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Internet-Draft An Oblivious Pseudorandom Function (OPRF) is a two-party protocol between client and server for computing the output of a Pseudorandom Function (PRF). The server provides the PRF secret key, and the client provides the PRF input. At the end of the protocol, the client learns the PRF output without learning anything about the PRF secret key, and the server learns neither the PRF input nor output. An OPRF can also satisfy a notion of 'verifiability', called a VOPRF. A VOPRF ensures clients can verify that the server used a specific private key during the execution of the protocol. A VOPRF can also be partially-oblivious, called a POPRF. A POPRF allows clients and servers to provide public input to the PRF computation. This document specifies an OPRF, VOPRF, and POPRF instantiated within standard prime-order groups, including elliptic curves. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. Discussion Venues Source for this draft and an issue tracker can be found at .

Introduction A Pseudorandom Function (PRF) F(k, x) is an efficiently computable function taking a private key k and a value x as input. This function is pseudorandom if the keyed function K(_) = F(k, _) is indistinguishable from a randomly sampled function acting on the same domain and range as K(). An Oblivious PRF (OPRF) is a two-party protocol between a server and a client, where the server holds a PRF key k and the client holds some input x. The protocol allows both parties to cooperate in computing F(k, x) such that the client learns F(k, x) without learning anything about k; and the server does not learn anything about x or F(k, x). A Verifiable OPRF (VOPRF) is an OPRF wherein the server also proves to the client that F(k, x) was produced by the key k corresponding to the server's public key the client knows. A Partially-Oblivious PRF (POPRF) is a variant of a VOPRF wherein client and server interact in computing F(k, x, y), for some PRF F with server-provided key k, client-provided input x, and public input y, and client receives proof that F(k, x, y) was computed using k corresponding to the public key that the client knows. A POPRF with fixed input y is functionally equivalent to a VOPRF. OPRFs have a variety of applications, including: password-protected secret sharing schemes , privacy-preserving password stores , and password-authenticated key exchange or PAKE . Verifiable OPRFs are necessary in some applications such as Privacy Pass . Verifiable OPRFs have also been used for password-protected secret sharing schemes such as that of . This document specifies OPRF, VOPRF, and POPRF protocols built upon prime-order groups. The document describes each protocol variant, along with application considerations, and their security properties. This document represents the consensus of the Crypto Forum Research Group (CFRG). It is not an IETF product and is not a standard.

Change log draft-17:

• Change how suites are identified and finalize test vectors.
• Apply editorial suggestions from IRTF chair review.

draft-16:

• Apply editorial suggestions from document shepherd.

draft-15:

• Apply editorial suggestions from CFRG RGLC.

draft-14:

• Correct current state of formal analysis for the VOPRF protocol variant.

draft-13:

• Editorial improvements based on Crypto Panel Review.

draft-12:

• Small editorial fixes

draft-11:

• Change Evaluate to BlindEvaluate, and add Evaluate for PRF evaluation

draft-10:

• Editorial improvements

draft-09:

• Split syntax for OPRF, VOPRF, and POPRF functionalities.
• Make Blind function fallible for invalid private and public inputs.
• Specify key generation.
• Remove serialization steps from core protocol functions.
• Refactor protocol presentation for clarity.
• Simplify security considerations.
• Update application interface considerations.
• Update test vectors.

draft-08:

• Adopt partially-oblivious PRF construction from .
• Update P-384 suite to use SHA-384 instead of SHA-512.
• Update test vectors.
• Apply various editorial changes.

draft-07:

• Bind blinding mechanism to mode (additive for verifiable mode and multiplicative for base mode).
• Add explicit errors for deserialization.
• Document explicit errors and API considerations.
• Adopt SHAKE-256 for decaf448 ciphersuite.
• Normalize HashToScalar functionality for all ciphersuites.
• Refactor and generalize DLEQ proof functionality and domain separation tags for use in other protocols.
• Update test vectors.
• Apply various editorial changes.

draft-06:

• Specify of group element and scalar serialization.
• Remove info parameter from the protocol API and update domain separation guidance.
• Fold Unblind function into Finalize.
• Optimize ComputeComposites for servers (using knowledge of the private key).
• Specify deterministic key generation method.
• Update test vectors.
• Apply various editorial changes.

draft-05:

• Move to ristretto255 and decaf448 ciphersuites.
• Clean up ciphersuite definitions.
• Pin domain separation tag construction to draft version.
• Move key generation outside of context construction functions.
• Editorial changes.

draft-04:

