[Docs] [txt|pdf] [Tracker] [Email] [Diff1] [Diff2] [Nits]

Versions: 00 01 02 03

Network Working Group                                        A. Davidson
Internet-Draft                                               N. Sullivan
Intended status: Informational                                Cloudflare
Expires: September 12, 2019                                      C. Wood
                                                              Apple Inc.
                                                          March 11, 2019


   Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups
                      draft-sullivan-cfrg-voprf-03

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 September 12, 2019.

Copyright Notice

   Copyright (c) 2019 IETF Trust and the persons identified as the
   document authors.  All rights reserved.





Davidson, et al.       Expires September 12, 2019               [Page 1]


Internet-Draft                    OPRFs                       March 2019


   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  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.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   4
     1.2.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   5
   2.  Background  . . . . . . . . . . . . . . . . . . . . . . . . .   5
   3.  Security Properties . . . . . . . . . . . . . . . . . . . . .   6
   4.  OPRF Protocol . . . . . . . . . . . . . . . . . . . . . . . .   6
     4.1.  Protocol correctness  . . . . . . . . . . . . . . . . . .   8
     4.2.  Instantiations of GG  . . . . . . . . . . . . . . . . . .   8
     4.3.  OPRF algorithms . . . . . . . . . . . . . . . . . . . . .   9
       4.3.1.  OPRF_Setup  . . . . . . . . . . . . . . . . . . . . .   9
       4.3.2.  OPRF_Blind  . . . . . . . . . . . . . . . . . . . . .  10
       4.3.3.  OPRF_Sign . . . . . . . . . . . . . . . . . . . . . .  10
       4.3.4.  OPRF_Unblind  . . . . . . . . . . . . . . . . . . . .  11
       4.3.5.  OPRF_Finalize . . . . . . . . . . . . . . . . . . . .  11
     4.4.  VOPRF algorithms  . . . . . . . . . . . . . . . . . . . .  11
       4.4.1.  VOPRF_Setup . . . . . . . . . . . . . . . . . . . . .  12
       4.4.2.  VOPRF_Blind . . . . . . . . . . . . . . . . . . . . .  12
       4.4.3.  VOPRF_Sign  . . . . . . . . . . . . . . . . . . . . .  13
       4.4.4.  VOPRF_Unblind . . . . . . . . . . . . . . . . . . . .  13
       4.4.5.  VOPRF_Finalize  . . . . . . . . . . . . . . . . . . .  13
     4.5.  Utility algorithms  . . . . . . . . . . . . . . . . . . .  14
       4.5.1.  bin2scalar  . . . . . . . . . . . . . . . . . . . . .  14
     4.6.  Efficiency gains with pre-processing and additive
           blinding  . . . . . . . . . . . . . . . . . . . . . . . .  14
       4.6.1.  OPRF_Preprocess . . . . . . . . . . . . . . . . . . .  15
       4.6.2.  OPRF_Blind  . . . . . . . . . . . . . . . . . . . . .  15
       4.6.3.  OPRF_Unblind  . . . . . . . . . . . . . . . . . . . .  16
   5.  NIZK Discrete Logarithm Equality Proof  . . . . . . . . . . .  16
     5.1.  DLEQ_Generate . . . . . . . . . . . . . . . . . . . . . .  16
     5.2.  DLEQ_Verify . . . . . . . . . . . . . . . . . . . . . . .  17
   6.  Batched VOPRF evaluation  . . . . . . . . . . . . . . . . . .  17
     6.1.  Batched DLEQ algorithms . . . . . . . . . . . . . . . . .  18
       6.1.1.  Batched_DLEQ_Generate . . . . . . . . . . . . . . . .  18
       6.1.2.  Batched_DLEQ_Verify . . . . . . . . . . . . . . . . .  19
     6.2.  Modified protocol execution . . . . . . . . . . . . . . .  20
     6.3.  PRNG and resampling . . . . . . . . . . . . . . . . . . .  20



Davidson, et al.       Expires September 12, 2019               [Page 2]


