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Versions: 00 01 draft-irtf-cfrg-vrf

Network Working Group                                        S. Goldberg
Internet-Draft                                         Boston University
Intended status: Standards Track                         D. Papadopoulos
Expires: September 14, 2017                       University of Maryland
                                                               J. Vcelak
                                                                     NS1
                                                          March 13, 2017


                   Verifiable Random Functions (VRFs)
                          draft-goldbe-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
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   Internet-Drafts are draft documents valid for a maximum of six months
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   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on September 14, 2017.

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
   (http://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents



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   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  . . . . . . . . . . . . . . . . . . . . . . .   3
   3.  VRF Security Properties . . . . . . . . . . . . . . . . . . .   4
     3.1.  Full Uniqueness or Trusted Uniqueness . . . . . . . . . .   4
     3.2.  Full Pseudorandomness or Selective Pseudorandomness . . .   5
     3.3.  Full Collison Resistance or Trusted Collision Resistance    5
   4.  RSA Full Domain Hash VRF (RSA-FDH-VRF)  . . . . . . . . . . .   6
     4.1.  RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . .   7
     4.2.  RSA-FDH-VRF Proof To Hash . . . . . . . . . . . . . . . .   7
     4.3.  RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . .   8
   5.  Elliptic Curve VRF (EC-VRF) . . . . . . . . . . . . . . . . .   8
     5.1.  EC-VRF Proving  . . . . . . . . . . . . . . . . . . . . .  10
     5.2.  EC-VRF Proof To Hash  . . . . . . . . . . . . . . . . . .  10
     5.3.  EC-VRF Verifying  . . . . . . . . . . . . . . . . . . . .  11
     5.4.  EC-VRF Auxiliary Functions  . . . . . . . . . . . . . . .  11
       5.4.1.  EC-VRF Hash To Curve  . . . . . . . . . . . . . . . .  11
       5.4.2.  EC-VRF Hash Points  . . . . . . . . . . . . . . . . .  13
       5.4.3.  EC-VRF Decode Proof . . . . . . . . . . . . . . . . .  13
     5.5.  EC-VRF Ciphersuites . . . . . . . . . . . . . . . . . . .  14
   6.  Implementation Status . . . . . . . . . . . . . . . . . . . .  15
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  15
     7.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  15
     7.2.  Proper randomness for EC-VRF  . . . . . . . . . . . . . .  16
     7.3.  Timing attacks  . . . . . . . . . . . . . . . . . . . . .  16
     7.4.  Selective vs Full Pseudorandomness  . . . . . . . . . . .  16
   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  17
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  17
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  17
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  17
     10.2.  Informative References . . . . . . . . . . . . . . . . .  18
   Appendix A.  Open Issues  . . . . . . . . . . . . . . . . . . . .  19
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  19








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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.

   The main application of the VRF is to protect the privacy of data
   records stored in a hash-based data structure against a querying
   adversary.  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 hashing, 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.

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:




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      beta = VRF_hash(SK, alpha)

   The VRF_hash algorithm MUST be deterministic, in the sense that it
   will always produce 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 anyone 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 states 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, even for a computationally-bounded adversary that knows the
   VRF secret key SK.

   More precisely, full uniqueness states that a computationally bounded
   adversary cannot choose a VRF public key PK, a VR 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




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   generated in a trustworthy manner.  In otherwords, uniqueness might
   not hold if keys were generated in an invalid manner.

3.2.  Full Pseudorandomness or Selective Pseudorandomness

   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
   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 a party that knows the private VRF key SK!  Such a party
   can easily distinguish beta from a random value by comparing it 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).

   Finally, 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 not generated with good randomness.)

3.3.  Full Collison Resistance or Trusted Collision Resistance

   Finally, like any cryprographic hash function, VRFs need to be
   collision resistant.  Specifically, it should be computationally
   infeasible for an adversary to find two distinct VRF inputs alpha1




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   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.

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

   The RSA Full Domain Hash VRF (RSA-FDH-VRF) is VRF that satisfies the
   trusted uniqueness, full pseudorandomness, and trusted collision
   resistance 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.

   Used parameters:

      (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

   Options constraints:

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

   Used primitives:



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      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]

      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



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   Output:

      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 the same, output "VALID"; else output
       "INVALID".

