)
  prover.Constrain(X1AuxVar, [(t1Var, genHVar)])
  # 5b. X1Aux = b * X1 (X1 = x1 * generatorH)
  prover.Constrain(X1AuxVar, [(bVar, X1Var)])

  # 6. X2Aux = b * x2 * generatorH
  # 6a. X2Aux = b * X2 (X2 = x2 * generatorH)
  prover.Constrain(X2AuxVar, [(bVar, X2Var)])
  # 6b. X2Aux = t2 * H (t2 = b * x2)
  prover.Constrain(X2AuxVar, [(t2Var, genHVar)])

  # 7. U = b * generatorG
  prover.Constrain(UVar, [(bVar, genGVar)])
  # 8. encUPrime = b * (X0 + x1 * Enc(m1) + x2 * Enc(m2))
  # simplified: encUPrime = b * X0 + t1 * m1Enc + t2 * m2Enc, since t1 = b * x1 and t2 = b * x2
  prover.Constrain(encUPrimeVar, [(bVar, X0Var), (t1Var, m1EncVar), (t2Var, m2EncVar)])

  return prover.Prove()

5.3.2. CredentialResponse Proof Verification 

validity = VerifyCredentialResponseProof(serverPublicKey, response, request)

Inputs:
- serverPublicKey:
  - X0: Element, server public key 0.
  - X1: Element, server public key 1.
  - X2: Element, server public key 2.
- response:
  - U: Element, a randomized generator for the response. `b*G`.
  - encUPrime: Element, encrypted UPrime.
  - X0Aux: Element, auxiliary point for X0.
  - X1Aux: Element, auxiliary point for X1.
  - X2Aux: Element, auxiliary point for X2.
  - HAux: Element, auxiliary point for generatorH.
  - responseProof: ZKProof, a proof of correct generation of U, encUPrime, server public keys, and auxiliary points.
    - challenge: Scalar, the challenge used in the proof of valid response.
    - response0: Scalar, the response corresponding to x0.
    - response1: Scalar, the response corresponding to x1.
    - response2: Scalar, the response corresponding to x2.
    - response3: Scalar, the response corresponding to x0Blinding.
    - response4: Scalar, the response corresponding to b.
    - response5: Scalar, the response corresponding to t1.
    - response6: Scalar, the response corresponding to t2.
- request:
  - m1Enc: Element, first encrypted secret.
  - m2Enc: Element, second encrypted secret.
  - requestProof: ZKProof, a proof of correct generation of m1Enc and m2Enc.

Outputs:
- validity: Boolean, True if the proof verifies correctly, False otherwise.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()

def VerifyCredentialResponseProof(serverPublicKey, response, request):
  verifier = Verifier(contextString + "CredentialResponse")

  x0Var = verifier.AppendScalar("x0")
  x1Var = verifier.AppendScalar("x1")
  x2Var = verifier.AppendScalar("x2")
  x0BlindingVar = verifier.AppendScalar("x0Blinding")
  bVar = verifier.AppendScalar("b", b)
  t1Var = verifier.AppendScalar("t1")
  t2Var = verifier.AppendScalar("t2")

  genGVar = verifier.AppendElement("genG", generatorG)
  genHVar = verifier.AppendElement("genH", generatorH)
  m1EncVar = verifier.AppendElement("m1Enc", request.m1Enc)
  m2EncVar = verifier.AppendElement("m2Enc", request.m2Enc)
  UVar = verifier.AppendElement("U", response.U)
  encUPrimeVar = verifier.AppendElement("encUPrime", response.encUPrime)
  X0Var = verifier.AppendElement("X0", serverPublicKey.X0)
  X1Var = verifier.AppendElement("X1", serverPublicKey.X1)
  X2Var = verifier.AppendElement("X2", serverPublicKey.X2)
  X0AuxVar = verifier.AppendElement("X0Aux", response.X0Aux)
  X1AuxVar = verifier.AppendElement("X1Aux", response.X1Aux)
  X2AuxVar = verifier.AppendElement("X2Aux", response.X2Aux)
  HAuxVar = verifier.AppendElement("HAux", response.HAux)