Internet-Draft                    OPRFs                       March 2019


   7.  Supported ciphersuites  . . . . . . . . . . . . . . . . . . .  20
     7.1.  ECVOPRF-P256-HKDF-SHA256-SSWU:  . . . . . . . . . . . . .  20
     7.2.  ECVOPRF-RISTRETTO-HKDF-SHA512-Elligator2: . . . . . . . .  21
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  21
     8.1.  Timing Leaks  . . . . . . . . . . . . . . . . . . . . . .  21
     8.2.  Hashing to curves . . . . . . . . . . . . . . . . . . . .  22
     8.3.  Verifiability (key consistency) . . . . . . . . . . . . .  22
   9.  Applications  . . . . . . . . . . . . . . . . . . . . . . . .  22
     9.1.  Privacy Pass  . . . . . . . . . . . . . . . . . . . . . .  22
     9.2.  Private Password Checker  . . . . . . . . . . . . . . . .  23
       9.2.1.  Parameter Commitments . . . . . . . . . . . . . . . .  23
   10. Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  23
   11. Normative References  . . . . . . . . . . . . . . . . . . . .  23
   Appendix A.  Test Vectors . . . . . . . . . . . . . . . . . . . .  25
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  27

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.

   OPRFs have been shown to be useful for constructing: password-
   protected secret sharing schemes [JKK14]; privacy-preserving password
   stores [SJKS17]; and password-authenticated key exchange or PAKE
   [OPAQUE].  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.  VOPRFs have also been used for password-protected secret
   sharing schemes e.g.  [JKKX16].

   This document introduces an OPRF protocol built in prime-order
   groups, applying to finite fields of prime-order and also elliptic
   curve (EC) settings.  The protocol has the option of being extended



Davidson, et al.       Expires September 12, 2019               [Page 3]


Internet-Draft                    OPRFs                       March 2019


   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.  In the EC setting, we will refer to the
   protocol as ECOPRF (or ECVOPRF if verifiability is concerned).  The
   document describes the protocol, its security properties, and
   provides preliminary test vectors for experimentation.  The rest of
   the document is structured as follows:

   o  Section Section 2: Describe background, related work, and use
      cases of OPRF/VOPRF protocols.

   o  Section Section 3: Discuss security properties of OPRFs/VOPRFs.

   o  Section Section 4: Specify an authentication protocol from OPRF
      functionality, based in prime-order groups (with an optional
      verifiable mode).  Algorithms are stated formally for OPRFs in
      Section 4.3 and for VOPRFs in Section 4.4.

   o  Section Section 5: Specify the NIZK discrete logarithm equality
      (DLEQ) construction used for constructing the VOPRF protocol.

   o  Section Section 6: Specifies how the DLEQ proof mechanism can be
      batched for multiple VOPRF invocations, and how this changes the
      protocol execution.

   o  Section Section 7: Considers explicit instantiations of the
      protocol in the elliptic curve setting.

   o  Section Section 8: Discusses the security considerations for the
      OPRF and VOPRF protocol.

   o  Section Section 9: Discusses some existing applications of OPRF
      and VOPRF protocols.

   o  Section Appendix A: Specifies test vectors for implementations in
      the elliptic curve setting.

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.




Davidson, et al.       Expires September 12, 2019               [Page 4]


Internet-Draft                    OPRFs                       March 2019


   o  ECVOPRF: A VOPRF built on Elliptic Curves.

   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].

2.  Background

   OPRFs are functionally related to RSA-based blind signature schemes,
   e.g., [ChaumBlindSignature].  Briefly, a blind signature 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 can be used to provide a symmetric equivalent to blind
   signatures.  Essentially the client learns y = PRF(k,x) for some
   input x of their choice, from a server that holds k.  Since the
   security of an OPRF means that x is hidden in the interaction, then
   the client can later reveal x to the server along with y.

   The server can verify that y is computed correctly by recomputing the
   PRF on x using k.  In doing so, the client provides knowledge of a
   'signature' y for their value x.  However, the verification procedure
   is symmetric since it requires knowledge of k.  This is discussed
   more in the following section.





Davidson, et al.       Expires September 12, 2019               [Page 5]


Internet-Draft                    OPRFs                       March 2019


3.  Security Properties

   The security properties of an OPRF 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  The output distribution of y = F(k, x) is indistinguishable from
      the uniform distribution in the domain of the function 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  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.

   Optionally, for any protocol that satisfies the above properties,
   there is an additional security property:

   o  Verifiable: V must only complete execution of the protocol if it
      can successfully assert that P used its secret key k.

   In practice, the notion of verifiability requires that P commits to
   the key k before the actual protocol execution takes place.  Then V
   verifies that P has used k in the protocol using this commitment.

4.  OPRF Protocol

   In this section we describe the OPRF 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 modelled as 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.  This public key is also referred to as a commitment to the key
   k.  Let x be the verifier's (V) input to the OPRF protocol.
   (Commonly, it is a random L-bit string, though this is not required.)

