Network Working Group A. Davidson Internet-Draft N. Sullivan Intended status: Informational C. Wood Expires: January 14, 2021 Cloudflare July 13, 2020 Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups draft-irtf-cfrg-voprf-04 Abstract An Oblivious Pseudorandom Function (OPRF) is a two-party protocol for computing the output of a PRF. One party (the server) holds the PRF secret key, and the other (the client) holds the PRF input. The 'obliviousness' property ensures that the server does not learn anything about the client's input during the evaluation. The client should also not learn anything about the server's secret PRF key. Optionally, OPRFs can also satisfy a notion 'verifiability' (VOPRF). In this setting, the client can verify that the server's output is indeed the result of evaluating the underlying PRF with just a public key. This document specifies OPRF and VOPRF constructions instantiated within prime-order groups, including elliptic curves. Status of This Memo This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79. Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet- Drafts is at https://datatracker.ietf.org/drafts/current/. Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress." This Internet-Draft will expire on January 14, 2021. Copyright Notice Copyright (c) 2020 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 Davidson, et al. Expires January 14, 2021 [Page 1]

Internet-Draft OPRFs July 2020 (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. Code Components extracted from this document must include Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 1.1. Change log . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Terminology . . . . . . . . . . . . . . . . . . . . . . . 5 1.3. Requirements . . . . . . . . . . . . . . . . . . . . . . 5 2. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1. Prime-order group API . . . . . . . . . . . . . . . . . . 5 2.2. Other conventions . . . . . . . . . . . . . . . . . . . . 6 3. OPRF Protocol . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1. Overview . . . . . . . . . . . . . . . . . . . . . . . . 8 3.2. Context Setup . . . . . . . . . . . . . . . . . . . . . . 8 3.3. Data Structures . . . . . . . . . . . . . . . . . . . . . 9 3.4. Context APIs . . . . . . . . . . . . . . . . . . . . . . 10 3.4.1. Server Context . . . . . . . . . . . . . . . . . . . 10 3.4.2. VerifiableServerContext . . . . . . . . . . . . . . . 12 3.4.3. Client Context . . . . . . . . . . . . . . . . . . . 16 3.4.4. VerifiableClientContext . . . . . . . . . . . . . . . 17 4. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 19 4.1. OPRF(curve25519, SHA-512) . . . . . . . . . . . . . . . . 20 4.2. OPRF(curve448, SHA-512) . . . . . . . . . . . . . . . . . 20 4.3. OPRF(P-256, SHA-512) . . . . . . . . . . . . . . . . . . 21 4.4. OPRF(P-384, SHA-512) . . . . . . . . . . . . . . . . . . 22 4.5. OPRF(P-521, SHA-512) . . . . . . . . . . . . . . . . . . 23 5. Security Considerations . . . . . . . . . . . . . . . . . . . 23 5.1. Security properties . . . . . . . . . . . . . . . . . . . 24 5.2. Cryptographic security . . . . . . . . . . . . . . . . . 25 5.2.1. Computational hardness assumptions . . . . . . . . . 25 5.2.2. Protocol security . . . . . . . . . . . . . . . . . . 25 5.2.3. Q-strong-DH oracle . . . . . . . . . . . . . . . . . 26 5.2.4. Implications for ciphersuite choices . . . . . . . . 27 5.3. Hashing to curve . . . . . . . . . . . . . . . . . . . . 27 5.4. Timing Leaks . . . . . . . . . . . . . . . . . . . . . . 27 5.5. Key rotation . . . . . . . . . . . . . . . . . . . . . . 28 6. Additive blinding . . . . . . . . . . . . . . . . . . . . . . 28 6.1. Preprocess . . . . . . . . . . . . . . . . . . . . . . . 29 6.2. Blind . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.3. Unblind . . . . . . . . . . . . . . . . . . . . . . . . . 30 6.3.1. Parameter Commitments . . . . . . . . . . . . . . . . 31 7. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 31 Davidson, et al. Expires January 14, 2021 [Page 2]

Internet-Draft OPRFs July 2020 8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 31 9. References . . . . . . . . . . . . . . . . . . . . . . . . . 31 9.1. Normative References . . . . . . . . . . . . . . . . . . 31 9.2. URIs . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 34 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, where the server holds a PRF key k and the client holds some input x. The protocol allows both parties to cooperate in computing F(k, x) such that: the client learns F(k, x) without learning anything about k; and the server does not learn anything about x. A Verifiable OPRF (VOPRF) is an OPRF wherein the server can prove to the client that F(k, x) was computed using the key k. The usage of OPRFs has been demonstrated in constructing a number of applications: password-protected secret sharing schemes [JKKX16]; privacy-preserving password stores [SJKS17]; and password- authenticated key exchange or PAKE [OPAQUE]. The usage of a VOPRF is necessary in some applications, e.g., the Privacy Pass protocol [draft-davidson-pp-protocol], wherein this VOPRF is used to generate one-time authentication tokens to bypass CAPTCHA challenges. VOPRFs have also been used for password-protected secret sharing schemes e.g. [JKK14]. This document introduces an OPRF protocol built in prime-order groups, applying to finite fields of prime-order and also elliptic curve (EC) groups. The protocol has the option of being extended to a VOPRF with the addition of a NIZK proof for proving discrete log equality relations. This proof demonstrates correctness of the computation, using a known public key that serves as a commitment to the server's secret key. The document describes the protocol, the public-facing API, and its security properties. 1.1. Change log draft-04 [1]: o Introduce Client and Server contexts for controlling verifiability and required functionality. o Condense API. Davidson, et al. Expires January 14, 2021 [Page 3]

Internet-Draft OPRFs July 2020 o Remove batching from standard functionality (included as an extension) o Add Curve25519 and P-256 ciphersuites for applications that prevent strong-DH oracle attacks. o Provide explicit prime-order group API and instantiation advice for each ciphersuite. o Proof-of-concept implementation in sage. o Remove privacy considerations advice as this depends on applications. draft-03 [2]: o Certify public key during VerifiableFinalize o Remove protocol integration advice o Add text discussing how to perform domain separation o Drop OPRF_/VOPRF_ prefix from algorithm names o Make prime-order group assumption explicit o Changes to algorithms accepting batched inputs o Changes to construction of batched DLEQ proofs o Updated ciphersuites to be consistent with hash-to-curve and added OPRF specific ciphersuites draft-02 [3]: o Added section discussing cryptographic security and static DH oracles o Updated batched proof algorithms draft-01 [4]: o Updated ciphersuites to be in line with https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-04 o Made some necessary modular reductions more explicit Davidson, et al. Expires January 14, 2021 [Page 4]