  # 1. X0 = x0 * generatorG + x0Blinding * generatorH
  verifier.Constrain(X0Var, [(x0Var, genGVar), (x0BlindingVar, genHVar)])
  # 2. X1 = x1 * generatorH
  verifier.Constrain(X1Var, [(x1Var, genHVar)])
  # 3. X2 = x2 * generatorH
  verifier.Constrain(X2Var, [(x2Var, genHVar)])

  # 4. X0Aux = b * x0Blinding * generatorH
  # 4a. HAux = b * generatorH
  verifier.Constrain(HAuxVar, [(bVar, genHVar)])
  # 4b: X0Aux = x0Blinding * HAux (= b * x0Blinding * generatorH)
  verifier.Constrain(X0AuxVar, [(x0BlindingVar, HAuxVar)])

  # 5. X1Aux = b * x1 * generatorH
  # 5a. X1Aux = t1 * generatorH (t1 = b * x1)
  verifier.Constrain(X1AuxVar, [(t1Var, genHVar)])
  # 5b. X1Aux = b * X1 (X1 = x1 * generatorH)
  verifier.Constrain(X1AuxVar, [(bVar, X1Var)])

  # 6. X2Aux = b * x2 * generatorH
  # 6a. X2Aux = b * X2 (X2 = x2 * generatorH)
  verifier.Constrain(X2AuxVar, [(bVar, X2Var)])
  # 6b. X2Aux = t2 * H (t2 = b * x2)
  verifier.Constrain(X2AuxVar, [(t2Var, genHVar)])

  # 7. U = b * generatorG
  verifier.Constrain(UVar, [(bVar, genGVar)])
  # 8. encUPrime = b * (X0 + x1 * Enc(m1) + x2 * Enc(m2))
  # simplified: encUPrime = b * X0 + t1 * m1Enc + t2 * m2Enc, since t1 = b * x1 and t2 = b * x2
  verifier.Constrain(encUPrimeVar, [(bVar, X0Var), (t1Var, m1EncVar), (t2Var, m2EncVar)])

  return verifier.Verify(response.proof)

5.4. Presentation Proof

The presentation proof is a proof of knowledge of (m1, r, z) used in the presentation, and a proof that the nonce used to make the tag is in the range of [0, presentationLimit).

Statements to prove:

1. m1Commit = m1 * U + z * generatorH
2. V = z * X1 - r * generatorG
3. G.HashToGroup(presentationContext, "Tag") = m1 * tag + nonce * tag
4. m1Tag = m1 * tag

5.4.1. Presentation Proof Creation 

presentationProof = MakePresentationProof(U, UPrimeCommit, m1Commit, tag, generatorT, credential, V, r, z, nonce, m1Tag)

Inputs:
- U: Element, re-randomized from the U in the response.
- UPrimeCommit: Element, a public key to the MACGGM output UPrime.
- m1Commit: Element, a public key to the client secret (m1).
- tag: Element, the tag element used for enforcing the presentation limit.
- generatorT: Element, used for presentation tag computation.
- credential:
  - m1: Scalar, client's first secret.
  - U: Element, a randomized generator for the response. `b*G`.
  - UPrime: Element, the MAC over the server's private keys and the client's secrets.
  - X1: Element, server public key 1.
- V: Element, a proof helper element.
- r: Scalar (private), a randomly generated element used in presentation.
- z: Scalar (private), a randomly generated element used in presentation.
- nonce: Int, the nonce associated with the presentation.
- m1Tag: Element, helper element for the proof.

Outputs:
- proof: ZKProof
  - challenge: Scalar, the challenge used in the proof of valid presentation.
  - response0: Scalar, the response corresponding to m1.
  - response1: Scalar, the response corresponding to z.
  - response2: Scalar, the response corresponding to -r.
  - response3: Scalar, the response corresponding to nonce.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()
- contextString: public input

def MakePresentationProof(U, UPrimeCommit, m1Commit, tag, generatorT, presentationContext, credential, V, r, z, nonce, m1Tag)
  prover = Prover(contextString + "CredentialPresentation")

  m1Var = prover.AppendScalar("m1", credential.m1)
  zVar = prover.AppendScalar("z", z)
  rNegVar = prover.AppendScalar("-r", -r)
  nonceVar = prover.AppendScalar("nonce", nonce)