   The OPRF protocol begins with V blinding its input for the signer
   such that it appears uniformly distributed GG.  The latter then
   applies its secret key to the blinded value and returns the result.



Davidson, et al.       Expires September 12, 2019               [Page 6]


Internet-Draft                    OPRFs                       March 2019


   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(k,G,Y,M,Z)]
                     Z[,D]
                   <-------
       [b = DLEQ_Verify(G,Y,M,Z,D)]
       N = Zr^(-1)
       Output H_2(x, N) [if b=1, else "error"]

   Steps that are enclosed in square brackets (DLEQ_Generate and
   DLEQ_Verify) are optional for achieving verifiability.  These are
   described in Section Section 5.  In the verifiable mode, we assume
   that P has previously committed to their choice of key k with some
   values (G,Y=kG) and these are publicly known by V.  Notice that
   revealing (G,Y) does not reveal k by the well-known hardness of the
   discrete log problem.

   Strictly speaking, the actual PRF function that is computed is:

   F(k, x) = N = kH_1(x)

   It is clear that this is a PRF H_1(x) maps x to a random element in
   GG, and GG is cyclic.  This output is computed when the client
   computes Zr^(-1) by the commutativity of the multiplication.  The
   client finishes the computation by outputting H_2(x,N).  Note that
   the output from P is not the PRF value because the actual input x is
   blinded by r.

   This protocol may be decomposed into a series of steps, as described
   below:

   o  OPRF_Setup(l): Generate am integer k of sufficient bit-length l
      and output k.

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

   o  OPRF_Sign(k,M,h): Sign input M using secret key k to produce Z,
      the input h is optional and equal to the cofactor of an elliptic
      curve.  If h is not provided then it defaults to 1.



Davidson, et al.       Expires September 12, 2019               [Page 7]


Internet-Draft                    OPRFs                       March 2019


   o  OPRF_Unblind(r,Z): Unblind blinded signature Z with blind r,
      yielding N and output N.

   o  OPRF_Finalize(x,N): Finalize N to produce the output H_2(x, N).

   For verifiability we modify the algorithms of VOPRF_Setup, VOPRF_Sign
   and VOPRF_Unblind to be the following:

   o  VOPRF_Setup(l): Generate an integer k of sufficient bit-length l
      and output (k, (G,Y)) where Y = kG for some generator G in GG.

   o  VOPRF_Sign(k,(G,Y),M,h): Sign input M using secret key k to
      produce Z.  Generate a NIZK proof D = DLEQ_Generate(k,G,Y,M,Z),
      and output (Z, D).  The optional cofactor h can also be provided
      as in OPRF_Sign.

   o  VOPRF_Unblind(r,G,Y,M,(Z,D)): Unblind blinded signature Z with
      blind r, yielding N.  Output N if 1 = DLEQ_Verify(G,Y,M,Z,D).
      Otherwise, output "error".

   We leave the rest of the OPRF algorithms unmodified.  When referring
   explicitly to VOPRF execution, we replace 'OPRF' in all method names
   with 'VOPRF'.

4.1.  Protocol correctness

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

   OPRF_Finalize(x, OPRF_Unblind(r,M,OPRF_Sign(k,M))) = H_2(x, F(k,x))

   with overwhelming probability.  Likewise, in the verifiable setting,
   we require that:

VOPRF_Finalize(x, VOPRF_Unblind(r,(G,Y),M,(VOPRF_Sign(k,(G,Y),M)))) = H_2(x, F(k,x))

   with overwhelming probability, where (r, M) = VOPRF_Blind(x).

4.2.  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



Davidson, et al.       Expires September 12, 2019               [Page 8]


Internet-Draft                    OPRFs                       March 2019


   cyclic, and we can pick a generator of this subgroup by picking g
   from FF_p (ignoring the identity element).

   For practicality of the protocol, it is preferable to focus on the
   cases where GG is an additive subgroup so that we can instantiate the
   OPRF in the elliptic curve setting.  This amounts to choosing GG to
   be a prime-order subgroup of an 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.

4.3.  OPRF algorithms

   This section provides algorithms for each step in the OPRF protocol.
   We describe the VOPRF analogues in Section 4.4.  We provide generic
   utility algorithms in Section 4.5.

