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Versions: (draft-sullivan-cfrg-voprf) 00 01 02 03 04 05

Network Working Group                                        A. Davidson
Internet-Draft                                          A. Faz-Hernandez
Intended status: Informational                               N. Sullivan
Expires: 6 May 2021                                            C.A. Wood
                                                              Cloudflare
                                                         2 November 2020


   Oblivious Pseudorandom Functions (OPRFs) using Prime-Order Groups
                        draft-irtf-cfrg-voprf-05

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.

Discussion Venues

   This note is to be removed before publishing as an RFC.

   Source for this draft and an issue tracker can be found at
   https://github.com/cfrg/draft-irtf-cfrg-voprf.

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 6 May 2021.



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Copyright Notice

   Copyright (c) 2020 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Simplified BSD License text
   as described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Change log  . . . . . . . . . . . . . . . . . . . . . . .   4
     1.2.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   5
     1.3.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   5
   2.  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . .   6
     2.1.  Prime-order group API . . . . . . . . . . . . . . . . . .   6
     2.2.  Other conventions . . . . . . . . . . . . . . . . . . . .   7
   3.  OPRF Protocol . . . . . . . . . . . . . . . . . . . . . . . .   8
     3.1.  Overview  . . . . . . . . . . . . . . . . . . . . . . . .   8
     3.2.  Context Setup . . . . . . . . . . . . . . . . . . . . . .   9
     3.3.  Data Structures . . . . . . . . . . . . . . . . . . . . .  10
     3.4.  Context APIs  . . . . . . . . . . . . . . . . . . . . . .  11
       3.4.1.  Server Context  . . . . . . . . . . . . . . . . . . .  11
       3.4.2.  VerifiableServerContext . . . . . . . . . . . . . . .  13
       3.4.3.  Client Context  . . . . . . . . . . . . . . . . . . .  16
       3.4.4.  VerifiableClientContext . . . . . . . . . . . . . . .  18
   4.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . . . .  20
     4.1.  OPRF(ristretto255, SHA-256) . . . . . . . . . . . . . . .  21
     4.2.  OPRF(decaf448, SHA-512) . . . . . . . . . . . . . . . . .  21
     4.3.  OPRF(P-256, SHA-256)  . . . . . . . . . . . . . . . . . .  22
     4.4.  OPRF(P-384, SHA-512)  . . . . . . . . . . . . . . . . . .  22
     4.5.  OPRF(P-521, SHA-512)  . . . . . . . . . . . . . . . . . .  22
   5.  Security Considerations . . . . . . . . . . . . . . . . . . .  23
     5.1.  Security properties . . . . . . . . . . . . . . . . . . .  23
     5.2.  Cryptographic security  . . . . . . . . . . . . . . . . .  24
       5.2.1.  Computational hardness assumptions  . . . . . . . . .  24
       5.2.2.  Protocol security . . . . . . . . . . . . . . . . . .  25
       5.2.3.  Q-strong-DH oracle  . . . . . . . . . . . . . . . . .  25
       5.2.4.  Implications for ciphersuite choices  . . . . . . . .  26
     5.3.  Hashing to curve  . . . . . . . . . . . . . . . . . . . .  26
     5.4.  Timing Leaks  . . . . . . . . . . . . . . . . . . . . . .  27
     5.5.  Key rotation  . . . . . . . . . . . . . . . . . . . . . .  27



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   6.  Additive blinding . . . . . . . . . . . . . . . . . . . . . .  27
     6.1.  Preprocess  . . . . . . . . . . . . . . . . . . . . . . .  28
     6.2.  Blind . . . . . . . . . . . . . . . . . . . . . . . . . .  28
     6.3.  Unblind . . . . . . . . . . . . . . . . . . . . . . . . .  29
       6.3.1.  Parameter Commitments . . . . . . . . . . . . . . . .  30
   7.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  30
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  30
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  30
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  30
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  32
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  32

1.  Introduction

   A pseudorandom function (PRF) F(k, x) is an efficiently computable
   function taking a private key k and a value x as input.  This
   function is pseudorandom if the keyed function K(_) = F(K, _) is
   indistinguishable from a randomly sampled function acting on the same
   domain and range as K().  An oblivious PRF (OPRF) is a two-party
   protocol between a server and a client, where the server holds a PRF
   key k and the client holds some input x.  The protocol allows both
   parties to cooperate in computing F(k, x) such that: the client
   learns F(k, x) without learning anything about k; and the server does
   not learn anything about x or F(k, x).  A Verifiable OPRF (VOPRF) is
   an OPRF wherein the server can prove to the client that F(k, x) was
   computed using the key k.

   The usage of OPRFs has been demonstrated in constructing a number of
   applications: password-protected secret sharing schemes [JKKX16];
   privacy-preserving password stores [SJKS17]; and password-
   authenticated key exchange or PAKE [I-D.irtf-cfrg-opaque].  A VOPRF
   is necessary in some applications, e.g., the Privacy Pass protocol
   [I-D.davidson-pp-protocol], wherein this VOPRF is used to generate
   one-time authentication tokens to bypass CAPTCHA challenges.  VOPRFs
   have also been used for password-protected secret sharing schemes
   e.g.  [JKK14].

   This document introduces an OPRF protocol built in prime-order
   groups, applying to finite fields of prime-order and also elliptic
   curve (EC) groups.  The protocol has the option of being extended to
   a VOPRF with the addition of a NIZK proof for proving discrete log
   equality relations.  This proof demonstrates correctness of the
   computation, using a known public key that serves as a commitment to
   the server's secret key.  The document describes the protocol, the
   public-facing API, and its security properties.






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1.1.  Change log

   draft-05 (https://tools.ietf.org/html/draft-irtf-cfrg-voprf-05):

   *  Move to ristretto255 and decaf448 ciphersuites.

   *  Clean up ciphersuite definitions.

   *  Pin domain separation tag construction to draft version.

   *  Move key generation outside of context construction functions.

   *  Editorial changes.

   draft-04 (https://tools.ietf.org/html/draft-irtf-cfrg-voprf-04):

   *  Introduce Client and Server contexts for controlling verifiability
      and required functionality.

   *  Condense API.