  genGVar = prover.AppendElement("genG", generatorG)
  genHVar = prover.AppendElement("genH", generatorH)
  UVar = prover.AppendElement("U", U)
  _ = prover.AppendElement("UPrimeCommit", UPrimeCommit)
  m1CommitVar = prover.AppendElement("m1Commit", m1Commit)
  VVar = prover.AppendElement("V", V)
  X1Var = prover.AppendElement("X1", credential.X1)
  tagVar = prover.AppendElement("tag", tag)
  genTVar = prover.AppendElement("genT", generatorT)
  m1TagVar = prover.AppendElement("m1Tag", m1Tag)

  # 1. m1Commit = m1 * U + z * generatorH
  prover.Constrain(m1CommitVar, [(m1Var, UVar), (zVar, genHVar)])
  # 2. V = z * X1 - r * generatorG
  prover.Constrain(VVar, [(zVar, X1Var), (rNegVar, genGVar)])
  # 3. G.HashToGroup(presentationContext, "Tag") = m1 * tag + nonce * tag
  prover.Constrain(genTVar, [(m1Var, tagVar), (nonceVar, tagVar)])
  # 4. m1Tag = m1 * tag
  prover.Constrain(m1TagVar, [(m1Var, tagVar)])

  return prover.Prove()

5.4.2. Presentation Proof Verification 

validity = VerifyPresentationProof(serverPrivateKey, serverPublicKey, requestContext, presentationContext, presentation, m1Tag)

Inputs:
- serverPrivateKey:
  - x0: Scalar (private), server private key 0.
  - x1: Scalar (private), server private key 1.
  - x2: Scalar (private), server private key 2.
  - x0Blinding: Scalar (private), blinding value for x0.
- serverPublicKey:
  - X0: Element, server public key 0.
  - X1: Element, server public key 1.
  - X2: Element, server public key 2.
- requestContext: Data, context for the credential request.
- presentationContext: Data (public), used for presentation tag computation.
- presentation:
  - U: Element, re-randomized from the U in the response.
  - UPrimeCommit: Element, a public key to the issued UPrime.
  - m1Commit: Element, a public key to the client secret (m1).
  - tag: Element, the tag element used for enforcing the presentation limit.
  - presentationProof: ZKProof, a proof of correct generation of the presentation.
    - challenge: Scalar, the challenge used in the proof of valid presentation.
    - response0: Scalar, the response corresponding to m1.
    - response1: Scalar, the response corresponding to z.
    - response2: Scalar, the response corresponding to -r.
    - response3: Scalar, the response corresponding to nonce.
- m1Tag: Element, helper to validate the presentation proof.

Outputs:
- validity: Boolean, True if the proof verifies correctly, False otherwise.

Parameters:
- G: Group
- generatorG: Element, equivalent to G.GeneratorG()
- generatorH: Element, equivalent to G.GeneratorH()
- contextString: public input

def VerifyPresentationProof(serverPrivateKey, serverPublicKey, requestContext, presentationContext, presentation, m1Tag):
  m2 = G.HashToScalar(requestContext, "requestContext")
  V = serverPrivateKey.x0 * presentation.U + serverPrivateKey.x1 * presentation.m1Commit + serverPrivateKey.x2 * m2 * presentation.U - presentation.UPrimeCommit
  generatorT = G.HashToGroup(presentationContext, "Tag")

  verifier = Verifier(contextString + "CredentialPresentation")

  m1Var = verifier.AppendScalar("m1")
  zVar = verifier.AppendScalar("z")
  rNegVar = verifier.AppendScalar("-r")
  nonceVar = verifier.AppendScalar("nonce")

  genGVar = verifier.AppendElement("genG", generatorG)
  genHVar = verifier.AppendElement("genH", generatorH)
  UVar = verifier.AppendElement("U", presentation.U)
  _ = verifier.AppendElement("UPrimeCommit", presentation.UPrimeCommit)
  m1CommitVar = verifier.AppendElement("m1Commit", presentation.m1Commit)
  VVar = verifier.AppendElement("V", presentation.V)
  X1Var = verifier.AppendElement("X1", serverPublicKey.X1)
  tagVar = prover.AppendElement("tag", presentation.tag)
  genTVar = verifier.AppendElement("genT", generatorT)
  m1TagVar = prover.AppendElement("m1Tag", m1Tag)

