RFC 9497 | OPRFs | December 2023
Davidson, et al. | Informational | [Page]

Stream:
  Internet Research Task Force (IRTF)
RFC:
  9497
Category:
  Informational
Published:
  December 2023
ISSN:
  2070-1721
Authors:
  A. Davidson
  Brave Software
  A. Faz-Hernandez
  Cloudflare, Inc.
  N. Sullivan
  Cloudflare, Inc.
  C. A. Wood
  Cloudflare, Inc.

# RFC 9497

# Oblivious Pseudorandom Functions (OPRFs) Using Prime-Order Groups

## Abstract

An Oblivious Pseudorandom Function (OPRF) is a two-party protocol between a client and a server for computing the output of a Pseudorandom Function (PRF). The server provides the PRF private 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 private 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.¶

## Status of This Memo

This document is not an Internet Standards Track specification; it is published for informational purposes.¶

This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internet-related research and development activities. These results might not be suitable for deployment. This RFC represents the consensus of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not candidates for any level of Internet Standard; see Section 2 of RFC 7841.¶

Information about the current status of this document, any errata, and how to provide feedback on it may be obtained at https://www.rfc-editor.org/info/rfc9497.¶

## Copyright Notice

Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.¶

This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.¶

## 1. 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, wherein 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, which the client knows. A Partially Oblivious PRF (POPRF) is a variant of a VOPRF, where the 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 the 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 [JKKX16], privacy-preserving password stores [SJKS17], and password-authenticated key exchange (PAKE) [OPAQUE]. Verifiable OPRFs are necessary in some applications, such as Privacy Pass [PRIVACY-PASS]. Verifiable OPRFs have also been used for password-protected secret sharing schemes, such as that of [JKK14].¶

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.¶

### 1.1. Requirements Language

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶

### 1.2. 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 nonnegative integer `x` into a byte array of specified length `xLen`, as described in [RFC8017]. 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 to 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 the client and server in the protocol. Both `PrivateInput` and `PublicInput` are opaque byte strings of arbitrary length no larger than 216 - 1 bytes. This length restriction exists because `PublicInput` and `PrivateInput` values are length-prefixed with two bytes before use throughout the protocol.¶

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 PRF evaluation as the output of the protocol.¶
Server:
  Computes the PRF using a private key. Learns nothing about the client's input or output.¶

## 2. Preliminaries

The protocols in this document have two primary dependencies:¶

`Group`:
  A prime-order group implementing the API described below in Section 2.1. See Section 4 for specific instances of groups.¶
`Hash`:
  A cryptographic hash function whose output length is `Nh` bytes.¶

Section 4 specifies ciphersuites as combinations of `Group` and `Hash`.¶

### 2.1. Prime-Order Group

In this document, we assume the construction of an additive, prime-order group, denoted `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. (However, different representations of the group for a single `p` may exist. Section 4 lists specific groups that indicate both the 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 is 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 Section 4. Security properties of this function are described in [RFC9380].¶
HashToScalar(x):
  Deterministically maps an array of bytes `x` to an element in GF(p). This function is optionally parameterized by a DST; see Section 4. Security properties of this function are described in [RFC9380], Section 10.5.¶
RandomScalar():
  Chooses at random a nonzero 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 4 for group-specific input validation steps.¶
SerializeScalar(s):
  Maps Scalar `s` to a canonical byte array `buf` of fixed-length `Ns`.¶
DeserializeScalar(buf):
  Attempts to map a byte array `buf` to Scalar `s`. This function can raise a DeserializeError if deserialization fails; see Section 4 for group-specific input validation steps.¶

Section 4 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 [RFC7748], which require accounting for cofactors, Section 4 describes required steps necessary to ensure the resulting group is of prime order.¶

### 2.2. 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 [ChaumPedersen] proof, which is shown to be zero-knowledge by Jarecki, et al. [JKK14] and is noninteractive after applying the Fiat-Shamir transform [FS00]. Furthermore, Davidson, et al. [DGSTV18] 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 Pseudorandom Number Generator (PRNG). The description is split into two subsections: 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.¶

#### 2.2.1. Proof Generation

Generating a proof is done with the `GenerateProof` function, as defined below. Given Element values A and B, two non-empty lists of Element values C and D of length `m`, and 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. We use the notation `proof[0]` and `proof[1]` to denote the first and second elements in this tuple, respectively.¶

`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.¶

    Input:

      Scalar k
      Element A
      Element B
      Element C[m]
      Element D[m]

    Output:

      Proof proof

    Parameters:

      Group G

    def GenerateProof(k, A, B, C, D):
      (M, Z) = ComputeCompositesFast(k, B, C, D)

      r = G.RandomScalar()
      t2 = r * A
      t3 = r * M

      Bm = G.SerializeElement(B)
      a0 = G.SerializeElement(M)
      a1 = G.SerializeElement(Z)
      a2 = G.SerializeElement(t2)
      a3 = G.SerializeElement(t3)

      challengeTranscript =
        I2OSP(len(Bm), 2) || Bm ||
        I2OSP(len(a0), 2) || a0 ||
        I2OSP(len(a1), 2) || a1 ||
        I2OSP(len(a2), 2) || a2 ||
        I2OSP(len(a3), 2) || a3 ||
        "Challenge"

      c = G.HashToScalar(challengeTranscript)
      s = r - c * k

      return [c, s]

¶

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.¶

    Input:

      Scalar k
      Element B
      Element C[m]
      Element D[m]

    Output:

      Element M
      Element Z

    Parameters:

      Group G
      PublicInput contextString

    def ComputeCompositesFast(k, B, C, D):
      Bm = G.SerializeElement(B)
      seedDST = "Seed-" || contextString
      seedTranscript =
        I2OSP(len(Bm), 2) || Bm ||
        I2OSP(len(seedDST), 2) || seedDST
      seed = Hash(seedTranscript)

      M = G.Identity()
      for i in range(m):
        Ci = G.SerializeElement(C[i])
        Di = G.SerializeElement(D[i])
        compositeTranscript =
          I2OSP(len(seed), 2) || seed || I2OSP(i, 2) ||
          I2OSP(len(Ci), 2) || Ci ||
          I2OSP(len(Di), 2) || Di ||
          "Composite"

        di = G.HashToScalar(compositeTranscript)
        M = di * C[i] + M

      Z = k * M

      return (M, Z)

¶

When used in the protocol described in Section 3, the parameter `contextString` is as defined in Section 3.2.¶

#### 2.2.2. Proof Verification

Verifying a proof is done with the `VerifyProof` function, as defined below. This function takes Element values A and B, two non-empty lists of Element values 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.¶

    Input:

      Element A
      Element B
      Element C[m]
      Element D[m]
      Proof proof

    Output:

      boolean verified

    Parameters:

      Group G

    def VerifyProof(A, B, C, D, proof):
      (M, Z) = ComputeComposites(B, C, D)
      c = proof[0]
      s = proof[1]

      t2 = ((s * A) + (c * B))
      t3 = ((s * M) + (c * Z))

      Bm = G.SerializeElement(B)
      a0 = G.SerializeElement(M)
      a1 = G.SerializeElement(Z)
      a2 = G.SerializeElement(t2)
      a3 = G.SerializeElement(t3)

      challengeTranscript =
        I2OSP(len(Bm), 2) || Bm ||
        I2OSP(len(a0), 2) || a0 ||
        I2OSP(len(a1), 2) || a1 ||
        I2OSP(len(a2), 2) || a2 ||
        I2OSP(len(a3), 2) || a3 ||
        "Challenge"

      expectedC = G.HashToScalar(challengeTranscript)
      verified = (expectedC == c)

      return verified

¶

The definition of `ComputeComposites` is given below.¶

    Input:

      Element B
      Element C[m]
      Element D[m]

    Output:

      Element M
      Element Z

    Parameters:

      Group G
      PublicInput contextString

    def ComputeComposites(B, C, D):
      Bm = G.SerializeElement(B)
      seedDST = "Seed-" || contextString
      seedTranscript =
        I2OSP(len(Bm), 2) || Bm ||
        I2OSP(len(seedDST), 2) || seedDST
      seed = Hash(seedTranscript)

      M = G.Identity()
      Z = G.Identity()
      for i in range(m):
        Ci = G.SerializeElement(C[i])
        Di = G.SerializeElement(D[i])
        compositeTranscript =
          I2OSP(len(seed), 2) || seed || I2OSP(i, 2) ||
          I2OSP(len(Ci), 2) || Ci ||
          I2OSP(len(Di), 2) || Di ||
          "Composite"

        di = G.HashToScalar(compositeTranscript)
        M = di * C[i] + M
        Z = di * D[i] + Z

      return (M, Z)

¶

When used in the protocol described in Section 3, the parameter `contextString` is as defined in Section 3.2.¶

## 3. Protocol

In this section, we define and describe three protocol variants referred to as the OPRF, VOPRF, and POPRF modes. Each of these variants involves two messages between the client and server, but they differ slightly in terms of the security properties; see Section 7.1 for more information. A high-level description of the functionality of each mode follows.¶

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 the `output` and the server learns nothing. This interaction is shown below.¶

        Client(input)                                        Server(skS)
      -------------------------------------------------------------------
      blind, blindedElement = Blind(input)

                                 blindedElement
                                   ---------->

                    evaluatedElement = BlindEvaluate(skS, blindedElement)

                                 evaluatedElement
                                   <----------

      output = Finalize(input, blind, evaluatedElement)

Figure 1: OPRF Protocol Overview

In the VOPRF mode, the client additionally receives proof that the server used `skS` in computing the function. To achieve verifiability, as in [JKK14], 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 is 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.¶

        Client(input, pkS)       <---- pkS ------        Server(skS, pkS)
      -------------------------------------------------------------------
      blind, blindedElement = Blind(input)

                                 blindedElement
                                   ---------->

                  evaluatedElement, proof = BlindEvaluate(skS, pkS,
                                                          blindedElement)

                             evaluatedElement, proof
                                   <----------

      output = Finalize(input, blind, evaluatedElement,
                        blindedElement, pkS, proof)

Figure 2: VOPRF Protocol Overview with Additional Proof

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

        Client(input, pkS, info) <---- pkS ------  Server(skS, pkS, info)
      -------------------------------------------------------------------
      blind, blindedElement, tweakedKey = Blind(input, info, pkS)

                                 blindedElement
                                   ---------->

             evaluatedElement, proof = BlindEvaluate(skS, blindedElement,
                                                     info)

                             evaluatedElement, proof
                                   <----------

      output = Finalize(input, blind, evaluatedElement,
                        blindedElement, proof, info, tweakedKey)

Figure 3: POPRF Protocol Overview with Additional Public Input

Each protocol consists of an offline setup phase and an online phase, as described in Sections 3.2 and 3.3, respectively. Configuration details for the offline phase are described in Section 3.1.¶