   1.  P samples a uniformly random key k <- {0,1}^l for sufficient
       length l, and interprets it as an integer.

   2.  V computes X = H_1(x) and a random element r (blinding factor)
       from GF(p), and computes M = rX.

   3.  V sends M to P.

   4.  P computes Z = kM = rkX.

   5.  In the elliptic curve setting, P multiplies Z by the cofactor
       (denoted h) of the elliptic curve.

   6.  P sends Z to V.

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

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

4.3.1.  OPRF_Setup









Davidson, et al.       Expires September 12, 2019               [Page 9]


Internet-Draft                    OPRFs                       March 2019


 Input:

  l: Some suitable choice of key-length (e.g. as described in {{NIST}}).

 Output:

  k: A key chosen from {0,1}^l and interpreted as an integer value.

 Steps:

  1. Sample k_bin <-$ {0,1}^l
  2. Output k <- bin2scalar(k_bin, l)

4.3.2.  OPRF_Blind

   Input:

    x: V's PRF input.

   Output:

    r: Random scalar in [1, p - 1].
    M: Blinded representation of x using blind r, an element in GG.

   Steps:

    1.  r <-$ GF(p)
    2.  M := rH_1(x)
    3.  Output (r, M)

4.3.3.  OPRF_Sign

   Input:

    k: Signer secret key.
    M: An element in GG.
    h: optional cofactor (defaults to 1).

   Output:

    Z: Scalar multiplication of the point M by k, element in GG.

   Steps:

    1. Z := kM
    2. Z <- hZ
    3. Output Z




Davidson, et al.       Expires September 12, 2019              [Page 10]


Internet-Draft                    OPRFs                       March 2019


4.3.4.  OPRF_Unblind

   Input:

    r: Random scalar in [1, p - 1].
    Z: An element in GG.

   Output:

    N: Unblinded signature, element in GG.

   Steps:

    1. N := (1/r)Z
    2. Output N

4.3.5.  OPRF_Finalize

   Input:

    x: PRF input string.
    N: An element in GG.

   Output:

    y: Random element in {0,1}^L.

   Steps:

    1. y := H_2(x, N)
    2. Output y

4.4.  VOPRF algorithms

   The steps in the VOPRF setting are written as:

   1.  P samples a uniformly random key k <- {0,1}^l for sufficient
       length l, and interprets it as an integer.

   2.  P commits to k by computing (G,Y) for Y=kG and where G is a
       generator of GG.  P makes (G,Y) publicly available.

   3.  V computes X = H_1(x) and a random element r (blinding factor)
       from GF(p), and computes M = rX.

   4.  V sends M to P.

   5.  P computes Z = kM = rkX, and D = DLEQ_Generate(k,G,Y,M,Z).



Davidson, et al.       Expires September 12, 2019              [Page 11]


Internet-Draft                    OPRFs                       March 2019


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

   7.  V ensures that 1 = DLEQ_Verify(G,Y,M,Z,D).  If not, V outputs an
       error.

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

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

4.4.1.  VOPRF_Setup

 Input:

  G: Public generator of GG.
  l: Some suitable choice of key-length (e.g. as described in {{NIST}}).

 Output:

  k: A key chosen from {0,1}^l and interpreted as an integer value.
  (G,Y): A pair of curve points, where Y=kG.

 Steps:

   1. k <- OPRF_Setup(l)
   2. Y := kG
   3. Output (k, (G,Y))

4.4.2.  VOPRF_Blind

   Input:

    x: V's PRF input.

   Output:

    r: Random scalar in [1, p - 1].
    M: Blinded representation of x using blind r, an element in GG.

   Steps:

    1.  r <-$ GF(p)
    2.  M := rH_1(x)
    3.  Output (r, M)








Davidson, et al.       Expires September 12, 2019              [Page 12]


Internet-Draft                    OPRFs                       March 2019


4.4.3.  VOPRF_Sign

   Input:

    k: Signer secret key.
    G: Public generator of group GG.
    Y: Signer public key (= kG).
    M: An element in GG.
    h: optional cofactor (defaults to 1).

   Output:

    Z: Scalar multiplication of the point M by k, element in GG.
    D: DLEQ proof that log_G(Y) == log_M(Z).

   Steps:

    1. Z := kM
    2. Z <- hZ
    3. D = DLEQ_Generate(k,G,Y,M,Z)
    4. Output (Z, D)

4.4.4.  VOPRF_Unblind

   Input:

    r: Random scalar in [1, p - 1].
    G: Public generator of group GG.
    Y: Signer public key.
    M: Blinded representation of x using blind r, an element in GG.
    Z: An element in GG.
    D: D = DLEQ_Generate(k,G,Y,M,Z).

   Output:

    N: Unblinded signature, element in GG.

