vrf.txt

CFRG                                                         S. Goldberg
Internet-Draft                                         Boston University
Intended status: Standards Track                               L. Reyzin
Expires: May 13, 2021                     Boston University and Algorand
                                                         D. Papadopoulos
                           Hong Kong University of Science and Techology
                                                               J. Vcelak
                                                                     NS1
                                                        November 9, 2020


                   Verifiable Random Functions (VRFs)
                         draft-irtf-cfrg-vrf-08

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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   This Internet-Draft will expire on May 13, 2021.

Copyright Notice

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

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents



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

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Rationale . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.2.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   3
     1.3.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  VRF Algorithms  . . . . . . . . . . . . . . . . . . . . . . .   4
   3.  VRF Security Properties . . . . . . . . . . . . . . . . . . .   5
     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.  A random-oracle-like unpredictability 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)  . . . . . . . . . . . . . . . . .  10
     5.1.  ECVRF Proving . . . . . . . . . . . . . . . . . . . . . .  12
     5.2.  ECVRF Proof To Hash . . . . . . . . . . . . . . . . . . .  12
     5.3.  ECVRF Verifying . . . . . . . . . . . . . . . . . . . . .  13
     5.4.  ECVRF Auxiliary Functions . . . . . . . . . . . . . . . .  14
       5.4.1.  ECVRF Hash To Curve . . . . . . . . . . . . . . . . .  14
       5.4.2.  ECVRF Nonce Generation  . . . . . . . . . . . . . . .  17
       5.4.3.  ECVRF Hash Points . . . . . . . . . . . . . . . . . .  18
       5.4.4.  ECVRF Decode Proof  . . . . . . . . . . . . . . . . .  19
     5.5.  ECVRF Ciphersuites  . . . . . . . . . . . . . . . . . . .  19
     5.6.  When the ECVRF Keys are Untrusted . . . . . . . . . . . .  22
       5.6.1.  ECVRF Validate Key  . . . . . . . . . . . . . . . . .  22
   6.  Implementation Status . . . . . . . . . . . . . . . . . . . .  24
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  25
     7.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  25
       7.1.1.  Uniqueness and collision resistance with untrusted
               keys  . . . . . . . . . . . . . . . . . . . . . . . .  25
       7.1.2.  Pseudorandomness with untrusted keys  . . . . . . . .  26
     7.2.  Selective vs Full Pseudorandomness  . . . . . . . . . . .  26
     7.3.  Proper pseudorandom nonce for ECVRF . . . . . . . . . . .  27
     7.4.  Side-channel attacks  . . . . . . . . . . . . . . . . . .  27
     7.5.  Proofs Provide No Secrecy for VRF Input . . . . . . . . .  27
     7.6.  Prehashing  . . . . . . . . . . . . . . . . . . . . . . .  28
     7.7.  Hash function domain separation and future-proofing . . .  28



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   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  29
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  30
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  30
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  31
     10.2.  Informative References . . . . . . . . . . . . . . . . .  32
   Appendix A.  Test Vectors for the ECVRFs  . . . . . . . . . . . .  33
     A.1.  ECVRF-P256-SHA256-TAI . . . . . . . . . . . . . . . . . .  33
     A.2.  ECVRF-P256-SHA256-SSWU  . . . . . . . . . . . . . . . . .  34
     A.3.  ECVRF-EDWARDS25519-SHA512-TAI . . . . . . . . . . . . . .  36
     A.4.  ECVRF-EDWARDS25519-SHA512-ELL2  . . . . . . . . . . . . .  37
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  39

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 or alpha_string:  The input to be hashed by the VRF.




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   beta or beta_string:  The VRF hash output.

   pi or pi_string:  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.

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_proof_to_hash(pi)

   Notice that this means that

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

   and thus this document will specify VRF_prove and VRF_proof_to_hash
   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, beta = VRF_proof_to_hash(pi)) if pi is valid,
   and INVALID otherwise.





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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 private key SK.

   More precisely, "full uniqueness" states that a computationally-
   bounded adversary cannot choose a VRF public key PK, a VRF input
   alpha, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha,
   pi1) outputs (VALID, beta1), VRF_verify(PK, alpha, pi2) outputs
   (VALID, beta2), and beta1 is not equal to beta2.

   A slightly weaker security property called "trusted uniqueness"
   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 precisely, "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 precisely, suppose the public and private VRF keys (PK, SK) were
   generated in a trustworthy manner.  Pseudorandomness ensures that the



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   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 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, beta).

   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.  A random-oracle-like unpredictability property

   Pseudorandomness, as defined in Section 3.3, does not hold if the VRF
   keys were generated adversarially.  For instance, if an adversary
   outputs VRF keys that are deterministically generated (or hard-coded
   and publicly known), then the outputs are easily derived by anyone.

   There is, however, a different type of unpredictability that is
   desirable in certain VRF applications (such as [GHMVZ17] and
   [DGKR18]).  This property is similar to the unpredictability achieved
   by an (ordinary, unkeyed) cryptographic hash function: if the input
   has enough entropy (i.e., cannot be predicted), then the correct
   output is indistinguishable from uniform.

   A formal definition of this property appears in Section 3.2 of
   [DGKR18].  The VRF schemes presented in this specification are



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   believed to satisfy this property if the public key was generated in
   a trustworthy manner.  Additionally, the ECVRF is believed to also
   satisify this property even if the public key was not generated in a
   trustworthy manner, as long as the public key satisfies the key
   validation procedure in Section 5.6.

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 [PWHVNRG17].

