Network Working Group A. Davidson Internet-Draft ISG, Royal Holloway, University of London Intended status: Informational N. Sullivan Expires: April 25, 2019 Cloudflare C. Wood Apple Inc. October 22, 2018 Verifiable Oblivious Pseudorandom Functions (VOPRFs) in Prime-Order Groups draft-sullivan-cfrg-voprf-02 Abstract A Verifiable Oblivious Pseudorandom Function (VOPRF) is a two-party protocol for computing the output of a PRF that is symmetrically verifiable. In summary, the PRF key holder learns nothing of the input while simultaneously providing proof that its private key was used during execution. VOPRFs are useful for computing one-time unlinkable tokens that are verifiable by secret key holders. This document specifies a VOPRF construction instantiated within prime- order subgroups, 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 http://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 April 25, 2019. Copyright Notice Copyright (c) 2018 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 April 25, 2019 [Page 1]

Internet-Draft VOPRFs October 2018 (http://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 . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3 1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 4 2. Background . . . . . . . . . . . . . . . . . . . . . . . . . 4 3. Security Properties . . . . . . . . . . . . . . . . . . . . . 4 4. VOPRF Protocol . . . . . . . . . . . . . . . . . . . . . . . 5 4.1. Instantiations of GG . . . . . . . . . . . . . . . . . . 6 4.2. Algorithmic Details . . . . . . . . . . . . . . . . . . . 7 4.2.1. VOPRF_Blind . . . . . . . . . . . . . . . . . . . . . 7 4.2.2. VOPRF_Sign . . . . . . . . . . . . . . . . . . . . . 8 4.2.3. VOPRF_Unblind . . . . . . . . . . . . . . . . . . . . 8 4.2.4. VOPRF_Finalize . . . . . . . . . . . . . . . . . . . 8 5. NIZK Discrete Logarithm Equality Proof . . . . . . . . . . . 9 5.1. DLEQ_Generate . . . . . . . . . . . . . . . . . . . . . . 9 5.2. DLEQ_Verify . . . . . . . . . . . . . . . . . . . . . . . 10 5.3. Elliptic Curve Group and Hash Function Instantiations . . 10 6. Security Considerations . . . . . . . . . . . . . . . . . . . 12 6.1. Timing Leaks . . . . . . . . . . . . . . . . . . . . . . 13 6.2. Hashing to curves . . . . . . . . . . . . . . . . . . . . 13 7. Privacy Considerations . . . . . . . . . . . . . . . . . . . 13 7.1. Key Consistency . . . . . . . . . . . . . . . . . . . . . 13 8. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 13 9. Normative References . . . . . . . . . . . . . . . . . . . . 13 Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 15 Appendix B. Applications . . . . . . . . . . . . . . . . . . . . 17 B.1. Privacy Pass . . . . . . . . . . . . . . . . . . . . . . 17 B.2. Private Password Checker . . . . . . . . . . . . . . . . 18 B.2.1. Parameter Commitments . . . . . . . . . . . . . . . . 18 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 18 1. Introduction A pseudorandom function (PRF) F(k, x) is an efficiently computable function with secret key k on input x. Roughly, F is pseudorandom if the output y = F(k, x) is indistinguishable from uniformly sampling any element in F's range for random choice of k. An oblivious PRF (OPRF) is a two-party protocol between a prover P and verifier V where P holds a PRF key k and V holds some input x. The protocol Davidson, et al. Expires April 25, 2019 [Page 2]

Internet-Draft VOPRFs October 2018 allows both parties to cooperate in computing F(k, x) with P's secret key k and V's input x such that: V learns F(k, x) without learning anything about k; and P does not learn anything about x. A Verifiable OPRF (VOPRF) is an OPRF wherein P can prove to V that F(k, x) was computed using key k, which is bound to a trusted public key Y = kG. Informally, this is done by presenting a non-interactive zero- knowledge (NIZK) proof of equality between (G, Y) and (Z, M), where Z = kM for some point M. VOPRFs are useful for producing tokens that are verifiable by V. This may be needed, for example, if V wants assurance that P did not use a unique key in its computation, i.e., if V wants key consistency from P. This property is necessary in some applications, e.g., the Privacy Pass protocol [PrivacyPass], wherein this VOPRF is used to generate one-time authentication tokens to bypass CAPTCHA challenges. This document introduces a VOPRF protocol built in prime-order groups. This applies to finite fields of prime-order and also elliptic curve (EC) settings. In the EC setting, we will refer to the protocol as ECVOPRF. The document describes the protocol, its security properties, and provides preliminary test vectors for experimentation. This rest of document is structured as follows: o Section Section 2: Describe background, related work, and use cases of VOPRF protocols. o Section Section 3: Discuss security properties of VOPRFs. o Section Section 4: Specify a VOPRF protocol based in prime-order groups. o Section Section 5: Specify the NIZK discrete logarithm equality construction used for verifying protocol outputs. 1.1. Terminology The following terms are used throughout this document. o PRF: Pseudorandom Function. o OPRF: Oblivious PRF. o VOPRF: Verifiable Oblivious Pseudorandom Function. o ECVOPRF: A VOPRF built on Elliptic Curves. o Verifier (V): Protocol initiator when computing F(k, x). Davidson, et al. Expires April 25, 2019 [Page 3]

