[Docs] [txt|pdf|xml|html] [Tracker] [WG] [Email] [Diff1] [Diff2] [Nits]

Versions: (draft-goldbe-vrf) 00 01 02 03 04 05

CFRG                                                         S. Goldberg
Internet-Draft                                                 L. Reyzin
Intended status: Standards Track                       Boston University
Expires: December 30, 2018                               D. Papadopoulos
                           Hong Kong University of Science and Techology
                                                               J. Vcelak
                                                                     NS1
                                                           June 28, 2018


                   Verifiable Random Functions (VRFs)
                         draft-irtf-cfrg-vrf-02

Abstract

   A Verifiable Random Function (VRF) is the public-key version of a
   keyed cryptographic hash.  Only the holder of the private key can
   compute the hash, but anyone with public key can verify the
   correctness of the hash.  VRFs are useful for preventing enumeration
   of hash-based data structures.  This document specifies several VRF
   constructions that are secure in the cryptographic random oracle
   model.  One VRF uses RSA and the other VRF uses Eliptic Curves (EC).

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 December 30, 2018.

Copyright Notice

   Copyright (c) 2018 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of



Goldberg, et al.        Expires December 30, 2018               [Page 1]


Internet-Draft                     VRF                         June 2018


   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.  Rationale . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.2.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   3
     1.3.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  VRF Algorithms  . . . . . . . . . . . . . . . . . . . . . . .   4
   3.  VRF Security Properties . . . . . . . . . . . . . . . . . . .   4
     3.1.  Full Uniqueness or Trusted Uniqueness . . . . . . . . . .   5
     3.2.  Full Collison Resistance or Trusted Collision Resistance    5
     3.3.  Full Pseudorandomness or Selective Pseudorandomness . . .   5
     3.4.  An additional pseudorandomness property . . . . . . . . .   6
   4.  RSA Full Domain Hash VRF (RSA-FDH-VRF)  . . . . . . . . . . .   7
     4.1.  RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . .   8
     4.2.  RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . .   8
     4.3.  RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . .   9
   5.  Elliptic Curve VRF (ECVRF)  . . . . . . . . . . . . . . . . .   9
     5.1.  ECVRF Proving . . . . . . . . . . . . . . . . . . . . . .  11
     5.2.  ECVRF Proof To Hash . . . . . . . . . . . . . . . . . . .  12
     5.3.  ECVRF Verifying . . . . . . . . . . . . . . . . . . . . .  13
     5.4.  ECVRF Auxiliary Functions . . . . . . . . . . . . . . . .  13
       5.4.1.  ECVRF Hash To Curve . . . . . . . . . . . . . . . . .  13
       5.4.2.  ECVRF Hash Points . . . . . . . . . . . . . . . . . .  17
       5.4.3.  ECVRF Decode Proof  . . . . . . . . . . . . . . . . .  17
     5.5.  ECVRF Ciphersuites  . . . . . . . . . . . . . . . . . . .  18
     5.6.  When the ECVRF Keys are Untrusted . . . . . . . . . . . .  19
       5.6.1.  ECVRF Validate Key  . . . . . . . . . . . . . . . . .  20
   6.  Implementation Status . . . . . . . . . . . . . . . . . . . .  20
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  21
     7.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  21
       7.1.1.  Uniqueness and collision resistance with untrusted
               keys  . . . . . . . . . . . . . . . . . . . . . . . .  21
       7.1.2.  Pseudorandomness with untrusted keys  . . . . . . . .  22
     7.2.  Selective vs Full Pseudorandomness  . . . . . . . . . . .  22
     7.3.  Proper pseudorandom nonce for ECVRF . . . . . . . . . . .  22
     7.4.  Timing attacks  . . . . . . . . . . . . . . . . . . . . .  23
     7.5.  Proofs Provide No Secrecy for VRF Input . . . . . . . . .  23
     7.6.  Prehashing  . . . . . . . . . . . . . . . . . . . . . . .  23
     7.7.  Hash function domain separation and future-proofing . . .  24
   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  24
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  25



Goldberg, et al.        Expires December 30, 2018               [Page 2]


Internet-Draft                     VRF                         June 2018


   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  25
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  25
     10.2.  Informative References . . . . . . . . . . . . . . . . .  26
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  27

1.  Introduction

1.1.  Rationale

   A Verifiable Random Function (VRF) [MRV99] is the public-key version
   of a keyed cryptographic hash.  Only the holder of the private VRF
   key can compute the hash, but anyone with corresponding public key
   can verify the correctness of the hash.

   A key application of the VRF is to provide privacy against offline
   enumeration (e.g. dictionary attacks) on data stored in a hash-based
   data structure.  In this application, a Prover holds the VRF private
   key and uses the VRF hashing to construct a hash-based data structure
   on the input data.  Due to the nature of the VRF, only the Prover can
   answer queries about whether or not some data is stored in the data
   structure.  Anyone who knows the public VRF key can verify that the
   Prover has answered the queries correctly.  However no offline
   inferences (i.e. inferences without querying the Prover) can be made
   about the data stored in the data strucuture.

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].

1.3.  Terminology

   The following terminology is used through this document:

   SK:  The private key for the VRF.

   PK:  The public key for the VRF.

   alpha:  The input to be hashed by the VRF.

   beta:  The VRF hash output.

   pi:  The VRF proof.

   Prover:  The Prover holds the private VRF key SK and public VRF key
      PK.




Goldberg, et al.        Expires December 30, 2018               [Page 3]


Internet-Draft                     VRF                         June 2018


   Verifier:  The Verifier holds the public VRF key PK.

2.  VRF Algorithms

   A VRF comes with a key generation algorithm that generates a public
   VRF key PK and private VRF key SK.

