Verifiable Oblivious Pseudorandom Functions (VOPRFs)
draft-sullivan-cfrg-voprf-00

Network Working Group                                        N. Sullivan
Internet-Draft                                                Cloudflare
Intended status: Informational                                   C. Wood
Expires: September 6, 2018                                    Apple Inc.
                                                          March 05, 2018

          Verifiable Oblivious Pseudorandom Functions (VOPRFs)
                    draft-sullivan-cfrg-voprf-00

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 based on Elliptic Curves.

Status of This Memo

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   include Simplified BSD License text as described in Section 4.e of
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   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   3
     1.2.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Background  . . . . . . . . . . . . . . . . . . . . . . . . .   4
   3.  Security Properties . . . . . . . . . . . . . . . . . . . . .   4
   4.  Elliptic Curve VOPRF Protocol . . . . . . . . . . . . . . . .   5
     4.1.  Algorithmic Details . . . . . . . . . . . . . . . . . . .   6
       4.1.1.  ECVOPRF_Blind . . . . . . . . . . . . . . . . . . . .   6
       4.1.2.  ECVOPRF_Sign  . . . . . . . . . . . . . . . . . . . .   7
       4.1.3.  ECVOPRF_Unblind . . . . . . . . . . . . . . . . . . .   7
       4.1.4.  ECVOPRF_Finalize  . . . . . . . . . . . . . . . . . .   8
   5.  NIZK Discrete Logarithm Equality Proof  . . . . . . . . . . .   8
     5.1.  DLEQ_Generate . . . . . . . . . . . . . . . . . . . . . .   9
     5.2.  DLEQ_Verify . . . . . . . . . . . . . . . . . . . . . . .   9
     5.3.  Group and Hash Function Instantiations  . . . . . . . . .   9
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .  11
     6.1.  Timing Leaks  . . . . . . . . . . . . . . . . . . . . . .  12
   7.  Privacy Considerations  . . . . . . . . . . . . . . . . . . .  12
     7.1.  Key Consistency . . . . . . . . . . . . . . . . . . . . .  12
   8.  Acknowledgments . . . . . . . . . . . . . . . . . . . . . . .  12
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  12
   10. Normative References  . . . . . . . . . . . . . . . . . . . .  12
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .  13
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  16

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
   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.

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   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 authentic tokens to bypass CAPTCHA challenges.

   This document introduces a VOPRF protocol built on Elliptic Curves,
   called ECVOPRF.  It 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 related, and use
      cases of VOPRF protocols.

   o  Section Section 3: Discuss security properties of VOPRFs.

   o  Section Section 4: Specify a VOPRF protocol based on elliptic
      curves.

   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  Verifier (V): Protocol initiator when computing F(k, x).

   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].

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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:

   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.

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4.  Elliptic Curve VOPRF Protocol

   In this section we describe the ECVOPRF protocol.  Let GG be a prime-
   order subgroup of an elliptic curve over base field GF(p) for prime
   p, 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.)  ECVOPRF 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 <-$ G
        M = rH_1(x)
                      M
                   ------->
                              Z = kM
                              D = DLEQ_Generate(Z/M == Y/G)
                      Z,D
                   <-------
       b = DLEQ_Verify(M, Z, D, Y)
       Output H_2(x, Zr^(-1)) if b=1, else "error"

   DLEQ(Z/M == Y/G) is 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 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:

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   o  ECVOPRF_Blind(x): Compute and return a blind, r, and blinded
      representation of x, denoted M.

   o  ECVOPRF_Sign(M): Sign input M using secret key k to produce Z,
      generate a proof D of DLEQ(Z/M == Y/G), and output (Z, D).

   o  ECVOPRF_Unblind((Z, D), r, Y, G, M): Unblind blinded signature Z
      with blind r, yielding N.  Output N if D is a valid proof.
      Otherwise, output an error.

   o  ECVOPRF_Finalize(N): Finalize N to produce PRF output F(k, x).

   Protocol correctness requires that, for any key k, input x, and (r,
   M) = ECVOPRF_Blind(x), it must be true that:

   ECVOPRF_Finalize(x, ECVOPRF_Unblind(ECVOPRF_Sign(M), M, r)) = F(k, x)

   with overwhelming probability.

4.1.  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(Z/M == Y/G).

   4.  P sends (Z, D) to V.

   5.  V verifies D using Y.  If invalid, V outputs an error.

   6.  V unblinds Z to compute N = r^(-1)Z = kX.

   7.  V outputs the pair H_2(x, N).

4.1.1.  ECVOPRF_Blind

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   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)

4.1.2.  ECVOPRF_Sign

   Input:

    M - Point in G.

   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(Y, G, M, Z)
    2. Output (Z, D)

4.1.3.  ECVOPRF_Unblind

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   Input:

    Z - Point in GG.
    D - DLEQ proof that log_G(Y) == log_M(Z).
    M - Blinded representation of x using blind r, a point in G.
    r - Random scalar in [1, p - 1].

   Output:

    N - Unblinded signature, point in GG.

   Steps:

    1. N := (-r)Z
    2. If DLEQ_Verify(G, Y, M, Z, D) output N
    3. Output "error"

4.1.4.  ECVOPRF_Finalize

   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.

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5.1.  DLEQ_Generate

Input:

  G: Generator of group GG with prime order p.
  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: Generator of group GG with prime order p.
     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,E,M,Z,A',B')
   4. Output c == c'

5.3.  Group and Hash Function Instantiations

   This section specifies supported VOPRF group and hash function
   instantiations.