Internet-Draft VOPRFs October 2018 o Prover (P): Holder of secret key k. o NIZK: Non-interactive zero knowledge. o DLEQ: Discrete Logarithm Equality. 1.2. 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. Background VOPRFs are functionally related to RSA-based blind signature schemes, e.g., [ChaumBlindSignature]. Such a scheme works as follows. Let m be a message to be signed by a server. It is assumed to be a member of the RSA group. Also, let N be the RSA modulus, and e and d be the public and private keys, respectively. A prover P and verifier V engage in the following protocol given input m. 1. V generates a random blinding element r from the RSA group, and compute m' = m^r (mod N). Send m' to the P. 2. P uses m' to compute s' = (m')^d (mod N), and sends s' to the V. 3. V removes the blinding factor r to obtain the original signature as s = (s')^(r^-1) (mod N). By the properties of RSA, s is clearly a valid signature for m. OPRF protocols are the symmetric equivalent to blind signatures in the same way that PRFs are the symmetric equivalent traditional digital signatures. This is discussed more in the following section. 3. Security Properties The security properties of a VOPRF protocol with functionality y = F(k, x) include those of a standard PRF. Specifically: o Given value x, it is infeasible to compute y = F(k, x) without knowledge of k. o Output y = F(k, x) is indistinguishable from a random value in the domain of F. Additionally, we require the following additional properties: Davidson, et al. Expires April 25, 2019 [Page 4]

Internet-Draft VOPRFs October 2018 o Non-malleable: Given (x, y = F(k, x)), V must not be able to generate (x', y') where x' != x and y' = F(k, x'). o Verifiable: V must only complete execution of the protocol if it asserts that P used its secret key k, associated with public key Y = kG, in execution. o Oblivious: P must learn nothing about V's input, and V must learn nothing about P's private key. o Unlinkable: If V reveals x to P, P cannot link x to the protocol instance in which y = F(k, x) was computed. 4. VOPRF Protocol In this section we describe the VOPRF protocol. Let GG be a prime- order additive subgroup, with two distinct hash functions H_1 and H_2, where H_1 maps arbitrary input onto GG and H_2 maps arbitrary input to a fixed-length output, e.g., SHA256. All hash functions in the protocol are assumed to be random oracles. Let L be the security parameter. Let k be the prover's (P) secret key, and Y = kG be its corresponding public key for some generator G taken from the group GG. Let x be the verifier's (V) input to the VOPRF protocol. (Commonly, it is a random L-bit string, though this is not required.) VOPRF begins with V randomly blinding its input for the signer. The latter then applies its secret key to the blinded value and returns the result. To finish the computation, V then removes its blind and hashes the result using H_2 to yield an output. This flow is illustrated below. Verifier Prover ------------------------------------ r <-$ GG M = rH_1(x) M -------> Z = kM D = DLEQ_Generate(G,Y,M,Z) Z,D <------- b = DLEQ_Verify(G,Y,M,Z,D) Output H_2(x, Zr^(-1)) if b=1, else "error" DLEQ_Generate(G,Y,M,Z) and DLEQ_Verify(G,Y,M,Z,D) are described in Section Section 5. Intuitively, the DLEQ proof allows P to prove to V in NIZK that the same key k is the exponent of both Y and M. In other words, computing the discrete logarithm of Y and Z (with Davidson, et al. Expires April 25, 2019 [Page 5]