   The prover hashes an input alpha using the private VRF key SK to
   obtain a VRF hash output beta

      beta = VRF_hash(SK, alpha)

   The VRF_hash algorithm is deterministic, in the sense that it always
   produces the same output beta given a pair of inputs (SK, alpha).
   The prover also uses the private key SK to construct a proof pi that
   beta is the correct hash output

      pi = VRF_prove(SK, alpha)

   The VRFs defined in this document allow anyone to deterministically
   obtain the VRF hash output beta directly from the proof value pi as

      beta = VRF_proof2hash(pi)

   Notice that this means that

      VRF_hash(SK, alpha) = VRF_proof2hash(VRF_prove(SK, alpha))

   and thus this document will specify VRF_prove and VRF_proof2hash
   rather than VRF_hash.

   The proof pi allows a Verifier holding the public key PK to verify
   that beta is the correct VRF hash of input alpha under key PK.  Thus,
   the VRF also comes with an algorithm

      VRF_verify(PK, alpha, pi)

   that outputs VALID if beta=VRF_proof2hash(pi) is the correct VRF hash
   of alpha under key PK, and outputs INVALID otherwise.

3.  VRF Security Properties

   VRFs are designed to ensure the following security properties.








Goldberg, et al.        Expires December 30, 2018               [Page 4]


Internet-Draft                     VRF                         June 2018


3.1.  Full Uniqueness or Trusted Uniqueness

   Uniqueness means that, for any fixed public VRF key and for any input
   alpha, there is a unique VRF output beta that can be proved to be
   valid.  Uniqueness must hold even for an adversarial Prover that
   knows the VRF private key SK.

   More percisely, "full uniqueness" states that a computationally-
   bounded adversary cannot choose a VRF public key PK, a VRF input
   alpha, two different VRF hash outputs beta1 and beta2, and two proofs
   pi1 and pi2 such that VRF_verify(PK, alpha, pi1) and VRF_verify(PK,
   alpha, pi2) both output VALID.

   A slightly weaker security property called "trusted uniquness"
   sufficies for many applications.  Trusted uniqueness is the same as
   full uniqueness, but it must hold only if the VRF keys PK and SK were
   generated in a trustworthy manner.  In other words, uniqueness might
   not hold if keys were generated in an invalid manner or with bad
   randomness.

3.2.  Full Collison Resistance or Trusted Collision Resistance

   Like any cryprographic hash function, VRFs need to be collision
   resistant.  Collison resistance must hold even for an adversarial
   Prover that knows the VRF private key SK.

   More percisely, "full collision resistance" states that it should be
   computationally infeasible for an adversary to find two distinct VRF
   inputs alpha1 and alpha2 that have the same VRF hash beta, even if
   that adversary knows the private VRF key SK.

   For most applications, a slightly weaker security property called
   "trusted collision resistance" suffices.  Trusted collision
   resistance is the same as collision resistance, but it holds only if
   PK and SK were generated in a trustworthy manner.

3.3.  Full Pseudorandomness or Selective Pseudorandomness

   Pseudorandomness ensures that when an adversarial Verifier sees a VRF
   hash output beta without its corresponding VRF proof pi, then beta is
   indistinguishable from a random value.

   More percisely, suppose the public and private VRF keys (PK, SK) were
   generated in a trustworthy manner.  Pseudorandomness ensures that the
   VRF hash output beta (without its corresponding VRF proof pi) on any
   adversarially-chosen "target" VRF input alpha looks indistinguishable
   from random for any computationally bounded adversary who does not
   know the private VRF key SK.  This holds even if the adversary also



Goldberg, et al.        Expires December 30, 2018               [Page 5]


Internet-Draft                     VRF                         June 2018


   gets to choose other VRF inputs alpha' and observe their
   corresponding VRF hash outputs beta' and proofs pi'.

   With "full pseudorandomness", the adversary is allowed to choose the
   "target" VRF input alpha at any time, even after it observes VRF
   outputs beta' and proofs pi' on a variety of chosen inputs alpha'.

   "Selective pseudorandomness" is a weaker security property which
   suffices in many applications.  Here, the adversary must choose the
   target VRF input alpha independently of the public VRF key PK, and
   before it observes VRF outputs beta' and proofs pi' on inputs alpha'
   of its choice.

   It is important to remember that the VRF output beta does not look
   random to the Prover, or to any other party that knows the private
   VRF key SK!  Such a party can easily distinguish beta from a random
   value by comparing beta to the result of VRF_hash(SK, alpha).

   Also, the VRF output beta does not look random to any party that
   knows valid VRF proof pi corresponding to the VRF input alpha, even
   if this party does not know the private VRF key SK.  Such a party can
   easily distinguish beta from a random value by checking whether
   VRF_verify(PK, alpha, pi) returns "VALID" and beta =
   VRF_proof2hash(pi).

   Also, the VRF output beta may not look random if VRF key generation
   was not done in a trustworthy fashion.  (For example, if VRF keys
   were generated with bad randomness.)

3.4.  An additional pseudorandomness property

   [TODO: This property is not needed for applications that use VRFs to
   prevent enumeration of hash-based data structures.  However, we
   noticed that some other applications of VRF (e.g.  Algorand,
   Oroborus) rely on this property.  We are waiting on a formal
   definition of this property in the literature, and a proof that our
   ECVRF scheme can satisfy this property.  Preliminary analysis
   suggests that acheiving this property requires ECVRF verifiers to run
   an VRF_validate_key() key function upon receipt of VRF public keys
   and the proof2hash function to be modified to take in (gamma, beta,
   pk) rather than just gamma.]

   Pseudorandomness, as defined in Section 3.3, does not hold if the VRF
   keys were generated adversarially.