   *  Remove batching from standard functionality (included as an
      extension)

   *  Add Curve25519 and P-256 ciphersuites for applications that
      prevent strong-DH oracle attacks.

   *  Provide explicit prime-order group API and instantiation advice
      for each ciphersuite.

   *  Proof-of-concept implementation in sage.

   *  Remove privacy considerations advice as this depends on
      applications.

   draft-03 (https://tools.ietf.org/html/draft-irtf-cfrg-voprf-03):

   *  Certify public key during VerifiableFinalize

   *  Remove protocol integration advice

   *  Add text discussing how to perform domain separation

   *  Drop OPRF_/VOPRF_ prefix from algorithm names

   *  Make prime-order group assumption explicit

   *  Changes to algorithms accepting batched inputs



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   *  Changes to construction of batched DLEQ proofs

   *  Updated ciphersuites to be consistent with hash-to-curve and added
      OPRF specific ciphersuites

   draft-02 (https://tools.ietf.org/html/draft-irtf-cfrg-voprf-02):

   *  Added section discussing cryptographic security and static DH
      oracles

   *  Updated batched proof algorithms

   draft-01 (https://tools.ietf.org/html/draft-irtf-cfrg-voprf-01):

   *  Updated ciphersuites to be in line with
      https://tools.ietf.org/html/draft-irtf-cfrg-hash-to-curve-04

   *  Made some necessary modular reductions more explicit

1.2.  Terminology

   The following terms are used throughout this document.

   *  PRF: Pseudorandom Function.

   *  OPRF: Oblivious Pseudorandom Function.

   *  VOPRF: Verifiable Oblivious Pseudorandom Function.

   *  Client: Protocol initiator.  Learns pseudorandom function
      evaluation as the output of the protocol.

   *  Server: Computes the pseudorandom function over a secret key.
      Learns nothing about the client's input.

   *  NIZK: Non-interactive zero knowledge.

   *  DLEQ: Discrete Logarithm Equality.

1.3.  Requirements

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





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2.  Preliminaries

2.1.  Prime-order group API

   In this document, we assume the construction of an additive, prime-
   order group "GG" for performing all mathematical operations.  Such
   groups are uniquely determined by the choice of the prime "p" that
   defines the order of the group.  We use "GF(p)" to represent the
   finite field of order "p".  For the purpose of understanding and
   implementing this document, we take "GF(p)" to be equal to the set of
   integers defined by "{0, 1, ..., p-1}".

   The fundamental group operation is addition "+" with identity element
   "I".  For any elements "A" and "B" of the group "GG", "A + B = B + A"
   is also a member of "GG".  Also, for any "A" in "GG", there exists an
   element "-A" such that "A + (-A) = (-A) + A = I".  Scalar
   multiplication is equivalent to the repeated application of the group
   operation on an element A with itself "r-1" times, this is denoted as
   "r*A = A + ... + A".  For any element "A", the equality "p*A=I"
   holds.  Scalar base multiplication is equivalent to the repeated
   application of the group operation on the base point with itself
   "r-1" times, this is denoted as "ScalarBaseMult(r)".  The set of
   scalars corresponds to "GF(p)".

   We now detail a number of member functions that can be invoked on a
   prime-order group.

   *  Order(): Outputs the order of GG (i.e. "p").

   *  Identity(): Outputs the identity element of the group (i.e.  "I").

   *  Serialize(A): A member function of "GG" that maps a group element
      "A" to a unique byte array "buf".

   *  Deserialize(buf): A member function of "GG" that maps a byte array
      "buf" to a group element "A", or fails if the input is not a valid
      byte representation of an element.

   *  HashToGroup(x): A member function of "GG" that deterministically
      maps an array of bytes "x" to an element of "GG".  The map must
      ensure that, for any adversary receiving "R = HashToGroup(x)", it
      is computationally difficult to reverse the mapping.  Examples of
      hash to group functions satisfying this property are described for
      prime-order (sub)groups of elliptic curves, see
      [I-D.irtf-cfrg-hash-to-curve].






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   *  HashToScalar(x): A member function of "GG" that deterministically
      maps an array of bytes "x" to an element in GF(p).  A recommended
      method for its implementation is instantiating the hash to field
      function, defined in [I-D.irtf-cfrg-hash-to-curve] setting the
      target field to GF(p).

   *  RandomScalar(): A member function of "GG" that chooses at random a
      non-zero element in GF(p).

   It is convenient in cryptographic applications to instantiate such
   prime-order groups using elliptic curves, such as those detailed in
   [SEC2].  For some choices of elliptic curves (e.g. those detailed in
   [RFC7748], which require accounting for cofactors) there are some
   implementation issues that introduce inherent discrepancies between
   standard prime-order groups and the elliptic curve instantiation.  In
   this document, all algorithms that we detail assume that the group is
   a prime-order group, and this MUST be upheld by any implementer.
   That is, any curve instantiation should be written such that any
   discrepancies with a prime-order group instantiation are removed.
   See Section 4 for advice corresponding to implementation of this
   interface for specific definitions of elliptic curves.

2.2.  Other conventions

   *  We use the notation "x <-$ Q" to denote sampling "x" from the
      uniform distribution over the set "Q".

   *  For any object "x", we write "len(x)" to denote its length in
      bytes.

   *  For two byte arrays "x" and "y", write "x || y" to denote their
      concatenation.

   *  I2OSP and OS2IP: Convert a byte array to and from a non-negative
      integer as described in [RFC8017].  Note that these functions
      operate on byte arrays in big-endian byte order.

   All algorithm descriptions are written in a Python-like pseudocode.
   We use the "ABORT()" function for presentational clarity to denote
   the process of terminating the algorithm or returning an error
   accordingly.  We also use the "CT_EQUAL(a, b)" function to represent
   constant-time byte-wise equality between byte arrays "a" and "b".
   This function returns "true" if "a" and "b" are equal, and "false"
   otherwise.







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3.  OPRF Protocol

   In this section, we define two OPRF variants: a base mode and
   verifiable mode.  In the base mode, a client and server interact to
   compute y = F(skS, x), where x is the client's input, skS is the
   server's private key, and y is the OPRF output.  The client learns y
   and the server learns nothing.  In the verifiable mode, the client
   also gets proof that the server used skS in computing the function.