5.  Elliptic Curve VRF (EC-VRF)

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




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   The key pair generation primitive is specified in Section 3.2.1 of
   [SECG1].

   Fixed options:

      G - EC group

      q - prime order of group G

      g - generator of group G

      2n - ceil(log2(q)/8); where log2(x) is the binary logarithm of x
      and ceil(x) is the smallest integer larger than or equal to the
      real number x.

      Hash - cryptographic hash function

      hLen - output length in octets of function Hash

   Options constraints:

      Cryptographic security of Hash is at least as high as the
      cryptographic security of G

      hLen is equal to 2n

   Used parameters:

      g^x - EC public key

      x - EC private key

   Used primitives:

      "" - empty octet string

      || - octet string concatenation

      p^k - EC point multiplication

      p1*p2 - EC point addition

      h[i] - the i'th octet of octet string h

      ECP2OS - EC point to octet string conversion with point
      compression as specified in Section 2.3.3 of [SECG1]





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      OS2ECP - octet string to EC point conversion with point
      compression as specified in Section 2.3.4 of [SECG1]

5.1.  EC-VRF Proving

   ECVRF_prove(g^x, x, alpha)

   Input:

      g^x - EC public key

      x - EC private key

      alpha - VRF input, octet string

   Output:

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

   Steps:

   1.  h = ECVRF_hash_to_curve(alpha, g^x)

   2.  gamma = h^x

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

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

   5.  s = k - c*q mod q

   6.  pi = ECP2OS(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 5n+1

   Output:

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

   Steps:



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   1.  beta = pi[2] || pi[3] || ... pi[2n+1]

   2.  Output beta

5.3.  EC-VRF Verifying

   ECVRF_verify(g^x, pi, alpha)

   Input:

      g^x - EC public key

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

      alpha - VRF input, octet string

   Output:

      "VALID" or "INVALID"

   Steps:

   1.  gamma, c, s = ECVRF_decode_proof(pi)

   2.  If gamma is not a valid EC point in G, output "INVALID" and stop.

   3.  u = (g^x)^c * g^s

   4.  h = ECVRF_hash_to_curve(alpha, g^x)

   5.  v = gamma^c * h^s

   6.  c' = ECVRF_hash_points(g, h, g^x, gamma, u, v)

   7.  If c and c' are the same, 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(alpha, g^x) algorithm implements
   ECVRF_hash_to_curve in a simple and generic way that works for any
   elliptic curve that supports point compression.



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   However, this algorithm MUST NOT be used in applications where the
   VRF input alpha must be kept secret.  This is because the running
   time of the hashing algorithm depends on alpha, and so it is
   susceptible to timing attacks.  That said, the amount of information
   obtained from such a timing attack is likely to be small, since the
   algorithm is expected to find a valid curve point after only two
   attempts (i.e., when ctr=1) on average (see [Icart09]).

   ECVRF_hash_to_curve1(alpha, g^x)

   Input:

      alpha - value to be hashed, an octet string

      g^x - EC public key

   Output:

      h - hashed value, EC point in G

   Steps:

   1.  ctr = 0

   2.  pk = ECP2OS(g^x)

   3.  Repeat:

       A.  CTR = I2OSP(ctr, 4)

       B.  p = 0x02 || Hash(alpha || pk || CTR)

       C.  Goto step 3 if OS2ECP(p) is valid EC point in G

       D.  p = 0x03 || Hash(alpha || pk || CTR)

       E.  Goto step 3 if OS2ECP(p) is valid EC point in G

       F.  ctr = ctr + 1

   4.  h = OS2ECP(p)

   5.  Output h

   The initial octet 0x02 in the octet string created in step B
   represents that the point in compressed form has positive
   y-coefficient [SECG1].  Similarly, the 0x03 octet in step D
   represents negative y-coefficient.