Internet-Draft OPRFs July 2020 1.2. Terminology The following terms are used throughout this document. o PRF: Pseudorandom Function. o OPRF: Oblivious Pseudorandom Function. o VOPRF: Verifiable Oblivious Pseudorandom Function. o Client: Protocol initiator. Learns pseudorandom function evaluation as the output of the protocol. o Server: Computes the pseudorandom function over a secret key. Learns nothing about the client's input. o NIZK: Non-interactive zero knowledge. o DLEQ: Discrete Logarithm Equality. 1.3. Requirements The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in [RFC2119]. 2. Preliminaries 2.1. Prime-order group API In this document, we assume the construction of an additive, prime- order group "GG" for performing all mathematical operations. Such groups are uniquely determined by the choice of the prime "p" that defines the order of the group. We use "GF(p)" to represent the finite field of order "p". For the purpose of understanding and implementing this document, we take "GF(p)" to be equal to the set of integers defined by "{0, 1, ..., p-1}". The fundamental group operation is addition "+" with identity element "I". For any elements "A" and "B" of the group "GG", "A + B = B + A" is also a member of "GG". Also, for any "A" in "GG", it exists an element "-A" such that "A + (-A) = (-A) + A = I". Scalar multiplication 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". Any element "A" holds the equality "p*A=I". The set of scalars corresponds to "GF(p)". Davidson, et al. Expires January 14, 2021 [Page 5]

Internet-Draft OPRFs July 2020 We now detail a number of member functions that can be invoked on a prime-order group. o Order(): Outputs the order of the group (i.e. "p"). o Generator(): Outputs a fixed generator "G" for the group. o Identity(): Outputs the identity element of the group (i.e. "I"). o Serialize(A): A member function of "GG" that maps a group element "A" to a unique byte array "buf". o Deserialize(buf): A member function of "GG" that maps a byte array "buf" to a group element "A". o HashToGroup(x): A member function of "GG" that deterministically maps an array of bytes "x" to an element of "GG". The map must ensure that, for any adversary receiving "R = HashToGroup(x)", it is computationally difficult to reverse the mapping. Examples of hash to group functions satisfying this property are described for prime-order (sub)groups of elliptic curves, see [I-D.irtf-cfrg-hash-to-curve]. o HashToScalar(x): A member function of "GG" that deterministically maps an array of bytes "x" to a random element in GF(p). o RandomScalar(): A member function of "GG" that generates a random, non-zero element in GF(p). It is convenient in cryptographic applications to instantiate such prime-order groups using elliptic curves, such as those detailed in [SEC2]. For some choices of elliptic curves (e.g. those detailed in [RFC7748] require accounting for cofactors) there are some implementation issues that introduce inherent discrepancies between standard prime-order groups and the elliptic curve instantiation. In this document, all algorithms that we detail assume that the group is a prime-order group, and this MUST be upheld by any implementer. That is, any curve instantiation should be written such that any discrepancies with a prime-order group instantiation are removed. See Section 4 for advice corresponding to implementation of this interface for specific definitions of elliptic curves. 2.2. Other conventions o We use the notation "x <-$ Q" to denote sampling "x" from the uniform distribution over the set "Q". Davidson, et al. Expires January 14, 2021 [Page 6]

Internet-Draft OPRFs July 2020 o For any object "x", we write "len(x)" to denote its length in bytes. o For two byte arrays "x" and "y", write "x || y" to denote their concatenation. o We assume that all numbers are stored in big-endian orientation. o I2OSP and OS2IP: Convert a byte array to and from a non-negative integer as described in [RFC8017]. Note that these functions operate on byte arrays in big-endian byte order. All algorithm descriptions are written in a Python-like pseudocode. We use the "ABORT()" function for presentational clarity to denote the process of terminating the algorithm or returning an error accordingly. We also use the "CT_EQUAL(a, b)" function to represent constant-time byte-wise equality between byte arrays "a" and "b". This function returns a boolean "true" if "a" and "b" are equal, and "false" otherwise. 3. OPRF Protocol In this section, we define two OPRF variants: a base mode and verifiable mode. In the base mode, a client and server interact to compute y = F(skS, x), where x is the client's input, skS is the server's private key, and y is the OPRF output. The client learns y and the server learns nothing. In the verifiable mode, the client also gets proof that the server used skS in computing the function. To achieve verifiability, as in the original work of [JKK14], we provide a zero-knowledge proof that the key provided as input by the server in the "Evaluate" function is the same key as it used to produce their public key. As an example of the nature of attacks that this prevents, this ensures that the server uses the same private key for computing the VOPRF output and does not attempt to "tag" individual servers with select keys. This proof must not reveal the server's long-term private key to the client. The following one-byte values distinguish between these two modes: +----------+-------+---+----------------+------+ | Mode | Value | | | | +----------+-------+---+----------------+------+ | modeBase | 0x00 | | modeVerifiable | 0x01 | +----------+-------+---+----------------+------+ Davidson, et al. Expires January 14, 2021 [Page 7]