• Introduce Client and Server contexts for controlling verifiability and required functionality.
• Condense API.
• Remove batching from standard functionality (included as an extension)
• Add Curve25519 and P-256 ciphersuites for applications that prevent strong-DH oracle attacks.
• Provide explicit prime-order group API and instantiation advice for each ciphersuite.
• Proof-of-concept implementation in sage.
• Remove privacy considerations advice as this depends on applications.

draft-03:

• Certify public key during VerifiableFinalize.
• Remove protocol integration advice.
• Add text discussing how to perform domain separation.
• Drop OPRF_/VOPRF_ prefix from algorithm names.
• Make prime-order group assumption explicit.
• Changes to algorithms accepting batched inputs.
• Changes to construction of batched DLEQ proofs.
• Updated ciphersuites to be consistent with hash-to-curve and added OPRF specific ciphersuites.

draft-02:

• Added section discussing cryptographic security and static DH oracles.
• Updated batched proof algorithms.

draft-01:

• Updated ciphersuites to be in line with https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-04.
• Made some necessary modular reductions more explicit.

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

Notation and Terminology The following functions and notation are used throughout the document.

• For any object x, we write len(x) to denote its length in bytes.
• For two byte arrays x and y, write x || y to denote their concatenation.
• I2OSP(x, xLen): Converts a non-negative integer x into a byte array of specified length xLen as described in . Note that this function returns a byte array in big-endian byte order.
• 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.

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 PrivateInput data type refers to inputs that are known only to the client in the protocol, whereas the PublicInput data type refers to inputs that are known to both client and server in the protocol. Both PrivateInput and PublicInput are opaque byte strings of arbitrary length no larger than 2^13 octets. String values such as "DeriveKeyPair", "Seed-", and "Finalize" are ASCII string literals. The following terms are used throughout this document.

• PRF: Pseudorandom Function.
• OPRF: Oblivious Pseudorandom Function.
• VOPRF: Verifiable Oblivious Pseudorandom Function.
• POPRF: Partially Oblivious Pseudorandom Function.
• Client: Protocol initiator. Learns pseudorandom function evaluation as the output of the protocol.
• Server: Computes the pseudorandom function over a private key. Learns nothing about the client's input or output.

Preliminaries The protocols in this document have two primary dependencies:

• Group: A prime-order group implementing the API described below in . See for specific instances of groups.
• Hash: A cryptographic hash function whose output length is Nh bytes.

specifies ciphersuites as combinations of Group and Hash.

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. 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 generator element of the group.
• HashToGroup(x): Deterministically maps an array of bytes x to an element of Group. The map must ensure that, for any adversary receiving R = HashToGroup(x), it is computationally difficult to reverse the mapping. This function is optionally parameterized by a domain separation tag (DST); see . Security properties of this function are described in .
• HashToScalar(x): Deterministically maps an array of bytes x to an element in GF(p). This function is optionally parameterized by a DST; see . 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.

contains details for the implementation of this interface for different prime-order groups instantiated over elliptic curves. In particular, for some choices of elliptic curves, e.g., those detailed in , which require accounting for cofactors, describes required steps necessary to ensure the resulting group is of prime order.

Discrete Logarithm Equivalence Proofs A proof of knowledge allows a prover to convince a verifier that some statement is true. If the prover can generate a proof without interaction with the verifier, the proof is noninteractive. If the verifier learns nothing other than whether the statement claimed by the prover is true or false, the proof is zero-knowledge. This section describes a noninteractive zero-knowledge proof for discrete logarithm equivalence (DLEQ), which is used in the construction of VOPRF and POPRF. A DLEQ proof demonstrates that two pairs of group elements have the same discrete logarithm without revealing the discrete logarithm. The DLEQ proof resembles the Chaum-Pedersen proof, which is shown to be zero-knowledge by Jarecki, et al. and is noninteractive after applying the Fiat-Shamir transform . Furthermore, Davidson, et al. showed a proof system for batching DLEQ proofs that has constant-size proofs with respect to the number of inputs. The specific DLEQ proof system presented below follows this latter construction with two modifications: (1) the transcript used to generate the seed includes more context information, and (2) the individual challenges for each element in the proof is derived from a seed-prefixed hash-to-scalar invocation rather than being sampled from a seeded PRNG. The description is split into two sub-sections: one for generating the proof, which is done by servers in the verifiable protocols, and another for verifying the proof, which is done by clients in the protocol.