   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 (k must be less than
      2^32)

   Fixed options:

      Hash - cryptographic hash function

      hLen - output length in octets of hash function Hash

   Primitives used:

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

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



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      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 the hash function Hash as
      defined in Section B.2.1 of [RFC8017]

      || - octet string concatenation

4.1.  RSA-FDH-VRF Proving

   RSAFDHVRF_prove(K, alpha_string)

   Input:

      K - RSA private key

      alpha_string - VRF hash input, an octet string

   Output:

      pi_string - proof, an octet string of length k

   Steps:

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

   2.  EM = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) ||
       alpha_string, k - 1)

   3.  m = OS2IP(EM)

   4.  s = RSASP1(K, m)

   5.  pi_string = I2OSP(s, k)

   6.  Output pi_string

4.2.  RSA-FDH-VRF Proof To Hash

   RSAFDHVRF_proof_to_hash(pi_string)

   Input:

      pi_string - proof, an octet string of length k




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   Output:

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

   Important note:

      RSAFDHVRF_proof_to_hash should be run only on pi_string that is
      known to have been produced by RSAFDHVRF_prove, or from within
      RSAFDHVRF_verify as specified in Section 4.3.

   Steps:

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

   2.  beta_string = Hash(two_string || pi_string)

   3.  Output beta_string

4.3.  RSA-FDH-VRF Verifying

   RSAFDHVRF_verify((n, e), alpha_string, pi_string)

   Input:

      (n, e) - RSA public key

      alpha_string - VRF hash input, an octet string

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

   Output:

      ("VALID", beta_string), where beta_string is the VRF hash output,
      an octet string of length hLen; or
      "INVALID"

   Steps:

   1.  s = OS2IP(pi_string)

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

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

   4.  one_string = 0x01 = I2OSP(1, 1), a single octet with value 1

   5.  EM' = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) ||
       alpha_string, k - 1)



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   6.  If EM and EM' are equal, output ("VALID",
       RSAFDHVRF_proof_to_hash(pi_string)); 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
   [PWHVNRG17].

   To additionally satisfy "full uniqueness" and "full collision
   resistance", the Verifier MUST additionally perform the validation
   procedure specified in Section 5.6 upon receipt of the public VRF
   key.

   Notation used:

      Elliptic curve operations are written in additive notation, with
      P+Q denoting point addition and x*P denoting scalar multiplication
      of a point P by a scalar x

      x^y - x raised to the power y

      x*y - x multiplied by y

      s || t - concatenation of octet strings s and t

   Fixed options (specified in Section 5.5):

      F - finite field

      2n - length, in octets, of a field element in F, rounded up to the
      nearest even integer

      E - elliptic curve (EC) defined over F

      ptLen - 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

      qLen - length of q in octets, i.e., smallest integer such that
      2^(8qLen)>q (note that in the typical case, qLen equals 2n or is
      close to 2n)



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      cofactor - number of points on E divided by q

      B - generator of group G

      Hash - cryptographic hash function

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

      ECVRF_hash_to_curve - a function that hashes strings to an EC
      point.

      ECVRF_nonce_generation - a function that derives a pseudorandom
      nonce from SK and the input as part of ECVRF proving.

      suite_string - a single nonzero octet specifying the ECVRF
      ciphersuite, which determines the above options as well as type
      conversions and parameter generation

   Type conversions (specified in Section 5.5):

      int_to_string(a, len) - conversion of nonnegative integer a to to
      octet string of length len

      string_to_int(a_string) - conversion of an octet string a_string
      to a nonnegative integer

      point_to_string - conversion of EC point to an ptLen-octet string

      string_to_point - conversion of an ptLen-octet string to EC point.
      string_to_point returns INVALID if the octet string does not
      convert to a valid EC point.

      Note that with certain software libraries (for big integer and
      elliptic curve arithmetic), the int_to_string and point_to_string
      conversions are not needed.  For example, in some implementations,
      EC point operations will take octet strings as inputs and produce
      octet strings as outputs, without introducing a separate elliptic
      curve point type.

   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



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      Y = x*B - VRF public key, an EC point

5.1.  ECVRF Proving

   ECVRF_prove(SK, alpha_string)

   Input:

      SK - VRF private key

      alpha_string = input alpha, an octet string

   Output:

      pi_string - VRF proof, octet string of length ptLen+n+qLen

   Steps:

   1.  Use SK to derive the VRF secret scalar x and the VRF public key Y
       = x*B
       (this derivation depends on the ciphersuite, as per Section 5.5;
       these values can be cached, for example, after key generation,
       and need not be rederived each time)

   2.  H = ECVRF_hash_to_curve(Y, alpha_string)

   3.  h_string = point_to_string(H)

   4.  Gamma = x*H

   5.  k = ECVRF_nonce_generation(SK, h_string)

   6.  c = ECVRF_hash_points(H, Gamma, k*B, k*H) (see Section 5.4.3)

   7.  s = (k + c*x) mod q

   8.  pi_string = point_to_string(Gamma) || int_to_string(c, n) ||
       int_to_string(s, qLen)

   9.  Output pi_string

5.2.  ECVRF Proof To Hash

   ECVRF_proof_to_hash(pi_string)

   Input:

      pi_string - VRF proof, octet string of length ptLen+n+qLen



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   Output:

      "INVALID", or

      beta_string - VRF hash output, octet string of length hLen

   Important note:

      ECVRF_proof_to_hash should be run only on pi_string that is known
      to have been produced by ECVRF_prove, or from within ECVRF_verify
      as specified in Section 5.3.