Internet-Draft OPRFs July 2020 3.1. Overview Both participants agree on the mode and a choice of ciphersuite that is used before the protocol exchange. Once established, the core protocol runs to compute "output = F(skS, input)" as follows: Client(pkS, input, info) Server(skS, pkS) ---------------------------------------------------------- token, blindToken = Blind(input) blindToken ----------> evaluation = Evaluate(skS, pkS, blindToken) evaluation <---------- issuedToken = Unblind(pkS, token, blindToken, evaluation) output = Finalize(input, issuedToken, info) In "Blind" the client generates a token and blinding data. The server computes the (V)OPRF evaluation in "Evaluation" over the client's blinded token. In "Unblind" the client unblinds the server response (and verifies the server's proof if verifiability is required). In "Finalize", the client outputs a byte array corresponding to its input. Note that in the final output, the client computes Finalize over some auxiliary input data "info". This parameter SHOULD be used for domain separation in the (V)OPRF protocol. Specifically, any system which has multiple (V)OPRF applications should use separate auxiliary values to to ensure finalized outputs are separate. Guidance for constructing info can be found in [I-D.irtf-cfrg-hash-to-curve]; Section 3.1. 3.2. Context Setup Both modes of the OPRF involve an offline setup phase. In this phase, both the client and server create a context used for executing the online phase of the protocol. The base mode setup functions for creating client and server contexts are below: Davidson, et al. Expires January 14, 2021 [Page 8]

Internet-Draft OPRFs July 2020 def SetupBaseserver(suite): (skS, _) = KeyGen(GG) contextString = I2OSP(modeBase, 1) + I2OSP(suite.ID, 2) return ServerContext(contextString, skS) def SetupBaseClient(suite): contextString = I2OSP(modeBase, 1) + I2OSP(suite.ID, 2) return ClientContext(contextString) The "KeyGen" function used above takes a group "GG" and generates a private and public key pair (skX, pkX), where skX is a random, non- zero element in the scalar field "GG" and pkX is the product of skX and the group's fixed generator. The verifiable mode setup functions for creating client and server contexts are below. def SetupVerifiableserver(suite): (skS, pkS) = KeyGen(GG) contextString = I2OSP(modeVerifiable, 1) + I2OSP(suite.ID, 2) return VerifiableServerContext(contextString, skS), pkS def SetupVerifiableClient(suite, pkS): contextString = I2OSP(modeVerifiable, 1) + I2OSP(suite.ID, 2) return VerifiableClientContext(contextString, pkS) For verifiable modes, servers MUST make the resulting public key "pkS" accessible for clients. (Indeed, it is a required parameter when configuring a verifiable client context.) Each setup function takes a ciphersuite from the list defined in Section 4. Each ciphersuite has two-byte identifier, referred to as "suite.ID" in the pseudocode above, that identifies the suite. Section 4 lists these ciphersuite identifiers. 3.3. Data Structures The following is a list of data structures that are defined for providing inputs and outputs for each of the context interfaces defined in Section 3.4. The following types are a list of aliases that are used throughout the protocol. A "ClientInput" is a byte array. opaque ClientInput<1..2^16-1>; Davidson, et al. Expires January 14, 2021 [Page 9]

Internet-Draft OPRFs July 2020 A "SerializedElement" is also a byte array, representing the unique serialization of an "Element". opaque SerializedElement<1..2^16-1>; A "Token" is an object created by a client when constructing a (V)OPRF protocol input. It is stored so that it can be used after receiving the server response. struct { opaque data<1..2^16-1>; Scalar blind<1..2^16-1>; } Token; An "Evaluation" is the type output by the "Evaluate" algorithm. The member "proof" is added only in verifiable contexts. struct { SerializedElement element; Scalar proof<0...2^16-1>; /* only for modeVerifiable */ } Evaluation; Evaluations may also be combined in batches with a constant-size proof, producing a "BatchedEvaluation". These carry a list of "SerializedElement" values and proof. These evaluation types are only useful in verifiable contexts which carry proofs. struct { SerializedElement elements<1..2^16-1>; Scalar proof<0...2^16-1>; /* only for modeVerifiable */ } BatchedEvaluation; 3.4. Context APIs In this section, we detail the APIs available on the client and server OPRF contexts. This document uses the types "Element" and "Scalar" to denote elements of the group "GG" and its underlying scalar field, respectively. For notational clarity, "PublicKey" and "PrivateKey" are items of type "Element" and "Scalar", respectively. 3.4.1. Server Context The ServerContext encapsulates the context string constructed during setup and the OPRF key pair. It has two functions, "Evaluate" and "VerifyFinalize", described below. "Evaluate" takes as input serialized representations of blinded group elements from the client. "VerifyFinalize" takes ClientInput values and their corresponding output digests from "Verify" and returns true if the inputs match the Davidson, et al. Expires January 14, 2021 [Page 10]

Internet-Draft OPRFs July 2020 outputs. Note that "VerifyFinalize" is not used in the main OPRF protocol. It is exposed as an API for building higher-level protocols. 3.4.1.1. Evaluate Input: PrivateKey skS PublicKey pkS SerializedElement blindedToken Output: Evaluation Ev def Evaluate(skS, pkS, blindedToken): BT = GG.Deserialize(blindedToken) Z = skS * BT serializedElement = GG.Serialize(Z) Ev = Evaluation{ element: serializedElement } return Ev 3.4.1.2. VerifyFinalize Davidson, et al. Expires January 14, 2021 [Page 11]

Internet-Draft OPRFs July 2020 Input: PrivateKey skS PublicKey pkS ClientInput input opaque info<1..2^16-1> opaque output<1..2^16-1> Output: boolean valid def VerifyFinalize(skS, pkS, input, info, output): T = GG.HashToGroup(input) element = GG.Serialize(T) issuedElement = Evaluate(skS, pkS, [element]) E = GG.Serialize(issuedElement) finalizeDST = "RFCXXXX-Finalize-" + client.contextString hashInput = len(input) || input || len(E) || E || len(info) || info || len(finalizeDST) || finalizeDST digest = Hash(hashInput) return CT_EQUAL(digest, output) 3.4.2. VerifiableServerContext The VerifiableServerContext extends the base ServerContext with an augmented "Evaluate()" function. This function produces a proof that "skS" was used in computing the result. It makes use of the helper functions "ComputeComposites" and "GenerateProof", described below. 3.4.2.1. Evaluate Davidson, et al. Expires January 14, 2021 [Page 12]

Internet-Draft OPRFs July 2020 Input: PrivateKey skS PublicKey pkS SerializedElement blindedToken Output: Evaluation Ev def Evaluate(skS, pkS, blindedToken): BT = GG.Deserialize(blindedToken) Z = skS * BT serializedElement = GG.Serialize(Z) proof = GenerateProof(skS, pkS, blindedToken, serializedElement) Ev = Evaluation{ element: serializedElement, proof: proof } return Ev The helper functions "GenerateProof" and "ComputeComposites" are defined below. 3.4.2.2. GenerateProof Davidson, et al. Expires January 14, 2021 [Page 13]