Proof Generation Generating a proof is done with the GenerateProof function, defined below. Given elements A and B, two non-empty lists of elements C and D of length m, and a scalar k; this function produces a proof that k*A == B and k*C[i] == D[i] for each i in [0, ..., m - 1]. The output is a value of type Proof, which is a tuple of two Scalar values. GenerateProof accepts lists of inputs to amortize the cost of proof generation. Applications can take advantage of this functionality to produce a single, constant-sized proof for m DLEQ inputs, rather than m proofs for m DLEQ inputs. The helper function ComputeCompositesFast is as defined below, and is an optimization of the ComputeComposites function for servers since they have knowledge of the private key. When used in the protocol described in , the parameter contextString is as defined in .

Proof Verification Verifying a proof is done with the VerifyProof function, defined below. This function takes elements A and B, two non-empty lists of elements C and D of length m, and a Proof value output from GenerateProof. It outputs a single boolean value indicating whether or not the proof is valid for the given DLEQ inputs. Note this function can verify proofs on lists of inputs whenever the proof was generated as a batched DLEQ proof with the same inputs. The definition of ComputeComposites is given below. When used in the protocol described in , the parameter contextString is as defined in .

Protocol In this section, we define three protocol variants referred to as the OPRF, VOPRF, and POPRF modes with the following properties. In the OPRF mode, a client and server interact to compute output = F(skS, input), where input is the client's private input, skS is the server's private key, and output is the OPRF output. After the execution of the protocol, the client learns output and the server learns nothing. This interaction is shown below.

OPRF protocol overview evaluatedElement = BlindEvaluate(skS, blindedElement) evaluatedElement <---------- output = Finalize(input, blind, evaluatedElement) ]]>

In the VOPRF mode, the client additionally receives proof that the server used skS in computing the function. To achieve verifiability, as in , the server provides a zero-knowledge proof that the key provided as input by the server in the BlindEvaluate function is the same key as it used to produce the server's public key, pkS, which the client receives as input to the protocol. This proof does not reveal the server's private key to the client. This interaction is shown below.

VOPRF protocol overview with additional proof evaluatedElement, proof = BlindEvaluate(skS, pkS, blindedElement) evaluatedElement, proof <---------- output = Finalize(input, blind, evaluatedElement, blindedElement, pkS, proof) ]]>

The POPRF mode extends the VOPRF mode such that the client and server can additionally provide a public input info that is used in computing the pseudorandom function. That is, the client and server interact to compute output = F(skS, input, info) as is shown below.

POPRF protocol overview with additional public input evaluatedElement, proof = BlindEvaluate(skS, blindedElement, info) evaluatedElement, proof <---------- output = Finalize(input, blind, evaluatedElement, blindedElement, proof, info, tweakedKey) ]]>

Each protocol consists of an offline setup phase and an online phase, described in and , respectively. Configuration details for the offline phase are described in .

Configuration Each of the three protocol variants are identified with a one-byte value (in hexadecimal): Identifiers for protocol variants.

Mode	Value
modeOPRF	0x00
modeVOPRF	0x01
modePOPRF	0x02

Additionally, each protocol variant is instantiated with a ciphersuite, or suite. Each ciphersuite is identified with an ASCII string identifier, referred to as identifier; see for the set of initial ciphersuite values. The mode and ciphersuite identifier values are combined to create a "context string" used throughout the protocol with the following function:

Key Generation and Context Setup In the offline setup phase, the server key pair (skS, pkS) is generated using the following GenerateKeyPair function, which produces a randomly generate private and public key pair. lists a deterministic key generation function DeriveKeyPair that can be used to implement GenerateKeyPair as DeriveKeyPair(random(Ns), info), with optional (possibly empty) application-chosen public info string. Also during the offline setup phase, both the client and server create a context used for executing the online phase of the protocol after agreeing on a mode and ciphersuite identifier. The context, such as OPRFServerContext, is an implementation-specific data structure that stores a context string and the relevant key material for each party. The OPRF variant server and client contexts are created as follows: The VOPRF variant server and client contexts are created as follows: The POPRF variant server and client contexts are created as follows:

Deterministic Key Generation This section describes a deterministic key generation function, DeriveKeyPair. It accepts a randomly generated seed of length Ns bytes and an optional (possibly empty) public info string. The constant Ns corresponds to the size in bytes of a serialized Scalar and is defined in . 255: raise DeriveKeyPairError skS = G.HashToScalar(deriveInput || I2OSP(counter, 1), DST = "DeriveKeyPair" || contextString) counter = counter + 1 pkS = G.ScalarMultGen(skS) return skS, pkS ]]>

Online Protocol In the online phase, the client and server engage in a two message protocol to compute the protocol output. This section describes the protocol details for each protocol variant. Throughout each description the following parameters are assumed to exist:

• G, a prime-order Group implementing the API described in .
• contextString, a PublicInput domain separation tag constructed during context setup as created in .
• skS and pkS, a Scalar and Element representing the private and public keys configured for client and server in .