   Steps:

   1.  D = ECVRF_decode_proof(pi_string) (see Section 5.4.4)

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

   3.  (Gamma, c, s) = D

   4.  three_string = 0x03 = int_to_string(3, 1), a single octet with
       value 3

   5.  zero_string = 0x00 = int_to_string(0, 1), a single octet with
       value 0

   6.  beta_string = Hash(suite_string || three_string ||
       point_to_string(cofactor * Gamma) || zero_string)

   7.  Output beta_string

5.3.  ECVRF Verifying

   ECVRF_verify(Y, pi_string, alpha_string)

   Input:

      Y - public key, an EC point

      pi_string - VRF proof, octet string of length ptLen+n+qLen

      alpha_string - VRF input, octet string

   Output:

      ("VALID", beta_string), where beta_string is the VRF hash output,
      octet string of length hLen; or
      "INVALID"



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   Steps:

   1.  D = ECVRF_decode_proof(pi_string) (see Section 5.4.4)

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

   3.  (Gamma, c, s) = D

   4.  H = ECVRF_hash_to_curve(Y, alpha_string)

   5.  U = s*B - c*Y

   6.  V = s*H - c*Gamma

   7.  c' = ECVRF_hash_points(H, Gamma, U, V) (see Section 5.4.3)

   8.  If c and c' are equal, output ("VALID",
       ECVRF_proof_to_hash(pi_string)); 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
   the VRF input alpha is used for proving and verfying.  See
   Section 7.6 for further discussion.

   This section specifies a number of such algorithms, which are not
   compatible with each other.  The choice of a particular algorithm
   from the options specified in this section is made in Section 5.5.

5.4.1.1.  ECVRF_hash_to_curve_try_and_increment

   The following ECVRF_hash_to_curve_try_and_increment(Y, alpha_string)
   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_string.  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.

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

   ECVRF_hash_to_try_and_increment(Y, alpha_string)



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   Input:

      Y - public key, an EC point

      alpha_string - value to be hashed, an octet string

   Output:

      H - hashed value, a finite EC point in G

   Fixed option (specified in Section 5.5):

      arbitrary_string_to_point - conversion of an arbitrary octet
      string to an EC point.

   Steps:

   1.  ctr = 0

   2.  PK_string = point_to_string(Y)

   3.  one_string = 0x01 = int_to_string(1, 1), a single octet with
       value 1

   4.  zero_string = 0x00 = int_to_string(0, 1), a single octet with
       value 0

   5.  H = "INVALID"

   6.  While H is "INVALID" or H is EC point at infinity:

       A.  ctr_string = int_to_string(ctr, 1)

       B.  hash_string = Hash(suite_string || one_string || PK_string ||
           alpha_string || ctr_string || zero_string)

       C.  H = arbitrary_string_to_point(hash_string)

       D.  If H is not "INVALID" and cofactor > 1, set H = cofactor * H

       E.  ctr = ctr + 1

   7.  Output H








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5.4.1.2.  ECVRF_hash_to_curve_h2c_suite

   The ECVRF_hash_to_curve_h2c_suite(Y, alpha_string) algorithm
   implements ECVRF_hash_to_curve using one of the several hash-to-curve
   options defined in [I-D.irtf-cfrg-hash-to-curve].  The specific
   choice of the hash-to-curve option (called Suite ID in
   [I-D.irtf-cfrg-hash-to-curve]) is given by the h2c_suite_ID_string
   parameter.

   ECVRF_hash_to_curve_h2c_suite(Y, alpha_string)

   Input:

      alpha_string - value to be hashed, an octet string

      Y - public key, an EC point

   Output:

      H - hashed value, a finite EC point in G

   Fixed option (specified in Section 5.5):

      h2c_suite_ID_string - a hash-to-curve suite ID, encoded in ASCII
      (see discussion below)

   Steps

   1.  PK_string = point_to_string(Y)

   2.  string_to_hash = PK_string || alpha_string

   3.  H = encode(string_to_hash)
       (the encode function is discussed below)

   4.  Output H

   The encode function is provided by the hash-to-curve suite whose ID
   is h2c_suite_ID_string, as specified in
   [I-D.irtf-cfrg-hash-to-curve], Section 8.  The domain separation tag
   DST, a parameter to the hash-to-curve suite, SHALL be set to

      "ECVRF_" || h2c_suite_ID_string || suite_string

   where "ECVRF_" is represented as a 6-byte ASCII encoding (in
   hexadecimal, octets 45 43 56 52 46 5F).





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5.4.2.  ECVRF Nonce Generation

   The following algorithms generate the nonce value k in a
   deterministic pseudorandom fashion.  This section specifies a number
   of such algorithms, which are not compatible with each other.  The
   choice of a particular algorithm from the options specified in this
   section is made in Section 5.5.

5.4.2.1.  ECVRF Nonce Generation From RFC 6979

   ECVRF_nonce_generation_RFC6979(SK, h_string)

   Input:

      SK - an ECVRF secret key

      h_string - an octet string

   Output:

      k - an integer between 1 and q-1

   The ECVRF_nonce_generation function is as specified in [RFC6979]
   Section 3.2 where

      Input m is set equal to h_string

      The "suitable for DSA or ECDSA" check in step h.3 is omitted

      The hash function H is Hash and its output length hlen is set as
      hLen*8

      The secret key x is set equal to the VRF secret scalar x

      The prime q is the same as in this specification

      qlen is the binary length of q, i.e., the smallest integer such
      that 2^qlen > q

      All the other values and primitives as defined in [RFC6979]

5.4.2.2.  ECVRF Nonce Generation From RFC 8032

   The following is from Steps 2-3 of Section 5.1.6 in [RFC8032].