   Steps:

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

4.4.5.  VOPRF_Finalize







Davidson, et al.       Expires September 12, 2019              [Page 13]


Internet-Draft                    OPRFs                       March 2019


   Input:

    x: PRF input string.
    N: An element 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

4.5.  Utility algorithms

4.5.1.  bin2scalar

   This algorithm converts a binary string to an integer modulo p.

   Input:

    s: binary string (little-endian)
    l: length of binary string
    p: modulus

   Output:

    z: An integer modulo p

   Steps:

    1. sVec <- vec(s) (converts s to a column vector of dimension l)
    2. p2Vec <- (2^0, 2^1, ..., 2^{l-1}) (row vector of dimension l)
    3. z <- p2Vec * sVec (mod p)
    4. Output z

4.6.  Efficiency gains with pre-processing and additive blinding

   In the [OPAQUE] draft, 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, computing rH_1(x) is an example
   of multiplicative blinding.  A valid way of computing additive
   blinding would be to instead compute H_1(x)+rG, where G is the common
   generator for the group.





Davidson, et al.       Expires September 12, 2019              [Page 14]


Internet-Draft                    OPRFs                       March 2019


   If the client preprocesses values of the form rG, then computing
   H_1(x)+rG is more efficient than computing rH_1(x) (one addition
   against log_2(r)).  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 OPRF_Blind and OPRF_Unblind (and similarly for the VOPRF
   variants).

   We define the additive blinding variants of the above algorithms
   below along with a new algorithm OPRF_Preprocess that defines how
   preprocessing is carried out.  The equivalent algorithms for VOPRF
   are almost identical and so we do not redefine them here.  Notice
   that the only computation that changes is for V, the necessary
   computation of P does not change.

4.6.1.  OPRF_Preprocess

   Input:

    G: Public generator of GG

   Output:

    r: Random scalar in [1, p-1]
    rG: An element in GG.
    rY: An element in GG.

   Steps:

    1.  r <-$ GF(p)
    2.  Output (r, rG, rY)

4.6.2.  OPRF_Blind

   Input:

    x: V's PRF input.
    rG: Preprocessed element of GG.

   Output:

    M: Blinded representation of x using blind r, an element in GG.

   Steps:

    1.  M := H_1(x)+rG
    2.  Output M




Davidson, et al.       Expires September 12, 2019              [Page 15]


Internet-Draft                    OPRFs                       March 2019


4.6.3.  OPRF_Unblind

   Input:

    rY: Preprocessed element of GG.
    M: Blinded representation of x using rG, an element in GG.
    Z: An element in GG.

   Output:

    N: Unblinded signature, element in GG.

   Steps:

    1. N := Z-rY
    2. Output N

   Notice that OPRF_Unblind computes (Z-rY) = k(H_1(x)+rG) - rkG =
   kH_1(x) by the commutativity of scalar multiplication in GG.  This is
   the same output as in the original OPRF_Unblind algorithm.

5.  NIZK Discrete Logarithm Equality Proof

   For the VOPRF protocol we require that V is able to verify that P has
   used its private key k to evaluate the PRF.  We can do this by
   showing that the original commitment (G,Y) output by VOPRF_Setup(l)
   satisfies log_G(Y) == log_M(Z) where Z is the output of
   VOPRF_Sign(k,(G,Y),M).

   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, this allows extending the OPRF protocol 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 September 12, 2019              [Page 16]


Internet-Draft                    OPRFs                       March 2019


  Input:

   k: Signer secret key.
   G: Public generator of GG.
   Y: Signer public key (= kG).
   M: An element in GG.
   Z: An element in GG.
   H_3: A hash function from GG to {0,1}^L, modelled as a random oracle.

  Output:

   D: DLEQ proof (c, s).

  Steps:

   1. r <-$ GF(p)
   2. A := rG and B := rM.
   3. c <- H_3(G,Y,M,Z,A,B)
   4. s := (r - ck) (mod p)
   5. Output D := (c, s)

5.2.  DLEQ_Verify

   Input:

    G: Public generator of GG.
    Y: Signer public key.
    M: An element in GG.
    Z: An element 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'

6.  Batched VOPRF evaluation

   Common applications (e.g.  [PrivacyPass]) require V to obtain
   multiple PRF evaluations from P.  In the VOPRF case, this would also
   require generation and verification of a DLEQ proof for each Zi
   received by V.  This is costly, both in terms of computation and



Davidson, et al.       Expires September 12, 2019              [Page 17]


Internet-Draft                    OPRFs                       March 2019


   communication.  To get around this, applications use a 'batching'
   procedure for generating and verifying DLEQ proofs for a finite
   number of PRF evaluation pairs (Mi,Zi).  For n PRF evaluations:

   o  Proof generation is slightly more expensive from 2n modular
      exponentiations to 2n+2.

   o  Proof verification is much more efficient, from 4m modular
      exponentiations to 2n+4.

   o  Communications falls from 2n to 2 group elements.