Internet-Draft VOPRFs October 2018 respect to G and M, respectively) results in the same value. The committed value Y should be public before the protocol is initiated. The actual PRF function computed is as follows: F(k, x) = H_2(x, N) = H_2(x, kH_1(x)) Note that V finishes this computation upon receiving kH_1(x) from P. The output from P is not the PRF value. This protocol may be decomposed into a series of steps, as described below: o VOPRF_Blind(x): Compute and return a blind, r, and blinded representation of x, denoted M. o VOPRF_Sign(M): Sign input M using secret key k to produce Z, generate a proof D = DLEQ_Generate(G,Y,M,Z), and output (Z, D). o VOPRF_Unblind((Z, D), r, Y, G, M): Unblind blinded signature Z with blind r, yielding N. Output N if 1 = DLEQ_Verify(G,Y,M,Z,D). Otherwise, output "error". o VOPRF_Finalize(N): Finalize N to produce PRF output F(k, x). Protocol correctness requires that, for any key k, input x, and (r, M) = VOPRF_Blind(x), it must be true that: VOPRF_Finalize(x, VOPRF_Unblind(VOPRF_Sign(M), M, r)) = F(k, x) with overwhelming probability. 4.1. Instantiations of GG As we remarked above, GG is a subgroup with associated prime-order p. While we choose to write operations in the setting where GG comes equipped with an additive operation, we could also define the operations in the multiplicative setting. In the multiplicative setting we can choose GG to be a prime-order subgroup of a finite field FF_p. For example, let p be some large prime (e.g. > 2048 bits) where p = 2q+1 for some other prime q. Then the subgroup of squares of FF_p (elements u^2 where u is an element of FF_p) is cyclic, and we can pick a generator of this subgroup by picking g from FF_p (ignoring the identity element). In this document, however, we are going to focus on the cases where GG is indeed an additive subgroup. In the elliptic curve setting, this amounts to choosing GG to be a prime-order subgroup of an Davidson, et al. Expires April 25, 2019 [Page 6]

Internet-Draft VOPRFs October 2018 elliptic curve over base field GF(p) for prime p. There are also other settings where GG is a prime-order subgroup of an elliptic curve over a base field of non-prime order, these include the work of Ristretto [RISTRETTO] and Decaf [DECAF]. We will use p > 0 generally for constructing the base field GF(p), not just those where p is prime. To reiterate, we focus only on the additive case, and so we focus only on the cases where GF(p) is indeed the base field. 4.2. Algorithmic Details This section provides algorithms for each step in the VOPRF protocol. 1. V computes X = H_1(x) and a random element r (blinding factor) from GF(p), and computes M = rX. 2. V sends M to P. 3. P computes Z = kM = rkX, and D = DLEQ_Generate(G,Y,M,Z). 4. P sends (Z, D) to V. 5. V ensures that 1 = DLEQ_Verify(G,Y,M,Z,D). If not, V outputs an error. 6. V unblinds Z to compute N = r^(-1)Z = kX. 7. V outputs the pair H_2(x, N). 4.2.1. VOPRF_Blind Input: x - V's PRF input. Output: r - Random scalar in [1, p - 1]. M - Blinded representation of x using blind r, a point in GG. Steps: 1. r <-$ GF(p) 2. M := rH_1(x) 5. Output (r, M) Davidson, et al. Expires April 25, 2019 [Page 7]

Internet-Draft VOPRFs October 2018 4.2.2. VOPRF_Sign Input: G: Public generator of group GG. Y: Signer public key. M - Point in GG. Output: Z - Scalar multiplication of k and M, point in GG. D - DLEQ proof that log_G(Y) == log_M(Z). Steps: 1. Z := kM 2. D = DLEQ_Generate(G,Y,M,Z) 2. Output (Z, D) 4.2.3. VOPRF_Unblind Input: G: Public generator of group GG. Y: Signer public key. M - Blinded representation of x using blind r, a point in GG. Z - Point in GG. D - D = DLEQ_Generate(G,Y,M,Z). r - Random scalar in [1, p - 1]. Output: N - Unblinded signature, point in GG. Steps: 1. N := (-r)Z 2. If 1 = DLEQ_Verify(G,Y,M,Z,D), output N 3. Output "error" 4.2.4. VOPRF_Finalize Davidson, et al. Expires April 25, 2019 [Page 8]

Internet-Draft VOPRFs October 2018 Input: x - PRF input string. N - Point in GG, or "error". Output: y - Random element in {0,1}^L, or "error" Steps: 1. If N == "error", output "error". 2. y := H_2(x, N) 3. Output y 5. NIZK Discrete Logarithm Equality Proof In some cases, it may be desirable for the V to have proof that P used its private key to compute Z from M. This is done by proving log_G(Y) == log_M(Z). This may be used, for example, to ensure that P uses the same private key for computing the VOPRF output and does not attempt to "tag" individual verifiers with select keys. This proof must not reveal the P's long-term private key to V. Consequently, we extend the protocol in the previous section with a (non-interactive) discrete logarithm equality (DLEQ) algorithm built on a Chaum-Pedersen [ChaumPedersen] proof. This proof is divided into two procedures: DLEQ_Generate and DLEQ_Verify. These are specified below. 5.1. DLEQ_Generate Davidson, et al. Expires April 25, 2019 [Page 9]