   There is, however, a different type of pseudorandomness that could
   hold even if the VRF keys are generated adversarially, as long as the
   VRF input alpha is unpredictable.  This property is similar to the



Goldberg, et al.        Expires December 30, 2018               [Page 6]


Internet-Draft                     VRF                         June 2018


   pseudorandomness achieved by an (ordinary, unkeyed) cryptographic
   hash function.

   [TODO: Formal definition here.]

4.  RSA Full Domain Hash VRF (RSA-FDH-VRF)

   The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies
   the "trusted uniqueness", "trusted collision resistance", and "full
   pseudorandomness" properties defined in Section 3.  Its security
   follows from the standard RSA assumption in the random oracle model.
   Formal security proofs are in [nsec5ecc].

   The VRF computes the proof pi as a deterministic RSA signature on
   input alpha using the RSA Full Domain Hash Algorithm [RFC8017]
   parametrized with the selected hash algorithm.  RSA signature
   verification is used to verify the correctness of the proof.  The VRF
   hash output beta is simply obtained by hashing the proof pi with the
   selected hash algorithm.

   The key pair for RSA-FDH-VRF MUST be generated in a way that it
   satisfies the conditions specified in Section 3 of [RFC8017].

   In this document, the notation from [RFC8017] is used.

   Parameters used:

      (n, e) - RSA public key

      K - RSA private key

      k - length in octets of the RSA modulus n

   Fixed options:

      Hash - cryptographic hash function

      hLen - output length in octets of hash function Hash

   Constraints on options:

      Cryptographic security of Hash is at least as high as the
      cryptographic security level of the RSA key.

   Primitives used:

      I2OSP - Coversion of a nonnegative integer to an octet string as
      defined in Section 4.1 of [RFC8017]



Goldberg, et al.        Expires December 30, 2018               [Page 7]


Internet-Draft                     VRF                         June 2018


      OS2IP - Coversion of an octet string to a nonnegative integer as
      defined in Section 4.2 of [RFC8017]

      RSASP1 - RSA signature primitive as defined in Section 5.2.1 of
      [RFC8017]

      RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of
      [RFC8017]

      MGF1 - Mask Generation Function based on a hash function as
      defined in Section B.2.1 of [RFC8017]

4.1.  RSA-FDH-VRF Proving

   RSAFDHVRF_prove(K, alpha)

   Input:

      K - RSA private key

      alpha - VRF hash input, an octet string

   Output:

      pi - proof, an octet string of length k

   Steps:

   1.  EM = MGF1(alpha, k - 1)

   2.  m = OS2IP(EM)

   3.  s = RSASP1(K, m)

   4.  pi = I2OSP(s, k)

   5.  Output pi

4.2.  RSA-FDH-VRF Proof To Hash

   RSAFDHVRF_proof2hash(pi)

   Input:

      pi - proof, an octet string of length k

   Output:




Goldberg, et al.        Expires December 30, 2018               [Page 8]


Internet-Draft                     VRF                         June 2018


      beta - VRF hash output, an octet string of length hLen

   Steps:

   1.  beta = Hash(pi)

   2.  Output beta

4.3.  RSA-FDH-VRF Verifying

   RSAFDHVRF_verify((n, e), alpha, pi)

   Input:

      (n, e) - RSA public key

      alpha - VRF hash input, an octet string

      pi - proof to be verified, an octet string of length n

   Output:

      "VALID" or "INVALID"

   Steps:

   1.  s = OS2IP(pi)

   2.  m = RSAVP1((n, e), s)

   3.  EM = I2OSP(m, k - 1)

   4.  EM' = MGF1(alpha, k - 1)

   5.  If EM and EM' are equal, output "VALID"; else output "INVALID".

5.  Elliptic Curve VRF (ECVRF)

   The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that
   satisfies the trusted uniqueness, trusted collision resistance, and
   full pseudorandomness properties defined in Section 3.  The security
   of this VRF follows from the decisional Diffie-Hellman (DDH)
   assumption in the random oracle model.  Formal security proofs are in
   [nsec5ecc].

   To additionally satisfy "full uniqueness" and "full collision
   resistance", the Verifier MUST additionally perform the validation




Goldberg, et al.        Expires December 30, 2018               [Page 9]


Internet-Draft                     VRF                         June 2018


   procedure specified in Section 5.6 upon receipt of the public VRF
   key.

   Fixed options (specified in Section 5.5):

      F - finite field

      2n - length, in octets, of a field element in F; must be even

      E - elliptic curve (EC) defined over F

      m - length, in octets, of an EC point encoded as an octet string

      G - subgroup of E of large prime order

      q - prime order of group G; must be less than 2^{2n}

      cofactor - number of points on E divided by q

      g - generator of group G

      Hash - cryptographic hash function

      hLen - output length in octets of Hash; must be at least 2n

      suite - a single nonzero octet specifying the ECVRF ciphersuite,
      which determines the above options

   Notation and primitives used:

      p^k - When p is an EC point: point multiplication, i.e. k
      repetitions of the EC group operation applied to the EC point p.
      When p is an integer: exponentiation

      || - octet string concatenation

      I2OSP - nonnegative integer conversion to octet string as defined
      in Section 4.1 of [RFC8017]

      OS2IP - Coversion of an octet string to a nonnegative integer as
      defined in Section 4.2 of [RFC8017]

      EC2OSP - conversion of EC point to an m-octet string as specified
      in Section 5.5

      OS2ECP - conversion of an m-octet string to EC point as specified
      in Section 5.5.  OS2ECP returns INVALID if the octet string does
      not convert to a valid EC point.