### 3.1. Configuration

Each of the three protocol variants are identified with a one-byte value (in hexadecimal):¶

Table 1: 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 Section 4 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:¶

    def CreateContextString(mode, identifier):
      return "OPRFV1-" || I2OSP(mode, 1) || "-" || identifier

¶

### 3.2. Key Generation and Context Setup

In the offline setup phase, the server generates a fresh, random key pair (`skS`, `pkS`). There are two ways to generate this key pair. The first of which is using the `GenerateKeyPair` function described below.¶

    Input: None

    Output:

      Scalar skS
      Element pkS

    Parameters:

      Group G

    def GenerateKeyPair():
      skS = G.RandomScalar()
      pkS = G.ScalarMultGen(skS)
      return skS, pkS

¶

The second way to generate the key pair is via the deterministic key generation function `DeriveKeyPair`, as described in Section 3.2.1. Applications and implementations can use either method in practice.¶

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:¶

    def SetupOPRFServer(identifier, skS):
      contextString = CreateContextString(modeOPRF, identifier)
      return OPRFServerContext(contextString, skS)

    def SetupOPRFClient(identifier):
      contextString = CreateContextString(modeOPRF, identifier)
      return OPRFClientContext(contextString)

¶

The VOPRF variant server and client contexts are created as follows:¶

    def SetupVOPRFServer(identifier, skS):
      contextString = CreateContextString(modeVOPRF, identifier)
      return VOPRFServerContext(contextString, skS)

    def SetupVOPRFClient(identifier, pkS):
      contextString = CreateContextString(modeVOPRF, identifier)
      return VOPRFClientContext(contextString, pkS)

¶

The POPRF variant server and client contexts are created as follows:¶

    def SetupPOPRFServer(identifier, skS):
      contextString = CreateContextString(modePOPRF, identifier)
      return POPRFServerContext(contextString, skS)

    def SetupPOPRFClient(identifier, pkS):
      contextString = CreateContextString(modePOPRF, identifier)
      return POPRFClientContext(contextString, pkS)

¶

#### 3.2.1. Deterministic Key Generation

This section describes a deterministic key generation function, `DeriveKeyPair`. It accepts a seed of `32` bytes generated from a cryptographically secure random number generator and an optional (possibly empty) `info` string. Note that, by design, knowledge of `seed` and `info` is necessary to compute this function, which means that the secrecy of the output private key (`skS`) depends on the secrecy of `seed` (since the `info` string is public).¶

    Input:

      opaque seed[32]
      PublicInput info

    Output:

      Scalar skS
      Element pkS

    Parameters:

      Group G
      PublicInput contextString

    Errors: DeriveKeyPairError

    def DeriveKeyPair(seed, info):
      deriveInput = seed || I2OSP(len(info), 2) || info
      counter = 0
      skS = 0
      while skS == 0:
        if counter > 255:
          raise DeriveKeyPairError
        skS = G.HashToScalar(deriveInput || I2OSP(counter, 1),
                              DST = "DeriveKeyPair" || contextString)
        counter = counter + 1
      pkS = G.ScalarMultGen(skS)
      return skS, pkS

¶

### 3.3. 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 Section 2.1¶
contextString:
  a `PublicInput` domain separation tag constructed during context setup, as created in Section 3.1¶
skS and pkS:
  a Scalar and Element representing the private and public keys configured for the client and server in Section 3.2¶

Applications serialize protocol messages between the client and server for transmission. Element values and Scalar values are serialized to byte arrays, and values of type `Proof` are serialized as the concatenation of two serialized Scalar values. 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 Section 4 for more information and requirements on input validation for each ciphersuite.¶

#### 3.3.1. 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 Section 5.3.¶

    Input:

      PrivateInput input

    Output:

      Scalar blind
      Element blindedElement

    Parameters:

      Group G

    Errors: InvalidInputError

    def Blind(input):
      blind = G.RandomScalar()
      inputElement = G.HashToGroup(input)
      if inputElement == G.Identity():
        raise InvalidInputError
      blindedElement = blind * inputElement

      return blind, blindedElement

¶

Clients store `blind` locally and send `blindedElement` to the server for evaluation. Upon receipt, servers process `blindedElement` using the `BlindEvaluate` function described below.¶

    Input:

      Scalar skS
      Element blindedElement

    Output:

      Element evaluatedElement

    def BlindEvaluate(skS, blindedElement):
      evaluatedElement = skS * blindedElement
      return evaluatedElement

¶

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.¶

    Input:

      PrivateInput input
      Scalar blind
      Element evaluatedElement

    Output:

      opaque output[Nh]

    Parameters:

      Group G

    def Finalize(input, blind, evaluatedElement):
      N = G.ScalarInverse(blind) * evaluatedElement
      unblindedElement = G.SerializeElement(N)

      hashInput = I2OSP(len(input), 2) || input ||
                  I2OSP(len(unblindedElement), 2) || unblindedElement ||
                  "Finalize"
      return Hash(hashInput)

¶

An entity that knows both the private key and the input can compute the PRF result using the following `Evaluate` function.¶

    Input:

      Scalar skS
      PrivateInput input

    Output:

      opaque output[Nh]

    Parameters:

      Group G

    Errors: InvalidInputError

    def Evaluate(skS, input):
      inputElement = G.HashToGroup(input)
      if inputElement == G.Identity():
        raise InvalidInputError
      evaluatedElement = skS * inputElement
      issuedElement = G.SerializeElement(evaluatedElement)

      hashInput = I2OSP(len(input), 2) || input ||
                  I2OSP(len(issuedElement), 2) || issuedElement ||
                  "Finalize"
      return Hash(hashInput)

¶

#### 3.3.2. VOPRF Protocol

The VOPRF protocol begins with the client blinding its input, using the same `Blind` function as in Section 3.3.1. 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 a DLEQ proof using the following `BlindEvaluate` function.¶

    Input:

      Scalar skS
      Element pkS
      Element blindedElement

    Output:

      Element evaluatedElement
      Proof proof

    Parameters:

      Group G

    def BlindEvaluate(skS, pkS, blindedElement):
      evaluatedElement = skS * blindedElement
      blindedElements = [blindedElement]     // list of length 1
      evaluatedElements = [evaluatedElement] // list of length 1
      proof = GenerateProof(skS, G.Generator(), pkS,
                            blindedElements, evaluatedElements)
      return evaluatedElement, proof

¶

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.¶

    Input:

      PrivateInput input
      Scalar blind
      Element evaluatedElement
      Element blindedElement
      Element pkS
      Proof proof

    Output:

      opaque output[Nh]

    Parameters:

      Group G

    Errors: VerifyError

    def Finalize(input, blind, evaluatedElement,
                 blindedElement, pkS, proof):
      blindedElements = [blindedElement]     // list of length 1
      evaluatedElements = [evaluatedElement] // list of length 1
      if VerifyProof(G.Generator(), pkS, blindedElements,
                     evaluatedElements, proof) == false:
        raise VerifyError

      N = G.ScalarInverse(blind) * evaluatedElement
      unblindedElement = G.SerializeElement(N)

      hashInput = I2OSP(len(input), 2) || input ||
                  I2OSP(len(unblindedElement), 2) || unblindedElement ||
                  "Finalize"
      return Hash(hashInput)

¶

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 that knows both the private key and the input can compute the PRF result using the `Evaluate` function described in Section 3.3.1.¶

#### 3.3.3. 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 Section 5.3.¶

    Input:

      PrivateInput input
      PublicInput info
      Element pkS

    Output:

      Scalar blind
      Element blindedElement
      Element tweakedKey

    Parameters:

      Group G

    Errors: InvalidInputError

    def Blind(input, info, pkS):
      framedInfo = "Info" || I2OSP(len(info), 2) || info
      m = G.HashToScalar(framedInfo)
      T = G.ScalarMultGen(m)
      tweakedKey = T + pkS
      if tweakedKey == G.Identity():
        raise InvalidInputError

      blind = G.RandomScalar()
      inputElement = G.HashToGroup(input)
      if inputElement == G.Identity():
        raise InvalidInputError

      blindedElement = blind * inputElement

      return blind, blindedElement, tweakedKey

¶

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 a DLEQ proof using the following `BlindEvaluate` function.¶

    Input:

      Scalar skS
      Element blindedElement
      PublicInput info

    Output:

      Element evaluatedElement
      Proof proof

    Parameters:

      Group G

    Errors: InverseError

    def BlindEvaluate(skS, blindedElement, info):
      framedInfo = "Info" || I2OSP(len(info), 2) || info
      m = G.HashToScalar(framedInfo)
      t = skS + m
      if t == 0:
        raise InverseError

      evaluatedElement = G.ScalarInverse(t) * blindedElement

      tweakedKey = G.ScalarMultGen(t)
      evaluatedElements = [evaluatedElement] // list of length 1
      blindedElements = [blindedElement]     // list of length 1
      proof = GenerateProof(t, G.Generator(), tweakedKey,
                            evaluatedElements, blindedElements)

      return evaluatedElement, proof

¶

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 private key of the server. Thus, clients that cause this error should be assumed to know the server private key. Hence, this error can be a signal for the server to replace its private 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.¶

    Input:

      PrivateInput input
      Scalar blind
      Element evaluatedElement
      Element blindedElement
      Proof proof
      PublicInput info
      Element tweakedKey

    Output:

      opaque output[Nh]

    Parameters:

      Group G

    Errors: VerifyError

    def Finalize(input, blind, evaluatedElement, blindedElement,
                 proof, info, tweakedKey):
      evaluatedElements = [evaluatedElement] // list of length 1
      blindedElements = [blindedElement]     // list of length 1
      if VerifyProof(G.Generator(), tweakedKey, evaluatedElements,
                     blindedElements, proof) == false:
        raise VerifyError

      N = G.ScalarInverse(blind) * evaluatedElement
      unblindedElement = G.SerializeElement(N)

      hashInput = I2OSP(len(input), 2) || input ||
                  I2OSP(len(info), 2) || info ||
                  I2OSP(len(unblindedElement), 2) || unblindedElement ||
                  "Finalize"
      return Hash(hashInput)

¶

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 that knows both the private key and the input can compute the PRF result using the `Evaluate` function described below.¶

    Input:

      Scalar skS
      PrivateInput input
      PublicInput info

    Output:

      opaque output[Nh]

    Parameters:

      Group G

    Errors: InvalidInputError, InverseError

    def Evaluate(skS, input, info):
      inputElement = G.HashToGroup(input)
      if inputElement == G.Identity():
        raise InvalidInputError

      framedInfo = "Info" || I2OSP(len(info), 2) || info
      m = G.HashToScalar(framedInfo)
      t = skS + m
      if t == 0:
        raise InverseError
      evaluatedElement = G.ScalarInverse(t) * inputElement
      issuedElement = G.SerializeElement(evaluatedElement)

      hashInput = I2OSP(len(input), 2) || input ||
                  I2OSP(len(info), 2) || info ||
                  I2OSP(len(issuedElement), 2) || issuedElement ||
                  "Finalize"
      return Hash(hashInput)

