Anonymous Rate-Limited Credentials

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.

Conventions and Definitions

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 , 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.

Preliminaries The construction in this document has one primary dependency:

Group: A prime-order group implementing the API described below in . See for specific instances of groups.

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

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

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 .
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 .
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 .

This protocol bears resemblance to anonymous token protocols, such as those built on Blind RSA and Oblivious Pseudorandom Functions 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.

Key Generation In the key generation phase, the server generates its private and public keys, denoted ServerPrivateKey and ServerPublicKey, as follows. The server public keys can be serialized as follows: The length of this encoded response structure is NserverPublicKey = 3*Ne.

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:

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 for details about this step.
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 for details about this step.
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 for details about this step.

Each of these steps are described in the following subsections.

Credential Request Given a request context, the process for creating a credential request is as follows: See for more details on the generation of the credential request proof. The resulting request can be serialized as follows. The length of this encoded request structure is Nrequest = 2*Ne + 5*Ns.

Credential Response Given a credential request and server public and private keys, the process for creating a credential response is as follows. The resulting response can be serialized as follows. See for more details on the generation of the credential response proof. The length of this encoded response structure is Nresponse = 6*Ne + 8*Ns.

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.

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:

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.
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.
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.

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 ).

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. = 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 for more details on the generation of the presentation proof. The length of this structure is Npresentation = 4*Ne + 5*Ns.

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. presentationLimit: raise InvalidNonceError generatorT = G.HashToGroup(presentationContext, "Tag") m1Tag = generatorT - (nonce * presentation.tag) validity = VerifyPresentationProof( serverPrivateKey, serverPublicKey, requestContext, presentationContext, presentation, m1Tag) return validity, presentation.tag ]]> 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.

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

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 and , respectively.

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.

AppendScalar

AppendElement

Constrain

Prove The Prove function is defined below. 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.

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.

AppendScalar

Verify 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 ]]>

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

CredentialRequest Proof Creation

CredentialRequest Proof Verification

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:

CredentialResponse Proof Creation

CredentialResponse Proof Verification

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:

Presentation Proof Creation

Presentation Proof Verification

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:

Group: A prime-order Group exposing the API detailed in , with the generator element defined in the corresponding reference for each group. Each group also specifies HashToGroup, HashToScalar, and serialization functionalities. For HashToGroup, the domain separation tag (DST) is constructed in accordance with the recommendations in . For HashToScalar, each group specifies an integer order that is used in reducing integer values to a member of the corresponding scalar field.

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

ARC(P-256) This ciphersuite uses P-256 for the Group. The value of the ciphersuite identifier is "P256". The value of contextString is "ARCV1-P256".

Group: P-256 (secp256r1)
- Order(): Return 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551.
- Identity(): As defined in .
- Generator(): As defined in .
- RandomScalar(): Implemented by returning a uniformly random Scalar in the range [1, G.Order() - 1]. Refer to for implementation guidance.
- HashToGroup(x, info): Use hash_to_curve with suite P256_XMD:SHA-256_SSWU_RO_ , input x, and DST = "HashToGroup-" || contextString || info.
- HashToScalar(x, info): Use hash_to_field from using L = 48, expand_message_xmd with SHA-256, 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 ; Ne = 33.
- DeserializeElement(buf): Implemented by attempting to deserialize a 33-byte array to a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to , and then performs partial public-key validation as defined in section 5.6.2.3.4 of . 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 ; Ns = 32.
- DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 32-byte string using Octet-String-to-Field-Element from . This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1].

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

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 and is non-zero, return the result. Otherwise, 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 - 2^b| is less than 2^(b/2), then RandomScalar can simply return a uniformly random integer of at most b bits.

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 for the underlying derivation of L.

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

Credential Request 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 statistically hiding, and independent from other requests from the same client. More details about this unlinkability property can be found in and .

Credential Issuance Unlinkability The server commitment to x0 is defined as X0 = x0 * G.generatorG() + x0Blinding * G.GeneratorH(), following the definitions in . This is computationally binding to the secret key x0. This means that unless the discrete log is broken, the credentials issued under one server commitment X0, X1, ... will all be issued under the same private keys x0, x1, ... However, an adversary breaking the discrete log (e.g., a quantum adversary) can find pairs (x0, x0Blinding) and (x0', x0Blinding') both committing to X0 and use them to issue different credentials. This capability would let the adversary partitioning the client anonymity set by linking clients to the underlying secret used for credential issuance, i.e., x0 or x0'. This requires an active attack and therefore is not an immediate concern. Statistical anonymity is possible by committing to x0 and x0Blinding` separately, as in . However, the security of this construction requires additional analysis.

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 . 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.

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.

Alternatives considered ARC uses the MACGGM algebraic MAC as its underlying primitive, as detailed in and . This offers the benefit of having a lower credential size than MACDDH, which is an alternative algebraic MAC detailed in . The BBS anonymous credential scheme, as detailed in 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 and . However these keyed-verification BBS variants require more analysis, proofs of security properties, and review to be considered mature enough for safe deployment.

IANA Considerations This document has no IANA actions.

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

ARCV1-P256

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