  # 1. m1Commit = m1 * U + z * generatorH
  verifier.Constrain(m1CommitVar, [(m1Var, UVar), (zVar, genHVar)])
  # 2. V = z * X1 - r * generatorG
  verifier.Constrain(VVar, [(zVar, X1Var), (rNegVar, genGVar)])
  # 3. G.HashToGroup(presentationContext, "Tag") = m1 * tag + nonceVar * tag
  verifier.Constrain(genTVar, [(m1Var, tagVar), (nonceVar, tagVar)])
  # 4. m1Tag = m1 * tag
  prover.Constrain(m1TagVar, [(m1Var, tagVar)])

  return verifier.Verify(presentation.proof)

6. Ciphersuites

A ciphersuite (also referred to as 'suite' in this document) for the protocol wraps the functionality required for the protocol to take place. The ciphersuite should be available to both the client and server, and agreement on the specific instantiation is assumed throughout.

A ciphersuite contains an instantiation of the following functionality:

This section includes an initial set of ciphersuites with supported groups. It also includes implementation details for each ciphersuite, focusing on input validation.

6.1. ARC(P-384)

This ciphersuite uses P-384 [NISTCurves] for the Group. The value of the ciphersuite identifier is "P384". The value of contextString is "ARCV1-P384".

  • Group: P-384 (secp384r1) [NISTCurves]

    • Order(): Return 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973.

    • Identity(): As defined in [NISTCurves].

    • Generator(): As defined in [NISTCurves].

    • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to Section 6.2 for implementation guidance.

    • HashToGroup(x, info): Use hash_to_curve with suite P384_XMD:SHA-384_SSWU_RO_ [I-D.irtf-cfrg-hash-to-curve], input x, and DST = "HashToGroup-" || contextString || info.

    • HashToScalar(x, info): Use hash_to_field from [I-D.irtf-cfrg-hash-to-curve] using L = 72, expand_message_xmd with SHA-384, input x and DST = "HashToScalar-" || contextString || info, and prime modulus equal to Group.Order().

    • ScalarInverse(s): Returns the multiplicative inverse of input Scalar s mod Group.Order().

    • SerializeElement(A): Implemented using the compressed Elliptic-Curve-Point-to-Octet-String method according to [SEC1]; Ne = 49.

    • DeserializeElement(buf): Implemented by attempting to deserialize a 49-byte array to a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to [SEC1], and then performs partial public-key validation as defined in section 5.6.2.3.4 of [KEYAGREEMENT]. This includes checking that the coordinates of the resulting point are in the correct range, that the point is on the curve, and that the point is not the point at infinity. Additionally, this function validates that the resulting element is not the group identity element. If these checks fail, deserialization returns an InputValidationError error.

    • SerializeScalar(s): Implemented using the Field-Element-to-Octet-String conversion according to [SEC1]; Ns = 48.

    • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 48-byte string using Octet-String-to-Field-Element from [SEC1]. This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1].

6.2. Random Scalar Generation

Two popular algorithms for generating a random integer uniformly distributed in the range [0, G.Order() -1] are as follows:

6.2.1. Rejection Sampling

Generate a random byte array with Ns bytes, and attempt to map to a Scalar by calling DeserializeScalar in constant time. If it succeeds, return the result. If it fails, try again with another random byte array, until the procedure succeeds. Failure to implement DeserializeScalar in constant time can leak information about the underlying corresponding Scalar.

As an optimization, if the group order is very close to a power of 2, it is acceptable to omit the rejection test completely. In particular, if the group order is p, and there is an integer b such that |p - 2b| is less than 2(b/2), then RandomScalar can simply return a uniformly random integer of at most b bits.

6.2.2. Random Number Generation Using Extra Random Bits

Generate a random byte array with L = ceil(((3 * ceil(log2(G.Order()))) / 2) / 8) bytes, and interpret it as an integer; reduce the integer modulo G.Order() and return the result. See [I-D.irtf-cfrg-hash-to-curve], Section 5 for the underlying derivation of L.