Internet-Draft VOPRFs October 2018 Input: G: Public generator of group GG. Y: Signer public key. M: Point in GG. Z: Point in GG. H_3: A hash function from GG to a bitstring of length L modeled as a random oracle. Output: D: DLEQ proof (c, s). Steps: 1. r <-$ GF(p) 2. A = rG and B = rM. 2. c = H_3(G,Y,M,Z,A,B) 3. s = (r - ck) (mod p) 4. Output D = (c, s) 5.2. DLEQ_Verify Input: G: Public generator of group GG. Y: Signer public key. M: Point in GG. Z: Point in GG. D: DLEQ proof (c, s). Output: True if log_G(Y) == log_M(Z), False otherwise. Steps: 1. A' = (sG + cY) 2. B' = (sM + cZ) 3. c' = H_3(G,Y,M,Z,A',B') 4. Output c == c' 5.3. Elliptic Curve Group and Hash Function Instantiations This section specifies supported ECVOPRF group and hash function instantiations. We focus on the instantiations of the VOPRF in the elliptic curve setting for now. Eventually, we would like to provide instantiations based on curves over non-prime-order base fields. Davidson, et al. Expires April 25, 2019 [Page 10]

Internet-Draft VOPRFs October 2018 ECVOPRF-P256-SHA256: o G: P-256 o H_1: Simplified SWU encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA256 o H_3: SHA256 ECVOPRF-P256-SHA512: o G: P-256 o H_1: Simplified SWU encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA512 o H_3: SHA512 ECVOPRF-P384-SHA256: o G: P-384 o H_1: Icart encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA256 o H_3: SHA256 ECVOPRF-P384-SHA512: o G: P-384 o H_1: Icart encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA512 o H_3: SHA512 ECVOPRF-CURVE25519-SHA256: o G: Curve25519 [RFC7748] o H_1: Elligator2 encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA256 Davidson, et al. Expires April 25, 2019 [Page 11]

Internet-Draft VOPRFs October 2018 o H_3: SHA256 ECVOPRF-CURVE25519-SHA512: o G: Curve25519 [RFC7748] o H_1: Elligator2 encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA512 o H_3: SHA512 ECVOPRF-CURVE448-SHA256: o G: Curve448 [RFC7748] o H_1: Elligator2 encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA256 o H_3: SHA256 ECVOPRF-CURVE448-SHA512: o G: Curve448 [RFC7748] o H_1: Elligator2 encoding [I-D.irtf-cfrg-hash-to-curve] o H_2: SHA512 o H_3: SHA512 6. Security Considerations Security of the protocol depends on P's secrecy of k. Best practices recommend P regularly rotate k so as to keep its window of compromise small. Moreover, it each key should be generated from a source of safe, cryptographic randomness. Another critical aspect of this protocol is reliance on [I-D.irtf-cfrg-hash-to-curve] for mapping arbitrary input to points on a curve. Security requires this mapping be pre-image and collision resistant. Davidson, et al. Expires April 25, 2019 [Page 12]

Internet-Draft VOPRFs October 2018 6.1. 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: H_1() (hashing arbitrary strings to curves) and DLEQ_Generate(). [I-D.irtf-cfrg-hash-to-curve] describes various algorithms for constant-time implementations of H_1. 6.2. Hashing to curves We choose different encodings in relation to the elliptic curve that is used, all methods are illuminated precisely in [I-D.irtf-cfrg-hash-to-curve]. In summary, we use the simplified Shallue-Woestijne-Ulas algorithm for hashing binary strings to the P-256 curve; the Icart algorithm for hashing binary strings to P384; the Elligator2 algorithm for hashing binary strings to CURVE25519 and CURVE448. 7. Privacy Considerations 7.1. Key Consistency DLEQ proofs are essential to the protocol to allow V to check that P's designated private key was used in the computation. A side effect of this property is that it prevents P from using unique key for select verifiers as a way of "tagging" them. If all verifiers expect use of a certain private key, e.g., by locating P's public key key published from a trusted registry, then P cannot present unique keys to an individual verifier. 8. Acknowledgements This document resulted from the work of the Privacy Pass team [PrivacyPass]. 9. Normative References [ChaumBlindSignature] "Blind Signatures for Untraceable Payments", n.d., <http://sceweb.sce.uhcl.edu/yang/teaching/ csci5234WebSecurityFall2011/Chaum-blind-signatures.PDF>. [ChaumPedersen] "Wallet Databases with Observers", n.d., <https://chaum.com/publications/Wallet_Databases.pdf>. Davidson, et al. Expires April 25, 2019 [Page 13]

