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CFRG                                                         S. Goldberg
Internet-Draft                                         Boston University
Intended status: Informational                                 L. Reyzin
Expires: 17 December 2022                 Boston University and Algorand
                                                         D. Papadopoulos
                          Hong Kong University of Science and Technology
                                                               J. Vcelak
                                                                     NS1
                                                            15 June 2022


                   Verifiable Random Functions (VRFs)
                         draft-irtf-cfrg-vrf-13

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 the 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 based on RSA and Elliptic Curves that are secure in the
   cryptographic random oracle model.

   This document is a product of the Crypto Forum Research Group (CFRG)
   in the IRTF.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 17 December 2022.

Copyright Notice

   Copyright (c) 2022 IETF Trust and the persons identified as the
   document authors.  All rights reserved.



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

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Requirements  . . . . . . . . . . . . . . . . . . . . . .   4
     1.2.  Terminology . . . . . . . . . . . . . . . . . . . . . . .   4
   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  . . . . . . . . . . . . . . . . . . . . . . .   6
     3.3.  Full Pseudorandomness or Selective Pseudorandomness . . .   6
     3.4.  Some VRFs: Unpredictability Under Malicious Key
           Generation  . . . . . . . . . . . . . . . . . . . . . . .   7
   4.  RSA Full Domain Hash VRF (RSA-FDH-VRF)  . . . . . . . . . . .   8
     4.1.  RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . .   9
     4.2.  RSA-FDH-VRF Proof to Hash . . . . . . . . . . . . . . . .  10
     4.3.  RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . .  10
     4.4.  RSA-FDH-VRF Ciphersuites  . . . . . . . . . . . . . . . .  11
   5.  Elliptic Curve VRF (ECVRF)  . . . . . . . . . . . . . . . . .  12
     5.1.  ECVRF Proving . . . . . . . . . . . . . . . . . . . . . .  14
     5.2.  ECVRF Proof to Hash . . . . . . . . . . . . . . . . . . .  15
     5.3.  ECVRF Verifying . . . . . . . . . . . . . . . . . . . . .  16
     5.4.  ECVRF Auxiliary Functions . . . . . . . . . . . . . . . .  17
       5.4.1.  ECVRF Encode to Curve . . . . . . . . . . . . . . . .  17
       5.4.2.  ECVRF Nonce Generation  . . . . . . . . . . . . . . .  20
       5.4.3.  ECVRF Challenge Generation  . . . . . . . . . . . . .  21
       5.4.4.  ECVRF Decode Proof  . . . . . . . . . . . . . . . . .  22
       5.4.5.  ECVRF Validate Key  . . . . . . . . . . . . . . . . .  23
     5.5.  ECVRF Ciphersuites  . . . . . . . . . . . . . . . . . . .  25
   6.  Implementation Status . . . . . . . . . . . . . . . . . . . .  27
   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  28
     7.1.  Key Generation  . . . . . . . . . . . . . . . . . . . . .  28
       7.1.1.  Uniqueness and collision resistance with untrusted
               keys  . . . . . . . . . . . . . . . . . . . . . . . .  28
       7.1.2.  Pseudorandomness with untrusted keys  . . . . . . . .  29
     7.2.  Security Levels . . . . . . . . . . . . . . . . . . . . .  29
     7.3.  Selective vs. Full Pseudorandomness . . . . . . . . . . .  30
     7.4.  Proper pseudorandom nonce for ECVRF . . . . . . . . . . .  30
     7.5.  Side-channel attacks  . . . . . . . . . . . . . . . . . .  31



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     7.6.  Proofs provide no secrecy for the VRF input . . . . . . .  31
     7.7.  Prehashing  . . . . . . . . . . . . . . . . . . . . . . .  32
     7.8.  Hash function domain separation . . . . . . . . . . . . .  32
     7.9.  Hash function salting . . . . . . . . . . . . . . . . . .  33
     7.10. Futureproofing  . . . . . . . . . . . . . . . . . . . . .  33
   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  33
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  35
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  35
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  35
     10.2.  Informative References . . . . . . . . . . . . . . . . .  36
   Appendix A.  Test Vectors for the ECVRFs  . . . . . . . . . . . .  37
     A.1.  ECVRF-P256-SHA256-TAI . . . . . . . . . . . . . . . . . .  37
     A.2.  ECVRF-P256-SHA256-SSWU  . . . . . . . . . . . . . . . . .  39
     A.3.  ECVRF-EDWARDS25519-SHA512-TAI . . . . . . . . . . . . . .  41
     A.4.  ECVRF-EDWARDS25519-SHA512-ELL2  . . . . . . . . . . . . .  43
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  45

1.  Introduction

   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 the corresponding public
   key can verify the correctness of the hash.