7. Security Considerations

For arguments about correctness, unforgeability, anonymity, and blind issuance of the ARC protocol, see the "Formal Security Definitions for Keyed-Verification Anonymous Credentials" in [KVAC].

This section elaborates on unlinkability properties for ARC and other implementation details necessary for these properties to hold.

7.1. Credential Issuance Unlinkability

Client credential requests are constructed such that the server cannot distinguish between any two credential requests from the same client and two requests from different clients. We refer to this property as issuance unlinkability. This property is achieved by the way the credential requests are constructed. In particular, each credential request consists of two Pedersen commitments with fresh blinding factors, which are used to commit to a freshly generated client secret and request context. The resulting request is therefore perfectly hiding, and independent from other requests from the same client. More details about this unlinkability property can be found in [KVAC] and [REVISITING_KVAC].

7.2. Presentation Unlinkability

Client credential presentations are constructed so that all presentations are indistinguishable, even if coming from the same user. We refer to this property as presentation unlinkability. This property is achieved by the way the credential presentations are constructed. The presentation elements [U, UPrimeCommit, m1Commit] are indistinguishable from all other presentations made from credentials issued with the same server keys, as detailed in [KVAC].

The indistinguishability set for these presentation elements is sum_{i=0}^c(p_i), where c is the number of credentials issued with the same server keys, and p_i is the number of presentations made for each of those credentials.

The presentation elements [tag, nonce, presentationContext, presentationProof] are indistinguishable from all presentations made from credentials issued with the same server keys for that presentationContext, with the exception of presentations with the same nonce (since those presentations can be ascertained as being generated from different credentials, as long as the presentation tag is unique).

The indistinguishability set for those presentation elements is sum_{i=0}^c(p_i[presentationContext]) - k[presentationContext], where c is the number of credentials issued with the same server keys, p_i[presentationContext] is the number of presentations made for each of those credentials with the same presentationContext, and k is the number of presentations with the same nonce for that presentationContext. As long as the nonces are generated randomly from the range defined by the presentation limit, k[presentationContext] should be roughly equal to sum_{i=0}^c(p_i[presentationContext]) / n, where n is the presentation limit. Therefore, the indistinguishability set can be represented as sum_{i=0}^c(p_i[presentationContext])(1 - 1/n), where a larger presentation limit results in a larger indistinguishability set and therefore stronger unlinkability properties.

[[OPEN ISSUE: hide the nonce and replace the tag proof with a range proof built from something like Bulletproofs.]]

7.3. Timing Leaks

To ensure no information is leaked during protocol execution, all operations that use secret data MUST run in constant time. This includes all prime-order group operations and proof-specific operations that operate on secret data, including proof generation and verification.

8. Alternatives considered

ARC uses the MACGGM algebraic MAC as its underlying primitive, as detailed in [KVAC] and [REVISITING_KVAC]. This offers the benefit of having a lower credential size than MACDDH, which is an alternative algebraic MAC detailed in [KVAC].

The BBS anonymous credential scheme, as detailed in [BBS] and its variants, is efficient and publicly verifiable, but requires pairings for verification. This is problematic for adoption because pairings are not supported as widely in software and hardware as non-pairing elliptic curves.

It is possible to construct a keyed-verification variant of BBS which doesn't use pairings, as discussed in [BBDT17] and [REVISITING_KVAC]. However these keyed-verification BBS variants require more analysis, proofs of security properties, and review to be considered mature enough for safe deployment.

9. IANA Considerations

This document has no IANA actions.

10. Test Vectors

This section contains test vectors for the ARC ciphersuites specified in this document.