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   EC-VOPRF-P256-SHA256:

   o  G: P-256

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA256

   o  H_3: SHA256

   EC-VOPRF-P256-SHA512:

   o  G: P-256

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA512

   o  H_3: SHA512

   EC-VOPRF-P384-SHA256:

   o  G: P-384

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA256

   o  H_3: SHA256

   EC-VOPRF-P384-SHA512:

   o  G: P-384

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA512

   o  H_3: SHA512

   EC-VOPRF-CURVE25519-SHA256:

   o  G: Curve25519 [RFC7748]

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA256

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   o  H_3: SHA256

   EC-VOPRF-CURVE25519-SHA512:

   o  G: Curve25519 [RFC7748]

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA512

   o  H_3: SHA512

   EC-VOPRF-CURVE448-SHA256:

   o  G: Curve448 [RFC7748]

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-cfrg-hash-to-curve]

   o  H_2: SHA256

   o  H_3: SHA256

   EC-VOPRF-CURVE448-SHA512:

   o  G: Curve448 [RFC7748]

   o  H_1: ((TODO: choose from [I-D.draft-sullivan-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.draft-sullivan-cfrg-hash-to-curve] for mapping arbitrary input
   to points on a curve.  Security requires this mapping be pre-image
   and collision resistant.

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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.draft-sullivan-cfrg-hash-to-curve] describes various algorithms
   for constant-time implementations of H_1.

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.  Acknowledgments

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

9.  Contributors

   Alex Davidson contributed to earlier versions of this document.

10.  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>.

   [I-D.draft-sullivan-cfrg-hash-to-curve]
              "Hashing to Elliptic Curves", n.d.,
              <https://datatracker.ietf.org/doc/
              I-D.draft-sullivan-cfrg-hash-to-curve/>.

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   [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>.

Appendix A.  Test Vectors

   This section includes test vectors for the primary VOPRF 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

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Internet-Draft                   VOPRFs                       March 2018

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 \
   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

Sullivan & Wood         Expires September 6, 2018              [Page 14]
Internet-Draft                   VOPRFs                       March 2018

M: 04fd9b68973b0fcefcf4458b4faa1c3815bdad8975b7fb0bfc4c1db7e3f169fb3a26ddabe1b25 \
   c4a23cf8a2faeb12c18f06f2227e87ede6039f55a61ef0c89ca3c582e2864fe130ea0c709f92519d \
   c4a23cf8a2faeb12c18f06f2227e87ede6039f55a61ef0c89ca3c582e2864fe130ea0c709f92519d
k: bcc73da3b2adace9c4f4c32eeadef57436c97f8d395614e78aa91523e1e5d7f551ebb55e87da2 \
   k: bcc73da3b2adace9c4f4c32eeadef57436c97f8d395614e78aa91523e1e5d7f551ebb55e87da2
Z: 042d885d2945cde40e490dd8497975eaeb54e4e10c5986a9688c9de915b16cf43572fd155e159 \
   9e2233a75056a72b54d30092e30bb2edc70e0d90da934c27362e0e6303bafae12f13bf3d5be322e6 \
   9e2233a75056a72b54d30092e30bb2edc70e0d90da934c27362e0e6303bafae12f13bf3d5be322e6
Y: 044833fba4973c1c6eae6745850866ebbb23783ea0d4d8b867e2c93acb2f01fd3d36d9cb5c491 \
   ff9440c8c8e325db326bf88ddf0ba6008158a67999e18cd378d701d1f8a6a7b088dc261c85b6a78b \
   ff9440c8c8e325db326bf88ddf0ba6008158a67999e18cd378d701d1f8a6a7b088dc261c85b6a78b

P-521
X: 040039d290b20c604b5c59cb85dfacd90cbf9f5e275ee8c38a8ff80df0872e8e1dd214a9ec3b2 \
   2c8d75bf634739afdc09acc342542abacf35bf2a6488d084825c5d96003be29e201e75c1b78667f5 \
   a64cc7207722796b225b49edaaf457fcafff4f644252ebe8057291d317f30109f1526aacbfff2308 \
   a64cc7207722796b225b49edaaf457fcafff4f644252ebe8057291d317f30109f1526aacbfff2308
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 \

Sullivan & Wood         Expires September 6, 2018              [Page 15]
Internet-Draft                   VOPRFs                       March 2018

   d297a1edfb049efa7ce61c626c0ad99d8cf462abcce1ee1967d8a355011e2c5a7ce621fc822a7d95 \
   bf7359d938ee4a5c3431e7dd270b7fb6e95fda29cf460d89454763bb0db9b8b705503170a9ac1c7a \
   bf7359d938ee4a5c3431e7dd270b7fb6e95fda29cf460d89454763bb0db9b8b705503170a9ac1c7a
Y: 04006b0413e2686c4bb62340706de7723471080093422f02dd125c3e72f3507b9200d11481468 \
   74bbaa5b6108b834c892eeebab4e21f3707ee20c303ebc1e34fcd3c701f2171131ee7c5f07c1ccad \
   240183d777181259761741343959d476bbc2591a1af0a516e6403a6b81423234746d7a2e8c2ce60a \
   240183d777181259761741343959d476bbc2591a1af0a516e6403a6b81423234746d7a2e8c2ce60a

Authors' Addresses

   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

Sullivan & Wood         Expires September 6, 2018              [Page 16]