Internet-Draft OPRFs July 2020 Input: PrivateKey skS PublicKey pkS SerializedElement blindedToken SerializedElement element Output: Scalar proof[2] def GenerateProof(skS, pkS, blindedToken, element) G = GG.Generator() gen = GG.Serialize(G) blindTokenList = [blindedToken] elementList = [element] (a1, a2) = ComputeComposites(gen, pkS, blindTokenList, elementList) M = GG.Deserialize(a1) r = GG.RandomScalar() a3 = GG.Serialize(r * G) a4 = GG.Serialize(r * M) challengeDST = "RFCXXXX-challenge-" + self.contextString h2Input = I2OSP(len(gen), 2) || gen || I2OSP(len(pkS), 2) || pkS || I2OSP(len(a1), 2) || a1 || I2OSP(len(a2), 2) || a2 || I2OSP(len(a3), 2) || a3 || I2OSP(len(a4), 2) || a4 || I2OSP(len(challengeDST), 2) || challengeDST c = GG.HashToScalar(h2Input) s = (r - c * skS) mod p return (c, s) 3.4.2.2.1. Batching inputs Unlike other functions, "ComputeComposites" takes lists of inputs, rather than a single input. It is optimized to produce a constant- size output. This functionality lets applications batch inputs together to produce a constant-size proofs from "GenerateProof". Applications can take advantage of this functionality by invoking "GenerateProof" on batches of inputs. (Notice that in the pseudocode above, the single inputs "blindedToken" and "element" are translated into lists before invoking "ComputeComposites". A batched Davidson, et al. Expires January 14, 2021 [Page 14]

Internet-Draft OPRFs July 2020 "GenerateProof" variant would permit lists of inputs, and no list translation would be needed.) Note that using batched inputs creates a "BatchedEvaluation" object as the output of "Evaluate". 3.4.2.2.2. Fresh randomness We note here that it is essential that a different r value is used for every invocation. If this is not done, then this may leak "skS" as is possible in Schnorr or (EC)DSA scenarios where fresh randomness is not used. 3.4.2.3. ComputeComposites Input: SerializedElement gen PublicKey pkS SerializedElement blindedTokens[m] SerializedElement elements[m] Output: SerializedElement composites[2] def ComputeComposites(gen, pkS, blindedTokens, elements): seedDST = "RFCXXXX-seed-" + self.contextString compositeDST = "RFCXXXX-composite-" + self.contextString h1Input = I2OSP(len(gen), 2) || gen || I2OSP(len(pkS), 2) || pkS || I2OSP(len(blindedTokens), 2) || blindedTokens || I2OSP(len(elements), 2) || elements || I2OSP(len(seedDST), 2) || seedDST seed = Hash(h1Input) M = GG.Identity() Z = GG.Identity() for i = 0 to m: h2Input = I2OSP(len(seed), 2) || seed || I2OSP(i, 2) || I2OSP(len(compositeDST), 2) || compositeDST di = GG.HashToScalar(h2Input) Mi = GG.Deserialize(blindedTokens[i]) Zi = GG.Deserialize(elements[i]) M = di * Mi + M Z = di * Zi + Z return [GG.Serialize(M), GG.Serialize(Z)] Davidson, et al. Expires January 14, 2021 [Page 15]

Internet-Draft OPRFs July 2020 3.4.3. Client Context The ClientContext encapsulates the context string constructed during setup. It has three functions, "Blind()", "Unblind()", and "Finalize()", as described below. 3.4.3.1. Blind We note here that the blinding mechanism that we use can be modified slightly with the opportunity for making performance gains in some scenarios. We detail these modifications in Section 6. Input: ClientInput input Output: Token token SerializedElement blindedToken def Blind(input): r = GG.RandomScalar() P = GG.HashToGroup(input) blindedToken = GG.Serialize(r * P) token = Token{ data: input, blind: r } return (token, blindedToken) 3.4.3.2. Unblind Davidson, et al. Expires January 14, 2021 [Page 16]

Internet-Draft OPRFs July 2020 Input: PublicKey pkS Token token SerializedElement blindedToken Evaluation Ev Output: SerializedElement unblindedToken def Unblind(pkS, token, blindedToken, Ev): r = token.blind Z = GG.Deserialize(Ev.element) N = (r^(-1)) * Z unblindedToken = GG.Serialize(N) return unblindedToken 3.4.3.3. Finalize Input: Token T SerializedElement E opaque info<1..2^16-1> Output: opaque output<1..2^16-1> def Finalize(T, E, info): finalizeDST = "RFCXXXX-Finalize-" + self.contextString hashInput = len(T.data) || T.data || len(E) || E || len(info) || info || len(finalizeDST) || finalizeDST return Hash(hashInput) 3.4.4. VerifiableClientContext The VerifiableClientContext extends the base ClientContext with the desired server public key "pkS" with an augmented "Unblind()" function. This function verifies an evaluation proof using "pkS". It makes use of the helper function "ComputeComposites" described above. It has one helper function, "VerifyProof()", defined below. Davidson, et al. Expires January 14, 2021 [Page 17]

Internet-Draft OPRFs July 2020 3.4.4.1. VerifyProof This algorithm outputs a boolean "verified" which indicates whether the proof inside of the evaluation verifies correctly, or not. Input: PublicKey pkS SerializedElement blindedToken Evaluation Ev Output: boolean verified def VerifyProof(pkS, blindedToken, Ev): G = GG.Generator() gen = GG.Serialize(G) blindTokenList = [blindedToken] elementList = [Ev.element] (a1, a2) = ComputeComposites(gen, pkS, blindTokenList, elementList) A' = (Ev.proof[1] * G + Ev.proof[0] * pkS) B' = (Ev.proof[1] * M + Ev.proof[0] * Z) a3 = GG.Serialize(A') a4 = GG.Serialize(B') challengeDST = "RFCXXXX-challenge-" + self.contextString h2Input = I2OSP(len(gen), 2) || gen || I2OSP(len(pkS), 2) || pkS || I2OSP(len(a1), 2) || a1 || I2OSP(len(a2), 2) || a2 || I2OSP(len(a3), 2) || a3 || I2OSP(len(a4), 2) || a4 || I2OSP(len(challengeDST), 2) || challengeDST c = GG.HashToScalar(h2Input) return CT_EQUAL(c, Ev.proof[0]) 3.4.4.2. Unblind Davidson, et al. Expires January 14, 2021 [Page 18]