Goldberg, et al.        Expires December 30, 2018              [Page 10]


Internet-Draft                     VRF                         June 2018


      RS2ECP - conversion of a random 2n-octet string to an EC point as
      specified in Section 5.5

      ECVRF_hash_to_curve - collision resistant hash of strings to an EC
      point; options described in Section 5.4.1 and specified in
      Section 5.5.

      ECVRF_nonce_generation - derives a pseudorandom nonce from SK and
      the input as part of ECVRF proving.  Specified in Section 5.5

      ECVRF_hash_points - collision resistant hash of EC points to an
      n-octet string.  Specified in Section 5.4.2.

   Parameters used (the generation of these parameters is specified in
   Section 5.5):

      SK - VRF private key

      x - VRF secret scalar, an integer

         Note: depending on the ciphersuite used, the VRF secret scalar
         may be equal to SK; else, it is derived from SK

      y = g^x - VRF public key, an EC point

5.1.  ECVRF Proving

   Note: this function must have the VRF private key SK as input.  Below
   we make it more efficient by supplying it also with the secret scalar
   x and the public key y as additional inputs; however, each of these
   can be computed from SK if desired.

   ECVRF_prove(y, x, alpha)

   Input:

      SK - VRF private key

      x - VRF secret scalar

      y = g^x - VRF public key

   Output:

      pi - VRF proof, octet string of length m+3n

   Steps:




Goldberg, et al.        Expires December 30, 2018              [Page 11]


Internet-Draft                     VRF                         June 2018


   1.  h = ECVRF_hash_to_curve(suite, y, alpha)

   2.  h1 = EC2OSP(h)

   3.  gamma = h^x

   4.  k = ECVRF_nonce_generation(SK, h1)

   5.  c = ECVRF_hash_points(h, gamma, g^k, h^k)

   6.  s = (k + c*x) mod q (where * denotes integer multiplication)

   7.  pi = EC2OSP(gamma) || I2OSP(c, n) || I2OSP(s, 2n)

   8.  Output pi

5.2.  ECVRF Proof To Hash

   ECVRF_proof2hash(pi)

   Input:

      pi - VRF proof, octet string of length m+3n

   Output:

      "INVALID", or

      beta - VRF hash output, octet string of length 2n

   Steps:

   1.  D = ECVRF_decode_proof(pi)

   2.  If D is "INVALID", output "INVALID" and stop

   3.  (gamma, c, s) = D

   4.  three = 0x03 = I2OSP(3, 1), a single octet with value 3

   5.  preBeta = Hash(suite || three || EC2OSP(gamma^cofactor))

   6.  beta = first 2n octets of preBeta

   7.  Output beta






Goldberg, et al.        Expires December 30, 2018              [Page 12]


Internet-Draft                     VRF                         June 2018


5.3.  ECVRF Verifying

   ECVRF_verify(y, pi, alpha)

   Input:

      y - public key, an EC point

      pi - VRF proof, octet string of length 5n+1

      alpha - VRF input, octet string

   Output:

      "VALID" or "INVALID"

   Steps:

   1.  D = ECVRF_decode_proof(pi)

   2.  If D is "INVALID", output "INVALID" and stop

   3.  (gamma, c, s) = D

   4.  u = g^s / y^c (where / denotes EC point subtraction, i.e. the
       group operation applied to g^s and the inverse of y^c)

   5.  h = ECVRF_hash_to_curve(suite, y, alpha)

   6.  v = h^s / gamma^c (where / again denotes EC point subtraction)

   7.  c' = ECVRF_hash_points(h, gamma, u, v)

   8.  If c and c' are equal, output "VALID"; else output "INVALID"

5.4.  ECVRF Auxiliary Functions

5.4.1.  ECVRF Hash To Curve

   The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and
   converts it to h, an EC point in G.  This algorithm is the only place
   alpha is used in the proving and verfying.  See Section 7.6 for
   further discussion.

   The algorithms in this section are not compatible with each other;
   the choice of algorithm is made in Section 5.5.





Goldberg, et al.        Expires December 30, 2018              [Page 13]


Internet-Draft                     VRF                         June 2018


5.4.1.1.  ECVRF_hash_to_curve_try_and_increment

   The following ECVRF_hash_to_curve_try_and_increment(suite, y, alpha)
   algorithm implements ECVRF_hash_to_curve in a simple and generic way
   that works for any elliptic curve.

   The running time of this algorithm depends on alpha.  For the
   ciphersuites specified in Section 5.5, this algorithm is expected to
   find a valid curve point after approximately two attempts (i.e., when
   ctr=1) on average.  See also [Icart09].

   However, because the running time of algorithm depends on alpha, this
   algorithm SHOULD be avoided in applications where it is important
   that the VRF input alpha remain secret.

   ECVRF_hash_to_try_and_increment(suite, y, alpha)

   Input:

      suite - a single octet specifying ECVRF ciphersuite.

      y - public key, an EC point

      alpha - value to be hashed, an octet string

   Output:

      h - hashed value, a finite EC point in G

   Steps:

   1.  ctr = 0

   2.  pk = EC2OSP(y)

   3.  one = 0x01 = I2OSP(1, 1), a single octet with value 1

   4.  h = "INVALID"

   5.  While h is "INVALID" or h is EC point at infinity:

       A.  CTR = I2OSP(ctr, 4)

       B.  ctr = ctr + 1

       C.  hash = Hash(suite || one || pk || alpha || CTR)

       D.  attempted_hash = first 2n bits of hash



Goldberg, et al.        Expires December 30, 2018              [Page 14]


Internet-Draft                     VRF                         June 2018


       E.  h = RS2ECP(attempted_hash)

       F.  If h is not "INVALID" and cofactor > 1, set h = h^cofactor

   6.  Output h

5.4.1.2.  ECVRF_hash_to_curve_elligator2_25519

   The following ECVRF__hash_to_curve_elligator2_25519(suite, y, alpha)
   algorithm implements ECVRF_hash_to_curve using the elligator2
   algorithm exclusively for Curve25519.