Applications serialize protocol messages between client and server for transmission. Elements and scalars are serialized to byte arrays, and values of type Proof are serialized as the concatenation of two serialized scalars. Deserializing these values can fail, in which case the application MUST abort the protocol raising a DeserializeError failure. Applications MUST check that input Element values received over the wire are not the group identity element. This check is handled after deserializing Element values; see for more information and requirements on input validation for each ciphersuite.

OPRF Protocol The OPRF protocol begins with the client blinding its input, as described by the Blind function below. Note that this function can fail with an InvalidInputError error for certain inputs that map to the group identity element. Dealing with this failure is an application-specific decision; see . Clients store blind locally, and send blindedElement to the server for evaluation. Upon receipt, servers process blindedElement using the BlindEvaluate function described below. Servers send the output evaluatedElement to clients for processing. Recall that servers may process multiple client inputs by applying the BlindEvaluate function to each blindedElement received, and returning an array with the corresponding evaluatedElement values. Upon receipt of evaluatedElement, clients process it to complete the OPRF evaluation with the Finalize function described below. An entity which knows both the secret key and the input can compute the PRF result using the following Evaluate function.

VOPRF Protocol The VOPRF protocol begins with the client blinding its input, using the same Blind function as in . Clients store the output blind locally and send blindedElement to the server for evaluation. Upon receipt, servers process blindedElement to compute an evaluated element and DLEQ proof using the following BlindEvaluate function. In the description above, inputs to GenerateProof are one-item lists. Using larger lists allows servers to batch the evaluation of multiple elements while producing a single batched DLEQ proof for them. The server sends both evaluatedElement and proof back to the client. Upon receipt, the client processes both values to complete the VOPRF computation using the Finalize function below. As in BlindEvaluate, inputs to VerifyProof are one-item lists. Clients can verify multiple inputs at once whenever the server produced a batched DLEQ proof for them. Finally, an entity which knows both the secret key and the input can compute the PRF result using the Evaluate function described in .

POPRF Protocol The POPRF protocol begins with the client blinding its input, using the following modified Blind function. In this step, the client also binds a public info value, which produces an additional tweakedKey to be used later in the protocol. Note that this function can fail with an InvalidInputError error for certain private inputs that map to the group identity element, as well as certain public inputs that, if not detected at this point, will cause server evaluation to fail. Dealing with either failure is an application-specific decision; see . Clients store the outputs blind and tweakedKey locally and send blindedElement to the server for evaluation. Upon receipt, servers process blindedElement to compute an evaluated element and DLEQ proof using the following BlindEvaluate function. In the description above, inputs to GenerateProof are one-item lists. Using larger lists allows servers to batch the evaluation of multiple elements while producing a single batched DLEQ proof for them. BlindEvaluate triggers InverseError when the function is about to calculate the inverse of a zero scalar, which does not exist and therefore yields a failure in the protocol. This only occurs for info values that map to the secret key of the server. Thus, clients that observe this signal are assumed to know the server secret key. Hence, this error can be a signal for the server to replace its secret key. The server sends both evaluatedElement and proof back to the client. Upon receipt, the client processes both values to complete the POPRF computation using the Finalize function below. As in BlindEvaluate, inputs to VerifyProof are one-item lists. Clients can verify multiple inputs at once whenever the server produced a batched DLEQ proof for them. Finally, an entity which knows both the secret key and the input can compute the PRF result using the Evaluate function described below.

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 instantiations of the following functionalities:

• 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.
• Hash: A cryptographic hash function whose output length is Nh bytes long.

This section specifies an initial registry of ciphersuites with supported groups and hash functions. It also includes implementation details for each ciphersuite, focusing on input validation, as well as requirements for future ciphersuites. For each ciphersuite, contextString is that which is computed in the Setup functions. Applications should take caution in using ciphersuites targeting P-256 and ristretto255. See for related discussion.

OPRF(ristretto255, SHA-512) This ciphersuite uses ristretto255 for the Group and SHA-512 for the Hash function. The value of the ciphersuite identifier is "ristretto255-SHA512".