Internet-Draft OPRFs July 2020 Input: PublicKey pkS Token token SerializedElement blindedToken Evaluation Ev Output: SerializedElement unblindedToken def Unblind(pkS, token, blindedToken, Ev): if VerifyProof(pkS, blindedToken, Ev) == false: ABORT() r = token.blind Z = GG.Deserialize(Ev.element) N = (r^(-1)) * Z unblindedToken = GG.Serialize(N) return unblindedToken 4. Ciphersuites A ciphersuite for the protocol wraps the functionality required for the protocol to take place. This 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. o "GG": A prime-order group exposing the API detailed in Section 2.1. o "Hash": A cryptographic hash function that is indifferentiable from a Random Oracle. This section specifies supported VOPRF group and hash function instantiations. For each group, we specify the HashToGroup and Serialize functionalities. The Deserialize functionality is the inverse of the corresponding Serialize functionality. We only provide ciphersuites in the elliptic curve setting as these provide the most efficient way of instantiating the OPRF. Applications should take caution in using ciphersuites targeting P-256 and curve25519. See Section 5.2 for related discussion. Davidson, et al. Expires January 14, 2021 [Page 19]

Internet-Draft OPRFs July 2020 [[OPEN ISSUE: Replace Curve25519 and Curve448 with Ristretto/Decaf]] 4.1. OPRF(curve25519, SHA-512) o Group: * Elliptic curve: curve25519 [RFC7748] * Generator(): Return the point with the following affine coordinates: + x = "09" + y = "5F51E65E475F794B1FE122D388B72EB36DC2B28192839E4DD6163A5 D81312C14" * HashToGroup(): curve25519_XMD:SHA-512_ELL2_RO_ [I-D.irtf-cfrg-hash-to-curve] with DST "RFCXXXX- curve25519_XMD:SHA-512_ELL2_RO_" * Serialization: The standard 32-byte representation of the public key [RFC7748] * Order(): Returns "1000000000000000000000000000000014DEF9DEA2F79 CD65812631A5CF5D3ED" * Addition: Adding curve points directly corresponds to the group addition operation. * Deserialization: Implementers must check for each untrusted input point whether it's a member of the big prime-order subgroup of the curve. This can be done by scalar multiplying the point by Order() and checking whether it's zero. o Hash: SHA-512 o ID: 0x0001 4.2. OPRF(curve448, SHA-512) o Group: * Elliptic curve: curve448 [RFC7748] * Generator(): Return the point with the following affine coordinates: + x = "05" Davidson, et al. Expires January 14, 2021 [Page 20]

Internet-Draft OPRFs July 2020 + y = "7D235D1295F5B1F66C98AB6E58326FCECBAE5D34F55545D060F75DC 28DF3F6EDB8027E2346430D211312C4B150677AF76FD7223D457B5B1A" * HashToGroup(): curve448_XMD:SHA-512_ELL2_RO_ [I-D.irtf-cfrg-hash-to-curve] with DST "RFCXXXX- curve448_XMD:SHA-512_ELL2_RO_" * Serialization: The standard 56-byte representation of the public key [RFC7748] * Order(): Returns "3FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFF7CCA23E9C44EDB49AED63690216CC2728DC58F552378C292AB58 44F3" * Addition: Adding curve points directly corresponds to the group addition operation. * Deserialization: Implementers must check for each untrusted input point whether it's a member of the big prime-order subgroup of the curve. This can be done by scalar multiplying the point by Order() and checking whether it's zero. o Hash: SHA-512 o ID: 0x0002 4.3. OPRF(P-256, SHA-512) o Group: * Elliptic curve: P-256 (secp256r1) [x9.62] * Generator(): Return the point with the following affine coordinates: + x = "6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A1394 5D898C296" + y = "4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406 837BF51F5" * HashToGroup(): P256_XMD:SHA-256_SSWU_RO_ [I-D.irtf-cfrg-hash-to-curve] with DST "RFCXXXX-P256_XMD:SHA- 256_SSWU_RO_" * Serialization: The compressed point encoding for the curve [SEC1] consisting of 33 bytes. Davidson, et al. Expires January 14, 2021 [Page 21]

Internet-Draft OPRFs July 2020 * Order(): Returns "FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179 E84F3B9CAC2FC632551" * Addition: Adding curve points directly corresponds to the group addition operation. * Scalar multiplication: Scalar multiplication of curve points directly corresponds with scalar multiplication in the group. o Hash: SHA-512 o ID: 0x0003 4.4. OPRF(P-384, SHA-512) o Group: * Elliptic curve: P-384 (secp384r1) [x9.62] * Generator(): Return the point with the following affine coordinates: + x = "AA87CA22BE8B05378EB1C71EF320AD746E1D3B628BA79B9859F741E 082542A385502F25DBF55296C3A545E3872760AB7" + y = "3617DE4A96262C6F5D9E98BF9292DC29F8F41DBD289A147CE9DA311 3B5F0B8C00A60B1CE1D7E819D7A431D7C90EA0E5F" * HashToGroup(): P384_XMD:SHA-512_SSWU_RO_ [I-D.irtf-cfrg-hash-to-curve] with DST "RFCXXXX-P384_XMD:SHA- 512_SSWU_RO_" * Serialization: The compressed point encoding for the curve [SEC1] consisting of 49 bytes. * Order(): Returns "FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFC7634D81F4372DDF581A0DB248B0A77AECEC196ACCC52973" * Addition: Adding curve points directly corresponds to the group addition operation. * Scalar multiplication: Scalar multiplication of curve points directly corresponds with scalar multiplication in the group. o Hash: SHA-512 o ID: 0x0004 Davidson, et al. Expires January 14, 2021 [Page 22]