   ECVRF_hash_to_curve_elligator2_25519(y, alpha)

   Input:

      alpha - value to be hashed, an octet string

      y - public key, an EC point

      suite = a single octet specifying ECVRF ciphersuite.

   Output:

      h - hashed value, a finite EC point in G per the encoding in
      RFC8032 Section 5.1.2 [RFC8032]

   Fixed options:

      p = 2^255-19, the size of the finite field F, a prime, for
      Curve25519

      A = 486662, Montgomery curve constant for Curve25519

      cofactor = 8 , the cofactor for Curve25519

      * denotes integer multiplication

   Constraints on options:

      output length of Hash is at least 16n + 1 bits (i.e. greater than
      2n octets)

   Steps:

   1.   pk = EC2OSP(y)

   2.   one = 0x01 = I2OSP(1, 1), a single octet with value 1



Goldberg, et al.        Expires December 30, 2018              [Page 15]


Internet-Draft                     VRF                         June 2018


   3.   hash = Hash(suite || one || pk || alpha )

   4.   r = first 2n octets of hash

   5.   x_0 = next unused bit of hash

   6.   u = - A / (1 + 2*(r^2) ) mod p

   7.   v = u * (u^2 + A*u + 1) mod p

   8.   Let e equal the Legendre symbol of v modulo p

   9.   If e is equal to 1 then finalu = u; else finalu = (-A - u) mod p

   10.  y = (finalu - 1) / (finalu + 1) mod p

   11.  let h = (x_0,y), an EC point per the encoding in Section 5.1.2
        of [RFC8032]

   12.  h = h^cofactor

   In order to make this algorithm run in time that is (almost)
   independent of the input (so-called "constant-time"), implementers
   should pay particular attention to Steps 7 and 8 above.  These steps
   can be implemented using the following approach:

      e = v ^ ((p-1)/2) mod p

      finalu = (e*u + (e-1) * (A/2)) mod p

   The first step will produce a value e that is either 1 or p-1.  The
   second step is so fast compared to the first that that its dependence
   on e is likely to not matter; implementers can also ensure that it
   runs in the same amount of time regardless of e.

   If having this algorithm run in constant time is not important, then
   there are much faster algorithms to compute the Legendre symbol
   (which is the same as the Jacobi symbol because p is a prime).  See,
   for example, Section 12.3 of [ntb].

5.4.1.3.  ECVRF_hash_to_curve_other?

   [TODO: In addition to Elligator, it would be nice to have a generic
   ECVRF_hash_to_curve algorithm that runs in constant time and provides
   a generic way to hash an octet string onto any elliptic curve.  There
   is an an upcoming draft that deals with this [SW18].  Some other
   useful algorithms include [Icart09], and Shallue-Woestijne-Ulas
   algorithm from [BCIMRT10].]



Goldberg, et al.        Expires December 30, 2018              [Page 16]


Internet-Draft                     VRF                         June 2018


5.4.2.  ECVRF Hash Points

   ECVRF_hash_points(p_1, p_2, ..., p_j)

   Input:

      p_i - EC point in G

   Output:

      c - hash value, integer between 0 and 2^(8n)-1

   Steps:

   1.  two = 0x02 = I2OSP(2, 1), a single octet with value 2

   2.  Initialize str = suite || two

   3.  for p_i in [p_1, p_2, ... p_j]:
       str = str || EC2OSP(p_i)

   4.  c1 = Hash(str)

   5.  c2 = first n octets of c1

   6.  c = OS2IP(c2)

   7.  Output c

5.4.3.  ECVRF Decode Proof

   ECVRF_decode_proof(pi)

   Input:

      pi - VRF proof, octet string (m+3n octets)

   Output:

      "INVALID", or

      gamma - EC point

      c - integer between 0 and 2^(8n)-1

      s - integer between 0 and 2^(16n)-1

   Steps:



Goldberg, et al.        Expires December 30, 2018              [Page 17]


Internet-Draft                     VRF                         June 2018


   1.  let gamma', c', s' be pi split after m-th and m+n-th octet

   2.  gamma = OS2ECP(gamma')

   3.  if gamma = "INVALID" output "INVALID" and stop.

   4.  c = OS2IP(c')

   5.  s = OS2IP(s')

   6.  Output gamma, c, and s

5.5.  ECVRF Ciphersuites

   This document defines ECVRF-P256-SHA256 as follows:

   o  suite = 0x01 = I2OSP(1, 1).

   o  The EC group G is the NIST-P256 elliptic curve, with curve
      parameters as specified in [FIPS-186-3] (Section D.1.2.3) and
      [RFC5114]  (Section 2.6).  For this group, 2n = 32 and cofactor =
      1.

   o  The key pair generation primitive is specified in Section 3.2.1 of
      [SECG1] (q, g, SK, and PK in this document correspond to in n, G,
      d, and Q in Section 3.2.1 of [SECG1]).  In this ciphersuite, the
      secret scalar x is equal to the private key SK.

   o  The ECVRF_nonce_generation function is as specified in [RFC6979]
      Section 3.2 Steps b-h (omitting Step a and using h1 that is
      provided as input to ECVRF_nonce_generation), with the hash
      function SHA-256 as specified in [RFC6234]

   o  EC2OSP is specified in Section 2.3.3 of [SECG1] with point
      compression on.  This implies m = 2n + 1 = 33.

   o  OS2ECP is specified in Section 2.3.4 of [SECG1].

   o  RS2ECP(h) = OS2ECP(0x02 || h) (where 0x02 is a single octet with
      value 2, 0x02=I2OSP(2, 1)).  The input h is a 32-octet string and
      the output is either an EC point or "INVALID".

   o  The hash function Hash is SHA-256 as specified in [RFC6234].

   o  The ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.1.