• Group: ristretto255
  • Order(): Return 2^252 + 27742317777372353535851937790883648493 (see )
  • Identity(): As defined in .
  • Generator(): As defined in .
  • HashToGroup(): Use hash_to_ristretto255 with DST = "HashToGroup-" || contextString, and expand_message = expand_message_xmd using SHA-512.
  • HashToScalar(): Compute uniform_bytes using expand_message = expand_message_xmd, DST = "HashToScalar-" || contextString, and output length 64, interpret uniform_bytes as a 512-bit integer in little-endian order, and reduce the integer modulo Group.Order().
  • ScalarInverse(s): Returns the multiplicative inverse of input Scalar s mod Group.Order().
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to for implementation guidance.
  • SerializeElement(A): Implemented using the 'Encode' function from Section 4.3.2 of ; Ne = 32.
  • DeserializeElement(buf): Implemented using the 'Decode' function from Section 4.3.1 of . 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 by outputting the little-endian 32-byte encoding of the Scalar value with the top three bits set to zero; Ns = 32.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a little-endian 32-byte string. This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1]. Note that this means the top three bits of the input MUST be zero.
• Hash: SHA-512; Nh = 64.

OPRF(decaf448, SHAKE-256) This ciphersuite uses decaf448 for the Group and SHAKE-256 for the Hash function. The value of the ciphersuite identifier is "decaf448-SHAKE256".

• Group: decaf448
  • Order(): Return 2^446 - 13818066809895115352007386748515426880336692474882178609894547503885
  • Identity(): As defined in .
  • Generator(): As defined in .
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to for implementation guidance.
  • HashToGroup(): Use hash_to_decaf448 with DST = "HashToGroup-" || contextString, and expand_message = expand_message_xof using SHAKE-256.
  • HashToScalar(): Compute uniform_bytes using expand_message = expand_message_xof, DST = "HashToScalar-" || contextString, and output length 64, interpret uniform_bytes as a 512-bit integer in little-endian order, and reduce the integer modulo Group.Order().
  • ScalarInverse(s): Returns the multiplicative inverse of input Scalar s mod Group.Order().
  • SerializeElement(A): Implemented using the 'Encode' function from Section 5.3.2 of ; Ne = 56.
  • DeserializeElement(buf): Implemented using the 'Decode' function from Section 5.3.1 of . 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 by outputting the little-endian 56-byte encoding of the Scalar value; Ns = 56.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a little-endian 56-byte string. This function can fail if the input does not represent a Scalar in the range [0, G.Order() - 1].
• Hash: SHAKE-256; Nh = 64.

OPRF(P-256, SHA-256) This ciphersuite uses P-256 for the Group and SHA-256 for the Hash function. The value of the ciphersuite identifier is "P256-SHA256".

• 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 [0, G.Order() - 1]. Refer to for implementation guidance.
  • HashToGroup(): Use hash_to_curve with suite P256_XMD:SHA-256_SSWU_RO_ and DST = "HashToGroup-" || contextString.
  • HashToScalar(): Use hash_to_field from using L = 48, expand_message_xmd with SHA-256, DST = "HashToScalar-" || contextString, 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 input string 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 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].
• Hash: SHA-256; Nh = 32.

OPRF(P-384, SHA-384) This ciphersuite uses P-384 for the Group and SHA-384 for the Hash function. The value of the ciphersuite identifier is "P384-SHA384".

• Group: P-384 (secp384r1)
  • Order(): Return 0xffffffffffffffffffffffffffffffffffffffffffffffffc7634d81f4372ddf581a0db248b0a77aecec196accc52973.
  • Identity(): As defined in .
  • Generator(): As defined in .
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to for implementation guidance.
  • HashToGroup(): Use hash_to_curve with suite P384_XMD:SHA-384_SSWU_RO_ and DST = "HashToGroup-" || contextString.
  • HashToScalar(): Use hash_to_field from using L = 72, expand_message_xmd with SHA-384, DST = "HashToScalar-" || contextString, 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 = 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 , 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 = 48.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 48-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].
• Hash: SHA-384; Nh = 48.

OPRF(P-521, SHA-512) This ciphersuite uses P-521 for the Group and SHA-512 for the Hash function. The value of the ciphersuite identifier is "P521-SHA512".

• Group: P-521 (secp521r1)
  • Order(): Return 0x01fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409.
  • Identity(): As defined in .
  • Generator(): As defined in .
  • RandomScalar(): Implemented by returning a uniformly random Scalar in the range [0, G.Order() - 1]. Refer to for implementation guidance.
  • HashToGroup(): Use hash_to_curve with suite P521_XMD:SHA-512_SSWU_RO_ and DST = "HashToGroup-" || contextString.
  • HashToScalar(): Use hash_to_field from using L = 98, expand_message_xmd with SHA-512, DST = "HashToScalar-" || contextString, 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 = 67.
  • DeserializeElement(buf): Implemented by attempting to deserialize a 49 byte input string 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 = 66.
  • DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a 66-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].
• Hash: SHA-512; Nh = 64.