   To achieve verifiability, as in the original work of [JKK14], we
   provide a zero-knowledge proof that the key provided as input by the
   server in the "Evaluate" function is the same key as it used to
   produce their public key.  As an example of the nature of attacks
   that this prevents, this ensures that the server uses the same
   private key for computing the VOPRF output and does not attempt to
   "tag" individual servers with select keys.  This proof must not
   reveal the server's long-term private key to the client.

   The following one-byte values distinguish between these two modes:

                        +================+=======+
                        | Mode           | Value |
                        +================+=======+
                        | modeBase       | 0x00  |
                        +----------------+-------+
                        | modeVerifiable | 0x01  |
                        +----------------+-------+

                                 Table 1

3.1.  Overview

   Both participants agree on the mode and a choice of ciphersuite that
   is used before the protocol exchange.  Once established, the core
   protocol runs to compute "output = F(skS, input)" as follows:
















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      Client(pkS, input, info)                 Server(skS, pkS)
     ----------------------------------------------------------
       token, blindToken = Blind(input)

                            blindToken
                           ---------->

                            evaluation = Evaluate(skS, pkS, blindToken)

                            evaluation
                           <----------

       issuedToken = Unblind(pkS, token, blindToken, evaluation)
       output = Finalize(input, issuedToken, info)

   In "Blind" the client generates a token and blinding data.  The
   server computes the (V)OPRF evaluation in "Evaluation" over the
   client's blinded token.  In "Unblind" the client unblinds the server
   response (and verifies the server's proof if verifiability is
   required).  In "Finalize", the client produces a byte array
   corresponding to the output of the OPRF protocol.

   Note that in the final output, the client computes Finalize over some
   auxiliary input data "info".  This parameter SHOULD be used for
   domain separation in the (V)OPRF protocol.  Specifically, any system
   which has multiple (V)OPRF applications should use separate auxiliary
   values to ensure finalized outputs are separate.  Guidance for
   constructing info can be found in [I-D.irtf-cfrg-hash-to-curve];
   Section 3.1.

3.2.  Context Setup

   Both modes of the OPRF involve an offline setup phase.  In this
   phase, both the client and server create a context used for executing
   the online phase of the protocol.  Prior to this phase, keys ("skS",
   "pkS") should be generated by calling a "KeyGen" function.  "KeyGen"
   generates a private and public key pair ("skS", "pkS"), where "skS"
   is a non-zero element chosen at random from the scalar field of the
   corresponding group and "pkS = ScalarBaseMult(skS)".

   The base mode setup functions for creating client and server contexts
   are below:









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   def SetupBaseServer(suite, skS):
     contextString = I2OSP(modeBase, 1) || I2OSP(suite.ID, 2)
     return ServerContext(contextString, skS)

   def SetupBaseClient(suite):
     contextString = I2OSP(modeBase, 1) || I2OSP(suite.ID, 2)
     return ClientContext(contextString)

   The verifiable mode setup functions for creating client and server
   contexts are below:

   def SetupVerifiableServer(suite, skS, pkS):
     contextString = I2OSP(modeVerifiable, 1) || I2OSP(suite.ID, 2)
     return VerifiableServerContext(contextString, skS), pkS

   def SetupVerifiableClient(suite, pkS):
     contextString = I2OSP(modeVerifiable, 1) || I2OSP(suite.ID, 2)
     return VerifiableClientContext(contextString, pkS)

   Each setup function takes a ciphersuite from the list defined in
   Section 4.  Each ciphersuite has a two-byte field ID used to identify
   the suite.

3.3.  Data Structures

   The following is a list of data structures that are defined for
   providing inputs and outputs for each of the context interfaces
   defined in Section 3.4.  Each data structure description uses TLS
   notation (see [RFC8446], Section 3).

   The following types are a list of aliases that are used throughout
   the protocol.

   A "ClientInput" is a byte array.

   opaque ClientInput<1..2^16-1>;

   A "SerializedElement" is also a byte array, representing the unique
   serialization of an "Element".

   opaque SerializedElement<1..2^16-1>;

   A "Token" is an object created by a client when constructing a
   (V)OPRF protocol input.  It is stored so that it can be used after
   receiving the server response.






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   struct {
     opaque data<1..2^16-1>;
     Scalar blind<1..2^16-1>;
   } Token;

   An "Evaluation" is the type output by the "Evaluate" algorithm.  The
   member "proof" is added only in verifiable contexts.

   struct {
     SerializedElement element;
     Scalar proof<0...2^16-1>; /* only for modeVerifiable */
   } Evaluation;

   Evaluations may also be combined in batches with a constant-size
   proof, producing a "BatchedEvaluation".  These carry a list of
   "SerializedElement" values and proof.  These evaluation types are
   only useful in verifiable contexts which carry proofs.

   struct {
     SerializedElement elements<1..2^16-1>;
     Scalar proof<0...2^16-1>; /* only for modeVerifiable */
   } BatchedEvaluation;

3.4.  Context APIs

   In this section, we detail the APIs available on the client and
   server (V)OPRF contexts.  This document uses the types "Element" and
   "Scalar" to denote elements of the group "GG" and its underlying
   scalar field "GF(p)", respectively.  For notational clarity,
   "PublicKey" is an item of type "Element" and "PrivateKey" is an item
   of type "Scalar".

3.4.1.  Server Context

   The ServerContext encapsulates the context string constructed during
   setup and the (V)OPRF key pair.  It has three functions, "Evaluate",
   "FullEvaluate" and "VerifyFinalize" described below.  "Evaluate"
   takes serialized representations of blinded group elements from the
   client as inputs.

   "FullEvaluate" takes ClientInput values, and it is useful for
   applications that need to compute the whole OPRF protocol on the
   server side only.

   "VerifyFinalize" takes ClientInput values and their corresponding
   output digests from "Finalize" as input, and returns true if the
   inputs match the outputs.




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   Note that "VerifyFinalize" and "FullEvaluate" are not used in the
   main OPRF protocol.  They are exposed as an API for building higher-
   level protocols.