Internet-Draft VOPRFs October 2018 [DECAF] "Decaf, Eliminating cofactors through point compression", n.d., <https://www.shiftleft.org/papers/decaf/decaf.pdf>. [DGSTV18] "Privacy Pass, Bypassing Internet Challenges Anonymously", n.d., <https://www.degruyter.com/view/j/ popets.2018.2018.issue-3/popets-2018-0026/popets- 2018-0026.xml>. [I-D.irtf-cfrg-hash-to-curve] Scott, S., Sullivan, N., and C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg-hash-to-curve-01 (work in progress), July 2018. [JKK14] "Round-Optimal Password-Protected Secret Sharing and T-PAKE in the Password-Only model", n.d., <https://eprint.iacr.org/2014/650.pdf>. [JKKX16] "Highly-Efficient and Composable Password-Protected Secret Sharing (Or, How to Protect Your Bitcoin Wallet Online)", n.d., <https://eprint.iacr.org/2016/144>. [PrivacyPass] "Privacy Pass", n.d., <https://github.com/privacypass/ challenge-bypass-server>. [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>. [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>. [RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, January 2017, <https://www.rfc- editor.org/info/rfc8032>. [RISTRETTO] "The Ristretto Group", n.d., <https://ristretto.group/ ristretto.html>. [SJKS17] "SPHINX, A Password Store that Perfectly Hides from Itself", n.d., <http://webee.technion.ac.il/%7Ehugo/sphinx.pdf>. Davidson, et al. Expires April 25, 2019 [Page 14]

Internet-Draft VOPRFs October 2018 Appendix A. Test Vectors This section includes test vectors for the primary ECVOPRF protocol, excluding DLEQ output. ((TODO: add DLEQ vectors)) P-224 X: 0403cd8bc2f2f3c4c647e063627ca9c9ac246e3e3ec74ab76d32d3e999c522d60ff7aa1c8b0e4 \ X: 0403cd8bc2f2f3c4c647e063627ca9c9ac246e3e3ec74ab76d32d3e999c522d60ff7aa1c8b0e4 r: c4cf3c0b3a334f805d3ce3c3b4d007fbbdaf078a42a8cbdc833e54a9 M: 046b2b8482c36e65f87528415e210cff8561c1c8e07600a159893973365617ee2c1c33eb0662d \ M: 046b2b8482c36e65f87528415e210cff8561c1c8e07600a159893973365617ee2c1c33eb0662d k: a364119e1c932a534a8d440fef2169a0e4c458d702eca56746655845 Z: 04ed11656b4981e39242b170025bf8d5314bef75006e6c4c9afcdb9a85e21fb5fcf9055eb95d3 \ Z: 04ed11656b4981e39242b170025bf8d5314bef75006e6c4c9afcdb9a85e21fb5fcf9055eb95d3 Y: 04fd80db5301a54ee2cbc688d47cbcae9eb84f5d246e3da3e2b03e9be228ed6c57a936b6b5faf \ Y: 04fd80db5301a54ee2cbc688d47cbcae9eb84f5d246e3da3e2b03e9be228ed6c57a936b6b5faf P-224 X: 0429e41b7e1a58e178afc522d0fb4a6d17ae883e6fd439931cf1e81456ab7ed6445dbe0a231be \ X: 0429e41b7e1a58e178afc522d0fb4a6d17ae883e6fd439931cf1e81456ab7ed6445dbe0a231be r: 86a27e1bd51ac91eae32089015bf903fe21da8d79725edcc4dc30566 M: 04d8c8ffaa92b21aa1cc6056710bd445371e8afebd9ef0530c68cd0d09536423f78382e4f6b20 \ M: 04d8c8ffaa92b21aa1cc6056710bd445371e8afebd9ef0530c68cd0d09536423f78382e4f6b20 k: ab449c896261dc3bd1f20d87272e6c8184a2252a439f0b3140078c3d Z: 048ac9722189b596ffe5cb986332e89008361e68f77f12a931543f63eaa01fabf6f63d5d4b3b6 \ Z: 048ac9722189b596ffe5cb986332e89008361e68f77f12a931543f63eaa01fabf6f63d5d4b3b6 Y: 046e83dff2c9b6f9e88f1091f355ad6fa637bdbd829072411ea2d74a5bf3501ccf3bcc2789d48 \ Y: 046e83dff2c9b6f9e88f1091f355ad6fa637bdbd829072411ea2d74a5bf3501ccf3bcc2789d48 P-256 X: 041b0e84c521f8dcd530d59a692d4ffa1ca05b8ba7ce22a884a511f93919ac121cc91dd588228 \ X: 041b0e84c521f8dcd530d59a692d4ffa1ca05b8ba7ce22a884a511f93919ac121cc91dd588228 r: a3ec1dc3303a316fc06565ace0a8910da65cf498ea3884c4349b6c4fc9a2f99a M: 04794c5a54236782088594ccdb1975e93b05ff742674cc400cb101f55c0f37e877c5ada0d72fb \ M: 04794c5a54236782088594ccdb1975e93b05ff742674cc400cb101f55c0f37e877c5ada0d72fb k: 9c103b889808a8f4cb6d76ea8b634416a286be7fa4a14e94f1478ada7f172ec3 Z: 0484cfda0fdcba7693672fe5e78f4c429c096ece730789e8d00ec1f7be33a6515f186dcf7aa38 \ Z: 0484cfda0fdcba7693672fe5e78f4c429c096ece730789e8d00ec1f7be33a6515f186dcf7aa38 Y: 044ff2e31de9fda542c2c63314e2bce5ce2d5ccb8332dbe1115ff5740e5e60bb867994e196ead \ Y: 044ff2e31de9fda542c2c63314e2bce5ce2d5ccb8332dbe1115ff5740e5e60bb867994e196ead P-256 X: 043ea9d81b99ac0db002ad2823f7cab28af18f83419cce6800f3d786cc00b6fd030858d073916 \ X: 043ea9d81b99ac0db002ad2823f7cab28af18f83419cce6800f3d786cc00b6fd030858d073916 r: ed7294b42792760825645b635e9d92ef5a3baa70879dd59fdb1802d4a44271b2 M: 04ec894e496d0297756a17365f866d9449e6ebc51852ab1ffa57bc29c843ef003b116f5ef1f60 \ Davidson, et al. Expires April 25, 2019 [Page 15]