   This document defines ECVRF-ED25519-SHA512 as follows:



Goldberg, et al.        Expires December 30, 2018              [Page 18]


Internet-Draft                     VRF                         June 2018


   o  suite = 0x02 = I2OSP(2, 1).

   o  The EC group G is the Ed25519 elliptic curve with parameters
      defined in Table 1 of [RFC8032].  For this group, 2n = 32 and
      cofactor = 8.

   o  The private key and generation of the secret scalar and the public
      key are specified in Section 5.1.5 of [RFC8032]

   o  The ECVRF_nonce_generation function is as specified in [RFC8032]
      Section 5.1.6 Steps 1-2, with dom2(F, C) equal to the empty string
      and PH(M) equal to h1.

   o  EC2OSP is specified in Section 5.1.2 of [RFC8032].  This implies m
      = 2n = 32.

   o  OS2ECP is specified in Section 5.1.3 of [RFC8032].

   o  RS2ECP is equivalent to OS2ECP.

   o  The hash function Hash is SHA-512 as specified in [RFC6234].

   o  The ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.1.

   This document defines ECVRF-ED25519-SHA512-Elligator2 as follows:

   o  This ciphersuite is identical to ECVRF-ED25519-SHA512 except that
      the ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.2 and suite = 0x03 = I2OSP(3, 1).

5.6.  When the ECVRF Keys are Untrusted

   The ECVRF as specified above is a VRF that satisfies the "trusted
   uniqueness", "trusted collision resistance", and "full
   pseudorandomness" properties defined in Section 3.  In order to
   obtain "full uniqueness" and "full collision resistance" (which
   provide protection against a malicious VRF public key), the Verifier
   MUST perform the following additional validation procedure upon
   receipt of the public VRF key.  The public VRF key MUST NOT be used
   if this procedure returns "INVALID".

   Note that this procedure is not sufficient if the elliptic curve E or
   the point g, the generator of group G, is untrusted.  If the prover
   is untrusted, the Verifier MUST obtain E and g from a trusted source,
   such as a ciphersuite specification, rather than from the prover.





Goldberg, et al.        Expires December 30, 2018              [Page 19]


Internet-Draft                     VRF                         June 2018


   This procedure supposes that the public key provided to the Verifier
   is an octet string.  The procedure returns "INVALID" if the public
   key in invalid.  Otherwise, it returns y, the public key as an EC
   point.

5.6.1.  ECVRF Validate Key

   ECVRF_validate_key(PK)

   Input:

      PK - public key, an octet string

   Output:

      "INVALID", or

      y - public key, an EC point

   Steps:

   1.  y = OS2ECP(PK)

   2.  If y is "INVALID", output "INVALID" and stop

   3.  If y^cofactor is the EC point at infinty, output "INVALID" and
       stop

   4.  Output y

6.  Implementation Status

   An implementation of the RSA-FDH-VRF (SHA-256) and ECVRF-P256-SHA256
   was first developed as a part of the NSEC5 project [I-D.vcelak-nsec5]
   and is available at <http://github.com/fcelda/nsec5-crypto>.  The
   ECVRF implementation may be out of date as this spec has evolved.

   The Key Transparency project at Google uses a VRF implemention that
   is similar to the ECVRF-P256-SHA256, with a few minor changes
   including the use of SHA-512 instead of SHA-256.  Its implementation
   is available
   <https://github.com/google/keytransparency/blob/master/core/vrf/
   vrf.go>

   An implementation by Yahoo! similar to the ECVRF is available at
   <https://github.com/r2ishiguro/vrf>.





Goldberg, et al.        Expires December 30, 2018              [Page 20]


Internet-Draft                     VRF                         June 2018


   An implementation similar to ECVRF is available as part of the CONIKS
   implementation in Golang at <https://github.com/coniks-sys/coniks-
   go/tree/master/crypto/vrf>.

   Open Whisper Systems also uses a VRF very similar to ECVRF-
   ED25519-SHA512-Elligator, called VXEdDSA, and specified here:
   <https://whispersystems.org/docs/specifications/xeddsa/>

7.  Security Considerations

7.1.  Key Generation

   Applications that use the VRFs defined in this document MUST ensure
   that that the VRF key is generated correctly, using good randomness.

7.1.1.  Uniqueness and collision resistance with untrusted keys

   The ECVRF as specified in Section 5.1-Section 5.5 statisfies the
   "trusted uniqueness" and "trusted collision resistance" properties as
   long as the VRF keys are generated correctly, with good randomness.
   If the Verifier trusts the VRF keys are generated correctly, it MAY
   use the public key y as is.

   However, if the ECVRF uses keys that could be generated
   adversarially, then the the Verfier MUST first perform the validation
   procedure ECVRF_validate_key(PK) (specified in Section 5.6) upon
   receipt of the public key PK as an octet string.  If the validation
   procedure outputs "INVALID", then the public key MUST not be used.
   Otherwise, the procedure will output a valid public key y, and the
   ECVRF with public key y satisfies the "full uniqueness" and "full
   collision resistance" properties.

   The RSA-FDH-VRF statisfies the "trusted uniqueness" and "trusted
   collision resistance" properties as long as the VRF keys are
   generated correctly, with good randomness.  These properties may not
   hold if the keys are generated adversarially (e.g., if RSA is not
   permutation).  Meanwhile, the "full uniqueness" and "full collision
   resistance" are properties that hold even if VRF keys are generated
   by an adversary.  The RSA-FDH-VRF defined in this document does not
   have these properties.  However, if adversarial key generation is a
   concern, the RSA-FDH-VRF may be modifed to have these properties by
   adding additional cryptographic checks that its public key has the
   right form.  These modifications are left for future specification.








