Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-00

Versions: (draft-goldbe-vrf)   00                                       
CFRG                                                         S. Goldberg
Internet-Draft                                                 L. Reyzin
Intended status: Standards Track                       Boston University
Expires: March 16, 2018                                  D. Papadopoulos
                           Hong Kong University of Science and Techology
                                                               J. Vcelak
                                                                     NS1
                                                      September 12, 2017


                   Verifiable Random Functions (VRFs)
                         draft-irtf-cfrg-vrf-00

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 March 16, 2018.

Copyright Notice

   Copyright (c) 2017 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 March 16, 2018                 [Page 1]


Internet-Draft                     VRF                    September 2017


   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 . . . . . . . . . .   4
     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 (EC-VRF) . . . . . . . . . . . . . . . . .   9
     5.1.  EC-VRF Proving  . . . . . . . . . . . . . . . . . . . . .  11
     5.2.  EC-VRF Proof To Hash  . . . . . . . . . . . . . . . . . .  11
     5.3.  EC-VRF Verifying  . . . . . . . . . . . . . . . . . . . .  12
     5.4.  EC-VRF Auxiliary Functions  . . . . . . . . . . . . . . .  13
       5.4.1.  EC-VRF Hash To Curve  . . . . . . . . . . . . . . . .  13
       5.4.2.  EC-VRF Hash Points  . . . . . . . . . . . . . . . . .  14
       5.4.3.  EC-VRF Decode Proof . . . . . . . . . . . . . . . . .  15
     5.5.  EC-VRF Ciphersuites . . . . . . . . . . . . . . . . . . .  15
     5.6.  When the EC-VRF Keys are Untrusted  . . . . . . . . . . .  16
       5.6.1.  EC-VRF Validate Key . . . . . . . . . . . . . . . . .  17
   6.  Implementation Status . . . . . . . . . . . . . . . . . . . .  17
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  18
     7.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  18
       7.1.1.  Uniqueness and collision resistance with untrusted
               keys  . . . . . . . . . . . . . . . . . . . . . . . .  18
       7.1.2.  Pseudorandomness with untrusted keys  . . . . . . . .  19
     7.2.  Selective vs Full Pseudorandomness  . . . . . . . . . . .  19
     7.3.  Proper randomness for EC-VRF  . . . . . . . . . . . . . .  19
     7.4.  Timing attacks  . . . . . . . . . . . . . . . . . . . . .  20
   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  20
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  20
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  20
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  20
     10.2.  Informative References . . . . . . . . . . . . . . . . .  21



Goldberg, et al.         Expires March 16, 2018                 [Page 2]


Internet-Draft                     VRF                    September 2017


   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  22

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 secret
   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.

   Verifier:  The Verifier holds the public VRF key PK.





Goldberg, et al.         Expires March 16, 2018                 [Page 3]


Internet-Draft                     VRF                    September 2017


2.  VRF Algorithms

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

   A VRF 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 private key SK is also used 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))

   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 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.

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 secret key SK.

   "Full uniqueness" states that a computationally-bounded adversary
   cannot choose a VRF public key PK, a VRF input alpha, two different



Goldberg, et al.         Expires March 16, 2018                 [Page 4]


Internet-Draft                     VRF                    September 2017


   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 otherwords, 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 secret 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 secret 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
   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



Goldberg, et al.         Expires March 16, 2018                 [Page 5]


Internet-Draft                     VRF                    September 2017


   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: The following 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 rely on this property.
   As we have not yet found a formal definition of this property in the
   literature, we write it down here. ]

   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.  Suppose the VRF keys are generated
   by an adversary.  Then, a VRF hash output beta should look
   pseudorandom to the adversary as long as (1) its corresponding VRF
   hash alpha is chosen randomly and independently of the VRF key, (2)
   alpha is unknown to the adversary, (3) the corresponding proof pi is
   unknown to the adversary, and (4) the VRF public key chosen by the
   adversary is valid.

   [TODO: It should be possible to get the EC-VRF to satisfy this
   property, as long as verifiers run an VRF_validate_key() key function
   upon receipt of VRF public keys.  However, we need to work out
   exactly what properties are needed from the VRF public keys in order
   for this property to hold.  Some additional checks might need to be
   added to the ECVRF_validate_key() function.  Need to work out what
   are these checks.]