¶

## 4. 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 Section 2.1, 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 [RFC9380], Section 3.1. 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 includes an initial set of ciphersuites with supported groups and hash functions. It also includes implementation details for each ciphersuite, focusing on input validation. Future documents can specify additional ciphersuites as needed, provided they meet the requirements in Section 4.6.¶

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 Section 7.2 for related discussion.¶

### 4.1. OPRF(ristretto255, SHA-512)

This ciphersuite uses ristretto255 [RFC9496] for the Group and SHA-512 for the hash function. The value of the ciphersuite identifier is "ristretto255-SHA512".¶

Group:

  ristretto255 [RFC9496]¶

  Order():
    Return 2252 + 27742317777372353535851937790883648493 (see [RFC9496]).¶
  Identity():
    As defined in [RFC9496].¶
  Generator():
    As defined in [RFC9496].¶
  HashToGroup():
    Use hash_to_ristretto255 [RFC9380] 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 an output length of 64 bytes, 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 Section 4.7 for implementation guidance.¶
  SerializeElement(A):
    Implemented using the `Encode` function from Section 4.3.2 of [RFC9496]; Ne = 32.¶
  DeserializeElement(buf):
    Implemented using the `Decode` function from Section 4.3.1 of [RFC9496]. 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.¶

### 4.2. OPRF(decaf448, SHAKE-256)

This ciphersuite uses decaf448 [RFC9496] for the Group and SHAKE-256 for the hash function. The value of the ciphersuite identifier is "decaf448-SHAKE256".¶

Group:

  decaf448 [RFC9496]¶

  Order():
    Return 2446 - 13818066809895115352007386748515426880336692474882178609894547503885.¶
  Identity():
    As defined in [RFC9496].¶
  Generator():
    As defined in [RFC9496].¶
  RandomScalar():
    Implemented by returning a uniformly random Scalar in the range [0, `G.Order()` - 1]. Refer to Section 4.7 for implementation guidance.¶
  HashToGroup():
    Use hash_to_decaf448 [RFC9380] 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 [RFC9496]; Ne = 56.¶
  DeserializeElement(buf):
    Implemented using the `Decode` function from Section 5.3.1 of [RFC9496]. 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.¶

### 4.3. OPRF(P-256, SHA-256)

This ciphersuite uses P-256 [NISTCurves] for the Group and SHA-256 for the hash function. The value of the ciphersuite identifier is "P256-SHA256".¶

Group:

  P-256 (secp256r1) [NISTCurves]¶

  Order():
    Return 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551.¶
  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 4.7 for implementation guidance.¶
  HashToGroup():
    Use hash_to_curve with suite P256_XMD:SHA-256_SSWU_RO_ [RFC9380] and DST = "HashToGroup-" || contextString.¶
  HashToScalar():
    Use hash_to_field from [RFC9380] using L = 48, `expand_message_xmd` with SHA-256, DST = "HashToScalar-" || contextString, and a 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 = 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 [SEC1] and then performing 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 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 = 32.¶
  DeserializeScalar(buf):
    Implemented by attempting to deserialize a Scalar from a 32-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].¶
Hash:
  SHA-256; Nh = 32.¶

### 4.4. OPRF(P-384, SHA-384)

This ciphersuite uses P-384 [NISTCurves] for the Group and SHA-384 for the hash function. The value of the ciphersuite identifier is "P384-SHA384".¶

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 4.7 for implementation guidance.¶
  HashToGroup():
    Use hash_to_curve with suite P384_XMD:SHA-384_SSWU_RO_ [RFC9380] and DST = "HashToGroup-" || contextString.¶
  HashToScalar():
    Use hash_to_field from [RFC9380] using L = 72, `expand_message_xmd` with SHA-384, DST = "HashToScalar-" || contextString, and a 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 performing 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].¶
Hash:
  SHA-384; Nh = 48.¶

### 4.5. OPRF(P-521, SHA-512)

This ciphersuite uses P-521 [NISTCurves] for the Group and SHA-512 for the hash function. The value of the ciphersuite identifier is "P521-SHA512".¶

Group:

  P-521 (secp521r1) [NISTCurves]¶

  Order():
    Return 0x01fffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffffa51868783bf2f966b7fcc0148f709a5d03bb5c9b8899c47aebb6fb71e91386409.¶
  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 4.7 for implementation guidance.¶
  HashToGroup():
    Use hash_to_curve with suite P521_XMD:SHA-512_SSWU_RO_ [RFC9380] and DST = "HashToGroup-" || contextString.¶
  HashToScalar():
    Use hash_to_field from [RFC9380] using L = 98, `expand_message_xmd` with SHA-512, DST = "HashToScalar-" || contextString, and a 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 = 67.¶
  DeserializeElement(buf):
    Implemented by attempting to deserialize a 67-byte input string to a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to [SEC1] and then performing 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 = 66.¶
  DeserializeScalar(buf):
    Implemented by attempting to deserialize a Scalar from a 66-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].¶
Hash:
  SHA-512; Nh = 64.¶

### 4.6. Future Ciphersuites

A critical requirement of implementing the prime-order group using elliptic curves is a method to instantiate the function `HashToGroup`, which 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 Section 4, we give instantiations of this functionality based on the functions described in [RFC9380]. Consequently, any OPRF implementation must adhere to the implementation and security considerations discussed in [RFC9380] 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 Section 2.1. 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 Section 7.2.3 for additional details.¶

### 4.7. Random Scalar Generation

Two popular algorithms for generating a random integer uniformly distributed in the range [0, G.Order() - 1] are described in the following subsections.¶

#### 4.7.1. 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.¶

#### 4.7.2. 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 [RFC9380], Section 5 for the underlying derivation of `L`.¶

## 5. Application Considerations

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

### 5.1. Input Limits

Application inputs, expressed as `PrivateInput` or `PublicInput` values, MUST be smaller than 216 - 1 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 261 bytes.¶

### 5.2. External Interface Recommendations

In Section 3.3, the interface of the protocol functions allows that some inputs (and outputs) to be group Element and Scalar values. However, implementations can instead operate over Element and Scalar values internally and only expose interfaces that operate with an application-specific format of messages.¶

### 5.3. 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 (Sections 3.3.2 and 3.3.3).¶
`DeserializeError`:
  Group Element or Scalar deserialization failure (Sections 2.1 and 3.3).¶
`InputValidationError`:
  Validation of byte array inputs failed (Section 4).¶

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 (Sections 3.3.1 and 3.3.3).¶
`InverseError`:
  A tweaked private key is invalid, i.e., has no multiplicative inverse (Sections 2.1 and 3.3).¶

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.¶

### 5.4. 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 Section 7.2 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 [RFC9380], Section 10.4 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.¶

## 6. IANA Considerations

This document has no IANA actions.¶

## 7. 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.¶

### 7.1. 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 nonmalleable (i.e., we cannot compute a new evaluation of F from an existing evaluation). A genuinely random function will be nonmalleable with high probability, so a pseudorandom function must be nonmalleable 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 [DGSTV18].¶

Beyond client input secrecy, in the OPRF protocol, the server learns nothing about the output y of the function, nor does the client learn anything about the server's private key k.¶

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 the client's private and public input. Both the client and server know the public input, but similar to the OPRF and VOPRF protocols, the server learns nothing about the client's private input or the output of the function, and the client learns nothing about the server's private key.¶

This property becomes useful when dealing with key management operations, such as the rotation of the 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.¶

Since the POPRF variant has a different syntax than the OPRF and VOPRF variants, i.e., y = F(k, x, info), the pseudorandomness property is generalized:¶

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

### 7.2. Security Assumptions

Below, we discuss the cryptographic security of each protocol variant from Section 3, relative to the necessary cryptographic assumptions that need to be made.¶

#### 7.2.1. OPRF and VOPRF Assumptions

The OPRF and VOPRF protocol variants in this document are based on [JKK14]. In particular, the VOPRF construction is similar to the [JKK14] construction with the following distinguishing properties:¶

1. This document does not use session identifiers to differentiate different instances of the protocol.¶
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 [DGSTV18].¶

The pseudorandomness and input secrecy (and verifiability) of the OPRF (and VOPRF) protocols in [JKK14] are based on the One-More Gap Computational Diffie-Hellman assumption that is computationally difficult to solve in the corresponding prime-order group. In [JKK14], 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.¶

#### 7.2.2. POPRF Assumptions

The POPRF construction in this document is based on the construction known as 3HashSDHI, given by [TCRSTW21]. 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 [DGSTV18].¶

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 [TCRSTW21] is computationally difficult to solve in the corresponding prime-order group. Tyagi et al. [TCRSTW21] 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 [JKK14], which is a predecessor to the POPRF variant in Section 3.3.3.)¶

#### 7.2.3. 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 Diffie-Hellman (DH) samples; see [BG04] and [Cheon06]. 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 decaf448-SHAKE256), P-384 (used by ciphersuite P384-SHA384), or P-521 (used by ciphersuite P521-SHA512).¶

### 7.3. Domain Separation

Applications SHOULD construct input to the protocol to provide domain separation. Any system that has multiple OPRF applications should distinguish client inputs to ensure the OPRF results are separate. Guidance for constructing info can be found in [RFC9380], Section 3.1.¶

### 7.4. 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`.¶

## 8. References

### 8.1. Normative References

[KEYAGREEMENT]
  Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R. Davis, "Recommendation for pair-wise key-establishment schemes using discrete logarithm cryptography", NIST SP 800-56A (Rev. 3), DOI 10.6028/nist.sp.800-56ar3, April 2018, <https://doi.org/10.6028/nist.sp.800-56ar3>.
[RFC2119]
  Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.rfc-editor.org/info/rfc2119>.
[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, November 2016, <https://www.rfc-editor.org/info/rfc8017>.
[RFC8174]
  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, <https://www.rfc-editor.org/info/rfc8174>.
[RFC9380]
  Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380, DOI 10.17487/RFC9380, August 2023, <https://www.rfc-editor.org/info/rfc9380>.
[RFC9496]
  de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I., Tankersley, G., and F. Valsorda, "The ristretto255 and decaf448 Groups", RFC 9496, DOI 10.17487/RFC9496, December 2023, <https://www.rfc-editor.org/info/rfc9496>.