   A key application of the VRF is to provide privacy against offline
   dictionary attacks (also known as enumeration 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
   structure.

   This document defines a number of different VRFs based on RSA and
   elliptic curves.  The particular choice of the VRF depends on the
   desired security properties, the availability of cryptographically
   strong implementations, efficiency constraints, and the trust one
   places in RSA and elliptic curve Diffie-Hellman assumptions (and the
   trust in a particular choice of curve in case of elliptic curves).
   Differences in the security properties provided by the different
   options are discussed in Section 3 and Section 7.

   This document represents the consensus of the Crypto Forum Research
   Group (CFRG).




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1.1.  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 [RFC8174].

1.2.  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.

   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 the same 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 by
   using the function VRF_proof_to_hash:

      beta = VRF_proof_to_hash(pi)




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   Thus, for VRFs defined in this document, VRF_hash is defined as

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

   and therefore 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 VRFs defined in this document also come with an algorithm

      VRF_verify(PK, alpha, pi)

   that outputs (VALID, beta = VRF_proof_to_hash(pi)) if pi is valid,
   and INVALID otherwise.

3.  VRF Security Properties

   VRFs are designed to ensure the following security properties.

3.1.  Full Uniqueness or Trusted Uniqueness

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

   For many applications, a slightly weaker security property called
   "trusted uniqueness" suffices.  Trusted uniqueness is the same as
   full uniqueness, but it is guaranteed to hold only if the VRF keys PK
   and SK were generated in a trustworthy manner.

   As further discussed in Section 7.1.1, some VRFs specified in this
   document satisfy only trusted uniqueness, while others satisfy full
   uniqueness.  VRFs in this document that satisfy only trusted
   uniqueness but not full uniqueness MUST NOT be used if the key
   generation process cannot be trusted.








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3.2.  Full Collison Resistance or Trusted Collision Resistance

   Like any cryptographic 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 many applications, a slightly weaker security property called
   "trusted collision resistance" suffices.  Trusted collision
   resistance is the same as collision resistance, but it is guaranteed
   to hold only if the VRF keys PK and SK were generated in a
   trustworthy manner.

   As further discussed in Section 7.1.1, some VRFs specified in this
   document satisfy only trusted collision resistance, while others
   satisfy full collision resistance.  VRFs in this document that
   satisfy only trusted collision resistance but not full collision
   resistance MUST NOT be used if the key generation process cannot be
   trusted.

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
   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.



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   As further discussed in Section 7.3, VRFs specified in this document
   satisfy both full pseudorandomness and selective pseudorandomness,
   but their quantitative security against the selective
   pseudorandomness attack is stronger.

   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 a 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.  Some VRFs: Unpredictability Under Malicious Key Generation

   As explained in Section 3.3, pseudorandomness is guaranteed only if
   the VRF keys were generated in a trustworthy fashion.  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 and are therefore not pseudorandom.

   There is, however, a different type of unpredictability that is
   desirable in certain VRF applications (such as leader selection in
   the consensus protocols of [GHMVZ17] and [DGKR18]), called
   "unpredictability under malicious key generation".  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, no matter how the VRF keys are generated.

   A formal definition of this property appears in Section 3.2 of
   [DGKR18].  The RSA-FDH-VRF presented in this document does not
   satisfy this property.  The ECVRF presented in this document
   satisfies this property if validate_key parameter given to the
   ECVRF_verify is TRUE.









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

   The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that, for
   suitable key lengths, satisfies the "trusted uniqueness", "trusted
   collision resistance", and "full pseudorandomness" properties defined
   in Section 3, as further discussed in Section 7.  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 section, the notation from [RFC8017] is used.

   Parameters used:

      (n, e) - RSA public key

      K - RSA private key (its representation is implementation-
      dependent)

      k - length in octets of the RSA modulus n (k must be less than
      2^32)

   Fixed options (specified in Section 4.4):

      Hash - cryptographic hash function

      hLen - output length in octets of hash function Hash

      suite_string - an octet string specifying the RSA-FDH-VRF
      ciphersuite, which determines the above options

   Primitives used:

      I2OSP - Conversion of a nonnegative integer to an octet string as
      defined in Section 4.1 of [RFC8017] (given an integer and a length
      in octets, produces a big-endian representation of the integer,
      zero-padded to the desired length)





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      OS2IP - Conversion of an octet string to a nonnegative integer as
      defined in Section 4.2 of [RFC8017] (given a big-endian encoding
      of an integer, produces the integer)

      RSASP1 - RSA signature primitive as defined in Section 5.2.1 of
      [RFC8017] (given a private key and an input, raises the input to
      the private RSA exponent modulo n)

      RSAVP1 - RSA verification primitive as defined in Section 5.2.2 of
      [RFC8017] (given a public key and an input, raises the input to
      the public RSA exponent modulo n)