   ECVRF_nonce_generation_RFC8032(SK, h_string)

   Input:



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      SK - an ECVRF secret key

      h_string - an octet string

   Output:

      k - an integer between 0 and q-1

   Steps:

   1.  hashed_sk_string = Hash(SK)

   2.  truncated_hashed_sk_string =
       hashed_sk_string[32]...hashed_sk_string[63]

   3.  k_string = Hash(truncated_hashed_sk_string || h_string)

   4.  k = string_to_int(k_string) mod q

5.4.3.  ECVRF Hash Points

   ECVRF_hash_points(P1, P2, ..., PM)

   Input:

      P1...PM - EC points in G

   Output:

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

   Steps:

   1.  two_string = 0x02 = int_to_string(2, 1), a single octet with
       value 2

   2.  Initialize str = suite_string || two_string

   3.  for PJ in [P1, P2, ... PM]:
       str = str || point_to_string(PJ)

   4.  zero_string = 0x00 = int_to_string(0, 1), a single octet with
       value 0

   5.  str = str || zero_string

   6.  c_string = Hash(str)




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   7.  truncated_c_string = c_string[0]...c_string[n-1]

   8.  c = string_to_int(truncated_c_string)

   9.  Output c

5.4.4.  ECVRF Decode Proof

   ECVRF_decode_proof(pi_string)

   Input:

      pi_string - VRF proof, octet string (ptLen+n+qLen octets)

   Output:

      "INVALID", or

      Gamma - EC point

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

      s - integer between 0 and 2^(8qLen)-1

   Steps:

   1.  let gamma_string = pi_string[0]...p_string[ptLen-1]

   2.  let c_string = pi_string[ptLen]...pi_string[ptLen+n-1]

   3.  let s_string =pi_string[ptLen+n]...pi_string[ptLen+n+qLen-1]

   4.  Gamma = string_to_point(gamma_string)

   5.  if Gamma = "INVALID" output "INVALID" and stop.

   6.  c = string_to_int(c_string)

   7.  s = string_to_int(s_string)

   8.  Output Gamma, c, and s

5.5.  ECVRF Ciphersuites

   This document defines ECVRF-P256-SHA256-TAI as follows:

   o  suite_string = 0x01 = int_to_string(1, 1), a single octet with
      value 1.



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   o  The EC group G is the NIST P-256 elliptic curve, with curve
      parameters as specified in [FIPS-186-4] (Section D.1.2.3) and
      [RFC5114]  (Section 2.6).  For this group, 2n = qLen = 32 and
      cofactor = 1.

   o  The key pair generation primitive is specified in Section 3.2.1 of
      [SECG1] (q, B, 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
      Section 5.4.2.1.

   o  The int_to_string function is the I2OSP function specified in
      Section 4.1 of [RFC8017].  (This is big endian representation.)

   o  The string_to_int function is the OS2IP function specified in
      Section 4.2 of [RFC8017].  (This is big endian representation.)

   o  The point_to_string function converts an EC point to an octet
      string according to the encoding specified in Section 2.3.3 of
      [SECG1] with point compression on.  This implies ptLen = 2n + 1 =
      33.  (Note that certain software implementations do not introduce
      a separate elliptic curve point type and instead directly treat
      the EC point as an octet string per above encoding.  When using
      such an implementation, the point_to_string function can be
      treated as the identity function.)

   o  The string_to_point function converts an octet string to an EC
      point according to the encoding specified in Section 2.3.4 of
      [SECG1].  This function MUST output INVALID if the octet string
      does not decode to an EC point.

   o  The hash function Hash is SHA-256 as specified in [RFC6234], with
      hLen = 32.

   o  The ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.1, with arbitrary_string_to_point(s) =
      string_to_point(0x02 || s) (where 0x02 is a single octet with
      value 2, 0x02=int_to_string(2, 1)).  The input s to
      arbitrary_string_to_point is a 32-octet string and the output is
      either an EC point or "INVALID".

   This document defines ECVRF-P256-SHA256-SSWU as identical to ECVRF-
   P256-SHA256-TAI, except that:

   o  suite_string = 0x02 = int_to_string(2, 1), a single octet with
      value 2.



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   o  the ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.2 with h2c_suite_ID_string = P256_XMD:SHA-
      256_SSWU_NU_ (the suite is defined in
      [I-D.irtf-cfrg-hash-to-curve] Section 8.1)

   This document defines ECVRF-EDWARDS25519-SHA512-TAI as follows:

   o  suite_string = 0x03 = int_to_string(3, 1), a single octet with
      value 3.

   o  The EC group G is the edwards25519 elliptic curve with parameters
      defined in Table 1 of [RFC8032].  For this group, 2n = qLen = 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
      Section 5.4.2.2.

   o  The int_to_string function as specified in the first paragraph of
      Section 5.1.2 of [RFC8032].  (This is little endian
      representation.)

   o  The string_to_int function interprets the string as an integer in
      little-endian representation.

   o  The point_to_string function converts an EC point to an octect
      string according to the encoding specified in Section 5.1.2 of
      [RFC8032].  This implies ptLen = 2n = 32.  (Note that certain
      software implementations do not introduce a separate elliptic
      curve point type and instead directly treat the EC point as an
      octet string per above encoding.  When using such and
      implementation, the point_to_string function can be treated as the
      identity function.)

   o  The string_to_point function converts an octet string to an EC
      point according to the encoding specified in Section 5.1.3 of
      [RFC8032].  This function MUST output INVALID if the octet string
      does not decode to an EC point.

   o  The hash function Hash is SHA-512 as specified in [RFC6234], with
      hLen = 64.

   o  The ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.1, with arbitrary_string_to_point(s) =
      string_to_point(s[0]...s[31]).