### 8.2. Informative References

[BG04]
  Brown, D. and R. Gallant, "The Static Diffie-Hellman Problem", November 2004, <https://eprint.iacr.org/2004/306>.
[ChaumPedersen]
  Chaum, D. and T. Pedersen, "Wallet Databases with Observers", Advances in Cryptology - CRYPTO' 92, pp. 89-105, DOI 10.1007/3-540-48071-4_7, August 1992, <https://doi.org/10.1007/3-540-48071-4_7>.
[Cheon06]
  Cheon, J., "Security Analysis of the Strong Diffie-Hellman Problem", Advances in Cryptology - EUROCRYPT 2006, pp. 1-11, DOI 10.1007/11761679_1, 2006, <https://doi.org/10.1007/11761679_1>.
[DGSTV18]
  Davidson, A., Goldberg, I., Sullivan, N., Tankersley, G., and F. Valsorda, "Privacy Pass: Bypassing Internet Challenges Anonymously", Proceedings on Privacy Enhancing Technologies, vol. 2018, no. 3, pp. 164-180, DOI 10.1515/popets-2018-0026, April 2018, <https://doi.org/10.1515/popets-2018-0026>.
[FS00]
  Fiat, A. and A. Shamir, "How To Prove Yourself: Practical Solutions to Identification and Signature Problems", Advances in Cryptology - CRYPTO' 86, pp. 186-194, DOI 10.1007/3-540-47721-7_12, 1986, <https://doi.org/10.1007/3-540-47721-7_12>.
[JKK14]
  Jarecki, S., Kiayias, A., and H. Krawczyk, "Round-Optimal Password-Protected Secret Sharing and T-PAKE in the Password-Only Model", Lecture Notes in Computer Science, pp. 233-253, DOI 10.1007/978-3-662-45608-8_13, 2014, <https://doi.org/10.1007/978-3-662-45608-8_13>.
[JKKX16]
  Jarecki, S., Kiayias, A., Krawczyk, H., and J. Xu, "Highly-Efficient and Composable Password-Protected Secret Sharing (Or: How to Protect Your Bitcoin Wallet Online)", 2016 IEEE European Symposium on Security and Privacy (EuroS&P), DOI 10.1109/eurosp.2016.30, March 2016, <https://doi.org/10.1109/eurosp.2016.30>.
[NISTCurves]
  National Institute of Standards and Technology (NIST), "Digital Signature Standard (DSS)", FIPS PUB 186-5, DOI 10.6028/NIST.FIPS.186-5, February 2023, <https://doi.org/10.6028/NIST.FIPS.186-5>.
[OPAQUE]
  Bourdrez, D., Krawczyk, H., Lewi, K., and C. A. Wood, "The OPAQUE Asymmetric PAKE Protocol", Work in Progress, Internet-Draft, draft-irtf-cfrg-opaque-13, 18 December 2023, <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-opaque-13>.
[PRIVACY-PASS]
  Celi, S., Davidson, A., Valdez, S., and C. A. Wood, "Privacy Pass Issuance Protocol", Work in Progress, Internet-Draft, draft-ietf-privacypass-protocol-16, 3 October 2023, <https://datatracker.ietf.org/doc/html/draft-ietf-privacypass-protocol-16>.
[PrivacyPass]
  "Privacy Pass", commit 085380a, March 2018, <https://github.com/privacypass/team>.
[RFC7748]
  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves for Security", RFC 7748, DOI 10.17487/RFC7748, January 2016, <https://www.rfc-editor.org/info/rfc7748>.
[SEC1]
  Standards for Efficient Cryptography Group (SECG), "SEC 1: Elliptic Curve Cryptography", May 2009, <https://www.secg.org/sec1-v2.pdf>.
[SJKS17]
  Shirvanian, M., Jarecki, S., Krawczyk, H., and N. Saxena, "SPHINX: A Password Store that Perfectly Hides Passwords from Itself", 2017 IEEE 37th International Conference on Distributed Computing Systems (ICDCS), DOI 10.1109/ICDCS.2017.64, June 2017, <https://doi.org/10.1109/ICDCS.2017.64>.
[TCRSTW21]
  Tyagi, N., Celi, S., Ristenpart, T., Sullivan, N., Tessaro, S., and C. A. Wood, "A Fast and Simple Partially Oblivious PRF, with Applications", Advances in Cryptology - EUROCRYPT 2022 pp. 674-705, DOI 10.1007/978-3-031-07085-3_23, May 2022, <https://doi.org/10.1007/978-3-031-07085-3_23>.

## Appendix A. Test Vectors

This section includes test vectors for the protocol variants specified in this document. For each ciphersuite specified in Section 4, there is a set of test vectors for the protocol when running 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 `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 `Nh` bytes long.¶

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, denoted `Seed`, and info string, denoted `KeyInfo`, which are listed as well, using the `DeriveKeyPair` function from Section 3.2.¶

### A.1. ristretto255-SHA512

#### A.1.1. OPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 5ebcea5ee37023ccb9fc2d2019f9d7737be85591ae8652ffa9ef0f4d37063
    b0e

¶

##### A.1.1.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = 609a0ae68c15a3cf6903766461307e5c8bb2f95e7e6550e1ffa
    2dc99e412803c
    EvaluationElement = 7ec6578ae5120958eb2db1745758ff379e77cb64fe77b0b2
    d8cc917ea0869c7e
    Output = 527759c3d9366f277d8c6020418d96bb393ba2afb20ff90df23fb770826
    4e2f3ab9135e3bd69955851de4b1f9fe8a0973396719b7912ba9ee8aa7d0b5e24bcf
    6

¶

##### A.1.1.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = da27ef466870f5f15296299850aa088629945a17d1f5b7f5ff0
    43f76b3c06418
    EvaluationElement = b4cbf5a4f1eeda5a63ce7b77c7d23f461db3fcab0dd28e4e
    17cecb5c90d02c25
    Output = f4a74c9c592497375e796aa837e907b1a045d34306a749db9f34221f7e7
    50cb4f2a6413a6bf6fa5e19ba6348eb673934a722a7ede2e7621306d18951e7cf2c7
    3

¶

#### A.1.2. VOPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = e6f73f344b79b379f1a0dd37e07ff62e38d9f71345ce62ae3a9bc60b04ccd
    909
    pkSm = c803e2cc6b05fc15064549b5920659ca4a77b2cca6f04f6b357009335476a
    d4e

¶

##### A.1.2.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = 863f330cc1a1259ed5a5998a23acfd37fb4351a793a5b3c090b
    642ddc439b945
    EvaluationElement = aa8fa048764d5623868679402ff6108d2521884fa138cd7f
    9c7669a9a014267e
    Proof = ddef93772692e535d1a53903db24367355cc2cc78de93b3be5a8ffcc6985
    dd066d4346421d17bf5117a2a1ff0fcb2a759f58a539dfbe857a40bce4cf49ec600d
    ProofRandomScalar = 222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d98
    81aa6f61d645fc0e
    Output = b58cfbe118e0cb94d79b5fd6a6dafb98764dff49c14e1770b566e42402d
    a1a7da4d8527693914139caee5bd03903af43a491351d23b430948dd50cde10d32b3
    c

¶

##### A.1.2.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = cc0b2a350101881d8a4cba4c80241d74fb7dcbfde4a61fde2f9
    1443c2bf9ef0c
    EvaluationElement = 60a59a57208d48aca71e9e850d22674b611f752bed48b36f
    7a91b372bd7ad468
    Proof = 401a0da6264f8cf45bb2f5264bc31e109155600babb3cd4e5af7d181a2c9
    dc0a67154fabf031fd936051dec80b0b6ae29c9503493dde7393b722eafdf5a50b02
    ProofRandomScalar = 222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d98
    81aa6f61d645fc0e
    Output = 8a9a2f3c7f085b65933594309041fc1898d42d0858e59f90814ae90571a
    6df60356f4610bf816f27afdd84f47719e480906d27ecd994985890e5f539e7ea74b
    6

¶

##### A.1.2.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706,222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d9881aa6f61d645fc0
    e
    BlindedElement = 863f330cc1a1259ed5a5998a23acfd37fb4351a793a5b3c090b
    642ddc439b945,90a0145ea9da29254c3a56be4fe185465ebb3bf2a1801f7124bbba
    dac751e654
    EvaluationElement = aa8fa048764d5623868679402ff6108d2521884fa138cd7f
    9c7669a9a014267e,cc5ac221950a49ceaa73c8db41b82c20372a4c8d63e5dded2db
    920b7eee36a2a
    Proof = cc203910175d786927eeb44ea847328047892ddf8590e723c37205cb7460
    0b0a5ab5337c8eb4ceae0494c2cf89529dcf94572ed267473d567aeed6ab873dee08
    ProofRandomScalar = 419c4f4f5052c53c45f3da494d2b67b220d02118e0857cdb
    cf037f9ea84bbe0c
    Output = b58cfbe118e0cb94d79b5fd6a6dafb98764dff49c14e1770b566e42402d
    a1a7da4d8527693914139caee5bd03903af43a491351d23b430948dd50cde10d32b3
    c,8a9a2f3c7f085b65933594309041fc1898d42d0858e59f90814ae90571a6df6035
    6f4610bf816f27afdd84f47719e480906d27ecd994985890e5f539e7ea74b6

¶

#### A.1.3. POPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 145c79c108538421ac164ecbe131942136d5570b16d8bf41a24d4337da981
    e07
    pkSm = c647bef38497bc6ec077c22af65b696efa43bff3b4a1975a3e8e0a1c5a79d
    631

¶

##### A.1.3.1. Test Vector 1, Batch Size 1

    Input = 00
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = c8713aa89241d6989ac142f22dba30596db635c772cbf25021f
    dd8f3d461f715
    EvaluationElement = 1a4b860d808ff19624731e67b5eff20ceb2df3c3c03b906f
    5693e2078450d874
    Proof = 41ad1a291aa02c80b0915fbfbb0c0afa15a57e2970067a602ddb9e8fd6b7
    100de32e1ecff943a36f0b10e3dae6bd266cdeb8adf825d86ef27dbc6c0e30c52206
    ProofRandomScalar = 222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d98
    81aa6f61d645fc0e
    Output = ca688351e88afb1d841fde4401c79efebb2eb75e7998fa9737bd5a82a15
    2406d38bd29f680504e54fd4587eddcf2f37a2617ac2fbd2993f7bdf45442ace7d22
    1

¶

##### A.1.3.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706
    BlindedElement = f0f0b209dd4d5f1844dac679acc7761b91a2e704879656cb7c2
    01e82a99ab07d
    EvaluationElement = 8c3c9d064c334c6991e99f286ea2301d1bde170b54003fb9
    c44c6d7bd6fc1540
    Proof = 4c39992d55ffba38232cdac88fe583af8a85441fefd7d1d4a8d0394cd1de
    77018bf135c174f20281b3341ab1f453fe72b0293a7398703384bed822bfdeec8908
    ProofRandomScalar = 222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d98
    81aa6f61d645fc0e
    Output = 7c6557b276a137922a0bcfc2aa2b35dd78322bd500235eb6d6b6f91bc5b
    56a52de2d65612d503236b321f5d0bebcbc52b64b92e426f29c9b8b69f52de98ae50
    7