Goldberg, et al.         Expires March 16, 2018                 [Page 6]


Internet-Draft                     VRF                    September 2017


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]

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





Goldberg, et al.         Expires March 16, 2018                 [Page 7]


Internet-Draft                     VRF                    September 2017


      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:

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

   Steps:



Goldberg, et al.         Expires March 16, 2018                 [Page 8]


Internet-Draft                     VRF                    September 2017


   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 (EC-VRF)

   The Elliptic Curve Verifiable Random Function (EC-VRF) 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].

   Fixed options:

      F - finite field

      2n - length, in octets, of a field element in F

      E - elliptic curve (EC) defined over F



Goldberg, et al.         Expires March 16, 2018                 [Page 9]


Internet-Draft                     VRF                    September 2017


      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

      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

   Constraints on options:

      Field elements in F have bit lengths divisible by 16

      hLen is equal to 2n

   Parameters used:

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

      x - VRF private key, an integer where 0 < x < q

   Notation and primitives used:

      p^k - when p is an EC point: point multiplication, i.e. k
      repetitions of group operation on 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 March 16, 2018                [Page 10]


Internet-Draft                     VRF                    September 2017


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

5.1.  EC-VRF Proving

   Note: this function is made more efficient by taking in the public
   VRF key y, as well as the private VRF key x.

   ECVRF_prove(y, x, alpha)

   Input:

      y - public key, an EC point

      x - private key, an integer

      alpha - VRF input, an octet string

   Output:

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

   Steps:

   1.  h = ECVRF_hash_to_curve(y, alpha)

   2.  gamma = h^x

   3.  choose a random integer nonce k from [0, q-1]

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

   5.  s = k - c*x mod q (where * denotes integer multiplication)

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

   7.  Output pi

5.2.  EC-VRF Proof To Hash

   ECVRF_proof2hash(pi)

   Input:

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

   Output:




Goldberg, et al.         Expires March 16, 2018                [Page 11]


Internet-Draft                     VRF                    September 2017


      "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.  beta = Hash(EC2OSP(gamma^cofactor))

   5.  Output beta

5.3.  EC-VRF 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 = y^c * g^s (where * denotes EC point addition, i.e. a group
       operation on two EC points)

   5.  h = ECVRF_hash_to_curve(y, alpha)

   6.  v = gamma^c * h^s (where * denotes EC point addition)

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



Goldberg, et al.         Expires March 16, 2018                [Page 12]


Internet-Draft                     VRF                    September 2017


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

5.4.  EC-VRF Auxiliary Functions

5.4.1.  EC-VRF Hash To Curve

   The ECVRF_hash_to_curve algorithm takes in an octet string alpha and
   converts it to h, an EC point in G.

5.4.1.1.  ECVRF_hash_to_curve1

   The following ECVRF_hash_to_curve1(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_curve1(y, alpha)

   Input:

      alpha - value to be hashed, an octet string

      y - public key, an EC point

   Output:

      h - hashed value, a finite EC point in G

   Steps:

   1.  ctr = 0

   2.  pk = EC2OSP(y)

   3.  h = "INVALID"

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

       A.  CTR = I2OSP(ctr, 4)




Goldberg, et al.         Expires March 16, 2018                [Page 13]


Internet-Draft                     VRF                    September 2017


       B.  ctr = ctr + 1

       C.  attempted_hash = Hash(pk || alpha || CTR)

       D.  h = RS2ECP(attempted_hash)

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

   5.  Output h

5.4.1.2.  ECVRF_hash_to_curve2

   For applications where VRF input alpha must be kept secret, the
   following ECVRF_hash_to_curve algorithm MAY be used to used as
   generic way to hash an octet string onto any elliptic curve.

   [TODO: If there interest, we could look into specifying the generic
   deterministic time hash_to_curve algorithm from [Icart09].  Note also
   for the Ed25519 curve (but not the P256 curve), the Elligator
   algorithm could be used here.]

5.4.2.  EC-VRF Hash Points

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

   Input:

      p_i - EC point in G

   Output:

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

   Steps:

   1.  P = empty octet string

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

   3.  h1 = Hash(P)

   4.  h2 = first n octets of h1

   5.  h = OS2IP(h2)

   6.  Output h




Goldberg, et al.         Expires March 16, 2018                [Page 14]


Internet-Draft                     VRF                    September 2017


5.4.3.  EC-VRF 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:

   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.  EC-VRF Ciphersuites

   This document defines EC-VRF-P256-SHA256 as follows:

   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].