      MGF1 - Mask Generation Function based on the hash function Hash as
      defined in Section B.2.1 of [RFC8017] (given an input, produces a
      random-oracle-like output of desired length)

      || - octet string concatenation

4.1.  RSA-FDH-VRF Proving

   RSAFDHVRF_prove(K, alpha_string[, MGF_salt])

   Input:

      K - RSA private key

      alpha_string - VRF hash input, an octet string

   Optional Input:

      MGF_salt - a public octet string used as a hash function salt;
      this input is not used when MGF_salt is specified as part of the
      ciphersuite

   Output:

      pi_string - proof, an octet string of length k

   Steps:

   1.  mgf_domain_separator = 0x01

   2.  EM = MGF1(suite_string || mgf_domain_separator || MGF_salt ||
       alpha_string, k - 1)

   3.  m = OS2IP(EM)

   4.  s = RSASP1(K, m)



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   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

   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.  proof_to_hash_domain_separator = 0x02

   2.  beta_string = Hash(suite_string ||
       proof_to_hash_domain_separator || pi_string)

   3.  Output beta_string

4.3.  RSA-FDH-VRF Verifying

   RSAFDHVRF_verify((n, e), alpha_string, pi_string[, MGF_salt])

   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 k

   Optional Input:

      MGF_salt - a public octet string used as a hash function salt;
      this input is not used when MGF_salt is specified as part of the
      ciphersuite



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   Output:

   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); if RSAVP1 returns "signature
       representative out of range", output "INVALID" and stop.

   3.  mgf_domain_separator = 0x01

   4.  EM' = MGF1(suite_string || mgf_domain_separator || MGF_salt ||
       alpha_string, k - 1)

   5.  m' = OS2IP(EM')

   6.  If m and m' are equal, output ("VALID",
       RSAFDHVRF_proof_to_hash(pi_string)); else output "INVALID".

4.4.  RSA-FDH-VRF Ciphersuites

   This document defines RSA-FDH-VRF-SHA256 as follows:

   *  suite_string = 0x01

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

   *  MGF_salt = I2OSP(k, 4) || I2OSP(n, k)

   This document defines RSA-FDH-VRF-SHA384 as follows:

   *  suite_string = 0x02

   *  The hash function Hash is SHA-384 as specified in [RFC6234], with
      hLen = 48

   *  MGF_salt = I2OSP(k, 4) || I2OSP(n, k)

   This document defines RSA-FDH-VRF-SHA512 as follows:




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   *  suite_string = 0x03

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

   *  MGF_salt = I2OSP(k, 4) || I2OSP(n, k)

5.  Elliptic Curve VRF (ECVRF)

   The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that,
   for suitable parameter choices, satisfies the "full uniqueness",
   "trusted collision resistance", and "full pseudorandomness
   properties" defined in Section 3.  If validate_key parameter given to
   the ECVRF_verify is TRUE, then the ECVRF additionally satisfies "full
   collision resistance" and "unpredictability under malicious key
   generation".  See Section 7 for further discussion.  Formal security
   proofs are in [PWHVNRG17].

   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

      0xMN (where M and N are hexadecimal digits) - a single octet with
      value M*16+N; equivalently, int_to_string(M*16+N, 1), where
      int_to_string is as defined below.

   Fixed options (specified in Section 5.5):

      F - finite field

      fLen - length, in octets, of an element in F encoded as an octet
      string

      E - elliptic curve (EC) defined over F

      ptLen - length, in octets, of a point on E encoded as an octet
      string

      G - subgroup of E of large prime order




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      q - prime order of group G

      qLen - length of q in octets, i.e., smallest integer such that
      2^(8qLen)>q

      cLen - length, in octets, of a challenge value used by the VRF
      (note that in the typical case, cLen is qLen/2 or close to it)

      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 (hLen must be at least
      cLen; in the typical case, it is at least qLen)

      ECVRF_encode_to_curve - a function that hashes strings to points
      on E.

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

      suite_string - an octet string 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
      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 a point on E to an ptLen-octet
      string

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










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      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, when the libraries encode integers and
      EC points in the same way as required by the ciphersuites.  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

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

      PK_string = point_to_string(Y) - VRF public key represented as an
      octet string

      encode_to_curve_salt - a public value used as a hash function salt

5.1.  ECVRF Proving

   ECVRF_prove(SK, alpha_string[, encode_to_curve_salt])

   Input:

      SK - VRF private key

      alpha_string - input alpha, an octet string

   Optional input:

      encode_to_curve_salt - a public salt value, an octet string; this
      input is not used when encode_to_curve_salt is specified as part
      of the ciphersuite

   Output:

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

   Steps:

   1.  Use SK to derive the VRF secret scalar x and the VRF public key Y
       = x*B



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       (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_encode_to_curve(encode_to_curve_salt, alpha_string)
       (see Section 5.4.1)