10.1. ARCV1-P384 

// ServerKey
x0 = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562889
d89dbfa691d1cde91517fa222ed7ad364
x1 = 803d955f0e073a04aa5d92b3fb739f56f9db001266677f62c095021db018cd8
cbb55941d4073698ce45c405d1348b7b1
x2 = a097e722ed2427de86966910acba9f5c350e8040f828bf6ceca27405420cdf3
d63cb3aef005f40ba51943c8026877963
xb = 3a349db178a923aa7dff7d2d15d89756fd293126bb49a6d793cd77d7db960f5
692fec3b7ec07602c60cd32aee595dffd
X0 = 02caa111a43a5909de4af5cb836897334e5a34857ffc3565223cad95a20f1f3
2303eb8f7594b286238f243eca1a79c60b8
X1 = 03ef0f59c9b0cc51c9e603dfcaa9a3e3719e186252b64f9ce1ebec352c5b605
b805af308a9bd697df7c97b0f1147108c3a
X2 = 028a9547a39d925bdd054706aa5ff7616c28aca94c92041c678970c52ee6572
2f2c54d4f6cecba66abd721ecbcdb2b8a04

// CredentialRequest
request_context = 74657374207265717565737420636f6e74657874
m1 = 5a32aaf031be0555089356d299ce24b0eedfe7939e2382934ab5b0f76aae441
24955d2c5ebf9b41d88786259c34692d2
m2 = ae93d3ea7e5856d5d951a0ae87f8845d767df2e97dd669c8025715e604cb1c4
3569792b6864f592fed3abe29b9ebc950
r1 = df2c61be3c0b37bc73dc89fc386c96b3008035081690bfde3b1e68b91443c22
cc791d244340fe957d5aa44d7313740df
r2 = 10fcb00134739cc403f27a79588ca05ad59c5e6ff560cc597c2b8ca25256c72
0bceca2ab03921492c5e9e4ad3b558002
m1_enc = 030a5167977e0c038a98fce96e127fc228aa58526f71a920044b74b2f22
dd5839f0e1cf871e6419a1f522e94510ccf2d92
m2_enc = 03a0fea0d3e83ee67b36cb86d8380d4b8420a75e23eb5dcb560a79e74d1
36cf3c382bff7576f7e50c2cd3d247b56dbf56d
proof = ec5a68b4b89b7c411d0073ec384c698016ecfeee8903b7c4cc6e63651a85
44b4e06f2d61634a84304b4367bbca558609be6382768d7d070ea047831753b0c9ad
d8db1e3bd9476d2661ac6109218cdccbe5b4250d3785b431114d9b53d66ff7cbc44a
7ebbdf90dd5366187d7fbbb5d89f1e450ce8c87c75b94b726e931c81c963ba415eb0
334a73b73379abdf34284410e079f18c8200b035b11ade92e8d5e381c1cd5d4db9bb
a5aff1fe97ce1cebf29dc9389aea13dd64fd8fb1779b6bb549af335d4b9c4603dce2
a6bc5ca7b6d9c40287bf265f28c7b1d0b0e2300e5854361670617562aa405d861d91
70b08bc09e90

// CredentialResponse
b = b8b2e8c2103ad6f1970e873420d82a21e699140babbe599f7dd8f6e3e8f615e5
f201d1c2b0bc2f821f19e80a0a0e1e7b
U = 02be890d43908e52ed43ae7bc7098f3a7694617fe44a88c33c6fa4eb9e942c0b
2bb9d2fd56a44e1d6094fc7b9e8b949055
enc_U_prime = 027f43377c69a2ad931cc21a9cc4d6ea85f84d517d197db721c931
276a9ed543a78055ddeb9cac6be3af34c212bca5f403
X0_aux = 02890d3f4287e7878ce88b2bc1cc818b2c40fee0f93187af43acb479259
979cef1c39d609ea69cc7d6ba1e2a55d107653f
X1_aux = 0345f2be0dd21d49437a82b221f7a9f074b352e8698fe6ffa08aecad480
e96e93a25b6dacd4531fc961e78cfb5503f0e69
X2_aux = 020f31991b9a40be69bc06ef30c250d9353a824f4da88cc43e63bf92bc8
ac8bca7e26bffed33a32cc124fd1fb6c73f8b77
H_aux = 03c7714830e1d72604e52fec595c7a399fe0f3276766f84425fd1f98764a
c76dab631c6dfd05e0200c4ffe6d6967304882
proof = 1d5ea5448d22de8e4cb2193693a7c50874d37af3e2879cbf4484d26dd233
62e6f3765bbf894599e9551deb31e6f362693a5d799f07fbaf2376976608d73c401a
20db7385c89e4cd3b805c3ea9bdf03925b04bf24021246a778282adb1d740e82163f
b5037c3ad3221f55dd3ed5e3b3c6a0492827c410efde6d315e3b0b7835459292e818
05bea764e372fcd77776863c630b1f1509ef52e50a1fea5dc85b48523936408852d1
9461a2a8410e7aa49182905a444e089f0ad6b116f065debd09798121793487d7c84d
680331bacb1e8730928b897ac839b0b748cfd806833d79a3b50d736349b7bc674672
de02e375ee20348f5e349e30ab03ccb0f3a530935f4fada0e63ccb52b58785ecd948
3ea37cf3faf0ced66cc36ad33a80a94fa9b0041ff5be03e5fd1f74a748bd59866763
90427c27696d6591ec0443fac2415e9c0a4d6197d81a242dd71c739a5f9989a51d8b
245ecddb94ebb13c3da2bfc91d42ff992d4f0f1b567138ce1d1012053c4b70b10ed7
56e102faeb9a1d982227ab16c353