   Therefore, since P is usually a powerful server, we can tolerate a
   slight increase in proof generation complexity for much more
   efficient communication and proof verification.

   In this section, we describe algorithms for batching the DLEQ
   generation and verification procedure.  For these algorithms we
   require a pseudorandom generator PRNG: {0,1}^a x ZZ -> ({0,1}^b)^n
   that takes a seed of length a and an integer n as input, and outputs
   n elements in {0,1}^b.

6.1.  Batched DLEQ algorithms

6.1.1.  Batched_DLEQ_Generate


























Davidson, et al.       Expires September 12, 2019              [Page 18]


Internet-Draft                    OPRFs                       March 2019


Input:

 k: Signer secret key.
 G: Public generator of group GG.
 Y: Signer public key (= kG).
 n: Number of PRF evaluations.
 [Mi]: An array of points in GG of length n.
 [Zi]: An array of points in GG of length n.
 PRNG: A pseudorandom generator of the form above.
 salt: An integer salt value for each PRNG invocation
 info: A string value for splitting the domain of the PRNG
 H_4: A hash function from GG^(2n+2) to {0,1}^a, modelled as a random oracle.

Output:

 D: DLEQ proof (c, s).

Steps:

 1. seed <- H_4(G,Y,[Mi,Zi]))
 2. d1,...dn <- PRNG(seed,salt,info,n)
 3. c1,...,cn := (int)d1,...,(int)dn
 4. M := c1M1 + ... + cnMn
 5. Z := c1Z1 + ... + cnZn
 6. Output D <- DLEQ_Generate(k,G,Y,M,Z)

6.1.2.  Batched_DLEQ_Verify

 Input:

  G: Public generator of group GG.
  Y: Signer public key.
  [Mi]: An array of points in GG of length n.
  [Zi]: An array of points in GG of length n.
  D: DLEQ proof (c, s).

 Output:

  True if log_G(Y) == log_(Mi)(Zi) for each i in 1...n, False otherwise.

 Steps:

  1. seed <- H_4(G,Y,[Mi,Zi]))
  2. d1,...dn <- PRNG(seed,salt,info,n)
  3. c1,...,cn := (int)d1,...,(int)dn
  4. M := c1M1 + ... + cnMn
  5. Z := c1Z1 + ... + cnZn
  6. Output DLEQ_Verify(G,Y,M,Z,D)



Davidson, et al.       Expires September 12, 2019              [Page 19]


Internet-Draft                    OPRFs                       March 2019


6.2.  Modified protocol execution

   The VOPRF protocol from Section Section 4 changes to allow specifying
   multiple blinded PRF inputs [Mi] for i in 1...n.  Then P computes the
   array [Zi] and replaces DLEQ_Generate with Batched_DLEQ_Generate over
   these arrays.  The same applies to the algorithm VOPRF_Sign.  The
   same applies for replacing DLEQ_Verify with Batched_DLEQ_Verify when
   V verifies the response from P and during the algorithm VOPRF_Verify.

6.3.  PRNG and resampling

   Any function that satisfies the security properties of a pseudorandom
   number generator can be used for computing the batched DLEQ proof.
   For example, SHAKE-256 [SHAKE] or HKDF-SHA256 [RFC5869] would be
   reasonable choices for groups that have an order of 256 bits.

   We note that the PRNG outputs d1,...,dn must be smaller than the
   order of the group/curve that is being used.  Resampling can be
   achieved by increasing the value of the iterator that is used in the
   info field of the PRNG input.

7.  Supported ciphersuites

   This section specifies supported ECVOPRF group and hash function
   instantiations.  We only provide ciphersuites in the EC setting as
   these provide the most efficient way of instantiating the OPRF.  Our
   instantiation includes considerations for providing the DLEQ proofs
   that make the instantiation a VOPRF.  Supporting OPRF operations
   (ECOPRF) alone can be allowed by simply dropping the relevant
   components.  In addition, we currently only support ciphersuites
   demonstrating 128 bits of security.