   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].



Goldberg, et al.         Expires March 16, 2018                [Page 15]


Internet-Draft                     VRF                    September 2017


   o  RS2ECP(h) = OS2ECP(0x02 || h).  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 EC-VRF-ED25519-SHA256 as follows:

   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 key pair generation primitive is specified in Section 5.1.5 of
      [RFC8032]

   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-256 as specified in [RFC6234].

   o  The ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.1.

   [TODO: Should we add an EC-VRF-ED25519-SHA256-Elligator ciphersuite
   where the Elligator hash function is used for ECVRF_hash-to-curve?]

   [TODO: Add an Ed448 ciphersuite?]

   [NOTE: In the unlikely case that future versions of this spec use a
   elliptic curve group G that does not also come with a specification
   of the group generator g, then we can still have full uniqueness and
   full collision resistance by adding an check to
   ECVRF_validate_key(PK) that ensures that g is a point on the elliptic
   curve and g^cofactor is not the EC point at infinity.]

5.6.  When the EC-VRF Keys are Untrusted

   The EC-VRF as specified above is a VRF that satisfies the "trusted
   uniqueness", "trusted collision resistance", and "full
   pseudorandomness" properties defined in Section 3.  If the elliptic
   curve parameters (including the generator g) are trusted, but the VRF
   public key PK is not trusted, this VRF can be modified to



Goldberg, et al.         Expires March 16, 2018                [Page 16]


Internet-Draft                     VRF                    September 2017


   additionally satisfy "full uniqueness", and "full collision
   resistance".  This is done by additionally requiring the Verifier to
   perform the following validation procedure upon receipt of the public
   VRF key.

   The Verifier MUST perform this validation procedure when the entity
   that generated the public VRF key is untrusted.  The public key MUST
   NOT be used if this procedure returns "INVALID".  Note well that this
   procedure is not sufficient if the elliptic curve E or if g, the
   generator of group G, is untrusted.

   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.  EC-VRF 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 EC-VRF-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 EC-
   VRF implementation may be out of date as this spec has evolved.




Goldberg, et al.         Expires March 16, 2018                [Page 17]


Internet-Draft                     VRF                    September 2017


   The Key Transparency project at Google uses a VRF implemention that
   is similar to the EC-VRF-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 EC-VRF is available at
   <https://github.com/r2ishiguro/vrf>.

   An implementation similar to EC-VRF 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 EC-VRF-
   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 EC-VRF 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 EC-VRF 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
   EC-VRF 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



Goldberg, et al.         Expires March 16, 2018                [Page 18]


Internet-Draft                     VRF                    September 2017


   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.

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 EC-VRF, 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 EC-VRF, this means increasing the
      cryptographic strength of the EC group G.  For both RSA-FDH-VRF
      and EC-VRF the cryptographic strength of the hash function Hash
      may also potentially need to be increased.

7.3.  Proper randomness for EC-VRF

   Applications that use the EC-VRF defined in this document MUST ensure
   that the random nonce k used in the ECVRF_prove algorithm is chosen
   with proper randomness.  Otherwise, an adversary may be able to




Goldberg, et al.         Expires March 16, 2018                [Page 19]


Internet-Draft                     VRF                    September 2017


   recover the private VRF key x (and thus break pseudorandomness of the
   VRF) after observing several valid VRF proofs pi.

7.4.  Timing attacks

   The EC-VRF_hash_to_curve 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 EC-
   VRF_hash_to_curve 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.

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.

9.  Contributors

   Leonid Reyzin (Boston University) is a major contributor to this
   document.

   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) and David C.  Lawerence
   (Akamai) 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>.




Goldberg, et al.         Expires March 16, 2018                [Page 20]


Internet-Draft                     VRF                    September 2017


   [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>.

   [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

   [I-D.vcelak-nsec5]
              Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
              D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
              Existence", draft-vcelak-nsec5-05 (work in progress), July
              2017.

   [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>.







Goldberg, et al.         Expires March 16, 2018                [Page 21]


Internet-Draft                     VRF                    September 2017


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@cs.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 March 16, 2018                [Page 22]