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   This document defines ECVRF-EDWARDS25519-SHA512-ELL2 as identical to
   ECVRF-EDWARDS25519-SHA512-TAI, except:

   o  suite_string = 0x04 = int_to_string(4, 1), a single octet with
      value 4.

   o  the ECVRF_hash_to_curve function is as specified in
      Section 5.4.1.2 with h2c_suite_ID_string = edwards25519_XMD:SHA-
      512_ELL2_NU_ (the suite is defined in
      [I-D.irtf-cfrg-hash-to-curve] Section 8.4.)

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 B, the generator of group G, is untrusted.  If the prover
   is untrusted, the Verifier MUST obtain E and B from a trusted source,
   such as a ciphersuite specification, rather than from the prover.

   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_string)

   Input:

      PK_string - public key, an octet string

   Output:

      "INVALID", or

      Y - public key, an EC point

   Steps:




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   1.  Y = string_to_point(PK_string)

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

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

   4.  Output Y

   Note that if the cofactor = 1, then Step 3 need not multiply Y by the
   cofactor; instead, it suffices to output "INVALID" if Y is the point
   at infinity.  Moreover, when cofactor>1, it is not necessary to
   verify that Y is in the subgroup G; Step 3 suffices.  Therefore, if
   the cofactor is small, the total number of points that could cause
   Step 3 to output "INVALID" may be small, and it may be more efficient
   to simply check Y against a fixed list of such points.  For example,
   the following algorithm can be used for the edwards25519 curve:

   1.   Y = string_to_point(PK_string)

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

   3.   y_string = PK_string

   4.   oneTwentySeven_string = 0x7F = int_to_string(127, 1)
        (a single octet with value 127)

   5.   y_string[31] = y_string[31] & oneTwentySeven_string
        (this step clears the high-order bit of octet 31)

   6.   bad_pk[0] = int_to_string(0, 32)

   7.   bad_pk[1] = int_to_string(1, 32)

   8.   bad_y2 = 2707385501144840649318225287225658788936804267575313519
        463743609750303402022

   9.   bad_pk[2] = int_to_string(bad_y2, 32)

   10.  bad_pk[3] = int_to_string(p-bad_y2, 32)

   11.  bad_pk[4] = int_to_string(p-1, 32)

   12.  bad_pk[5] = int_to_string(p, 32)

   13.  bad_pk[6] = int_to_string(p+1, 32)





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   14.  If y_string is in bad_pk[0]...bad_pk[6], output "INVALID" and
        stop

   15.  Output Y

   (bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys,
   depending on the sign of the x-coordinate, which was cleared in step
   5, in order to make sure that it does not affect the comparison.
   bad_pk[1] and bad_pk[4] each match one bad public key, because
   x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6]
   are simply bad_pk[0] and bad_pk[1] shifted by p, in case the
   y-coordinate had not been modular reduced by p.  There is no need to
   shift the other bad_pk values by p, because they will exceed 2^255.
   These bad keys, which represent all points of order 1, 2, 4, and 8,
   have been obtained by converting the points specified in [X25519] to
   Edwards coordinates.)

6.  Implementation Status

   A reference C++ implementation of ECVRF-P256-SHA256-TAI, ECVRF-
   P256-SHA256-SSWU, ECVRF-EDWARDS25519-SHA512-TAI, and ECVRF-
   EDWARDS25519-SHA512-ELL2 is available at <https://github.com/reyzin/
   ecvrf>.  This implementation is neither secure nor especially
   effecient, but can be used to generate test vectors.

   A Python implementation of a previous version of ECVRF-
   EDWARDS25519-SHA512-ELL2 is available at
   <https://github.com/integritychain/draft-irtf-cfrg-vrf-05>.

   A C implementation of a previous version of ECVRF-
   EDWARDS25519-SHA512-ELL2 is available at
   <https://github.com/algorand/libsodium/tree/draft-irtf-cfrg-vrf-
   03/src/libsodium/crypto_vrf/ietfdraft03>.

   A Rust implemention of a previous version of ECVRF-P256-SHA256-TAI,
   as well as variants for the sect163k1 and secp256k1 curves, is
   available at <https://crates.io/crates/vrf>.

   A C implemention of a variant of this VRF for the secp256k1 curve is
   available at <https://github.com/aergoio/secp256k1-vrf>.

   An implementation of an earlier, slightly different, version of RSA-
   FDH-VRF (SHA-256) and ECVRF-P256-SHA256-TAI 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 Key Transparency project at Google uses a VRF implemention that
   is similar to the ECVRF-P256-SHA256-TAI, with a few minor changes



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   including the use of SHA-512 instead of SHA-256.  Its implementation
   is available at
   <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>.

   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-
   EDWARDS25519-SHA512-ELL2, called VXEdDSA, and specified here
   <https://whispersystems.org/docs/specifications/xeddsa/> and here
   <https://moderncrypto.org/mail-archive/curves/2017/000925.html>.
   Implementations in C and Java are available at
   <https://github.com/signalapp/curve25519-java> and
   <https://github.com/wavesplatform/curve25519-java>.

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



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   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.