Internet-Draft OPRFs July 2020 4.5. OPRF(P-521, SHA-512) o Group: * Elliptic curve: P-521 (secp521r1) [x9.62] * Generator(): Return the point with the following affine coordinates: + x = "00C6858E06B70404E9CD9E3ECB662395B4429C648139053FB521F82 8AF606B4D3DBAA14B5E77EFE75928FE1DC127A2FFA8DE3348B3C1856A429 BF97E7E31C2E5BD66" + y = "011839296A789A3BC0045C8A5FB42C7D1BD998F54449579B446817A FBD17273E662C97EE72995EF42640C550B9013FAD0761353C7086A272C24 088BE94769FD16650" * HashToGroup(): P521_XMD:SHA-512_SSWU_RO_ [I-D.irtf-cfrg-hash-to-curve] with DST "RFCXXXX-P521_XMD:SHA- 512_SSWU_RO_" * Serialization: The compressed point encoding for the curve [SEC1] consisting of 67 bytes. * Order(): Returns "1FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF FFFFFFFFFFFFFFFFFFFFFA51868783BF2F966B7FCC0148F709A5D03BB5C9B88 99C47AEBB6FB71E91386409" * Addition: Adding curve points directly corresponds to the group addition operation. * Scalar multiplication: Scalar multiplication of curve points directly corresponds with scalar multiplication in the group. o Hash: SHA-512 o ID: 0x0005 5. Security Considerations This section discusses the cryptographic security of our protocol, along with some suggestions and trade-offs that arise from the implementation of an OPRF. Davidson, et al. Expires January 14, 2021 [Page 23]

Internet-Draft OPRFs July 2020 5.1. Security properties The security properties of an OPRF protocol with functionality y = F(k, x) include those of a standard PRF. Specifically: o Pseudorandomness: 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, for a random sampling of k. In other words, consider an adversary that picks inputs x from the domain of F and evaluates F on (k,x) (without knowledge of randomly sampled k). Then the output distribution F(k,x) is indistinguishable from the output distribution of a randomly chosen function with the same domain and range. A consequence of showing that a function is pseudorandom, is that it is necessarily non-malleable (i.e. we cannot compute a new evaluation of F from an existing evaluation). A genuinely random function will be non-malleable with high probability, and so a pseudorandom function must be non-malleable to maintain indistinguishability. An OPRF protocol must also satisfy the following property: o Oblivious: The server must learn nothing about the client's input or the output of the function. In addition, the client must learn nothing about the server's private key. Essentially, obliviousness tells us that, even if the server learns the client's input x at some point in the future, then the server will not be able to link any particular OPRF evaluation to x. This property is also known as unlinkability [DGSTV18]. Optionally, for any protocol that satisfies the above properties, there is an additional security property: o Verifiable: The client must only complete execution of the protocol if it can successfully assert that the OPRF output it computes is correct. This is taken with respect to the OPRF key held by the server. Any OPRF that satisfies the 'verifiable' security property is known as a verifiable OPRF, or VOPRF for short. 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. Davidson, et al. Expires January 14, 2021 [Page 24]

Internet-Draft OPRFs July 2020 5.2. Cryptographic security Below, we discuss the cryptographic security of the (V)OPRF protocol from Section 3, relative to the necessary cryptographic assumptions that need to be made. 5.2.1. Computational hardness assumptions Each assumption states that the problems specified below are computationally difficult to solve in relation to a particular choice of security parameter "sp". Let GG = GG(sp) be a group with prime-order p, and let FFp be the finite field of order p. 5.2.1.1. Discrete-log (DL) problem Given G, a generator of GG, and H = hG for some h in FFp; output h. 5.2.1.2. Decisional Diffie-Hellman (DDH) problem Sample a uniformly random bit d in {0,1}. Given (G, aG, bG, C), where: o G is a generator of GG; o a,b are elements of FFp; o if d == 0: C = abG; else: C is sampled uniformly GG(sp). Output d' == d. 5.2.2. Protocol security Our OPRF construction is based on the VOPRF construction known as 2HashDH-NIZK given by [JKK14]; essentially without providing zero- knowledge proofs that verify that the output is correct. Our VOPRF construction is identical to the [JKK14] construction, except that we can optionally perform multiple VOPRF evaluations in one go, whilst only constructing one NIZK proof object. This is enabled using an established batching technique. Consequently the cryptographic security of our construction is based on the assumption that the One-More Gap DH is computationally difficult to solve. The (N,Q)-One-More Gap DH (OMDH) problem asks the following. Davidson, et al. Expires January 14, 2021 [Page 25]

Internet-Draft OPRFs July 2020 Given: - G, k * G, G_1, ... , G_N where G, G_1, ... G_N are elements of GG; - oracle access to an OPRF functionality using the key k; - oracle access to DDH solvers. Find Q+1 pairs of the form below: (G_{j_s}, k * G_{j_s}) where the following conditions hold: - s is a number between 1 and Q+1; - j_s is a number between 1 and N for each s; - Q is the number of allowed queries. The original paper [JKK14] gives a security proof that the 2HashDH- NIZK construction satisfies the security guarantees of a VOPRF protocol Section 5.1 under the OMDH assumption in the universal composability (UC) security model. 5.2.3. Q-strong-DH oracle A side-effect of our OPRF design is that it allows instantiation of a oracle for constructing Q-strong-DH (Q-sDH) samples. The Q-Strong-DH problem asks the following. Given G1, G2, h*G2, (h^2)*G2, ..., (h^Q)*G2; for G1 and G2 generators of GG. Output ( (1/(k+c))*G1, c ) where c is an element of FFp The assumption that this problem is hard was first introduced in [BB04]. Since then, there have been a number of cryptanalytic studies that have reduced the security of the assumption below that implied by the group instantiation (for example, [BG04] and [Cheon06]). In summary, the attacks reduce the security of the group instantiation by log_2(Q) bits. As an example, suppose that a group instantiation is used that provides 128 bits of security against discrete log cryptanalysis. Then an adversary with access to a Q-sDH oracle and makes Q=2^20 queries can reduce the security of the instantiation by log_2(2^20) = 20 bits. Notice that it is easy to instantiate a Q-sDH oracle using the OPRF functionality that we provide. A client can just submit sequential queries of the form (G, k * G, (k^2)G, ..., (k^(Q-1))G), where each query is the output of the previous interaction. This means that any client that submit Q queries to the OPRF can use the aforementioned Davidson, et al. Expires January 14, 2021 [Page 26]