3.4.1.1.  Evaluate

   Input:

     PrivateKey skS
     SerializedElement blindToken

   Output:

     Evaluation Ev

   def Evaluate(skS, blindToken):
     BT = GG.Deserialize(blindToken)
     Z = skS * BT
     serializedElement = GG.Serialize(Z)

     Ev = Evaluation{ element: serializedElement }

     return Ev

3.4.1.2.  FullEvaluate

   Input:

     PrivateKey skS
     ClientInput input
     opaque info<1..2^16-1>

   Output:

     opaque output<1..2^16-1>

   def FullEvaluate(skS, input, info):
     P = GG.HashToGroup(input)
     T = skS * P
     issuedToken = GG.serialize(T)

     finalizeDST = "VOPRF05-Finalize-" || self.contextString
     hashInput = I2OSP(len(input), 2) || input ||
                 I2OSP(len(issuedToken), 2) || issuedToken ||
                 I2OSP(len(info), 2) || info ||
                 I2OSP(len(finalizeDST), 2) || finalizeDST

     return Hash(hashInput)



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3.4.1.3.  VerifyFinalize

   Input:

     PrivateKey skS
     ClientInput input
     opaque info<1..2^16-1>
     opaque output<1..2^16-1>

   Output:

     boolean valid

   def VerifyFinalize(skS, input, info, output):
     T = GG.HashToGroup(input)
     element = GG.Serialize(T)
     issuedElement = Evaluate(skS, [element])
     E = GG.Serialize(issuedElement)

     finalizeDST = "VOPRF05-Finalize-" || self.contextString
     hashInput = I2OSP(len(input), 2) || input ||
                 I2OSP(len(E), 2) || E ||
                 I2OSP(len(info), 2) || info ||
                 I2OSP(len(finalizeDST), 2) || finalizeDST

     digest = Hash(hashInput)

     return CT_EQUAL(digest, output)

   [[RFC editor: please change "VOPRF05" to "RFCXXXX", where XXXX is the
   final number, here and elsewhere before publication.]]

3.4.2.  VerifiableServerContext

   The VerifiableServerContext extends the base ServerContext with an
   augmented "Evaluate()" function.  This function produces a proof that
   "skS" was used in computing the result.  It makes use of the helper
   functions "GenerateProof" and "ComputeComposites", described below.

3.4.2.1.  Evaluate











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   Input:

     PrivateKey skS
     PublicKey pkS
     SerializedElement blindToken

   Output:

     Evaluation Ev

   def Evaluate(skS, pkS, blindToken):
     BT = GG.Deserialize(blindToken)
     Z = skS * BT
     serializedElement = GG.Serialize(Z)

     proof = GenerateProof(skS, pkS, blindToken, serializedElement)
     Ev = Evaluation{ element: serializedElement, proof: proof }

     return Ev

   The helper functions "GenerateProof" and "ComputeComposites" are
   defined below.

3.4.2.2.  GenerateProof



























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   Input:

     PrivateKey skS
     PublicKey pkS
     SerializedElement blindToken
     SerializedElement element

   Output:

     Scalar proof[2]

   def GenerateProof(skS, pkS, blindToken, element)
     blindTokenList = [blindToken]
     elementList = [element]

     a = ComputeComposites(pkS, blindTokenList, elementList)

     M = GG.Deserialize(a[0])
     r = GG.RandomScalar()
     a2 = GG.Serialize(ScalarBaseMult(r))
     a3 = GG.Serialize(r * M)

     challengeDST = "VOPRF05-challenge-" || self.contextString
     h2Input = I2OSP(len(pkS), 2) || pkS ||
               I2OSP(len(a[0]), 2) || a[0] ||
               I2OSP(len(a[1]), 2) || a[1] ||
               I2OSP(len(a2), 2) || a2 ||
               I2OSP(len(a3), 2) || a3 ||
               I2OSP(len(challengeDST), 2) || challengeDST

     c = GG.HashToScalar(h2Input)
     s = (r - c * skS) mod p

     return [c, s]

3.4.2.2.1.  Batching inputs

   Unlike other functions, "ComputeComposites" takes lists of inputs,
   rather than a single input.  It is optimized to produce a constant-
   size output.  This functionality lets applications batch inputs
   together to produce a constant-size proofs from "GenerateProof".
   Applications can take advantage of this functionality by invoking
   "GenerateProof" on batches of inputs.  (Notice that in the pseudocode
   above, the single inputs "blindToken" and "element" are translated
   into lists before invoking "ComputeComposites".  A batched
   "GenerateProof" variant would permit lists of inputs, and no list
   translation would be needed.)




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   Note that using batched inputs creates a "BatchedEvaluation" object
   as the output of "Evaluate".

3.4.2.2.2.  Fresh randomness

   We note here that it is essential that a different "r" value is used
   for every invocation.  If this is not done, then this may leak "skS"
   as is possible in Schnorr or (EC)DSA scenarios where fresh randomness
   is not used.

3.4.2.3.  ComputeComposites

   Input:

     PublicKey pkS
     SerializedElement blindTokens[m]
     SerializedElement elements[m]

   Output:

     SerializedElement composites[2]

   def ComputeComposites(pkS, blindTokens, elements):
     seedDST = "VOPRF05-seed-" || self.contextString
     compositeDST = "VOPRF05-composite-" || self.contextString
     h1Input = I2OSP(len(pkS), 2) || pkS ||
               I2OSP(len(blindTokens), 2) || blindTokens ||
               I2OSP(len(elements), 2) || elements ||
               I2OSP(len(seedDST), 2) || seedDST

     seed = Hash(h1Input)
     M = GG.Identity()
     Z = GG.Identity()
     for i = 0 to m-1:
       h2Input = I2OSP(len(seed), 2) || seed || I2OSP(i, 2) ||
                 I2OSP(len(compositeDST), 2) || compositeDST
       di = GG.HashToScalar(h2Input)
       Mi = GG.Deserialize(blindTokens[i])
       Zi = GG.Deserialize(elements[i])
       M = di * Mi + M
       Z = di * Zi + Z
    return [GG.Serialize(M), GG.Serialize(Z)]

3.4.3.  Client Context

   The ClientContext encapsulates the context string constructed during
   setup.  It has three functions, "Blind()", "Unblind()", and
   "Finalize()", as described below.