Internet-Draft VOPRFs October 2018 M: 04ec894e496d0297756a17365f866d9449e6ebc51852ab1ffa57bc29c843ef003b116f5ef1f60 k: a324338a7254415dbedcd1855abd2503b4e5268124358d014dac4fc2c722cd67 Z: 04a477c5fefd9bc6bcd8e893a1b0c6dc73b0bd23ebe952dcad810de73b8a99f5e1e216a833b32 \ Z: 04a477c5fefd9bc6bcd8e893a1b0c6dc73b0bd23ebe952dcad810de73b8a99f5e1e216a833b32 Y: 04ffe55e2a95a21e1605c1ed11ac6bea93f00fa15a6b27e90adad470ad27f0e0fe5b8607b4689 \ Y: 04ffe55e2a95a21e1605c1ed11ac6bea93f00fa15a6b27e90adad470ad27f0e0fe5b8607b4689 P-384 X: 04c0b51e5dcd6a309c77bb5720bf9850279e8142b6127952595ab9092578de810a13795bceff3 \ d358f0480a61469f17ad62ebaecd0f817c1e9c7d41d536ab410e7a2b5d7a7905d1bef5499b654b0e \ d358f0480a61469f17ad62ebaecd0f817c1e9c7d41d536ab410e7a2b5d7a7905d1bef5499b654b0e r: 889b5e4812d683c4df735971240741ff869ccf77e10c2e97bef67d6fe6b8350abe59ec8fe2bfa \ r: 889b5e4812d683c4df735971240741ff869ccf77e10c2e97bef67d6fe6b8350abe59ec8fe2bfa M: 044e2d86fa6e53ebba7f2a9b661a2de884a8ccc68e29b68586d517eb66e8b4b7dac334c6e769d \ 485d672fac3a0311877572254754e318077aec3631208c6b503c5cdfe57716e1232da64cebe46f0d \ 485d672fac3a0311877572254754e318077aec3631208c6b503c5cdfe57716e1232da64cebe46f0d k: b8c854a33c8c564d0598b1ac179546acdccad671265cff1ea5a329279272e8d21c94b7e5b6bea \ k: b8c854a33c8c564d0598b1ac179546acdccad671265cff1ea5a329279272e8d21c94b7e5b6bea Z: 047bf23eef00e83e6cb6fb9ade5e5995cf81abb8dc73a570ff4cb7be48f21281edfed9bf76cc2 \ 88b35d2df615ff711ed2a1fb85cd0b22812438665cdd300039685b3f593f4b574f9e8b294982c2a2 \ 88b35d2df615ff711ed2a1fb85cd0b22812438665cdd300039685b3f593f4b574f9e8b294982c2a2 Y: 04ab4886ecf7e489a0be8529ff4b537941c95ba4ce570db537dcfad5cabc064c43f1b0a1d1b89 \ 101facd93f2f9a8b5f28431489be4664f446af8a51cc7c4221f633adb4f8f2f2a073dfd83ddf8d77 \ 101facd93f2f9a8b5f28431489be4664f446af8a51cc7c4221f633adb4f8f2f2a073dfd83ddf8d77 P-384 X: 047511a846277a2009f37b3583f14c8ea3af17e3a146e0e737fdc1260b6d4a18ff01f21ec3bbc \ e39e1cade76d455feadc1bb16f65cd54042e1bc5aba1dee2434f59d00698a963b902148750240f8f \ e39e1cade76d455feadc1bb16f65cd54042e1bc5aba1dee2434f59d00698a963b902148750240f8f r: e514ef9b3ea87eafdb78da48e642daa79f036ac00228997ab8da6ac198fb888cd2fec84d52010 \ r: e514ef9b3ea87eafdb78da48e642daa79f036ac00228997ab8da6ac198fb888cd2fec84d52010 M: 04fd9b68973b0fcefcf4458b4faa1c3815bdad8975b7fb0bfc4c1db7e3f169fb3a26ddabe1b25 \ c4a23cf8a2faeb12c18f06f2227e87ede6039f55a61ef0c89ca3c582e2864fe130ea0c709f92519d \ c4a23cf8a2faeb12c18f06f2227e87ede6039f55a61ef0c89ca3c582e2864fe130ea0c709f92519d k: bcc73da3b2adace9c4f4c32eeadef57436c97f8d395614e78aa91523e1e5d7f551ebb55e87da2 \ k: bcc73da3b2adace9c4f4c32eeadef57436c97f8d395614e78aa91523e1e5d7f551ebb55e87da2 Z: 042d885d2945cde40e490dd8497975eaeb54e4e10c5986a9688c9de915b16cf43572fd155e159 \ 9e2233a75056a72b54d30092e30bb2edc70e0d90da934c27362e0e6303bafae12f13bf3d5be322e6 \ 9e2233a75056a72b54d30092e30bb2edc70e0d90da934c27362e0e6303bafae12f13bf3d5be322e6 Y: 044833fba4973c1c6eae6745850866ebbb23783ea0d4d8b867e2c93acb2f01fd3d36d9cb5c491 \ ff9440c8c8e325db326bf88ddf0ba6008158a67999e18cd378d701d1f8a6a7b088dc261c85b6a78b \ ff9440c8c8e325db326bf88ddf0ba6008158a67999e18cd378d701d1f8a6a7b088dc261c85b6a78b P-521 X: 040039d290b20c604b5c59cb85dfacd90cbf9f5e275ee8c38a8ff80df0872e8e1dd214a9ec3b2 \ 2c8d75bf634739afdc09acc342542abacf35bf2a6488d084825c5d96003be29e201e75c1b78667f5 \ a64cc7207722796b225b49edaaf457fcafff4f644252ebe8057291d317f30109f1526aacbfff2308 \ a64cc7207722796b225b49edaaf457fcafff4f644252ebe8057291d317f30109f1526aacbfff2308 Davidson, et al. Expires April 25, 2019 [Page 16]