Future Ciphersuites A critical requirement of implementing the prime-order group using elliptic curves is a method to instantiate the function HashToGroup, that maps inputs to group elements. In the elliptic curve setting, this deterministically maps inputs (as byte arrays) to uniformly chosen points on the curve. In the security proof of the construction Hash is modeled as a random oracle. This implies that any instantiation of HashToGroup must be pre-image and collision resistant. In we give instantiations of this functionality based on the functions described in . Consequently, any OPRF implementation must adhere to the implementation and security considerations discussed in when instantiating the function. The DeserializeElement and DeserializeScalar functions instantiated for a particular prime-order group corresponding to a ciphersuite MUST adhere to the description in . Future ciphersuites MUST describe how input validation is done for DeserializeElement and DeserializeScalar. Additionally, future ciphersuites must take care when choosing the security level of the group. See for additional details.

Random Scalar Generation Two popular algorithms for generating a random integer uniformly distributed in the range [0, 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, 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 - 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.

Application Considerations This section describes considerations for applications, including external interface recommendations, explicit error treatment, and public input representation for the POPRF protocol variant.

Input Limits Application inputs, expressed as PrivateInput or PublicInput values, MUST be smaller than 2^13 bytes in length. Applications that require longer inputs can use a cryptographic hash function to map these longer inputs to a fixed-length input that fits within the PublicInput or PrivateInput length bounds. Note that some cryptographic hash functions have input length restrictions themselves, but these limits are often large enough to not be a concern in practice. For example, SHA-256 has an input limit of 2^61 bytes.

External Interface Recommendations In , the interface of the protocol functions allows that some inputs (and outputs) to be group elements and scalars. However, implementations can instead operate over group elements and scalars internally, and only expose interfaces that operate with an application-specific format of messages.

Error Considerations Some OPRF variants specified in this document have fallible operations. For example, Finalize and BlindEvaluate can fail if any element received from the peer fails input validation. The explicit errors generated throughout this specification, along with the conditions that lead to each error, are as follows:

• VerifyError: Verifiable OPRF proof verification failed; and .
• DeserializeError: Group Element or Scalar deserialization failure; and .
• InputValidationError: Validation of byte array inputs failed; .

There are other explicit errors generated in this specification; however, they occur with negligible probability in practice. We note them here for completeness.

• InvalidInputError: OPRF Blind input produces an invalid output element; and .
• InverseError: A tweaked private key is invalid (has no multiplicative inverse); and .

In general, the errors in this document are meant as a guide to implementors. They are not an exhaustive list of all the errors an implementation might emit. For example, implementations might run out of memory and return a corresponding error.

POPRF Public Input Functionally, the VOPRF and POPRF variants differ in that the POPRF variant admits public input, whereas the VOPRF variant does not. Public input allows clients and servers to cryptographically bind additional data to the POPRF output. A POPRF with fixed public input is functionally equivalent to a VOPRF. However, there are differences in the underlying security assumptions made about each variant; see for more details. This public input is known to both parties at the start of the protocol. It is RECOMMENDED that this public input be constructed with some type of higher-level domain separation to avoid cross protocol attacks or related issues. For example, protocols using this construction might ensure that the public input uses a unique, prefix-free encoding. See for further discussion on constructing domain separation values. Implementations of the POPRF may choose to not let applications control info in cases where this value is fixed or otherwise not useful to the application. In this case, the resulting protocol is functionally equivalent to the VOPRF, which does not admit public input.

IANA considerations This document has no IANA actions.

Security Considerations This section discusses the security of the protocols defined in this specification, along with some suggestions and trade-offs that arise from the implementation of the protocol variants in this document. Note that the syntax of the POPRF variant is different from that of the OPRF and VOPRF variants since it admits an additional public input, but the same security considerations apply.

Security Properties The security properties of an OPRF protocol with functionality y = F(k, x) include those of a standard PRF. Specifically:

• Pseudorandomness: For a random sampling of k, F is pseudorandom if the output y = F(k, x) on any input x is indistinguishable from uniformly sampling any element in F's range.

In other words, consider an adversary that picks inputs x from the domain of F and evaluates F on (k, x) (without knowledge of randomly sampled k). Then the output distribution F(k, x) is indistinguishable from the output distribution of a randomly chosen function with the same domain and range. A consequence of showing that a function is pseudorandom is that it is necessarily non-malleable (i.e. we cannot compute a new evaluation of F from an existing evaluation). A genuinely random function will be non-malleable with high probability, and so a pseudorandom function must be non-malleable to maintain indistinguishability.