¶

##### A.1.3.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec4c1f
    6706,222a5e897cf59db8145db8d16e597e8facb80ae7d4e26d9881aa6f61d645fc0
    e
    BlindedElement = c8713aa89241d6989ac142f22dba30596db635c772cbf25021f
    dd8f3d461f715,423a01c072e06eb1cce96d23acce06e1ea64a609d7ec9e9023f304
    9f2d64e50c
    EvaluationElement = 1a4b860d808ff19624731e67b5eff20ceb2df3c3c03b906f
    5693e2078450d874,aa1f16e903841036e38075da8a46655c94fc92341887eb5819f
    46312adfc0504
    Proof = 43fdb53be399cbd3561186ae480320caa2b9f36cca0e5b160c4a677b8bbf
    4301b28f12c36aa8e11e5a7ef551da0781e863a6dc8c0b2bf5a149c9e00621f02006
    ProofRandomScalar = 419c4f4f5052c53c45f3da494d2b67b220d02118e0857cdb
    cf037f9ea84bbe0c
    Output = ca688351e88afb1d841fde4401c79efebb2eb75e7998fa9737bd5a82a15
    2406d38bd29f680504e54fd4587eddcf2f37a2617ac2fbd2993f7bdf45442ace7d22
    1,7c6557b276a137922a0bcfc2aa2b35dd78322bd500235eb6d6b6f91bc5b56a52de
    2d65612d503236b321f5d0bebcbc52b64b92e426f29c9b8b69f52de98ae507

¶

### A.2. decaf448-SHAKE256

#### A.2.1. OPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = e8b1375371fd11ebeb224f832dcc16d371b4188951c438f751425699ed29e
    cc80c6c13e558ccd67634fd82eac94aa8d1f0d7fee990695d1e

¶

##### A.2.1.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = e0ae01c4095f08e03b19baf47ffdc19cb7d98e583160522a3c7
    d6a0b2111cd93a126a46b7b41b730cd7fc943d4e28e590ed33ae475885f6c
    EvaluationElement = 50ce4e60eed006e22e7027454b5a4b8319eb2bc8ced609eb
    19eb3ad42fb19e06ba12d382cbe7ae342a0cad6ead0ef8f91f00bb7f0cd9c0a2
    Output = 37d3f7922d9388a15b561de5829bbf654c4089ede89c0ce0f3f85bcdba0
    9e382ce0ab3507e021f9e79706a1798ffeac68ebd5cf62e5eb9838c7068351d97ae3
    7

¶

##### A.2.1.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = 86a88dc5c6331ecfcb1d9aacb50a68213803c462e377577cacc
    00af28e15f0ddbc2e3d716f2f39ef95f3ec1314a2c64d940a9f295d8f13bb
    EvaluationElement = 162e9fa6e9d527c3cd734a31bf122a34dbd5bcb7bb23651f
    1768a7a9274cc116c03b58afa6f0dede3994a60066c76370e7328e7062fd5819
    Output = a2a652290055cb0f6f8637a249ee45e32ef4667db0b4c80c0a70d2a6416
    4d01525cfdad5d870a694ec77972b9b6ec5d2596a5223e5336913f945101f0137f55
    e

¶

#### A.2.2. VOPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = e3c01519a076a326a0eb566343e9b21c115fa18e6e85577ddbe890b33104f
    cc2835ddfb14a928dc3f5d79b936e17c76b99e0bf6a1680930e
    pkSm = 945fc518c47695cf65217ace04b86ac5e4cbe26ca649d52854bb16c494ce0
    9069d6add96b20d4b0ae311a87c9a73e3a146b525763ab2f955

¶

##### A.2.2.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = 7261bbc335c664ba788f1b1a1a4cd5190cc30e787ef277665ac
    1d314f8861e3ec11854ce3ddd42035d9e0f5cddde324c332d8c880abc00eb
    EvaluationElement = ca1491a526c28d880806cf0fb0122222392cf495657be6e4
    c9d203bceffa46c86406caf8217859d3fb259077af68e5d41b3699410781f467
    Proof = f84bbeee47aedf43558dae4b95b3853635a9fc1a9ea7eac9b454c64c66c4
    f49cd1c72711c7ac2e06c681e16ea693d5500bbd7b56455df52f69e00b76b4126961
    e1562fdbaaac40b7701065cbeece3febbfe09e00160f81775d36daed99d8a2a10be0
    759e01b7ee81217203416c9db208
    ProofRandomScalar = b1b748135d405ce48c6973401d9455bb8ccd18b01d0295c0
    627f67661200dbf9569f73fbb3925daa043a070e5f953d80bb464ea369e5522b
    Output = e2ac40b634f36cccd8262b285adff7c9dcc19cd308564a5f4e581d1a853
    5773b86fa4fc9f2203c370763695c5093aea4a7aedec4488b1340ba3bf663a23098c
    1

¶

##### A.2.2.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = 88287e553939090b888ddc15913e1807dc4757215555e1c3a79
    488ef311594729c7fa74c772a732b78440b7d66d0aa35f3bb316f1d93e1b2
    EvaluationElement = c00978c73e8e4ee1d447ab0d3ad1754055e72cc85c08e3a0
    db170909a9c61cbff1f1e7015f289e3038b0f341faea5d7780c130106065c231
    Proof = 7a2831a6b237e11ac1657d440df93bc5ce00f552e6020a99d5c956ffc4d0
    7b5ade3e82ecdc257fd53d76239e733e0a1313e84ce16cc0d82734806092a693d7e8
    d3c420c2cb6ccd5d0ca32514fb78e9ad0973ebdcb52eba438fc73948d76339ee7101
    21d83e2fe6f001cfdf551aff9f36
    ProofRandomScalar = b1b748135d405ce48c6973401d9455bb8ccd18b01d0295c0
    627f67661200dbf9569f73fbb3925daa043a070e5f953d80bb464ea369e5522b
    Output = 862952380e07ec840d9f6e6f909c5a25d16c3dacb586d89a181b4aa7380
    c959baa8c480fe8e6c64e089d68ea7aeeb5817bd524d7577905b5bab487690048c94
    1

¶

##### A.2.2.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112,b1b748135d405ce
    48c6973401d9455bb8ccd18b01d0295c0627f67661200dbf9569f73fbb3925daa043
    a070e5f953d80bb464ea369e5522b
    BlindedElement = 7261bbc335c664ba788f1b1a1a4cd5190cc30e787ef277665ac
    1d314f8861e3ec11854ce3ddd42035d9e0f5cddde324c332d8c880abc00eb,2e15f3
    93c035492a1573627a3606e528c6294c767c8d43b8c691ef70a52cc7dc7d1b53fe45
    8350a270abb7c231b87ba58266f89164f714d9
    EvaluationElement = ca1491a526c28d880806cf0fb0122222392cf495657be6e4
    c9d203bceffa46c86406caf8217859d3fb259077af68e5d41b3699410781f467,8ec
    68e9871b296e81c55647ce64a04fe75d19932f1400544cd601468c60f998408bbb54
    6601d4a636e8be279e558d70b95c8d4a4f61892be
    Proof = 167d922f0a6ffa845eed07f8aa97b6ac746d902ecbeb18f49c009adc0521
    eab1e4d275b74a2dc266b7a194c854e85e7eb54a9a36376dfc04ec7f3bd55fc9618c
    3970cb548e064f8a2f06183a5702933dbc3e4c25a73438f2108ee1981c306181003c
    7ea92fce963ec7b4ba4f270e6d38
    ProofRandomScalar = 63798726803c9451ba405f00ef3acb633ddf0c420574a2ec
    6cbf28f840800e355c9fbaac10699686de2724ed22e797a00f3bd93d105a7f23
    Output = e2ac40b634f36cccd8262b285adff7c9dcc19cd308564a5f4e581d1a853
    5773b86fa4fc9f2203c370763695c5093aea4a7aedec4488b1340ba3bf663a23098c
    1,862952380e07ec840d9f6e6f909c5a25d16c3dacb586d89a181b4aa7380c959baa
    8c480fe8e6c64e089d68ea7aeeb5817bd524d7577905b5bab487690048c941

¶

#### A.2.3. POPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 792a10dcbd3ba4a52a054f6f39186623208695301e7adb9634b74709ab22d
    e402990eb143fd7c67ac66be75e0609705ecea800992aac8e19
    pkSm = 6c9d12723a5bbcf305522cc04b4a34d9ced2e12831826018ea7b5dcf54526
    47ad262113059bf0f6e4354319951b9d513c74f29cb0eec38c1

¶

##### A.2.3.1. Test Vector 1, Batch Size 1

    Input = 00
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = 161183c13c6cb33b0e4f9b7365f8c5c12d13c72f8b62d276ca0
    9368d093dce9b42198276b9e9d870ac392dda53efd28d1b7e6e8c060cdc42
    EvaluationElement = 06ec89dfde25bb2a6f0145ac84b91ac277b35de39ad1d6f4
    02a8e46414952ce0d9ea1311a4ece283e2b01558c7078b040cfaa40dd63b3e6c
    Proof = 66caee75bf2460429f620f6ad3e811d524cb8ddd848a435fc5d89af48877
    abf6506ee341a0b6f67c2d76cd021e5f3d1c9abe5aa9f0dce016da746135fedba2af
    41ed1d01659bfd6180d96bc1b7f320c0cb6926011ce392ecca748662564892bae665
    16acaac6ca39aadf6fcca95af406
    ProofRandomScalar = b1b748135d405ce48c6973401d9455bb8ccd18b01d0295c0
    627f67661200dbf9569f73fbb3925daa043a070e5f953d80bb464ea369e5522b
    Output = 4423f6dcc1740688ea201de57d76824d59cd6b859e1f9884b7eebc49b0b
    971358cf9cb075df1536a8ea31bcf55c3e31c2ba9cfa8efe54448d17091daeb9924e
    d

¶

##### A.2.3.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112
    BlindedElement = 12082b6a381c6c51e85d00f2a3d828cdeab3f5cb19a10b9c014
    c33826764ab7e7cfb8b4ff6f411bddb2d64e62a472af1cd816e5b712790c6
    EvaluationElement = f2919b7eedc05ab807c221fce2b12c4ae9e19e6909c47845
    64b690d1972d2994ca623f273afc67444d84ea40cbc58fcdab7945f321a52848
    Proof = a295677c54d1bc4286330907fc2490a7de163da26f9ce03a462a452fea42
    2b19ade296ba031359b3b6841e48455d20519ad01b4ac4f0b92e76d3cf16fbef0a3f
    72791a8401ef2d7081d361e502e96b2c60608b9fa566f43d4611c2f161d83aabef7f
    8017332b26ed1daaf80440772022
    ProofRandomScalar = b1b748135d405ce48c6973401d9455bb8ccd18b01d0295c0
    627f67661200dbf9569f73fbb3925daa043a070e5f953d80bb464ea369e5522b
    Output = 8691905500510843902c44bdd9730ab9dc3925aa58ff9dd42765a2baf63
    3126de0c3adb93bef5652f38e5827b6396e87643960163a560fc4ac9738c8de4e4a8
    d