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

   [PWHVNRG17] 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.








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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.

7.4.  Side-channel attacks

   Side channel attacks on cryptographic primatives are an important
   issue.  Here we discuss only one such side channel: timing attacks
   that can be used to leak information about the VRF input alpha.
   Implementers should take care to avoid side-channel attacks that leak
   information about the VRF private key SK (and the nonce k used in the
   ECVRF).

   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.

   ECVRF-P256-SHA256-SSWU and ECVRF-EDWARDS25519-SHA512-ELL2 can be made
   to run in time independent of alpha, following recommendations in
   [I-D.irtf-cfrg-hash-to-curve].

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.






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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_string octet and the public key are hashed together with
   alpha in ECVRF_hash_to_curve, which ensures that the curve (including
   the generator B) and the public key are included indirectly into
   subsequent hashes.

7.7.  Hash function domain separation and future-proofing

   Hashing is used for different purposes in the two VRFs (namely, in
   the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in
   hash_to_curve, nonce_generation, hash_points, and proof_to_hash).
   The theoretical analysis assumes each of these 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 RSA-
   FDH-VRF defined in this document, because the first octets of the
   input to the hash function used in MGF1 and in proof_to_hash are
   different.

   This is also 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 inputs used in hash_to_curve, proof_to_hash, 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 octets of the inputs to the hash function used in
      proof_to_hash, hash_points, and
      ECVRF_hash_to_curve_try_and_increment are all different

   o  the last octet of the input to the hash function used in
      proof_to_hash, hash_points, and
      ECVRF_hash_to_curve_try_and_increment is always zero, and
      therefore different from the last octet of the input to the hash



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      function used in ECVRF_hash_to_curve_h2c_suite, which is set equal
      to the nonzero length of the domain separation tag by
      [I-D.irtf-cfrg-hash-to-curve].

   For the RSA VRF, if future designs need to specify variants of the
   design in this document, such variants should use different first
   octets in inputs to MGF1 and to the hash funciton used in
   proof_to_hash, in order to avoid the possibility that an adversary
   can obtain a VRF output under one variant, and then claim it was
   obtained under another variant

   For the elliptic curve VRF, 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_string
   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_string 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
      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 B and PK=Y from
      ECVRF_hash_points because they are already present via H, which is
      computed via hash_to_curve using the suite_string (which
      identifies B) and Y.



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      03 - Changed Ed25519 conversions to little-endian, to match RFC
      8032; added simple key validation for Ed25519; added Simple SWU
      cipher suite; clarified Elligator and removed the extra x0 bit, to
      make Montgomery and Edwards Elligator the same; added domain
      separation for RSA VRF; improved notation throughout; added nonce
      generation as a section; changed counter in try-and-increment from
      four bytes to one, to avoid endian issues; renamed try-and-
      increment ciphersuites to -TAI; added qLen as a separate
      paremeter; changed output length to hLen for ECVRF, to match
      RSAVRF; made Verify return beta so unverified proofs don't end up
      in proof_to_hash; added test vectors.

      04 - Clarified handling of optional arguments x and PK in
      ECVRF_prove.  Edited implementation status to bring it up to date.

      05 - Renamed ed25519 into the more commonly used edwards25519.
      Corrected ECVRF_nonce_generation_RFC6979 (thanks to Gorka Irazoqui
      Apecechea and Mario Cao Cueto for finding the problem) and
      corresponding test vectors for the P256 suites.  Added a reference
      to the Rust implementation.

      06 - Made some variable names more descriptive.  Added a few
      implementation references.

      07 - Incorporated hash-to-curve draft by reference to replace our
      own Elligator2 and Simple SWU.  Clarified discussion of EC
      parameters and functions.  Added a 0 octet to all hashing to
      enforce domain separation from hashing done inside hash-to-curve.

      08 - Changed Reyzin's affiliation.

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, David C.  Lawerence, Trevor Perrin,
   Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova, Tony
   Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood,
   Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, Gary Belvinm,
   Piotr Nojszewski, Gorka Irazoqui Apecechea, and Mario Cao Cueto
   provided valuable input to this draft.  Riad Wahby was very helpful
   with the integration of the hash-to-curve draft.

10.  References







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10.1.  Normative References

   [FIPS-186-4]
              National Institute for Standards and Technology, "Digital
              Signature Standard (DSS)", FIPS PUB 186-4, July 2013,
              <https://csrc.nist.gov/publications/detail/fips/186/4/
              final>.

   [I-D.irtf-cfrg-hash-to-curve]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R., and
              C. Wood, "Hashing to Elliptic Curves", draft-irtf-cfrg-
              hash-to-curve-07 (work in progress), April 2020.

   [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>.

   [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>.





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10.2.  Informative References

   [ANSI.X9-62-2005]
              American National Standards Institute, "Public Key
              Cryptography for the Financial Services Industry: The
              Elliptic Curve Digital Signature Algorithm (ECDSA)",  ANSI
              X9.62, 2005.

   [DGKR18]   David, B., Gazi, P., Kiayias, A., and A. Russell,
              "Ouroboros Praos: An adaptively-secure, semi-synchronous
              proof-of-stake protocol", in Advances in Cryptology -
              EUROCRYPT, 2018, <https://eprint.iacr.org/2017/573>.

   [GHMVZ17]  Gilad, Y., Hemo, R., Micali, Y., Vlachos, Y., and Y.
              Zeldovich, "Algorand: Scaling Byzantine Agreements for
              Cryptocurrencies", in Proceedings of the 26th Symposium on
              Operating Systems Principles (SOSP), 2017,
              <https://eprint.iacr.org/2017/454>.