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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]. ]

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 = ""

   2.  for p_i in [p_1, p_2, ... p_j]:
       P = P || ECP2OS(p_i)

   3.  h' = Hash(P)

   4.  h = OS2IP(h'[1] || h'[2] || ... h'[n])

   5.  Output h

5.4.3.  EC-VRF Decode Proof

   ECVRF_decode_proof(pi)

   Input:

      pi - VRF proof, octet string (5n+1 octets)

   Output:

      gamma - EC point

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




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

   Steps:

   1.  let gamma', c', s' be pi split after (2n+1)-th and (3n+1)-th
       octet

   2.  gamma = OS2ECP(gamma')

   3.  c = OS2IP(c')

   4.  s = OS2IP(s')

   5.  Output gamma, c, and s

5.5.  EC-VRF Ciphersuites

   [Seeking feedback on this section!]

   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, the length in octets of
      a single coordinate of an EC point is 2n = 32.

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

   o  The ECVRF_hash_to_curve function is ECVRF_hash_to_curve1, 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 [RFC7748] (Section 4.1).  For this group, the length in
      octets of a single coordinate of an EC point is 2n = 32.

   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?]





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6.  Implementation Status

   An implementation of the RSA-FDH-VRF (SHA-256) and EC-VRF-P256-SHA256
   was developed as a part of the NSEC5 project [I-D.vcelak-nsec5] and
   is available at <http://github.com/fcelda/nsec5-crypto>.

   The Key Transparency project at Google uses a VRF implemention that
   is almost identical to the EC-VRF-P256-SHA256 specified here, 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>

   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.
   Without good randomness, pseudorandomness properties of the VRF may
   not hold.  Also, trusted uniqueness and trusted collision-resistance
   may also not hold if the keys are generated adversarially (e.g., the
   RSA modulus is not a product of two primes for the RSA-FDH-VRF or the
   public key g^x is not valid point in the prime-order group G for the
   EC).

   Full uniqueness and full collision-resistance (as opposed to trusted
   uniqueness and trusted collision-resistance) are properties that hold
   even if VRF keys are generated by an adversary.  The VRFs defined in
   this document do not have these properties.  However, they may be
   modifed to have these properties if adversarial key generation is a
   concern.  The modification consists of additional cryptographic
   proofs that keys have of the correct form.  These modifications are
   left for future specification.

   Note that for the RSA-FDH-VRF, it might be possible to construct such
   a proof using the [GQ88] identification protocol made non-interactive
   using the Fiat-Shamir heuristic in the random oracle model.

   However, 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.




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7.2.  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
   recover the private VRF key x (and thus break pseudorandomness of the
   VRF) after observing several valid VRF proofs pi.

7.3.  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.

7.4.  Selective vs Full Pseudorandomness

   [nsec5ecc] presents cryptographic reductions to an underlying hard
   problem (e.g.  Decisional Diffie Hellman, or the standard RSA
   assumption) 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 therefor 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.








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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-vcelak-nsec5-04.  Cleaned up
      the definitions of VRF algorithms.  Added security definitions for
      VRF and security considerations.  Parameterized EC-VRF so it could
      support curves other than P-256 and Ed25519.

9.  Contributors

   Leonid Reyzin (Boston University) made major contributions 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

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

   [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,
              <http://www.rfc-editor.org/info/rfc8017>.

   [RFC5114]  Lepinski, M. and S. Kent, "Additional Diffie-Hellman
              Groups for Use with IETF Standards", RFC 5114,
              DOI 10.17487/RFC5114, January 2008,
              <http://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,
              <http://www.rfc-editor.org/info/rfc6234>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <http://www.rfc-editor.org/info/rfc7748>.






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   [I-D.vcelak-nsec5]
              Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
              D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
              Existence", draft-vcelak-nsec5-04 (work in progress),
              March 2017.

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

   [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

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

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

   [CP92]     Chaum, D. and C. Pederson, "Wallet databases with
              observers", in FOCS, 1992.

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

   [GQ88]     Guillou, L. and JJ. Quisquater, "A Practical Zero-
              Knowledge Protocol Fitted to Security Microprocessor
              Minimizing Both Transmission and Memory", in Advances in
              Cryptology - EUROCRYPT '88, 1988.
















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Appendix A.  Open Issues

   Note to RFC Editor: please remove this appendix before publication as
   an RFC.

   1.  Open issues

Authors' Addresses

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

   EMail: goldbe@cs.bu.edu


   Dimitrios Papadopoulos
   University of Maryland
   8223 Paint Branch Dr
   College Park, MD  20740
   USA

   EMail: dipapado@bu.edu


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

   EMail: jvcelak@ns1.com

















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