¶

##### A.2.3.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 64d37aed22a27f5191de1c1d69fadb899d8862b58eb4220029e036ec65fa
    3833a26e9388336361686ff1f83df55046504dfecad8549ba112,b1b748135d405ce
    48c6973401d9455bb8ccd18b01d0295c0627f67661200dbf9569f73fbb3925daa043
    a070e5f953d80bb464ea369e5522b
    BlindedElement = 161183c13c6cb33b0e4f9b7365f8c5c12d13c72f8b62d276ca0
    9368d093dce9b42198276b9e9d870ac392dda53efd28d1b7e6e8c060cdc42,fc8847
    d43fb4cea4e408f585661a8f2867533fa91d22155d3127a22f18d3b007add480f7d3
    00bca93fa47fe87ae06a57b7d0f0d4c30b12f0
    EvaluationElement = 06ec89dfde25bb2a6f0145ac84b91ac277b35de39ad1d6f4
    02a8e46414952ce0d9ea1311a4ece283e2b01558c7078b040cfaa40dd63b3e6c,2e7
    4c626d07de49b1c8c21d87120fd78105f485e36816af9bde3e3efbeef76815326062
    fd333925b66c5ce5a20f100bf01770c16609f990a
    Proof = fd94db736f97ea4efe9d0d4ad2933072697a6bbeb32834057b23edf7c700
    9f011dfa72157f05d2a507c2bbf0b54cad99ab99de05921c021fda7d70e65bcecdb0
    5f9a30154127ace983c74d10fd910b554c5e95f6bd1565fd1f3dbbe3c523ece5c72d
    57a559b7be1368c4786db4a3c910
    ProofRandomScalar = 63798726803c9451ba405f00ef3acb633ddf0c420574a2ec
    6cbf28f840800e355c9fbaac10699686de2724ed22e797a00f3bd93d105a7f23
    Output = 4423f6dcc1740688ea201de57d76824d59cd6b859e1f9884b7eebc49b0b
    971358cf9cb075df1536a8ea31bcf55c3e31c2ba9cfa8efe54448d17091daeb9924e
    d,8691905500510843902c44bdd9730ab9dc3925aa58ff9dd42765a2baf633126de0
    c3adb93bef5652f38e5827b6396e87643960163a560fc4ac9738c8de4e4a8d

¶

### A.3. P256-SHA256

#### A.3.1. OPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 159749d750713afe245d2d39ccfaae8381c53ce92d098a9375ee70739c7ac
    0bf

¶

##### A.3.1.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 03723a1e5c09b8b9c18d1dcbca29e8007e95f14f4732d9346d4
    90ffc195110368d
    EvaluationElement = 030de02ffec47a1fd53efcdd1c6faf5bdc270912b8749e78
    3c7ca75bb412958832
    Output = a0b34de5fa4c5b6da07e72af73cc507cceeb48981b97b7285fc375345fe
    495dd

¶

##### A.3.1.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 03cc1df781f1c2240a64d1c297b3f3d16262ef5d4cf10273488
    2675c26231b0838
    EvaluationElement = 03a0395fe3828f2476ffcd1f4fe540e5a8489322d398be3c
    4e5a869db7fcb7c52c
    Output = c748ca6dd327f0ce85f4ae3a8cd6d4d5390bbb804c9e12dcf94f853fece
    3dcce

¶

#### A.3.2. VOPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = ca5d94c8807817669a51b196c34c1b7f8442fde4334a7121ae4736364312f
    ca6
    pkSm = 03e17e70604bcabe198882c0a1f27a92441e774224ed9c702e51dd17038b1
    02462

¶

##### A.3.2.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 02dd05901038bb31a6fae01828fd8d0e49e35a486b5c5d4b499
    4013648c01277da
    EvaluationElement = 0209f33cab60cf8fe69239b0afbcfcd261af4c1c5632624f
    2e9ba29b90ae83e4a2
    Proof = e7c2b3c5c954c035949f1f74e6bce2ed539a3be267d1481e9ddb178533df
    4c2664f69d065c604a4fd953e100b856ad83804eb3845189babfa5a702090d6fc5fa
    ProofRandomScalar = f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 0412e8f78b02c415ab3a288e228978376f99927767ff37c5718d420010a
    645a1

¶

##### A.3.2.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 03cd0f033e791c4d79dfa9c6ed750f2ac009ec46cd4195ca6fd
    3800d1e9b887dbd
    EvaluationElement = 030d2985865c693bf7af47ba4d3a3813176576383d19aff0
    03ef7b0784a0d83cf1
    Proof = 2787d729c57e3d9512d3aa9e8708ad226bc48e0f1750b0767aaff73482c4
    4b8d2873d74ec88aebd3504961acea16790a05c542d9fbff4fe269a77510db00abab
    ProofRandomScalar = f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 771e10dcd6bcd3664e23b8f2a710cfaaa8357747c4a8cbba03133967b5c
    24f18

¶

##### A.3.2.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364,f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b
    1
    BlindedElement = 02dd05901038bb31a6fae01828fd8d0e49e35a486b5c5d4b499
    4013648c01277da,03462e9ae64cae5b83ba98a6b360d942266389ac369b923eb3d5
    57213b1922f8ab
    EvaluationElement = 0209f33cab60cf8fe69239b0afbcfcd261af4c1c5632624f
    2e9ba29b90ae83e4a2,02bb24f4d838414aef052a8f044a6771230ca69c0a5677540
    fff738dd31bb69771
    Proof = bdcc351707d02a72ce49511c7db990566d29d6153ad6f8982fad2b435d6c
    e4d60da1e6b3fa740811bde34dd4fe0aa1b5fe6600d0440c9ddee95ea7fad7a60cf2
    ProofRandomScalar = 350e8040f828bf6ceca27405420cdf3d63cb3aef005f40ba
    51943c8026877963
    Output = 0412e8f78b02c415ab3a288e228978376f99927767ff37c5718d420010a
    645a1,771e10dcd6bcd3664e23b8f2a710cfaaa8357747c4a8cbba03133967b5c24f
    18

¶

#### A.3.3. POPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 6ad2173efa689ef2c27772566ad7ff6e2d59b3b196f00219451fb2c89ee4d
    ae2
    pkSm = 030d7ff077fddeec965db14b794f0cc1ba9019b04a2f4fcc1fa525dedf72e
    2a3e3

¶

##### A.3.3.1. Test Vector 1, Batch Size 1

    Input = 00
    Info = 7465737420696e666f
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 031563e127099a8f61ed51eeede05d747a8da2be329b40ba1f0
    db0b2bd9dd4e2c0
    EvaluationElement = 02c5e5300c2d9e6ba7f3f4ad60500ad93a0157e6288eb04b
    67e125db024a2c74d2
    Proof = f8a33690b87736c854eadfcaab58a59b8d9c03b569110b6f31f8bf7577f3
    fbb85a8a0c38468ccde1ba942be501654adb106167c8eb178703ccb42bccffb9231a
    ProofRandomScalar = f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 193a92520bd8fd1f37accb918040a57108daa110dc4f659abe212636d24
    5c592

¶

##### A.3.3.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 021a440ace8ca667f261c10ac7686adc66a12be31e3520fca31
    7643a1eee9dcd4d
    EvaluationElement = 0208ca109cbae44f4774fc0bdd2783efdcb868cb4523d521
    96f700210e777c5de3
    Proof = 043a8fb7fc7fd31e35770cabda4753c5bf0ecc1e88c68d7d35a62bf2631e
    875af4613641be2d1875c31d1319d191c4bbc0d04875f4fd03c31d3d17dd8e069b69
    ProofRandomScalar = f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 1e6d164cfd835d88a31401623549bf6b9b306628ef03a7962921d62bc5f
    fce8c

¶

##### A.3.3.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 3338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364,f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b
    1
    BlindedElement = 031563e127099a8f61ed51eeede05d747a8da2be329b40ba1f0
    db0b2bd9dd4e2c0,03ca4ff41c12fadd7a0bc92cf856732b21df652e01a3abdf0fa8
    847da053db213c
    EvaluationElement = 02c5e5300c2d9e6ba7f3f4ad60500ad93a0157e6288eb04b
    67e125db024a2c74d2,02f0b6bcd467343a8d8555a99dc2eed0215c71898c5edb77a
    3d97ddd0dbad478e8
    Proof = 8fbd85a32c13aba79db4b42e762c00687d6dbf9c8cb97b2a225645ccb00d
    9d7580b383c885cdfd07df448d55e06f50f6173405eee5506c0ed0851ff718d13e68
    ProofRandomScalar = 350e8040f828bf6ceca27405420cdf3d63cb3aef005f40ba
    51943c8026877963
    Output = 193a92520bd8fd1f37accb918040a57108daa110dc4f659abe212636d24
    5c592,1e6d164cfd835d88a31401623549bf6b9b306628ef03a7962921d62bc5ffce
    8c

¶

### A.4. P384-SHA384

#### A.4.1. OPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = dfe7ddc41a4646901184f2b432616c8ba6d452f9bcd0c4f75a5150ef2b2ed
    02ef40b8b92f60ae591bcabd72a6518f188

¶

##### A.4.1.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 02a36bc90e6db34096346eaf8b7bc40ee1113582155ad379700
    3ce614c835a874343701d3f2debbd80d97cbe45de6e5f1f
    EvaluationElement = 03af2a4fc94770d7a7bf3187ca9cc4faf3732049eded2442
    ee50fbddda58b70ae2999366f72498cdbc43e6f2fc184afe30
    Output = ed84ad3f31a552f0456e58935fcc0a3039db42e7f356dcb32aa6d487b6b
    815a07d5813641fb1398c03ddab5763874357

¶

##### A.4.1.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 02def6f418e3484f67a124a2ce1bfb19de7a4af568ede6a1ebb
    2733882510ddd43d05f2b1ab5187936a55e50a847a8b900
    EvaluationElement = 034e9b9a2960b536f2ef47d8608b21597ba400d5abfa1825
    fd21c36b75f927f396bf3716c96129d1fa4a77fa1d479c8d7b
    Output = dd4f29da869ab9355d60617b60da0991e22aaab243a3460601e48b07585
    9d1c526d36597326f1b985778f781a1682e75