Internet-Draft OPRFs July 2020 attacks to reduce security of the group instantiation by log_2(Q) bits. Recall that from a malicious client's perspective, the adversary wins if they can distinguish the OPRF interaction from a protocol that computes the ideal functionality provided by the PRF. 5.2.4. Implications for ciphersuite choices The OPRF instantiations that we recommend in this document are informed by the cryptanalytic discussion above. In particular, choosing elliptic curves configurations that describe 128-bit group instantiations would appear to in fact instantiate an OPRF with 128-log_2(Q) bits of security. 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. We see this possibility as something that may result in problems in the future. For applications that cannot tolerate discrete logarithm security of lower than 128 bits, we recommend only implementing ciphersuites with IDs: 0x0002, 0x0004, and 0x0005. 5.3. Hashing to curve A critical requirement of implementing the prime-order group using elliptic curves is a method to instantiate the function "GG.HashToGroup", that maps inputs to group elements. In the elliptic curve setting, this deterministically maps inputs x (as byte arrays) to uniformly chosen points in the curve. In the security proof of the construction Hash is modeled as a random oracle. This implies that any instantiation of "GG.HashToGroup" must be pre-image and collision resistant. In Section 4 we give instantiations of this functionality based on the functions described in [I-D.irtf-cfrg-hash-to-curve]. Consequently, any OPRF implementation must adhere to the implementation and security considerations discussed in [I-D.irtf-cfrg-hash-to-curve] when instantiating the function. 5.4. Timing Leaks To ensure no information is leaked during protocol execution, all operations that use secret data MUST be constant time. Operations that SHOULD be constant time include all prime-order group operations and proof-specific operations ("GenerateProof()" and "VerifyProof()"). Davidson, et al. Expires January 14, 2021 [Page 27]

Internet-Draft OPRFs July 2020 5.5. Key rotation Since the server's key is critical to security, the longer it is exposed by performing (V)OPRF operations on client inputs, the longer it is possible that the key can be compromised. For example,if the key is kept in circulation for a long period of time, then it also allows the clients to make enough queries to launch more powerful variants of the Q-sDH attacks from Section 5.2.3. To combat attacks of this nature, regular key rotation should be employed on the server-side. A suitable key-cycle for a key used to compute (V)OPRF evaluations would be between one week and six months. 6. Additive blinding Let "H" refer to the function "GG.HashToGroup", in Section 2.1 we assume that the client-side blinding is carried out directly on the output of "H(x)", i.e. computing "r * H(x)" for some "r <-$ GF(p)". In the [OPAQUE] document, it is noted that it may be more efficient to use additive blinding (rather than multiplicative) if the client can preprocess some values. For example, a valid way of computing additive blinding would be to instead compute "H(x) + (r * G)", where "G" is the fixed generator for the group "GG". The advantage of additive blinding is that it allows the client to pre-process tables of blinded scalar multiplications for "G". This may give it a computational efficiency advantage (due to the fact that a fixed-base multiplication can be calculated faster than a variable-base multiplication). Pre-processing also reduces the amount of computation that needs to be done in the online exchange. Choosing one of these values "r * G" (where "r" is the scalar value that is used), then computing "H(x) + (r * G)" is more efficient than computing "r * H(x)". Therefore, it may be advantageous to define the OPRF and VOPRF protocols using additive blinding (rather than multiplicative) blinding. In fact, the only algorithms that need to change are Blind and Unblind (and similarly for the VOPRF variants). We define the variants of the algorithms in Section 3.4 for performing additive blinding below, along with a new algorithm "Preprocess". The "Preprocess" algorithm can take place offline and before the rest of the OPRF protocol. The Blind algorithm takes the preprocessed values as inputs, but the signature of Unblind remains the same. Davidson, et al. Expires January 14, 2021 [Page 28]

Internet-Draft OPRFs July 2020 6.1. Preprocess struct { Scalar blind; SerializedElement blindedGenerator; SerializedElement blindedPublicKey; } PreprocessedBlind; Input: PublicKey pkS Output: PrepocessedBlind preproc def Preprocess(pkS): PK = GG.Deserialize(pkS) r = GG.RandomScalar() blindedGenerator = GG.Serialize(r * GG.Generator()) blindedPublicKey = GG.Serialize(r * PK) preproc = PrepocessedBlind{ blind: r, blindedGenerator: blindedGenerator, blindedPublicKey: blindedPublicKey, } return preproc 6.2. Blind Davidson, et al. Expires January 14, 2021 [Page 29]

Internet-Draft OPRFs July 2020 Input: ClientInput input PreprocessedBlind preproc Output: Token token SerializedElement blindedToken def Blind(input, preproc): Q = GG.Deserialize(preproc.blindedGenerator) /* Q = r * G */ P = GG.HashToGroup(input) token = Token{ data: input, blind: preproc.blindedPublicKey } blindedToken = GG.Serialize(P + Q) /* P + r * G */ return (token, blindedToken) 6.3. Unblind Input: Token token Evaluation ev SerializedElement blindedToken Output: SerializedElement unblinded def Unblind(token, ev, blindedToken): PKR = GG.Deserialize(token.blind) Z = GG.Deserialize(ev.element) N := Z - PKR unblindedToken = GG.Serialize(N) return unblindedToken Let "P = GG.HashToGroup(x)". Notice that Unblind computes: Z - PKR = k * (P + r * G) - r * pkS = k * P + k * (r * G) - r * (k * G) = k * P Davidson, et al. Expires January 14, 2021 [Page 30]