Internet-Draft VOPRFs October 2018 r: 010606612666705556ac3c28dde30f134e930b0c31bfc9715f0812e0b99f0212dc427e344cb97 \ r: 010606612666705556ac3c28dde30f134e930b0c31bfc9715f0812e0b99f0212dc427e344cb97 M: 040065366112a0598e4e5997e79e42f287f7202e5d956bef29890e963169d9eaab8d21501283c \ 47dd37aca1710c8b5f456b1c044c8582ba6feef3edc997fecef7d4f40180ceb9bbbe3ab1907ea2d1 \ 21ec00156848e04e323744d86444111fc09a21ca316df2cae925a0bb079d0faa2474ec8d5a96e6fa \ 21ec00156848e04e323744d86444111fc09a21ca316df2cae925a0bb079d0faa2474ec8d5a96e6fa k: 01297d92cfe6895269aa5406f2ba6cbfffbba66a11ab0db34655213624fa238c50e27177aea5d \ k: 01297d92cfe6895269aa5406f2ba6cbfffbba66a11ab0db34655213624fa238c50e27177aea5d Z: 040151d2dc5290ebd47065680dcb4db350c4d81346680c5589f94acfb1e28418585e7f2cbfa11 \ 5945d9f7b98157ea8c2ab190c6a47b939502c2f793b77ceff671f5e60086fdd1ebf960f29bf5d590 \ f8f7511d248df22d964637e2286adab4654991d338691f4673a006ff116e61afe65c914b27c3ef4c \ f8f7511d248df22d964637e2286adab4654991d338691f4673a006ff116e61afe65c914b27c3ef4c Y: 04009534bd720bd4ebe703968a8496eec352711a81b7fe594a72ef318c2ce582b41880262a1c6 \ 05079231de91e71b1301d1be4e9618e96081ccfd4f6cab92f52b860e01beec2c58cb01713e941035 \ adbe882ab4f3eaa31e27a96d210d35f6161b1487dd28d8da4a11a915182752b1450a89aad2a013c2 \ adbe882ab4f3eaa31e27a96d210d35f6161b1487dd28d8da4a11a915182752b1450a89aad2a013c2 P-521 X: 04012ea416842dfad169a9eb860b0af2af3c0140e1918ccd043650d83ad261633f20c5ca02c1b \ ffb857ab72814cf46cfc16ac9ba79887044709f72480358c4b990e46010a62336bb57b87b494b064 \ 4d2b6a385f3d5b5da29e22cae33c624f561513a5e8e6669b4e99704c56157dde83994a3c0800a64b \ 4d2b6a385f3d5b5da29e22cae33c624f561513a5e8e6669b4e99704c56157dde83994a3c0800a64b r: 019d02efd97add5facc5defbb63fd74daaacda04ae7321abec0da1551b4cc980b8ce6855a28a1 \ r: 019d02efd97add5facc5defbb63fd74daaacda04ae7321abec0da1551b4cc980b8ce6855a28a1 M: 040066e3d0b5b9758c9288a725ce6724fdc3bd797a8222f07233897a5916dc167531ebc6a4710 \ cbb240684c9a02eb82214b009d636f24abb8e409e78ff1f02a1dbfb90069056693e96acd760887f9 \ 6c9b1f487441b7142fb13a67deb7332194ff454b3aac89f9cf02c338dae69a700bd26844881e6106 \ 6c9b1f487441b7142fb13a67deb7332194ff454b3aac89f9cf02c338dae69a700bd26844881e6106 k: 018eeea896de104bf1e772155836f6ceddab0b4c2e3e4c33ba08a6fd6db0291cfb15faff0b3c7 \ k: 018eeea896de104bf1e772155836f6ceddab0b4c2e3e4c33ba08a6fd6db0291cfb15faff0b3c7 Z: 04016825ea754324d5761ada130a1b87b03b5e2a6b0f403343925c67df39bbf85bc782909124d \ d297a1edfb049efa7ce61c626c0ad99d8cf462abcce1ee1967d8a355011e2c5a7ce621fc822a7d95 \ bf7359d938ee4a5c3431e7dd270b7fb6e95fda29cf460d89454763bb0db9b8b705503170a9ac1c7a \ bf7359d938ee4a5c3431e7dd270b7fb6e95fda29cf460d89454763bb0db9b8b705503170a9ac1c7a Y: 04006b0413e2686c4bb62340706de7723471080093422f02dd125c3e72f3507b9200d11481468 \ 74bbaa5b6108b834c892eeebab4e21f3707ee20c303ebc1e34fcd3c701f2171131ee7c5f07c1ccad \ 240183d777181259761741343959d476bbc2591a1af0a516e6403a6b81423234746d7a2e8c2ce60a \ 240183d777181259761741343959d476bbc2591a1af0a516e6403a6b81423234746d7a2e8c2ce60a Appendix B. Applications This section describes various applications of the VOPRF protocol. B.1. Privacy Pass This VOPRF protocol is used by Privacy Pass system to help Tor users bypass CAPTCHA challenges. Their system works as follows. Client C connects - through Tor - to an edge server E serving content. Upon Davidson, et al. Expires April 25, 2019 [Page 17]