• Unconditional input secrecy: The server does not learn anything about the client input x, even with unbounded computation.

In other words, an attacker with infinite computing power cannot recover any information about the client's private input x from an invocation of the protocol. Essentially, input secrecy is the property that, even if the server learns the client's private input x at some point in the future, the server cannot link any particular PRF evaluation to x. This property is also known as unlinkability . For the VOPRF and POPRF protocol variants, there is an additional security property:

• Verifiable: The client must only complete execution of the protocol if it can successfully assert that the output it computes is correct. This is taken with respect to the private key held by the server.

Any VOPRF or POPRF that satisfies the 'verifiable' security property is known as 'verifiable'. In practice, the notion of verifiability requires that the server commits to the key before the actual protocol execution takes place. Then the client verifies that the server has used the key in the protocol using this commitment. In the following, we may also refer to this commitment as a public key. Finally, the POPRF variant also has the following security property:

• Partial obliviousness: The client and server must be able to perform the PRF on client's private input and public input. The server must learn nothing about the client's private input or the output of the function. In addition, the client must learn nothing about the server's private key.

This property becomes useful when dealing with key management operations such as the rotation of server's keys. Note that partial obliviousness only applies to the POPRF variant because neither the OPRF nor VOPRF variants accept public input to the protocol.

Security Assumptions Below, we discuss the cryptographic security of each protocol variant from , relative to the necessary cryptographic assumptions that need to be made.

OPRF and VOPRF Assumptions The OPRF and VOPRF protocol variants in this document are based on . In particular, the VOPRF construction is similar to the construction with the following distinguishing properties:

1. This document does not use session identifiers to differentiate different instances of the protocol; and
2. This document supports batching so that multiple evaluations can happen at once whilst only constructing one DLEQ proof object. This is enabled using an established batching technique .

The pseudorandomness and input secrecy (and verifiability) of the OPRF (and VOPRF) protocols in are based on the One-More Gap Computational Diffie Hellman assumption that is computationally difficult to solve in the corresponding prime-order group. In , these properties are proven for one instance (i.e., one key) of the VOPRF protocol, and without batching. There is currently no security analysis available for the VOPRF protocol described in this document in a setting with multiple server keys or batching.

POPRF Assumptions The POPRF construction in this document is based on the construction known as 3HashSDHI given by . The construction is identical to 3HashSDHI, except that this design can optionally perform multiple POPRF evaluations in one batch, whilst only constructing one DLEQ proof object. This is enabled using an established batching technique . Pseudorandomness, input secrecy, verifiability, and partial obliviousness of the POPRF variant is based on the assumption that the One-More Gap Strong Diffie-Hellman Inversion (SDHI) assumption from is computationally difficult to solve in the corresponding prime-order group. Tyagi et al. show that both the One-More Gap Computational Diffie Hellman assumption and the One-More Gap SDHI assumption reduce to the q-DL (Discrete Log) assumption in the algebraic group model, for some q number of BlindEvaluate queries. (The One-More Gap Computational Diffie Hellman assumption was the hardness assumption used to evaluate the OPRF and VOPRF designs based on , which is a predecessor to the POPRF variant in .)

Static Diffie Hellman Attack and Security Limits A side-effect of the OPRF protocol variants in this document is that they allow instantiation of an oracle for constructing static DH samples; see and . These attacks are meant to recover (bits of) the server private key. Best-known attacks reduce the security of the prime-order group instantiation by log_2(Q)/2 bits, where Q is the number of BlindEvaluate calls made by the attacker. As a result of this class of attacks, choosing prime-order groups with a 128-bit security level instantiates an OPRF with a reduced security level of 128-(log_2(Q)/2) bits of security. Moreover, such attacks are only possible for those certain applications where the adversary can query the OPRF directly. Applications can mitigate against this problem in a variety of ways, e.g., by rate-limiting client queries to BlindEvaluate or by rotating private keys. In applications where such an oracle is not made available this security loss does not apply. In most cases, it would require an informed and persistent attacker to launch a highly expensive attack to reduce security to anything much below 100 bits of security. Applications that admit the aforementioned oracle functionality, and that cannot tolerate discrete logarithm security of lower than 128 bits, are RECOMMENDED to choose groups that target a higher security level, such as decaf448 (used by ciphersuite 0x0002), P-384 (used by 0x0004), or P-521 (used by 0x0005).