   [I-D.vcelak-nsec5]
              Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
              D. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
              Existence", draft-vcelak-nsec5-08 (work in progress),
              December 2018.

   [MRV99]    Michali, S., Rabin, M., and S. Vadhan, "Verifiable Random
              Functions", in FOCS, 1999,
              <https://dash.harvard.edu/handle/1/5028196>.

   [PWHVNRG17]
              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>.

   [X25519]   Bernstein, D., "How do I validate Curve25519 public
              keys?", 2006, <https://cr.yp.to/ecdh.html#validate>.













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Appendix A.  Test Vectors for the ECVRFs

   The test vectors in this section were genereated using the reference
   implementation at <https://github.com/reyzin/ecvrf>.

A.1.  ECVRF-P256-SHA256-TAI

   These two example secret keys and messages are taken from
   Appendix A.2.5 of [RFC6979].

   SK = x =
   c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
   PK =
   0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
   alpha = 73616d706c65 (ASCII "sample")
   try_and_increment succeded on ctr = 1
   H =
   0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4
   k = 0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77
   U = k*B =
   02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672
   V = k*H =
   02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394
   pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5071
   b498e7c291a16dafb9ccff8c2ae1f039fa92a328d5f7e0d483ee18353067a13f69994
   4a78892ff24939bcd044827eef884
   beta =
   a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e

   SK = x =
   c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
   PK =
   0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
   alpha = 74657374 (ASCII "test")
   try_and_increment succeded on ctr = 3
   H =
   02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974
   k = 5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961
   U = k*B =
   022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48e
   V = k*H =
   033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119
   pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080ddb
   56c874cc95b7d29a6a65cb518fe6f4418256385f12b1eccbad023c901bb983ff707b1
   09b3a3b526ca3a1e8661f7b8481a2
   beta =
   a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d




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   This example secret key is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   SK = x =
   2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
   PK =
   03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
   alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417
   070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
   (ASCII "Example using ECDSA key from Appendix L.4.2 of
   ANSI.X9-62-2005")
   try_and_increment succeded on ctr = 1
   H =
   0258055c26c4b01d01c00fb57567955f7d39cd6f6e85fd37c58f696cc6b7aa761d
   k = 5689e2e08e1110b4dda293ac21667eac6db5de4a46a519c73d533f69be2f4da3
   U = k*B =
   020f465cd0ec74d2e23af0abde4c07e866ae4e5138bded5dd1196b8843f380db84
   V = k*H =
   036cb6f811428fc4904370b86c488f60c280fa5b496d2f34ff8772f60ed24b2d1d
   pi = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b8c6
   679d92353c8a4fdfddb2a8540094b686cb5fb50f730d833a098a0399ccad32f3fec4d
   a2299891fc75ebda42baeb65e8c11
   beta =
   90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19

A.2.  ECVRF-P256-SHA256-SSWU

   These two example secret keys and messages are taken from
   Appendix A.2.5 of [RFC6979].

   SK = x =
   c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
   PK =
   0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
   alpha = 73616d706c65 (ASCII "sample")
   In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c2a4
   b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7e
   In SSWU: u =
   df565615a2372e8b31b8771f7503bafc144e48b05688b97958cc27ce29a8d810
   In SSWU: x1 =
   e7e39eb8a4c982426fcff629e55a3e13516cfeb62c02c369b1e750316f5e94eb
   In SSWU: gx1 is a nonsquare
   H =
   02b31973e872d4a097e2cfae9f37af9f9d73428fde74ac537dda93b5f18dbc5842
   k = e92820035a0a8afe132826c6312662b6ea733fc1a0d33737945016de54d02dd8
   U = k*B =
   031490f49d0355ffcdf66e40df788bee93861917ee713acff79be40d20cc91a30a




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   V = k*H =
   03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3
   pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394c78
   88a39dfe7432f119228473f37db3f87ca470c63b0237432a791f18f823c1215e276b7
   ac0962725ba8daec2bf90c0ccc91a
   beta =
   21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b

   SK = x =
   c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
   PK =
   0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
   alpha = 74657374 (ASCII "test")
   In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109afb24
   3b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3
   In SSWU: u =
   d8b0107f7e7aa36390240d834852f8703a6dc407019d6196bda5861b8fc00181
   In SSWU: x1 =
   ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7
   In SSWU: gx1 is a square
   H =
   03ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7
   k = febc3451ea7639fde2cf41ffd03f463124ecb3b5a79913db1ed069147c8a7dea
   U = k*B =
   031200f9900e96f811d1247d353573f47e0d9da601fc992566234fc1a5b37749ae
   V = k*H =
   02d3715dcfee136c7ae50e95ffca76f4ca6c29ddfb92a39c31a0d48e75c6605cd1
   pi = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa8db
   2721e0499b7cecd68027a82f6095da076625a5f2f62908f1c283d5ee9b9e852d85bed
   f64f2452a4e5094729e101824443e
   beta =
   8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd

   This example secret key is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   SK = x =
   2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
   PK =
   03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
   alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417
   070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
   (ASCII "Example using ECDSA key from Appendix L.4.2 of
   ANSI.X9-62-2005")
   In SSWU: uniform_bytes = 9b81d55a242d3e8438d3bcfb1bee985a87fd144802c9
   268cf9adeee160e6e9ff765569797a0f701cb4316018de2e7dd4
   In SSWU: u =
   e43c98c2ae06d13839fedb0303e5ee815896beda39be83fb11325b97976efdce