Internet-Draft VOPRFs October 2018 receipt, E serves a CAPTCHA to C, who then solves the CAPTCHA and supplies, in response, n blinded points. E verifies the CAPTCHA response and, if valid, signs (at most) n blinded points, which are then returned to C. When C attempts to connect to E again and is prompted with a CAPTCHA, C uses one of the unblinded and signed points, or tokens, to derive a shared symmetric key sk used to MAC the CAPTCHA challenge. C sends the CAPTCHA, MAC, and token input x to E, who can use x to derive sk and verify the CAPTCHA MAC. Thus, each token is used at most once by the system. The Privacy Pass implementation uses the P-256 instantiation of the VOPRF protocol. For more details, see [DGSTV18]. B.2. Private Password Checker In this application, let D be a collection of plaintext passwords obtained by prover P. For each password p in D, P computes VOPRF_Sign(H_1(p)), where H_1 is as described above, and stores the result in a separate collection D'. P then publishes D' with Y, its public key. If a client C wishes to query D' for a password p', it runs the VOPRF protocol using p as input x to obtain output y. By construction, y will be the signature of p hashed onto the curve. C can then search D' for y to determine if there is a match. Examples of such password checkers already exist, for example: [JKKX16], [JKK14] and [SJKS17]. B.2.1. Parameter Commitments For some applications, it may be desirable for P to bind tokens to certain parameters, e.g., protocol versions, ciphersuites, etc. To accomplish this, P should use a distinct scalar for each parameter combination. Upon redemption of a token T from V, P can later verify that T was generated using the scalar associated with the corresponding parameters. Authors' Addresses Alex Davidson ISG, Royal Holloway, University of London Egham Hill Twickenham, TW20 0EX United Kingdom Email: alex.davidson.2014@rhul.ac.uk Davidson, et al. Expires April 25, 2019 [Page 18]

```
Internet-Draft VOPRFs October 2018
Nick Sullivan
Cloudflare
101 Townsend St
San Francisco
United States of America
Email: nick@cloudflare.com
Christopher A. Wood
Apple Inc.
One Apple Park Way
Cupertino, California 95014
United States of America
Email: cawood@apple.com
Davidson, et al. Expires April 25, 2019 [Page 19]
```