Domain Separation Applications SHOULD construct input to the protocol to provide domain separation. Any system which has multiple OPRF applications should distinguish client inputs to ensure the OPRF results are separate. Guidance for constructing info can be found in .

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 GenerateProof and BlindEvaluate.

Acknowledgements This document resulted from the work of the Privacy Pass team . The authors would also like to acknowledge helpful conversations with Hugo Krawczyk. Eli-Shaoul Khedouri provided additional review and comments on key consistency. Daniel Bourdrez, Tatiana Bradley, Sofia Celi, Frank Denis, Julia Hesse, Russ Housley, Kevin Lewi, Christopher Patton, and Bas Westerbaan also provided helpful input and contributions to the document.

References Normative References In many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. RFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings. This document provides recommendations for the implementation of public-key cryptography based on the RSA algorithm, covering cryptographic primitives, encryption schemes, signature schemes with appendix, and ASN.1 syntax for representing keys and for identifying the schemes. This document represents a republication of PKCS #1 v2.2 from RSA Laboratories' Public-Key Cryptography Standards (PKCS) series. By publishing this RFC, change control is transferred to the IETF. This document also obsoletes RFC 3447. Cloudflare, Inc. Cornell Tech Cloudflare, Inc. Stanford University Cloudflare, Inc. This document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. This memo specifies two prime-order groups, ristretto255 and decaf448, suitable for safely implementing higher-level and complex cryptographic protocols. The ristretto255 group can be implemented using Curve25519, allowing existing Curve25519 implementations to be reused and extended to provide a prime-order group. Likewise, the decaf448 group can be implemented using edwards448. Informative References This memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128-bit and ~224-bit security level, respectively, and are generated deterministically based on a list of required properties. Certicom Research Certicom Research Royal Holloway, University of London (work completed during an internship at Cloudflare), London , UK University of Waterloo, Waterloo , Belgium Cloudflare, San Francisco, California , USA Algorand Foundation Novi Research Cloudflare, Inc. This document describes the OPAQUE protocol, a secure asymmetric password-authenticated key exchange (aPAKE) that supports mutual authentication in a client-server setting without reliance on PKI and with security against pre-computation attacks upon server compromise. In addition, the protocol provides forward secrecy and the ability to hide the password from the server, even during password registration. This document specifies the core OPAQUE protocol and one instantiation based on 3DH. Brave Software Brave Software Cloudflare Google LLC Cloudflare This document specifies two variants of the the two-message issuance protocol for Privacy Pass tokens: one that produces tokens that are privately verifiable using the issuance private key, and another that produces tokens that are publicly verifiable using the issuance public key.

Test Vectors This section includes test vectors for the protocol variants specified in this document. For each ciphersuite specified in , there is a set of test vectors for the protocol when run the OPRF, VOPRF, and POPRF modes. Each test vector lists the batch size for the evaluation. Each test vector value is encoded as a hexadecimal byte string. The fields of each test vector are described below.

• "Input": The private client input, an opaque byte string.
• "Info": The public info, an opaque byte string. Only present for POPRF test vectors.
• "Blind": The blind value output by Blind(), a serialized Scalar of Ns bytes long.
• "BlindedElement": The blinded value output by Blind(), a serialized Element of Ne bytes long.
• "EvaluatedElement": The evaluated element output by BlindEvaluate(), a serialized Element of Ne bytes long.
• "Proof": The serialized Proof output from GenerateProof() composed of two serialized Scalar values each of Ns bytes long. Only present for VOPRF and POPRF test vectors.
• "ProofRandomScalar": The random scalar r computed in GenerateProof(), a serialized Scalar of Ns bytes long. Only present for VOPRF and POPRF test vectors.
• "Output": The protocol output, an opaque byte string of length Nh bytes.

Test vectors with batch size B > 1 have inputs separated by a comma ",". Applicable test vectors will have B different values for the "Input", "Blind", "BlindedElement", "EvaluationElement", and "Output" fields. The server key material, pkSm and skSm, are listed under the mode for each ciphersuite. Both pkSm and skSm are the serialized values of pkS and skS, respectively, as used in the protocol. Each key pair is derived from a seed Seed and info string KeyInfo, which are listed as well, using the DeriveKeyPair function from .

ristretto255-SHA512

OPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

VOPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

POPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

decaf448-SHAKE256

OPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

VOPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

POPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

P256-SHA256

OPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

VOPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

POPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

P384-SHA384

OPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

VOPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

POPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

P521-SHA512

OPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

VOPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2

POPRF Mode

Test Vector 1, Batch Size 1

Test Vector 2, Batch Size 1

Test Vector 3, Batch Size 2