7.1.  ECVOPRF-P256-HKDF-SHA256-SSWU:

   o  GG: SECP256K1 curve [SEC2]

   o  H_1: H2C-P256-SHA256-SSWU- [I-D.irtf-cfrg-hash-to-curve]

      *  label: voprf_h2c

   o  H_2: SHA256

   o  H_3: SHA256

   o  H_4: SHA256

   o  PRNG: HKDF-SHA256




Davidson, et al.       Expires September 12, 2019              [Page 20]


Internet-Draft                    OPRFs                       March 2019


7.2.  ECVOPRF-RISTRETTO-HKDF-SHA512-Elligator2:

   o  GG: Ristretto [RISTRETTO]

   o  H_1: H2C-Curve25519-SHA512-Elligator2-Clear
      [I-D.irtf-cfrg-hash-to-curve]

      *  label: voprf_h2c

   o  H_2: SHA512

   o  H_3: SHA512

   o  H_4: SHA512

   o  PRNG: HKDF-SHA512

   In the case of Ristretto, internal point representations are
   represented by Ed25519 [RFC7748] points.  As a result, we can use the
   same hash-to-curve encoding as we would use for Ed25519
   [I-D.irtf-cfrg-hash-to-curve].  We remark that the 'label' field is
   necessary for domain separation of the hash-to-curve functionality.

8.  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 inputs x to
   points on a curve.  Security requires this mapping be pre-image and
   collision resistant.

8.1.  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.








Davidson, et al.       Expires September 12, 2019              [Page 21]


Internet-Draft                    OPRFs                       March 2019


8.2.  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.

8.3.  Verifiability (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 a 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
   published from a trusted registry, then P cannot present unique keys
   to an individual verifier.

   For this side effect to hold, P must also be prevented from using
   other techniques to manipulate their public key within the trusted
   registry to reduce client anonymity.  For example, if P's public key
   is rotated too frequently then this may stratify the user base into
   small anonymity groups (those with VOPRF_Sign outputs taken from a
   given key epoch).  In this case, it may become practical to link
   VOPRF sessions for a given user and thus compromises their privacy.

   Similarly, if P can publish N public keys to a trusted registry then
   P may be able to control presentation of these keys in such a way
   that V is retroactively identified by V's key choice across multiple
   requests.

9.  Applications

   This section describes various applications of the VOPRF protocol.

9.1.  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
   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 along with a batched DLEQ proof.  C stores the
   tokens if the batched proof verifies correctly.  When C attempts to
   connect to E again and is prompted with a CAPTCHA, C uses one of the



Davidson, et al.       Expires September 12, 2019              [Page 22]


Internet-Draft                    OPRFs                       March 2019


   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].

9.2.  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 on 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].

9.2.1.  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.

10.  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.

11.  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 September 12, 2019              [Page 23]


Internet-Draft                    OPRFs                       March 2019


   [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-02 (work in
              progress), October 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>.

   [NIST]     "Keylength - NIST Report on Cryptographic Key Length and
              Cryptoperiod (2016)", n.d.,
              <https://www.keylength.com/en/4/>.

   [OPAQUE]   "The OPAQUE Asymmetric PAKE Protocol", n.d.,
              <https://tools.ietf.org/html/
              draft-krawczyk-cfrg-opaque-01>.

   [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>.

   [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>.





Davidson, et al.       Expires September 12, 2019              [Page 24]


Internet-Draft                    OPRFs                       March 2019


   [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 ristretto255 Group", n.d.,
              <https://tools.ietf.org/html/
              draft-hdevalence-cfrg-ristretto-00>.

   [SEC2]     Standards for Efficient Cryptography Group (SECG), ., "SEC
              2: Recommended Elliptic Curve Domain Parameters", n.d.,
              <http://www.secg.org/sec2-v2.pdf>.

   [SHAKE]    "SHA-3 Standard, Permutation-Based Hash and Extendable-
              Output Functions", n.d.,
              <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", n.d.,
              <http://webee.technion.ac.il/%7Ehugo/sphinx.pdf>.

Appendix A.  Test Vectors

   This section includes test vectors for the ECVOPRF-P256-HKDF-SHA256
   VOPRF ciphersuite, including batched DLEQ output.