¶

#### A.4.2. VOPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 051646b9e6e7a71ae27c1e1d0b87b4381db6d3595eeeb1adb41579adbf992
    f4278f9016eafc944edaa2b43183581779d
    pkSm = 031d689686c611991b55f1a1d8f4305ccd6cb719446f660a30db61b7aa87b
    46acf59b7c0d4a9077b3da21c25dd482229a0

¶

##### A.4.2.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 02d338c05cbecb82de13d6700f09cb61190543a7b7e2c6cd4fc
    a56887e564ea82653b27fdad383995ea6d02cf26d0e24d9
    EvaluationElement = 02a7bba589b3e8672aa19e8fd258de2e6aae20101c8d7612
    46de97a6b5ee9cf105febce4327a326255a3c604f63f600ef6
    Proof = bfc6cf3859127f5fe25548859856d6b7fa1c7459f0ba5712a806fc091a30
    00c42d8ba34ff45f32a52e40533efd2a03bc87f3bf4f9f58028297ccb9ccb18ae718
    2bcd1ef239df77e3be65ef147f3acf8bc9cbfc5524b702263414f043e3b7ca2e
    ProofRandomScalar = 803d955f0e073a04aa5d92b3fb739f56f9db001266677f62
    c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    Output = 3333230886b562ffb8329a8be08fea8025755372817ec969d114d1203d0
    26b4a622beab60220bf19078bca35a529b35c

¶

##### A.4.2.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 02f27469e059886f221be5f2cca03d2bdc61e55221721c3b3e5
    6fc012e36d31ae5f8dc058109591556a6dbd3a8c69c433b
    EvaluationElement = 03f16f903947035400e96b7f531a38d4a07ac89a80f89d86
    a1bf089c525a92c7f4733729ca30c56ce78b1ab4f7d92db8b4
    Proof = d005d6daaad7571414c1e0c75f7e57f2113ca9f4604e84bc90f9be52da89
    6fff3bee496dcde2a578ae9df315032585f801fb21c6080ac05672b291e575a40295
    b306d967717b28e08fcc8ad1cab47845d16af73b3e643ddcc191208e71c64630
    ProofRandomScalar = 803d955f0e073a04aa5d92b3fb739f56f9db001266677f62
    c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    Output = b91c70ea3d4d62ba922eb8a7d03809a441e1c3c7af915cbc2226f485213
    e895942cd0f8580e6d99f82221e66c40d274f

¶

##### A.4.2.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364,803d955f0e073a04aa5d92b3fb739f5
    6f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    BlindedElement = 02d338c05cbecb82de13d6700f09cb61190543a7b7e2c6cd4fc
    a56887e564ea82653b27fdad383995ea6d02cf26d0e24d9,02fa02470d7f151018b4
    1e82223c32fad824de6ad4b5ce9f8e9f98083c9a726de9a1fc39d7a0cb6f4f188dd9
    cea01474cd
    EvaluationElement = 02a7bba589b3e8672aa19e8fd258de2e6aae20101c8d7612
    46de97a6b5ee9cf105febce4327a326255a3c604f63f600ef6,028e9e115625ff4c2
    f07bf87ce3fd73fc77994a7a0c1df03d2a630a3d845930e2e63a165b114d98fe34e6
    1b68d23c0b50a
    Proof = 6d8dcbd2fc95550a02211fb78afd013933f307d21e7d855b0b1ed0af7807
    6d8137ad8b0a1bfa05676d325249c1dbb9a52bd81b1c2b7b0efc77cf7b278e1c947f
    6283f1d4c513053fc0ad19e026fb0c30654b53d9cea4b87b037271b5d2e2d0ea
    ProofRandomScalar = a097e722ed2427de86966910acba9f5c350e8040f828bf6c
    eca27405420cdf3d63cb3aef005f40ba51943c8026877963
    Output = 3333230886b562ffb8329a8be08fea8025755372817ec969d114d1203d0
    26b4a622beab60220bf19078bca35a529b35c,b91c70ea3d4d62ba922eb8a7d03809
    a441e1c3c7af915cbc2226f485213e895942cd0f8580e6d99f82221e66c40d274f

¶

#### A.4.3. POPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 5b2690d6954b8fbb159f19935d64133f12770c00b68422559c65431942d72
    1ff79d47d7a75906c30b7818ec0f38b7fb2
    pkSm = 02f00f0f1de81e5d6cf18140d4926ffdc9b1898c48dc49657ae36eb1e45de
    b8b951aaf1f10c82d2eaa6d02aafa3f10d2b6

¶

##### A.4.3.1. Test Vector 1, Batch Size 1

    Input = 00
    Info = 7465737420696e666f
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 03859b36b95e6564faa85cd3801175eda2949707f6aa0640ad0
    93cbf8ad2f58e762f08b56b2a1b42a64953aaf49cbf1ae3
    EvaluationElement = 0220710e2e00306453f5b4f574cb6a512453f35c45080d09
    373e190c19ce5b185914fbf36582d7e0754bb7c8b683205b91
    Proof = 82a17ef41c8b57f1e3122311b4d5cd39a63df0f67443ef18d961f9b659c1
    601ced8d3c64b294f604319ca80230380d437a49c7af0d620e22116669c008ebb767
    d90283d573b49cdb49e3725889620924c2c4b047a2a6225a3ba27e640ebddd33
    ProofRandomScalar = 803d955f0e073a04aa5d92b3fb739f56f9db001266677f62
    c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    Output = 0188653cfec38119a6c7dd7948b0f0720460b4310e40824e048bf82a165
    27303ed449a08caf84272c3bbc972ede797df

¶

##### A.4.3.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364
    BlindedElement = 03f7efcb4aaf000263369d8a0621cb96b81b3206e99876de2a0
    0699ed4c45acf3969cd6e2319215395955d3f8d8cc1c712
    EvaluationElement = 034993c818369927e74b77c400376fd1ae29b6ac6c6ddb77
    6cf10e4fbc487826531b3cf0b7c8ca4d92c7af90c9def85ce6
    Proof = 693471b5dff0cd6a5c00ea34d7bf127b2795164e3bdb5f39a1e5edfbd13e
    443bc516061cd5b8449a473c2ceeccada9f3e5b57302e3d7bc5e28d38d6e3a3056e1
    e73b6cc030f5180f8a1ffa45aa923ee66d2ad0a07b500f2acc7fb99b5506465c
    ProofRandomScalar = 803d955f0e073a04aa5d92b3fb739f56f9db001266677f62
    c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    Output = ff2a527a21cc43b251a567382677f078c6e356336aec069dea8ba369953
    43ca3b33bb5d6cf15be4d31a7e6d75b30d3f5

¶

##### A.4.3.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 504650f53df8f16f6861633388936ea23338fa65ec36e0290022b48eb562
    889d89dbfa691d1cde91517fa222ed7ad364,803d955f0e073a04aa5d92b3fb739f5
    6f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b1
    BlindedElement = 03859b36b95e6564faa85cd3801175eda2949707f6aa0640ad0
    93cbf8ad2f58e762f08b56b2a1b42a64953aaf49cbf1ae3,021a65d618d645f1a20b
    c33b06deaa7e73d6d634c8a56a3d02b53a732b69a5c53c5a207ea33d5afdcde9a22d
    59726bce51
    EvaluationElement = 0220710e2e00306453f5b4f574cb6a512453f35c45080d09
    373e190c19ce5b185914fbf36582d7e0754bb7c8b683205b91,02017657b315ec65e
    f861505e596c8645d94685dd7602cdd092a8f1c1c0194a5d0485fe47d071d972ab51
    4370174cc23f5
    Proof = 4a0b2fe96d5b2a046a0447fe079b77859ef11a39a3520d6ff7c626aad9b4
    73b724fb0cf188974ec961710a62162a83e97e0baa9eeada73397032d928b3e97b1e
    a92ad9458208302be3681b8ba78bcc17745bac00f84e0fdc98a6a8cba009c080
    ProofRandomScalar = a097e722ed2427de86966910acba9f5c350e8040f828bf6c
    eca27405420cdf3d63cb3aef005f40ba51943c8026877963
    Output = 0188653cfec38119a6c7dd7948b0f0720460b4310e40824e048bf82a165
    27303ed449a08caf84272c3bbc972ede797df,ff2a527a21cc43b251a567382677f0
    78c6e356336aec069dea8ba36995343ca3b33bb5d6cf15be4d31a7e6d75b30d3f5

¶

### A.5. P521-SHA512

#### A.5.1. OPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 0153441b8faedb0340439036d6aed06d1217b34c42f17f8db4c5cc610a4a9
    55d698a688831b16d0dc7713a1aa3611ec60703bffc7dc9c84e3ed673b3dbe1d5fcc
    ea6

¶

##### A.5.1.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 0300e78bf846b0e1e1a3c320e353d758583cd876df56100a3a1
    e62bacba470fa6e0991be1be80b721c50c5fd0c672ba764457acc18c6200704e9294
    fbf28859d916351
    EvaluationElement = 030166371cf827cb2fb9b581f97907121a16e2dc5d8b10ce
    9f0ede7f7d76a0d047657735e8ad07bcda824907b3e5479bd72cdef6b839b967ba5c
    58b118b84d26f2ba07
    Output = 26232de6fff83f812adadadb6cc05d7bbeee5dca043dbb16b03488abb99
    81d0a1ef4351fad52dbd7e759649af393348f7b9717566c19a6b8856284d69375c80
    9

¶

##### A.5.1.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 0300c28e57e74361d87e0c1874e5f7cc1cc796d61f9cad50427
    cf54655cdb455613368d42b27f94bf66f59f53c816db3e95e68e1b113443d66a99b3
    693bab88afb556b
    EvaluationElement = 0301ad453607e12d0cc11a3359332a40c3a254eaa1afc642
    96528d55bed07ba322e72e22cf3bcb50570fd913cb54f7f09c17aff8787af75f6a7f
    af5640cbb2d9620a6e
    Output = ad1f76ef939042175e007738906ac0336bbd1d51e287ebaa66901abdd32
    4ea3ffa40bfc5a68e7939c2845e0fd37a5a6e76dadb9907c6cc8579629757fd4d04b
    a

¶

#### A.5.2. VOPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 015c7fc1b4a0b1390925bae915bd9f3d72009d44d9241b962428aad5d13f2
    2803311e7102632a39addc61ea440810222715c9d2f61f03ea424ec9ab1fe5e31cf9
    238
    pkSm = 0301505d646f6e4c9102451eb39730c4ba1c4087618641edbdba4a60896b0
    7fd0c9414ce553cbf25b81dfcca50a8f6724ab7a2bc4d0cf736967a287bb6084cc06
    78ac0