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3.4.3.1.  Blind

   We note here that the blinding mechanism that we use can be modified
   slightly with the opportunity for making performance gains in some
   scenarios.  We detail these modifications in Section 6.

   Input:

     ClientInput input

   Output:

     Token token
     SerializedElement blindToken

   def Blind(input):
     r = GG.RandomScalar()
     P = GG.HashToGroup(input)
     blindToken = GG.Serialize(r * P)

     token = Token{ data: input, blind: r }

     return (token, blindToken)

3.4.3.2.  Unblind

   Input:

     Token token
     Evaluation Ev

   Output:

     SerializedElement issuedToken

   def Unblind(token, Ev):
     r = token.blind
     Z = GG.Deserialize(Ev.element)
     N = (r^(-1)) * Z

     issuedToken = GG.Serialize(N)

     return issuedToken

3.4.3.3.  Finalize






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   Input:

     Token token
     SerializedElement issuedToken
     opaque info<1..2^16-1>

   Output:

     opaque output<1..2^16-1>

   def Finalize(token, issuedToken, info):
     finalizeDST = "VOPRF05-Finalize-" || self.contextString
     hashInput = I2OSP(len(token.data), 2) || token.data ||
                 I2OSP(len(issuedToken), 2) || issuedToken ||
                 I2OSP(len(info), 2) || info ||
                 I2OSP(len(finalizeDST), 2) || finalizeDST
     return Hash(hashInput)

3.4.4.  VerifiableClientContext

   The VerifiableClientContext extends the base ClientContext with the
   desired server public key "pkS" with an augmented "Unblind()"
   function.  This function verifies an evaluation proof using "pkS".
   It makes use of the helper function "ComputeComposites" described
   above.  It has one helper function, "VerifyProof()", defined below.

3.4.4.1.  VerifyProof

   This algorithm outputs a boolean "verified" which indicates whether
   the proof inside of the evaluation verifies correctly, or not.





















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   Input:

     PublicKey pkS
     SerializedElement blindToken
     Evaluation Ev

   Output:

     boolean verified

   def VerifyProof(pkS, blindToken, Ev):
     blindTokenList = [blindToken]
     elementList = [Ev.element]

     a = ComputeComposites(pkS, blindTokenList, elementList)

     A' = (ScalarBaseMult(Ev.proof[1]) + Ev.proof[0] * pkS)
     B' = (Ev.proof[1] * M + Ev.proof[0] * Z)
     a2 = GG.Serialize(A')
     a3 = GG.Serialize(B')

     challengeDST = "VOPRF05-challenge-" || self.contextString
     h2Input = I2OSP(len(pkS), 2) || pkS ||
               I2OSP(len(a[0]), 2) || a[0] ||
               I2OSP(len(a[1]), 2) || a[1] ||
               I2OSP(len(a2), 2) || a2 ||
               I2OSP(len(a3), 2) || a3 ||
               I2OSP(len(challengeDST), 2) || challengeDST

     c  = GG.HashToScalar(h2Input)

     return CT_EQUAL(c, Ev.proof[0])

3.4.4.2.  Unblind

















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   Input:

     PublicKey pkS
     Token token
     SerializedElement blindToken
     Evaluation Ev

   Output:

     SerializedElement issuedToken

   def Unblind(pkS, token, blindToken, Ev):
     if VerifyProof(pkS, blindToken, Ev) == false:
       ABORT()

     r = token.blind
     Z = GG.Deserialize(Ev.element)
     N = (r^(-1)) * Z

     issuedToken = GG.Serialize(N)

     return issuedToken

4.  Ciphersuites

   A ciphersuite (also referred to as 'suite' in this document) for the
   protocol wraps the functionality required for the protocol to take
   place.  This ciphersuite should be available to both the client and
   server, and agreement on the specific instantiation is assumed
   throughout.  A ciphersuite contains instantiations of the following
   functionalities:

   *  "GG": A prime-order group exposing the API detailed in
      Section 2.1, with base point defined in the corresponding
      reference for each group.

   *  "Hash": A cryptographic hash function that is indifferentiable
      from a Random Oracle.

   This section specifies supported VOPRF group and hash function
   instantiations.  For each group, we specify the HashToGroup,
   HashToScalar, and serialization functionalities.

   Applications should take caution in using ciphersuites targeting
   P-256 and ristretto255.  See Section 5.2 for related discussion.






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4.1.  OPRF(ristretto255, SHA-256)

   *  Group: ristretto255 [RISTRETTO]

      -  HashToGroup(): hash_to_ristretto255
         [I-D.irtf-cfrg-hash-to-curve] with DST = "VOPRF05-" ||
         contextString, where contextString is that which is computed in
         the Setup functions, and "expand_message" =
         "expand_message_xmd" using SHA-256.

      -  HashToScalar(): Use hash_to_field from
         [I-D.irtf-cfrg-hash-to-curve] using Order() as the prime
         modulus, with L=48, and expand_message_xmd with SHA-256.

      -  Serialization: Serialization converts group elements to 32-byte
         strings using the 'Encode' function from [RISTRETTO].
         Deserialization converts 32-byte strings to group elements
         using the 'Decode' function from [RISTRETTO].

   *  Hash: SHA-256

   *  ID: 0x0001

4.2.  OPRF(decaf448, SHA-512)

   *  Group: decaf448 [RISTRETTO]

      -  HashToGroup(): hash_to_decaf448 [I-D.irtf-cfrg-hash-to-curve]
         with DST = "VOPRF05-" || contextString, where contextString is
         that which is computed in the Setup functions, and
         "expand_message" = "expand_message_xmd" using SHA-512.

      -  HashToScalar(): Use hash_to_field from
         [I-D.irtf-cfrg-hash-to-curve] using Order() as the prime
         modulus, with L=84, and "expand_message_xmd" with SHA-512.

      -  Serialization: Serialization converts group elements to 56-byte
         strings using the 'Encode' function from [RISTRETTO].
         Deserialization converts 56-byte strings to group elements
         using the 'Decode' function from [RISTRETTO].