Goldberg, et al.        Expires December 30, 2018              [Page 21]


Internet-Draft                     VRF                         June 2018


7.1.2.  Pseudorandomness with untrusted keys

   Without good randomness, the "pseudorandomness" properties of the VRF
   may not hold.  Note that it is not possible to guarantee
   pseudorandomness in the face of adversarially generated VRF keys.
   This is because an adversary can always use bad randomness to
   generate the VRF keys, and thus, the VRF output may not be
   pseudorandom.

7.2.  Selective vs Full Pseudorandomness

   [nsec5ecc] presents cryptographic reductions to an underlying hard
   problem (e.g.  Decisional Diffie Hellman for the ECVRF, or the
   standard RSA assumption for RSA-FDH-VRF) that prove the VRFs
   specificied in this document possess full pseudorandomness as well as
   selective pseudorandomness.  However, the cryptographic reductions
   are tighter for selective pseudorandomness than for full
   pseudorandomness.  This means the the VRFs have quantitavely stronger
   security guarentees for selective pseudorandomness.

   Applications that are concerned about tightness of cryptographic
   reductions therefore have two options.

   o  They may choose to ensure that selective pseudorandomness is
      sufficient for the application.  That is, that pseudorandomness of
      outputs matters only for inputs that are chosen independently of
      the VRF key.

   o  If full pseudorandomness is required for the application, the
      application may increase security parameters to make up for the
      loose security reduction.  For RSA-FDH-VRF, this means increasing
      the RSA key length.  For ECVRF, this means increasing the
      cryptographic strength of the EC group G.  For both RSA-FDH-VRF
      and ECVRF the cryptographic strength of the hash function Hash may
      also potentially need to be increased.

7.3.  Proper pseudorandom nonce for ECVRF

   The security of the ECVRF defined in this document relies on the fact
   that nonce k used in the ECVRF_prove algorithm is chosen uniformly
   and pseudorandomly modulo q, and is unknown to the advesrary.
   Otherwise, an adversary may be able to recover the private VRF key x
   (and thus break pseudorandomness of the VRF) after observing several
   valid VRF proofs pi.  The nonce generation methods specified in the
   ECVRF ciphersuites of Section 5.5 are designed with this requirement
   in mind.





Goldberg, et al.        Expires December 30, 2018              [Page 22]


Internet-Draft                     VRF                         June 2018


7.4.  Timing attacks

   The ECVRF_hash_to_curve_try_and_increment algorithm defined in
   Section 5.4.1.1 SHOULD NOT be used in applications where the VRF
   input alpha is secret and is hashed by the VRF on-the-fly.  This is
   because the algorithm's running time depends on the VRF input alpha,
   and thus creates a timing channel that can be used to learn
   information about alpha.  That said, for most inputs the amount of
   information obtained from such a timing attack is likely to be small
   (1 bit, on average), since the algorithm is expected to find a valid
   curve point after only two attempts.  However, there might be inputs
   which cause the algorithm to make many attempts before it finds a
   valid curve point; for such inputs, the information leaked in a
   timing attack will be more than 1 bit.

   Meanwhile, ECVRF-ED25519-SHA512-Elligator2 runs in constant time if
   the implementation of the ECVRF_hash_to_curve function specified in
   Section 5.4.1.2 also runs in constant time.

7.5.  Proofs Provide No Secrecy for VRF Input

   The VRF proof pi is not designed to provide secrecy and, in general,
   may reveal the VRF input alpha.  Anyone who knows PK and pi is able
   to perform an offline dictionary attack to search for alpha, by
   verifying guesses for alpha using VRF_verify.  This is in contrast to
   the VRF hash output beta which, without the proof, is pseudorandom
   and thus is designed to reveal no information about alpha.

7.6.  Prehashing

   The VRFs specified in this document allow for read-once access to the
   input alpha for both signing and verifying.  Thus, additional
   prehashing of alpha (as specified, for example, in [RFC8032] for
   EdDSA signatures) is not needed, even for applications that need to
   handle long alpha or to support the Initialized-Update-Finalize (IUF)
   interface (in such an interface, alpha is not supplied all at once,
   but rather in pieces by a sequence of calls to Update).  The ECVRF,
   in particular, uses alpha only in ECVRF_hash_to_curve.  The curve
   point h becomes the representative of alpha thereafter.  Note that
   the suite octet and the public key are hashed together with alpha in
   ECVRF_hash_to_curve, which ensures that the curve (including the
   generator g) and the public key are included indirectly into
   subsequent hashes.








Goldberg, et al.        Expires December 30, 2018              [Page 23]


Internet-Draft                     VRF                         June 2018


7.7.  Hash function domain separation and future-proofing

   Hashing is used for four different purposes in ECVRF (namely, in
   hash_to_curve, nonce_generation, hash_points, and proof2hash).  The
   theoretical analysis assumes each of these four functions is a
   separate random oracle.  This analysis still holds even if the same
   hash function is used, as long as the four queries made to the hash
   function for a given SK and alpha are overwhelmingly unlikely to
   equal each other or to any queries made to the hash function for the
   same SK and different alpha.  This is indeed the case for the ECVRF
   ciphersuites defined in this document, because:

   o  inputs to the hash function used during nonce_generation are
      unlikely to equal to inputs given to hash_to_curve, proof2hash,
      and hash_points.  This follows since nonce_generation inputs a
      secret to the hash function that is not used by honest parties as
      input to any other hash function, and is not available to the
      adversary

   o  the second octet of the input to the hash function used in
      hash_to_curve, proof2hash, and hash_points are all different

   If future designs need to specify variants (e.g., additional
   ciphersuites) of the design in this document, then, to avoid the
   possibility that an adversary can obtain a VRF output under one
   variant, and then claim it was obtained under another variant, they
   should specify a different suite constant.  This way, the inputs to
   the hash_to_curve hash function used in producing h are guaranteed to
   be different; since all the other hashing done by the prover depends
   on h, inputs all the hash functions used by the prover will also be
   different as long as hash_to_curve is collision resistant.