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   In SSWU: x1 =
   be9e195a50f175d3563aed8dc2d9f513a5536c1e9aee1757d86c08d32d582a86
   In SSWU: gx1 is a nonsquare
   H =
   022dd5150e5a2a24c66feab2f68532be1486e28e07f1b9a055cf38ccc16f6595ff
   k = 8e29221f33564f3f66f858ba2b0c14766e1057adbd422c3e7d0d99d5e142b613
   U = k*B =
   03a8823ff9fd16bf879261c740b9c7792b77fee0830f21314117e441784667958d
   V = k*H =
   02d48fbb45921c755b73b25be2f23379e3ce69294f6cee9279815f57f4b422659d
   pi = 039f8d9cdc162c89be2871cbcb1435144739431db7fab437ab7bc4e2651a9e99
   d5288aac70a5e4bd07df303c1d460eb6336bb5fa95436a07c2f6b7aec6fef7cc4846e
   a901ee1e238dee12bf752029b0b2e
   beta =
   4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a

A.3.  ECVRF-EDWARDS25519-SHA512-TAI

   These three example secret keys and messages are taken from
   Section 7.1 of [RFC8032].

   SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
   PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
   alpha = (the empty string)
   x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
   try_and_increment succeded on ctr = 0
   H = 91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318
   k = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106f345b8594b
   6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2a6afae4
   U = k*B =
   aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071
   V = k*H =
   5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f
   pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f723f
   5e8bd1839b414219e8626d393787a192241fc442e6569e96c462f62b8079b9ed83ff2
   ee21c90c7c398802fdeebea4001
   beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757876
   ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd8ae

   SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
   PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
   alpha = 72 (1 byte)
   x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
   try_and_increment succeded on ctr = 1
   H = 5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551
   k = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b2342acb9c167c
   ab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc16cc5ce




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   U = k*B =
   1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3
   V = k*H =
   fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423
   pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed593
   f7eaf3eb2f1a968cba3f6e23b386aeeaab7b1ea44a256e811892e13eeae7c9f6ea899
   2557453eac11c4d5476b1f35a08
   beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496310
   eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5031

   SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
   PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
   alpha = af82 (2 bytes)
   x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
   try_and_increment succeded on ctr = 0
   H = bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a
   k = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59fc31a0febc5
   9d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6bb6d6c3a
   U = k*B =
   2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5
   V = k*H =
   88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506
   pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccbf80
   e29dc513c01c3a980e0e545bcd848222d08a6c3e3665ff5a4cab13a643bef812e284c
   6b2ee063a2cb4f456794723ad0a
   beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c61c
   452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df19f

A.4.  ECVRF-EDWARDS25519-SHA512-ELL2

   These three example secret keys and messages are taken from
   Section 7.1 of [RFC8032].

   SK = 9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
   PK = d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
   alpha = (the empty string)
   x = 307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
   In Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca529
   8b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269ae
   In Elligator2: u =
   30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55c
   In Elligator2: gx1 =
   8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08
   In Elligator2: gx1 is a square
   H = b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9
   k = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a42458f817bed7a
   386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88dab8d41c




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   U = k*B =
   762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9
   V = k*H =
   4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af
   pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab46f
   25898f6bd7d4ed4c75f0282b0f7bb9d0e61b387b76db60b3cbf34bf09109ccb33fab7
   42a8bddc0c8ba3caf5c0b75bb04
   beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80c53
   cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6c54

   SK = 4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
   PK = 3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
   alpha = 72 (1 byte)
   x = 68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
   In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd77bb
   99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6d
   In Elligator2: u =
   3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773d
   In Elligator2: gx1 =
   25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464d
   In Elligator2: gx1 is a nonsquare
   H = 76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57
   k = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37c4c37f1d41
   e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab7544e444d
   U = k*B =
   8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061
   V = k*H =
   fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8b
   pi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c509072ef
   9bf1a234f833f72d8fff36075fd9b836da28b5569e74caa418bae7ef521f2ddd35f57
   27d271ecc70b4a83c1fc8ebc40c
   beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4635
   987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc735

   SK = c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
   PK = fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
   alpha = af82 (2 bytes)
   x = 909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
   In Elligator2: uniform_bytes = be0aed556e36cdfddf8f1eeddbb7356a24fad6
   4cf95a922a098038f215588b216beabbfe6acf20256188e883292b7a3a
   In Elligator2: u =
   f6675dc6d17fc790d4b3f1c6acf689a13d8b5815f23880092a925af94cd6fa24
   In Elligator2: gx1 =
   a63d48e3247c903e22fdfb88fd9295e396712a5fe576af335dbe16f99f0af26c
   In Elligator2: gx1 is a square
   H = 13d2a8b5ca32db7e98094a61f656a08c6c964344e058879a386a947a4e189ed1
   k = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5aaf79cf576
   087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861b3d683b



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   U = k*B =
   a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60
   V = k*H =
   fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25a
   pi = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57fdce
   6187befa109606682503b3a1424f0f729ca0418099fbd86a48093e6a8de26307b8d93
   e02da927e6dd5b73c8f119aee0f
   beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5cd09
   0a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314a58

Authors' Addresses

   Sharon Goldberg
   Boston University
   111 Cummington Mall
   Boston, MA  02215
   USA

   EMail: goldbe@cs.bu.edu


   Leonid Reyzin
   Boston University and Algorand
   111 Cummington Mall
   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.hk


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

   EMail: jvcelak@ns1.com






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