// Credential
m1 = 5a32aaf031be0555089356d299ce24b0eedfe7939e2382934ab5b0f76aae441
24955d2c5ebf9b41d88786259c34692d2
U = 02be890d43908e52ed43ae7bc7098f3a7694617fe44a88c33c6fa4eb9e942c0b
2bb9d2fd56a44e1d6094fc7b9e8b949055
U_prime = 02bec70edf38a3f5c77d5c6f39afd5f94cd266f958c804a954f6104b57
a2c8310862a790cbc6b519f8db989d59aebaf081
X1 = 03ef0f59c9b0cc51c9e603dfcaa9a3e3719e186252b64f9ce1ebec352c5b605
b805af308a9bd697df7c97b0f1147108c3a

// Presentation1
presentation_context = 746573742070726573656e746174696f6e20636f6e746
57874
a = ad00ec0f71b7a8fb7c0aef35c7243e17b78e57df8f0a95d102ff12bbb97e15ed
35c23e54f9b4483d30b76772ee60d886
r = e5cab8c896187b61abe017e57022528742252210dd60ddbbf1a57e3b144e26dd
693b7644a9626a8c36896ede53d12930
z = f61f4c924d3ef04b31e9196935ff27c5f5a4bbcf14e55e357df9f5ccb5ded37b
2b14bc2e1a68e31f86416f0606ee75d1
U = 03383b2ad2831739bc86c0c98119f256e54c9d89a762a9fc91b3904eb3aee726
0350a19085ea093a8059369219f03da2c3
U_prime_commit = 034af7c09ee5150fc914a3bb0adf17f7e90af3c4d9246ec8c51
1f938467174113513b7577329cecd2a7bff0b97e43a9808
m1_commit = 02fb95e1d8010da0c63d38ca212c1f76d768cecc8aa26ab07e775570
70c8343e1da571230f071a15a03973cd57dc33ebf4
nonce = 0x0
tag = 0247e3fd325bc774c27329a78a62f616f5e409d3a4857609cecde3251140f2
bb101905c4cbe66fe06a779e44f5d9e97f08
proof = 91a355f96a4917bc81aa7e5573da2a79bc9e026508e8d161cdbba91f7c3f
3f4a1a313adfe78c9e0025640720b1cd973b36fce60a4f82aaedc5cefc4c32760fce
c573a2609dafaddf4283949445451b614331172be50abc83c32aaf5cb10b4aee06d2
d1240cc7c4fb10491a96d125571dc4317c7583ad54c0971e22cb13b2c040e1d4791d
ba0a25dd3d1b785e484e8a1e8f24ca91f0f6d0cddf8456a21fa42a06748fa865cd00
8f3fe91e40a47126a47e54a321bec5c7b73700fbcb335c8a09880000000000000000
00000000000000000000000000000000000000000000000000000000000000000000
000000000004