   *  Hash: SHA-512

   *  ID: 0x0002







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

   *  Group: P-256 (secp256r1) [x9.62]

      -  HashToGroup(): P256_XMD:SHA-256_SSWU_RO_
         [I-D.irtf-cfrg-hash-to-curve] with DST = "VOPRF05-" ||
         contextString, where contextString is that which is computed in
         the Setup functions.

      -  HashToScalar(): Use hash_to_field from
         [I-D.irtf-cfrg-hash-to-curve] using Order() as the prime
         modulus, with L=48, and "expand_message_xmd" with SHA-256.

      -  Serialization: The compressed point encoding for the curve
         [SEC1] consisting of 33 bytes.

   *  Hash: SHA-256

   *  ID: 0x0003

4.4.  OPRF(P-384, SHA-512)

   *  Group: P-384 (secp384r1) [x9.62]

      -  HashToGroup(): P384_XMD:SHA-512_SSWU_RO_
         [I-D.irtf-cfrg-hash-to-curve] with DST = "VOPRF05-" ||
         contextString, where contextString is that which is computed in
         the Setup functions.

      -  HashToScalar(): Use hash_to_field from
         [I-D.irtf-cfrg-hash-to-curve] using Order() as the prime
         modulus, with L=72, and "expand_message_xmd" with SHA-512.

      -  Serialization: The compressed point encoding for the curve
         [SEC1] consisting of 49 bytes.

   *  Hash: SHA-512

   *  ID: 0x0004

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

   *  Group: P-521 (secp521r1) [x9.62]

      -  HashToGroup(): P521_XMD:SHA-512_SSWU_RO_
         [I-D.irtf-cfrg-hash-to-curve] with DST = "VOPRF05-" ||
         contextString, where contextString is that which is computed in
         the Setup functions.



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      -  HashToScalar(): Use hash_to_field from
         [I-D.irtf-cfrg-hash-to-curve] using Order() as the prime
         modulus, with L=98, and "expand_message_xmd" with SHA-512.

      -  Serialization: The compressed point encoding for the curve
         [SEC1] consisting of 67 bytes.

   *  Hash: SHA-512

   *  ID: 0x0005

5.  Security Considerations

   This section discusses the cryptographic security of our protocol,
   along with some suggestions and trade-offs that arise from the
   implementation of an OPRF.

5.1.  Security properties

   The security properties of an OPRF protocol with functionality y =
   F(k, x) include those of a standard PRF.  Specifically:

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

   In other words, consider an adversary that picks inputs x from the
   domain of F and evaluates F on (k,x) (without knowledge of randomly
   sampled k).  Then the output distribution F(k,x) is indistinguishable
   from the output distribution of a randomly chosen function with the
   same domain and range.

   A consequence of showing that a function is pseudorandom, is that it
   is necessarily non-malleable (i.e. we cannot compute a new evaluation
   of F from an existing evaluation).  A genuinely random function will
   be non-malleable with high probability, and so a pseudorandom
   function must be non-malleable to maintain indistinguishability.

   An OPRF protocol must also satisfy the following property:

   *  Oblivious: The server must learn nothing about the client's input
      or the output of the function.  In addition, the client must learn
      nothing about the server's private key.

   Essentially, obliviousness tells us that, even if the server learns
   the client's input x at some point in the future, then the server
   will not be able to link any particular OPRF evaluation to x.  This
   property is also known as unlinkability [DGSTV18].



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

   *  Verifiable: The client must only complete execution of the
      protocol if it can successfully assert that the OPRF output it
      computes is correct.  This is taken with respect to the OPRF key
      held by the server.

   Any OPRF that satisfies the 'verifiable' security property is known
   as a verifiable OPRF, or VOPRF for short.  In practice, the notion of
   verifiability requires that the server commits to the key before the
   actual protocol execution takes place.  Then the client verifies that
   the server has used the key in the protocol using this commitment.
   In the following, we may also refer to this commitment as a public
   key.

5.2.  Cryptographic security

   Below, we discuss the cryptographic security of the (V)OPRF protocol
   from Section 3, relative to the necessary cryptographic assumptions
   that need to be made.

5.2.1.  Computational hardness assumptions

   Each assumption states that the problems specified below are
   computationally difficult to solve in relation to a particular choice
   of security parameter "sp".

   Let GG = GG(sp) be a group with prime-order p, and let FFp be the
   finite field of order p.

5.2.1.1.  Discrete-log (DL) problem

   Given G, a generator of GG, and H = hG for some h in FFp; output h.

5.2.1.2.  Decisional Diffie-Hellman (DDH) problem

   Sample a uniformly random bit d in {0,1}. Given (G, aG, bG, C),
   where:

   *  G is a generator of GG;

   *  a,b are elements of FFp;

   *  if d == 0: C = abG; else: C is sampled uniformly GG(sp).

   Output d' == d.




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5.2.2.  Protocol security

   Our OPRF construction is based on the VOPRF construction known as
   2HashDH-NIZK given by [JKK14]; essentially without providing zero-
   knowledge proofs that verify that the output is correct.  Our VOPRF
   construction is identical to the [JKK14] construction, except that we
   can optionally perform multiple VOPRF evaluations in one go, whilst
   only constructing one NIZK proof object.  This is enabled using an
   established batching technique.

   Consequently the cryptographic security of our construction is based
   on the assumption that the One-More Gap DH is computationally
   difficult to solve.

   The (N,Q)-One-More Gap DH (OMDH) problem asks the following.

    Given:
    - G, k * G, G_1, ... , G_N where G, G_1, ... G_N are elements of GG;
    - oracle access to an OPRF functionality using the key k;
    - oracle access to DDH solvers.

    Find Q+1 pairs of the form below:

    (G_{j_s}, k * G_{j_s})

    where the following conditions hold:
      - s is a number between 1 and Q+1;
      - j_s is a number between 1 and N for each s;
      - Q is the number of allowed queries.

   The original paper [JKK14] gives a security proof that the 2HashDH-
   NIZK construction satisfies the security guarantees of a VOPRF
   protocol Section 5.1 under the OMDH assumption in the universal
   composability (UC) security model.

5.2.3.  Q-strong-DH oracle

   A side-effect of our OPRF design is that it allows instantiation of a
   oracle for constructing Q-strong-DH (Q-sDH) samples.  The Q-Strong-DH
   problem asks the following.