¶

##### A.5.2.1. Test Vector 1, Batch Size 1

    Input = 00
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 0301d6e4fb545e043ddb6aee5d5ceeee1b44102615ab04430c2
    7dd0f56988dedcb1df32ef384f160e0e76e718605f14f3f582f9357553d153b99679
    5b4b3628a4f6380
    EvaluationElement = 03013fdeaf887f3d3d283a79e696a54b66ff0edcb559265e
    204a958acf840e0930cc147e2a6835148d8199eebc26c03e9394c9762a1c991dde40
    bca0f8ca003eefb045
    Proof = 0077fcc8ec6d059d7759b0a61f871e7c1dadc65333502e09a51994328f79
    e5bda3357b9a4f410a1760a3612c2f8f27cb7cb032951c047cc66da60da583df7b24
    7edd0188e5eb99c71799af1d80d643af16ffa1545acd9e9233fbb370455b10eb257e
    a12a1667c1b4ee5b0ab7c93d50ae89602006960f083ca9adc4f6276c0ad60440393c
    ProofRandomScalar = 015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e07
    3a04aa5d92b3fb739f56f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 5e003d9b2fb540b3d4bab5fedd154912246da1ee5e557afd8f56415faa1
    a0fadff6517da802ee254437e4f60907b4cda146e7ba19e249eef7be405549f62954
    b

¶

##### A.5.2.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 03005b05e656cb609ce5ff5faf063bb746d662d67bbd07c0626
    38396f52f0392180cf2365cabb0ece8e19048961d35eeae5d5fa872328dce98df076
    ee154dd191c615e
    EvaluationElement = 0301b19fcf482b1fff04754e282292ed736c5f0aa080d4f4
    2663cd3a416c6596f03129e8e096d8671fe5b0d19838312c511d2ce08d431e43e3ef
    06199d8cab7426238d
    Proof = 01ec9fece444caa6a57032e8963df0e945286f88fbdf233fb5101f0924f7
    ea89c47023f5f72f240e61991fd33a299b5b38c45a5e2dd1a67b072e59dfe86708a3
    59c701e38d383c60cf6969463bcf13251bedad47b7941f52e409a3591398e2792441
    0b18a301c0e19f527cad504fa08388050ac634e1b05c5216d337742f2754e1fc502f
    ProofRandomScalar = 015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e07
    3a04aa5d92b3fb739f56f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = fa15eebba81ecf40954f7135cb76f69ef22c6bae394d1a4362f9b03066b
    54b6604d39f2e53369ca6762a3d9787e230e832aa85955af40ecb8deebb009a8cf47
    4

¶

##### A.5.2.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364,015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e073a04aa5d92b3fb7
    39f56f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b
    1
    BlindedElement = 0301d6e4fb545e043ddb6aee5d5ceeee1b44102615ab04430c2
    7dd0f56988dedcb1df32ef384f160e0e76e718605f14f3f582f9357553d153b99679
    5b4b3628a4f6380,0301403b597538b939b450c93586ba275f9711ba07e42364bac1
    d5769c6824a8b55be6f9a536df46d952b11ab2188363b3d6737635d9543d4dba14a6
    e19421b9245bf5
    EvaluationElement = 03013fdeaf887f3d3d283a79e696a54b66ff0edcb559265e
    204a958acf840e0930cc147e2a6835148d8199eebc26c03e9394c9762a1c991dde40
    bca0f8ca003eefb045,03001f96424497e38c46c904978c2fa1636c5c3dd2e634a85
    d8a7265977c5dce1f02c7e6c118479f0751767b91a39cce6561998258591b5d7c1bb
    02445a9e08e4f3e8d
    Proof = 00b4d215c8405e57c7a4b53398caf55f1f1623aaeb22408ddb9ea2913090
    9b3f95dbb1ff366e81e86e918f9f2fd8b80dbb344cd498c9499d112905e585417e00
    68c600fe5dea18b389ef6c4cc062935607b8ccbbb9a84fba3143868a3e8a58efa0bf
    6ca642804d09dc06e980f64837811227c4267b217f1099a4e28b0854f4e5ee659796
    ProofRandomScalar = 01ec21c7bb69b0734cb48dfd68433dd93b0fa097e722ed24
    27de86966910acba9f5c350e8040f828bf6ceca27405420cdf3d63cb3aef005f40ba
    51943c8026877963
    Output = 5e003d9b2fb540b3d4bab5fedd154912246da1ee5e557afd8f56415faa1
    a0fadff6517da802ee254437e4f60907b4cda146e7ba19e249eef7be405549f62954
    b,fa15eebba81ecf40954f7135cb76f69ef22c6bae394d1a4362f9b03066b54b6604
    d39f2e53369ca6762a3d9787e230e832aa85955af40ecb8deebb009a8cf474

¶

#### A.5.3. POPRF Mode

    Seed = a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a3a
    3a3
    KeyInfo = 74657374206b6579
    skSm = 014893130030ce69cf714f536498a02ff6b396888f9bb507985c32928c442
    7d6d39de10ef509aca4240e8569e3a88debc0d392e3361bcd934cb9bdd59e339dff7
    b27
    pkSm = 0301de8ceb9ffe9237b1bba87c320ea0bebcfc3447fe6f278065c6c69886d
    692d1126b79b6844f829940ace9b52a5e26882cf7cbc9e57503d4cca3cd834584729
    f812a

¶

##### A.5.3.1. Test Vector 1, Batch Size 1

    Input = 00
    Info = 7465737420696e666f
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 020095cff9d7ecf65bdfee4ea92d6e748d60b02de34ad98094f
    82e25d33a8bf50138ccc2cc633556f1a97d7ea9438cbb394df612f041c485a515849
    d5ebb2238f2f0e2
    EvaluationElement = 0301408e9c5be3ffcc1c16e5ae8f8aa68446223b0804b119
    62e856af5a6d1c65ebbb5db7278c21db4e8cc06d89a35b6804fb1738a295b691638a
    f77aa1327253f26d01
    Proof = 0106a89a61eee9dd2417d2849a8e2167bc5f56e3aed5a3ff23e22511fa1b
    37a29ed44d1bbfd6907d99cfbc558a56aec709282415a864a281e49dc53792a4a638
    a0660034306d64be12a94dcea5a6d664cf76681911c8b9a84d49bf12d4893307ec14
    436bd05f791f82446c0de4be6c582d373627b51886f76c4788256e3da7ec8fa18a86
    ProofRandomScalar = 015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e07
    3a04aa5d92b3fb739f56f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 808ae5b87662eaaf0b39151dd85991b94c96ef214cb14a68bf5c1439548
    82d330da8953a80eea20788e552bc8bbbfff3100e89f9d6e341197b122c46a208733
    b

¶

##### A.5.3.2. Test Vector 2, Batch Size 1

    Input = 5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364
    BlindedElement = 030112ea89cf9cf589496189eafc5f9eb13c9f9e170d6ecde7c
    5b940541cb1a9c5cfeec908b67efe16b81ca00d0ce216e34b3d5f46a658d3fd8573d
    671bdb6515ed508
    EvaluationElement = 0200ebc49df1e6fa61f412e6c391e6f074400ecdd2f56c4a
    8c03fe0f91d9b551f40d4b5258fd891952e8c9b28003bcfa365122e54a5714c8949d
    5d202767b31b4bf1f6
    Proof = 0082162c71a7765005cae202d4bd14b84dae63c29067e886b82506992bd9
    94a1c3aac0c1c5309222fe1af8287b6443ed6df5c2e0b0991faddd3564c73c7597ae
    cd9a003b1f1e3c65f28e58ab4e767cfb4adbcaf512441645f4c2aed8bf67d132d966
    006d35fa71a34145414bf3572c1de1a46c266a344dd9e22e7fb1e90ffba1caf556d9
    ProofRandomScalar = 015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e07
    3a04aa5d92b3fb739f56f9db001266677f62c095021db018cd8cbb55941d4073698c
    e45c405d1348b7b1
    Output = 27032e24b1a52a82ab7f4646f3c5df0f070f499db98b9c5df33972bd5af
    5762c3638afae7912a6c1acdb1ae2ab2fa670bd5486c645a0e55412e08d33a4a0d6e
    3

¶

##### A.5.3.3. Test Vector 3, Batch Size 2

    Input = 00,5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a5a
    Info = 7465737420696e666f
    Blind = 00d1dccf7a51bafaf75d4a866d53d8cafe4d504650f53df8f16f68616333
    88936ea23338fa65ec36e0290022b48eb562889d89dbfa691d1cde91517fa222ed7a
    d364,015e80ae32363b32cb76ad4b95a5a34e46bb803d955f0e073a04aa5d92b3fb7
    39f56f9db001266677f62c095021db018cd8cbb55941d4073698ce45c405d1348b7b
    1
    BlindedElement = 020095cff9d7ecf65bdfee4ea92d6e748d60b02de34ad98094f
    82e25d33a8bf50138ccc2cc633556f1a97d7ea9438cbb394df612f041c485a515849
    d5ebb2238f2f0e2,0201a328cf9f3fdeb86b6db242dd4cbb436b3a488b70b72d2fbb
    d1e5f50d7b0878b157d6f278c6a95c488f3ad52d6898a421658a82fe7ceb000b01ae
    dea7967522d525
    EvaluationElement = 0301408e9c5be3ffcc1c16e5ae8f8aa68446223b0804b119
    62e856af5a6d1c65ebbb5db7278c21db4e8cc06d89a35b6804fb1738a295b691638a
    f77aa1327253f26d01,020062ab51ac3aa829e0f5b7ae50688bcf5f63a18a83a6e0d
    a538666b8d50c7ea2b4ef31f4ac669302318dbebe46660acdda695da30c22cee7ca2
    1f6984a720504502e
    Proof = 00731738844f739bca0cca9d1c8bea204bed4fd00285785738b985763741
    de5cdfa275152d52b6a2fdf7792ef3779f39ba34581e56d62f78ecad5b7f8083f384
    961501cd4b43713253c022692669cf076b1d382ecd8293c1de69ea569737f37a2477
    2ab73517983c1e3db5818754ba1f008076267b8058b6481949ae346cdc17a8455fe2
    ProofRandomScalar = 01ec21c7bb69b0734cb48dfd68433dd93b0fa097e722ed24
    27de86966910acba9f5c350e8040f828bf6ceca27405420cdf3d63cb3aef005f40ba
    51943c8026877963
    Output = 808ae5b87662eaaf0b39151dd85991b94c96ef214cb14a68bf5c1439548
    82d330da8953a80eea20788e552bc8bbbfff3100e89f9d6e341197b122c46a208733
    b,27032e24b1a52a82ab7f4646f3c5df0f070f499db98b9c5df33972bd5af5762c36
    38afae7912a6c1acdb1ae2ab2fa670bd5486c645a0e55412e08d33a4a0d6e3

¶

## Acknowledgements

This document resulted from the work of the Privacy Pass team [PrivacyPass]. 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.¶

## Authors' Addresses