Davidson, et al.       Expires September 12, 2019              [Page 25]


Internet-Draft                    OPRFs                       March 2019


P-256
X: 04b14b08f954f5b6ab1d014b1398f03881d70842acdf06194eb96a6d08186f8cb985c1c5521 \
    f4ee19e290745331f7eb89a4053de0673dc8ef14cfe9bf8226c6b31
r: b72265c85b1ba42cfed7caaf00d2ccac0b1a99259ba0dbb5a1fc2941526a6849
M: 046025a41f81a160c648cfe8fdcaa42e5f7da7a71055f8e23f1dc7e4204ab84b705043ba5c7 \
    000123e1fd058150a4d3797008f57a8b2537766d9419c7396ba5279
k: f84e197c8b712cdf452d2cff52dec1bd96220ed7b9a6f66ed28c67503ae62133
Z: 043ab5ccb690d844dcb780b2d9e59126d62bc853ba01b2c339ba1c1b78c03e4b6adc5402f77 \
    9fc29f639edc138012f0e61960e1784973b37f864e4dc8abbc68e0b
N: 04e8aa6792d859075821e2fba28500d6974ba776fe230ba47ef7e42be1d967654ce776f889e \
    e1f374ffa0bce904408aaa4ed8a19c6cc7801022b7848031f4e442a
D: { s: faddfaf6b5d6b4b6357adf856fc1e0044614ebf9dafdb4c6541c1c9e61243c5b,
     c: 8b403e170b56c915cc18864b3ab3c2502bd8f5ca25301bc03ab5138343040c7b }

P-256
X: 047e8d567e854e6bdc95727d48b40cbb5569299e0a4e339b6d707b2da3508eb6c238d3d4cb4 \
    68afc6ffc82fccbda8051478d1d2c9b21ffdfd628506c873ebb1249
r: f222dfe530fdbfcb02eb851867bfa8a6da1664dfc7cee4a51eb6ff83c901e15e
M: 04e2efdc73747e15e38b7a1bb90fe5e4ef964b3b8dccfda428f85a431420c84efca02f0f09c \
    83a8241b44572a059ab49c080a39d0bce2d5d0b44ff5d012b5184e7
k: fb164de0a87e601fd4435c0d7441ff822b5fa5975d0c68035beac05a82c41118
Z: 049d01e1c555bd3324e8ce93a13946b98bdcc765298e6d60808f93c00bdfba2ebf48eef8f28 \
    d8c91c903ad6bea3d840f3b9631424a6cc543a0a0e1f2d487192d5b
N: 04723880e480b60b4415ca627585d1715ab5965570d30c94391a8b023f8854ac26f76c1d6ab \
    bb38688a5affbcadad50ecbf7c93ef33ddfd735003b5a4b1a21ba14
D: { s: dfdf6ae40d141b61d5b2d72cf39c4a6c88db6ac5b12044a70c212e2bf80255b4,
     c: 271979a6b51d5f71719127102621fe250e3235867cfcf8dea749c3e253b81997 }

Batched DLEQ (P256)
M_0: 046025a41f81a160c648cfe8fdcaa42e5f7da7a71055f8e23f1dc7e4204ab84b705043ba5c\
    7000123e1fd058150a4d3797008f57a8b2537766d9419c7396ba5279
M_1: 04e2efdc73747e15e38b7a1bb90fe5e4ef964b3b8dccfda428f85a431420c84efca02f0f09\
    c83a8241b44572a059ab49c080a39d0bce2d5d0b44ff5d012b5184e7
Z_0: 043ab5ccb690d844dcb780b2d9e59126d62bc853ba01b2c339ba1c1b78c03e4b6adc5402f7\
    79fc29f639edc138012f0e61960e1784973b37f864e4dc8abbc68e0b
Z_1: 04647e1ab7946b10c1c1c92dd333e2fc9e93e85fdef5939bf2f376ae859248513e0cd91115\
    e48c6852d8dd173956aec7a81401c3f63a133934898d177f2a237eeb
k: f84e197c8b712cdf452d2cff52dec1bd96220ed7b9a6f66ed28c67503ae62133
PRNG: HKDF-SHA256
salt: "DLEQ_PROOF"
info: an iterator i for invoking the PRNG on M_i and Z_i
D: { s: b2123044e633d4721894d573decebc9366869fe3c6b4b79a00311ecfa46c9e34,
     c: 3506df9008e60130fcddf86fdb02cbfe4ceb88ff73f66953b1606f6603309862 }








Davidson, et al.       Expires September 12, 2019              [Page 26]


Internet-Draft                    OPRFs                       March 2019


Authors' Addresses

   Alex Davidson
   Cloudflare
   County Hall
   London, SE1 7GP
   United Kingdom

   Email: adavidson@cloudflare.com


   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 September 12, 2019              [Page 27]


Html markup produced by rfcmarkup 1.129c, available from https://tools.ietf.org/tools/rfcmarkup/