       Given G1, G2, h*G2, (h^2)*G2, ..., (h^Q)*G2; for G1 and G2
       generators of GG.

       Output ( (1/(k+c))*G1, c ) where c is an element of FFp






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   The assumption that this problem is hard was first introduced in
   [BB04].  Since then, there have been a number of cryptanalytic
   studies that have reduced the security of the assumption below that
   implied by the group instantiation (for example, [BG04] and
   [Cheon06]).  In summary, the attacks reduce the security of the group
   instantiation by log_2(Q) bits.

   As an example, suppose that a group instantiation is used that
   provides 128 bits of security against discrete log cryptanalysis.
   Then an adversary with access to a Q-sDH oracle and makes Q=2^20
   queries can reduce the security of the instantiation by log_2(2^20) =
   20 bits.

   Notice that it is easy to instantiate a Q-sDH oracle using the OPRF
   functionality that we provide.  A client can just submit sequential
   queries of the form (G, k * G, (k^2)G, ..., (k^(Q-1))G), where each
   query is the output of the previous interaction.  This means that any
   client that submit Q queries to the OPRF can use the aforementioned
   attacks to reduce security of the group instantiation by log_2(Q)
   bits.

   Recall that from a malicious client's perspective, the adversary wins
   if they can distinguish the OPRF interaction from a protocol that
   computes the ideal functionality provided by the PRF.

5.2.4.  Implications for ciphersuite choices

   The OPRF instantiations that we recommend in this document are
   informed by the cryptanalytic discussion above.  In particular,
   choosing elliptic curves configurations that describe 128-bit group
   instantiations would appear to in fact instantiate an OPRF with
   128-log_2(Q) bits of security.

   In most cases, it would require an informed and persistent attacker
   to launch a highly expensive attack to reduce security to anything
   much below 100 bits of security.  We see this possibility as
   something that may result in problems in the future.  For
   applications that cannot tolerate discrete logarithm security of
   lower than 128 bits, we recommend only implementing ciphersuites with
   IDs: 0x0002, 0x0004, and 0x0005.

5.3.  Hashing to curve

   A critical requirement of implementing the prime-order group using
   elliptic curves is a method to instantiate the function
   "GG.HashToGroup", that maps inputs to group elements.  In the
   elliptic curve setting, this deterministically maps inputs x (as byte
   arrays) to uniformly chosen points on the curve.



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   In the security proof of the construction Hash is modeled as a random
   oracle.  This implies that any instantiation of "GG.HashToGroup" must
   be pre-image and collision resistant.  In Section 4 we give
   instantiations of this functionality based on the functions described
   in [I-D.irtf-cfrg-hash-to-curve].  Consequently, any OPRF
   implementation must adhere to the implementation and security
   considerations discussed in [I-D.irtf-cfrg-hash-to-curve] when
   instantiating the function.

5.4.  Timing Leaks

   To ensure no information is leaked during protocol execution, all
   operations that use secret data MUST run in constant time.
   Operations that SHOULD run in constant time include all prime-order
   group operations and proof-specific operations ("GenerateProof()" and
   "VerifyProof()").

5.5.  Key rotation

   Since the server's key is critical to security, the longer it is
   exposed by performing (V)OPRF operations on client inputs, the longer
   it is possible that the key can be compromised.  For example,if the
   key is kept in circulation for a long period of time, then it also
   allows the clients to make enough queries to launch more powerful
   variants of the Q-sDH attacks from Section 5.2.3.

   To combat attacks of this nature, regular key rotation should be
   employed on the server-side.  A suitable key-cycle for a key used to
   compute (V)OPRF evaluations would be between one week and six months.

6.  Additive blinding

   Let "H" refer to the function "GG.HashToGroup", in Section 2.1 we
   assume that the client-side blinding is carried out directly on the
   output of "H(x)", i.e. computing "r * H(x)" for some "r <-$ GF(p)".
   In the [I-D.irtf-cfrg-opaque] document, it is noted that it may be
   more efficient to use additive blinding (rather than multiplicative)
   if the client can preprocess some values.  For example, a valid way
   of computing additive blinding would be to instead compute "H(x) + (r
   * G)", where "G" is the fixed generator for the group "GG".

   The advantage of additive blinding is that it allows the client to
   pre-process tables of blinded scalar multiplications for "G".  This
   may give it a computational efficiency advantage (due to the fact
   that a fixed-base multiplication can be calculated faster than a
   variable-base multiplication).  Pre-processing also reduces the
   amount of computation that needs to be done in the online exchange.
   Choosing one of these values "r * G" (where "r" is the scalar value



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   that is used), then computing "H(x) + (r * G)" is more efficient than
   computing "r * H(x)".  Therefore, it may be advantageous to define
   the OPRF and VOPRF protocols using additive blinding (rather than
   multiplicative) blinding.  In fact, the only algorithms that need to
   change are Blind and Unblind (and similarly for the VOPRF variants).

   We define the variants of the algorithms in Section 3.4 for
   performing additive blinding below, along with a new algorithm
   "Preprocess".  The "Preprocess" algorithm can take place offline and
   before the rest of the OPRF protocol.  The Blind algorithm takes the
   preprocessed values as inputs, but the signature of Unblind remains
   the same.