8.  Change Log

   Note to RFC Editor: if this document does not obsolete an existing
   RFC, please remove this appendix before publication as an RFC.

      00 - Forked this document from draft-goldbe-vrf-01.

      01 - Minor updates, mostly highlighting TODO items.

      02 - Added specification of elligator2 for Curve25519, along with
      ciphersuites for ECVRF-ED25519-SHA512-Elligator.  Changed ECVRF-
      ED25519-SHA256 suite to ECVRF-ED25519-SHA512.  (This change made
      because Ed25519 in [RFC8032] signatures use SHA512 and not
      SHA256.)  Made ECVRF nonce generation a separate component, so
      that nonces are determinsitic.  In ECVRF proving, changed + to -
      (and made corresponding verification changes) in order to be



Goldberg, et al.        Expires December 30, 2018              [Page 24]


Internet-Draft                     VRF                         June 2018


      consistent with EdDSA and ECDSA.  Highlighted that
      ECVRF_hash_to_curve acts like a prehash.  Added "suites" variable
      to ECVRF for future-proofing.  Ensured domain separation for hash
      functions by modifying hash_points and added discussion about
      domain separation.  Updated todos in the "additional
      pseudorandomness property" section.  Added an discussion of
      secrecy into security considerations.  Removed g and PK=y from
      ECVRF_hash_points because they are already present via h, which is
      computed via hash_to_curve using the suite (which identifies g)
      and y.

9.  Contributors

   This document also would not be possible without the work of Moni
   Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf
   Ziv (Facebook).  Shumon Huque (Salesforce), David C.  Lawerence
   (Akamai), and Trevor Perrin provided valuable input to this draft.

10.  References

10.1.  Normative References

   [FIPS-186-3]
              National Institute for Standards and Technology, "Digital
              Signature Standard (DSS)", FIPS PUB 186-3, June 2009.

   [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>.

   [RFC5114]  Lepinski, M. and S. Kent, "Additional Diffie-Hellman
              Groups for Use with IETF Standards", RFC 5114,
              DOI 10.17487/RFC5114, January 2008,
              <https://www.rfc-editor.org/info/rfc5114>.

   [RFC6234]  Eastlake 3rd, D. and T. Hansen, "US Secure Hash Algorithms
              (SHA and SHA-based HMAC and HKDF)", RFC 6234,
              DOI 10.17487/RFC6234, May 2011,
              <https://www.rfc-editor.org/info/rfc6234>.

   [RFC6979]  Pornin, T., "Deterministic Usage of the Digital Signature
              Algorithm (DSA) and Elliptic Curve Digital Signature
              Algorithm (ECDSA)", RFC 6979, DOI 10.17487/RFC6979, August
              2013, <https://www.rfc-editor.org/info/rfc6979>.






Goldberg, et al.        Expires December 30, 2018              [Page 25]


Internet-Draft                     VRF                         June 2018


   [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>.

   [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>.

   [SECG1]    Standards for Efficient Cryptography Group (SECG), "SEC 1:
              Elliptic Curve Cryptography", Version 2.0, May 2009,
              <http://www.secg.org/sec1-v2.pdf>.

10.2.  Informative References

   [BCIMRT10]
              Brier, E., Coron, J., Icart, T., Madore, D., Randriam, H.,
              and M. Tibouchi, "Efficient Indifferentiable Hashing into
              Ordinary Elliptic Curves", in CRYPTO, 2010.

   [I-D.vcelak-nsec5]
              Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
              D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
              Existence", draft-vcelak-nsec5-07 (work in progress), June
              2018.

   [Icart09]  Icart, T., "How to Hash into Elliptic Curves", in CRYPTO,
              2009.

   [MRV99]    Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random
              Functions", in FOCS, 1999.

   [nsec5ecc]
              Papadopoulos, D., Wessels, D., Huque, S., Vcelak, J.,
              Naor, M., Reyzin, L., and S. Goldberg, "Making NSEC5
              Practical for DNSSEC", in ePrint Cryptology Archive
              2017/099, February 2017,
              <https://eprint.iacr.org/2017/099.pdf>.

   [ntb]      Shoup, V., "A Computational Introduction to Number Theory
              and Algebra", 2008, <http://www.shoup.net/ntb/ntb-v2.pdf>.

   [SW18]     Sullivan, E. and C. Wood, "Hashing to Elliptic Curves",
              2018.






Goldberg, et al.        Expires December 30, 2018              [Page 26]


Internet-Draft                     VRF                         June 2018


Authors' Addresses

   Sharon Goldberg
   Boston University
   111 Cummington St, MCS135
   Boston, MA  02215
   USA

   EMail: goldbe@cs.bu.edu


   Leonid Reyzin
   Boston University
   111 Cummington St, MCS135
   Boston, MA  02215
   USA

   EMail: reyzin@bu.edu


   Dimitrios Papadopoulos
   Hong Kong University of Science and Techology
   Clearwater Bay
   Hong Kong

   EMail: dipapado@cse.ust.hkbu.edu


   Jan Vcelak
   NS1
   16 Beaver St
   New York, NY  10004
   USA

   EMail: jvcelak@ns1.com
















Goldberg, et al.        Expires December 30, 2018              [Page 27]


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