Internet-Draft OPRFs July 2020 by the commutativity of scalar multiplication in GG. This is the same output as in the "Unblind" algorithm for multiplicative blinding. Note that the verifiable variant of "Unblind" works as above but includes the step to "VerifyProof", as specified in Section 3.4.4. 6.3.1. Parameter Commitments For some applications, it may be desirable for server to bind tokens to certain parameters, e.g., protocol versions, ciphersuites, etc. To accomplish this, server should use a distinct scalar for each parameter combination. Upon redemption of a token T from the client, server can later verify that T was generated using the scalar associated with the corresponding parameters. 7. Contributors o Alex Davidson (alex.davidson92@gmail.com) o Nick Sullivan (nick@cloudflare.com) o Chris Wood (caw@heapingbits.net) o Eli-Shaoul Khedouri (eli@intuitionmachines.com) o Armando Faz Hernandez (armfazh@cloudflare.com) 8. Acknowledgements This document resulted from the work of the Privacy Pass team [PrivacyPass]. The authors would also like to acknowledge the helpful conversations with Hugo Krawczyk. Eli-Shaoul Khedouri provided additional review and comments on key consistency. 9. References 9.1. Normative References [BB04] "Short Signatures Without Random Oracles", <http://ai.stanford.edu/~xb/eurocrypt04a/bbsigs.pdf>. [BG04] "The Static Diffie-Hellman Problem", <https://eprint.iacr.org/2004/306>. Davidson, et al. Expires January 14, 2021 [Page 31]

Internet-Draft OPRFs July 2020 [ChaumBlindSignature] "Blind Signatures for Untraceable Payments", <http://sceweb.sce.uhcl.edu/yang/teaching/ csci5234WebSecurityFall2011/Chaum-blind-signatures.PDF>. [ChaumPedersen] "Wallet Databases with Observers", <https://chaum.com/publications/Wallet_Databases.pdf>. [Cheon06] "Security Analysis of the Strong Diffie-Hellman Problem", <https://www.iacr.org/archive/ eurocrypt2006/40040001/40040001.pdf>. [DECAF] "Decaf, Eliminating cofactors through point compression", <https://www.shiftleft.org/papers/decaf/decaf.pdf>. [DGSTV18] "Privacy Pass, Bypassing Internet Challenges Anonymously", <https://www.degruyter.com/view/j/popets.2018.2018.issue- 3/popets-2018-0026/popets-2018-0026.xml>. [draft-davidson-pp-protocol] Davidson, A., "Privacy Pass: The Protocol", n.d., <https://tools.ietf.org/html/draft-davidson-pp-protocol- 00>. [I-D.irtf-cfrg-hash-to-curve] Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg- hash-to-curve-09 (work in progress), June 2020. [JKK14] "Round-Optimal Password-Protected Secret Sharing and T-PAKE in the Password-Only model", <https://eprint.iacr.org/2014/650>. [JKKX16] "Highly-Efficient and Composable Password-Protected Secret Sharing (Or, How to Protect Your Bitcoin Wallet Online)", <https://eprint.iacr.org/2016/144>. [JKKX17] "TOPPSS: Cost-minimal Password-Protected Secret Sharing based on Threshold OPRF", <https://eprint.iacr.org/2017/363>. [keytrans] "Security Through Transparency", <https://security.googleblog.com/2017/01/security-through- transparency.html>. Davidson, et al. Expires January 14, 2021 [Page 32]

Internet-Draft OPRFs July 2020 [NIST] "Keylength - NIST Report on Cryptographic Key Length and Cryptoperiod (2016)", <https://www.keylength.com/en/4/>. [OPAQUE] "The OPAQUE Asymmetric PAKE Protocol", <https://tools.ietf.org/html/draft-krawczyk-cfrg-opaque- 02>. [PrivacyPass] "Privacy Pass", <https://github.com/privacypass/challenge-bypass-server>. [RFC2104] Krawczyk, H., Bellare, M., and R. Canetti, "HMAC: Keyed- Hashing for Message Authentication", RFC 2104, DOI 10.17487/RFC2104, February 1997, <https://www.rfc-editor.org/info/rfc2104>. [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>. [RFC5869] Krawczyk, H. and P. Eronen, "HMAC-based Extract-and-Expand Key Derivation Function (HKDF)", RFC 5869, DOI 10.17487/RFC5869, May 2010, <https://www.rfc-editor.org/info/rfc5869>. [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>. [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>. [RISTRETTO] "The ristretto255 Group", <https://tools.ietf.org/html/ draft-hdevalence-cfrg-ristretto-01>. [SEC1] Standards for Efficient Cryptography Group (SECG), ., "SEC 1: Elliptic Curve Cryptography", <https://www.secg.org/sec1-v2.pdff>. [SEC2] Standards for Efficient Cryptography Group (SECG), ., "SEC 2: Recommended Elliptic Curve Domain Parameters", <http://www.secg.org/sec2-v2.pdf>. Davidson, et al. Expires January 14, 2021 [Page 33]

Internet-Draft OPRFs July 2020 [SHAKE] "SHA-3 Standard, Permutation-Based Hash and Extendable- Output Functions", <https://www.nist.gov/publications/sha- 3-standard-permutation-based-hash-and-extendable-output- functions?pub_id=919061>. [SJKS17] "SPHINX, A Password Store that Perfectly Hides from Itself", <https://eprint.iacr.org/2018/695>. [x9.62] ANSI, "Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI X9.62-1998, September 1998. 9.2. URIs [1] https://tools.ietf.org/html/draft-irtf-cfrg-voprf-04 [2] https://tools.ietf.org/html/draft-irtf-cfrg-voprf-03 [3] https://tools.ietf.org/html/draft-irtf-cfrg-voprf-02 [4] https://tools.ietf.org/html/draft-irtf-cfrg-voprf-01 Authors' Addresses Alex Davidson Cloudflare County Hall London, SE1 7GP United Kingdom Email: alex.davidson92@gmail.com Nick Sullivan Cloudflare 101 Townsend St San Francisco United States of America Email: nick@cloudflare.com Davidson, et al. Expires January 14, 2021 [Page 34]

Internet-Draft OPRFs July 2020 Christopher A. Wood Cloudflare 101 Townsend St San Francisco United States of America Email: caw@heapingbits.net Davidson, et al. Expires January 14, 2021 [Page 35]