6.1.  Preprocess

   struct {
     Scalar blind;
     SerializedElement blindedGenerator;
     SerializedElement blindedPublicKey;
   } PreprocessedBlind;

   Input:

     PublicKey pkS

   Output:

     PrepocessedBlind preproc

   def Preprocess(pkS):
     PK = GG.Deserialize(pkS)
     r = GG.RandomScalar()
     blindedGenerator = GG.Serialize(ScalarBaseMult(r))
     blindedPublicKey = GG.Serialize(r * PK)

     preproc = PrepocessedBlind{
       blind: r,
       blindedGenerator: blindedGenerator,
       blindedPublicKey: blindedPublicKey,
     }

     return preproc

6.2.  Blind







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Input:

  ClientInput input
  PreprocessedBlind preproc

Output:

  Token token
  SerializedElement blindToken

def Blind(input, preproc):
  Q = GG.Deserialize(preproc.blindedGenerator) /* Q = ScalarBaseMult(r) */
  P = GG.HashToGroup(input)

  token = Token{
    data: input,
    blind: preproc.blindedPublicKey
  }
  blindToken = GG.Serialize(P + Q)           /* P + ScalarBaseMult(r) */

  return (token, blindToken)

6.3.  Unblind

   Input:

     Token token
     Evaluation ev
     SerializedElement blindToken

   Output:

    SerializedElement unblinded

   def Unblind(token, ev, blindToken):
     PKR = GG.Deserialize(token.blind)
     Z = GG.Deserialize(ev.element)
     N := Z - PKR

     issuedToken = GG.Serialize(N)

     return issuedToken

   Let "P = GG.HashToGroup(x)".  Notice that Unblind computes:

   Z - PKR = k * (P + r * G) - r * pkS
           = k * P + k * (r * G) - r * (k * G)
           = k * P



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   by the commutativity of scalar multiplication in GG.  This is the
   same output as in the "Unblind" algorithm for multiplicative
   blinding.

   Note that the verifiable variant of "Unblind" works as above but
   includes the step to "VerifyProof", as specified in Section 3.4.4.

6.3.1.  Parameter Commitments

   For some applications, it may be desirable for server to bind tokens
   to certain parameters, e.g., protocol versions, ciphersuites, etc.
   To accomplish this, server should use a distinct scalar for each
   parameter combination.  Upon redemption of a token T from the client,
   server can later verify that T was generated using the scalar
   associated with the corresponding parameters.

7.  Contributors

   *  Sofia Celi (cherenkov@riseup.net)

   *  Alex Davidson (alex.davidson92@gmail.com)

   *  Armando Faz Hernandez (armfazh@cloudflare.com)

   *  Eli-Shaoul Khedouri (eli@intuitionmachines.com)

   *  Nick Sullivan (nick@cloudflare.com)

   *  Christopher A.  Wood (caw@heapingbits.net)

8.  Acknowledgements

   This document resulted from the work of the Privacy Pass team
   [PrivacyPass].  The authors would also like to acknowledge the
   helpful conversations with Hugo Krawczyk.  Eli-Shaoul Khedouri
   provided additional review and comments on key consistency.

9.  References

9.1.  Normative References

   [BB04]     "Short Signatures Without Random Oracles",
              <http://ai.stanford.edu/~xb/eurocrypt04a/bbsigs.pdf>.

   [BG04]     "The Static Diffie-Hellman Problem",
              <https://eprint.iacr.org/2004/306>.





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   [Cheon06]  "Security Analysis of the Strong Diffie-Hellman Problem",
              <https://www.iacr.org/archive/
              eurocrypt2006/40040001/40040001.pdf>.

   [DGSTV18]  "Privacy Pass, Bypassing Internet Challenges Anonymously",
              <https://www.degruyter.com/view/j/popets.2018.2018.issue-
              3/popets-2018-0026/popets-2018-0026.xml>.

   [I-D.davidson-pp-protocol]
              Davidson, A., "Privacy Pass: The Protocol", Work in
              Progress, Internet-Draft, draft-davidson-pp-protocol-01,
              13 July 2020, <http://www.ietf.org/internet-drafts/draft-
              davidson-pp-protocol-01.txt>.

   [I-D.irtf-cfrg-hash-to-curve]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and
              C. Wood, "Hashing to Elliptic Curves", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-hash-to-curve-10, 11
              October 2020, <http://www.ietf.org/internet-drafts/draft-
              irtf-cfrg-hash-to-curve-10.txt>.

   [I-D.irtf-cfrg-opaque]
              Krawczyk, H., Lewi, K., and C. Wood, "The OPAQUE
              Asymmetric PAKE Protocol", Work in Progress, Internet-
              Draft, draft-irtf-cfrg-opaque-00, 28 September 2020,
              <http://www.ietf.org/internet-drafts/draft-irtf-cfrg-
              opaque-00.txt>.

   [JKK14]    "Round-Optimal Password-Protected Secret Sharing and
              T-PAKE in the Password-Only model",
              <https://eprint.iacr.org/2014/650>.

   [JKKX16]   "Highly-Efficient and Composable Password-Protected Secret
              Sharing (Or, How to Protect Your Bitcoin Wallet Online)",
              <https://eprint.iacr.org/2016/144>.

   [PrivacyPass]
              "Privacy Pass",
              <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>.



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   [RFC8017]  Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
              "PKCS #1: RSA Cryptography Specifications Version 2.2",
              RFC 8017, DOI 10.17487/RFC8017, November 2016,
              <https://www.rfc-editor.org/info/rfc8017>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RISTRETTO]
              Valence, H., Grigg, J., Tankersley, G., Valsorda, F.,
              Lovecruft, I., and M. Hamburg, "The ristretto255 and
              decaf448 Groups", Work in Progress, Internet-Draft, draft-
              irtf-cfrg-ristretto255-decaf448-00, 5 October 2020,
              <http://www.ietf.org/internet-drafts/draft-irtf-cfrg-
              ristretto255-decaf448-00.txt>.

   [SEC1]     Standards for Efficient Cryptography Group (SECG), ., "SEC
              1: Elliptic Curve Cryptography",
              <https://www.secg.org/sec1-v2.pdff>.

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

   [SJKS17]   "SPHINX, A Password Store that Perfectly Hides from
              Itself", <https://eprint.iacr.org/2018/695>.

   [x9.62]    ANSI, "Public Key Cryptography for the Financial Services
              Industry: the Elliptic Curve Digital Signature Algorithm
              (ECDSA)", ANSI X9.62-1998, September 1998.

9.2.  Informative References

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

Authors' Addresses

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

   Email: alex.davidson92@gmail.com




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   Armando Faz-Hernandez
   Cloudflare
   101 Townsend St
   San Francisco,
   United States of America

   Email: armfazh@cloudflare.com


   Nick Sullivan
   Cloudflare
   101 Townsend St
   San Francisco,
   United States of America

   Email: nick@cloudflare.com


   Christopher A. Wood
   Cloudflare
   101 Townsend St
   San Francisco,
   United States of America

   Email: caw@heapingbits.net


























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