// Presentation2
presentation_context = 746573742070726573656e746174696f6e20636f6e746
57874
a = a8e630484ba024bf9363805bc7a45f1695bcf45150a61f5c44a6cfbf343cd9e0
f593f127b6f49bec0f9b20f0550504a2
r = df6d39c3c0716d7cf8093073168bf967d7ed72750b6d366ed0febdc539b52d89
434f468a578c59d7ca9015b7da240ad6
z = 1b8e374ed4390e5a9023b309ecb94c0791eedfb168c556ff5ca3b89d047f482c
9279b47f584aab6c7f895f7674251771
U = 020c627c7ced92dd621860017ed29361bb78c4a17c8f7deb79f0c49a47723898
99a7e3b7b21e6a6c73abfca1332dc7df6e
U_prime_commit = 023d7eb948df3f49abe39e8ef32f4bb1bcca0f13f04836efd8b
7bc9bd0a73f915531ce845dc8c334d03c13647e5e4cc908
m1_commit = 036eea98df5b8248262fc5b511eef49bef1c2ec2a724df3e3a811296
fcf7891298d99a22f05ef0b08a2d00857117d88ac7
nonce = 0x1
tag = 03f9ceb1690ef6cd9c1b7d4c29dc86cf25565e4045ae431f8d28029e0405f9
ac251ef5a9e873f4a038ecd5a1e43d56bf5d
proof = 5547662ff1c63f14292cc998b24f0c74c20149cd89accde1c7a02485ebac
2c7888fd2ffbbafd539f79c8c4677dfcf79acf12a70bd0032c89c70e096620430169
9a3c50d1a778d55d812f7d4ef71d29c8b8377607d6d8d5884a31ec8386c909145ee7
aa6bc17558be1e2dece1329a9473782d97a7ea5819a60d03990563dd2efccd1fd1a6
d4376b2f900a4092a73266e3fec867feae45e645e368178df894c3e1353acd57f375
7faabf651b2ef04b87b426c1a695bed9002a0657b3752fdeaee4aab899d00e39c0eb
d6d336674db0f38b3dfeb6327653321dffc328fc088b0166cf1cddb68db353db7323
55034ec831dd

11. Acknowledgments

The authors would like to acknowledge helpful conversations with Tommy Pauly about rate limiting and Privacy Pass integration.

12. References

12.1. Normative References

[I-D.irtf-cfrg-hash-to-curve]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", Work in Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-16, , <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-hash-to-curve-16>.
[KEYAGREEMENT]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R. Davis, "Recommendation for pair-wise key-establishment schemes using discrete logarithm cryptography", National Institute of Standards and Technology, DOI 10.6028/nist.sp.800-56ar3, , <https://doi.org/10.6028/nist.sp.800-56ar3>.
[RFC8017]
Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, "PKCS #1: RSA Cryptography Specifications Version 2.2", RFC 8017, DOI 10.17487/RFC8017, , <https://www.rfc-editor.org/rfc/rfc8017>.

12.2. Informative References

[BBDT17]
"Improved Algebraic MACs and Practical Keyed-Verification Anonymous Credentials", n.d., <https://link.springer.com/chapter/10.1007/978-3-319-69453-5_20>.
[BBS]
"Short Group Signatures", n.d., <https://eprint.iacr.org/2004/174>.
[BLIND-RSA]
Denis, F., Jacobs, F., and C. A. Wood, "RSA Blind Signatures", RFC 9474, DOI 10.17487/RFC9474, , <https://www.rfc-editor.org/rfc/rfc9474>.
[KVAC]
"Keyed-Verification Anonymous Credentials from Algebraic MACs", n.d., <https://eprint.iacr.org/2013/516>.
[NISTCurves]
"Digital signature standard (DSS)", National Institute of Standards and Technology (U.S.), DOI 10.6028/nist.fips.186-4, , <https://doi.org/10.6028/nist.fips.186-4>.
[OPRFS]
Davidson, A., Faz-Hernandez, A., Sullivan, N., and C. A. Wood, "Oblivious Pseudorandom Functions (OPRFs) Using Prime-Order Groups", RFC 9497, DOI 10.17487/RFC9497, , <https://www.rfc-editor.org/rfc/rfc9497>.
[REVISITING_KVAC]
"Revisiting Keyed-Verification Anonymous Credentials", n.d., <https://eprint.iacr.org/2024/1552>.
[SEC1]
Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", <https://www.secg.org/sec1-v2.pdf>.

Authors' Addresses

Cathie Yun
Apple, Inc.
Christopher A. Wood
Apple, Inc.