   3.  h_string = point_to_string(H)

   4.  Gamma = x*H

   5.  k = ECVRF_nonce_generation(SK, h_string) (see Section 5.4.2)

   6.  c = ECVRF_challenge_generation(Y, 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, cLen) ||
       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+cLen+qLen

   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)




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

   3.  (Gamma, c, s) = D

   4.  proof_to_hash_domain_separator_front = 0x03

   5.  proof_to_hash_domain_separator_back = 0x00

   6.  beta_string = Hash(suite_string ||
       proof_to_hash_domain_separator_front || point_to_string(cofactor
       * Gamma) || proof_to_hash_domain_separator_back)

   7.  Output beta_string

5.3.  ECVRF Verifying

   ECVRF_verify(PK_string, alpha_string, pi_string[,
   encode_to_curve_salt, validate_key])

   Input:

      PK_string - public key, an octet string

      alpha_string - VRF input, octet string

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

   Optional input:

      encode_to_curve_salt - a public salt value, an octet string; this
      input is not used when encode_to_curve_salt is specified as part
      of the ciphersuite

      validate_key - a boolean.  An implementation MAY support only the
      option of validate_key = TRUE, or only the option of validate_key
      = FALSE, in which case this input is not needed.  If an
      implementation supports only one option, it MUST specify which
      option is supports.

   Output:

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

      "INVALID"

   Steps:




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   1.   Y = string_to_point(PK_string)

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

   3.   If validate_key, run ECVRF_validate_key(Y) (Section 5.4.5); if
        it outputs "INVALID", output "INVALID" and stop

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

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

   6.   (Gamma, c, s) = D

   7.   H = ECVRF_encode_to_curve(encode_to_curve_salt, alpha_string)
        (see Section 5.4.1)

   8.   U = s*B - c*Y

   9.   V = s*H - c*Gamma

   10.  c' = ECVRF_challenge_generation(Y, H, Gamma, U, V) (see
        Section 5.4.3)

   11.  If c and c' are equal, output ("VALID",
        ECVRF_proof_to_hash(pi_string)); else output "INVALID"

   Note that the first three steps need to be performed only once for a
   given public key.

5.4.  ECVRF Auxiliary Functions

5.4.1.  ECVRF Encode to Curve

   The ECVRF_encode_to_curve algorithm takes a public salt (see
   Section 7.9) and 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 verifying.  See Section 7.7 for further
   discussion.

   This section specifies a number of such algorithms, which are not
   compatible with each other and are intended to use with various
   ciphersuites specified in Section 5.5.

   Input:

      encode_to_curve_salt - public salt value, an octet string

      alpha_string - value to be hashed, an octet string



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   Output:

      H - hashed value, a point in G

5.4.1.1.  ECVRF_encode_to_curve_try_and_increment

   The following
   ECVRF_encode_to_curve_try_and_increment(encode_to_curve_salt,
   alpha_string) algorithm implements ECVRF_encode_to_curve in a simple
   and generic way that works for any elliptic curve.  To use this
   algorithm, hLen MUST be at least fLen.

   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_encode_to_curve_try_and_increment(encode_to_curve_salt,
   alpha_string)

   Fixed option (specified in Section 5.5):

      interpret_hash_value_as_a_point - a function that attempts to
      convert a cryptographic hash value to a point on E; may output
      INVALID.

   Steps:

   1.  ctr = 0

   2.  encode_to_curve_domain_separator_front = 0x01

   3.  encode_to_curve_domain_separator_back = 0x00

   4.  H = "INVALID"

   5.  While H is "INVALID" or H is the identity element of the elliptic
       curve group:

       a.  ctr_string = int_to_string(ctr, 1)







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       b.  hash_string = Hash(suite_string ||
           encode_to_curve_domain_separator_front ||
           encode_to_curve_salt || alpha_string || ctr_string ||
           encode_to_curve_domain_separator_back)

       c.  H = interpret_hash_value_as_a_point(hash_string)

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

       e.  ctr = ctr + 1

   6.  Output H

   Note even though the loop is infinite as written, and
   int_to_string(ctr,1) may fail when ctr reaches 256,
   interpret_hash_value_as_a_point functions specified in Section 5.5
   will succeed on roughly half hash_string values.  Thus the loop is
   expected to stop after two iterations, and ctr is overwhelmingly
   unlikely (probability about 2^-256) to reach 256.

5.4.1.2.  ECVRF_encode_to_curve_h2c_suite

   The ECVRF_encode_to_curve_h2c_suite(encode_to_curve_salt,
   alpha_string) algorithm implements ECVRF_encode_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_encode_to_curve_h2c_suite(encode_to_curve_salt, alpha_string)

   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.  string_to_be_hashed = encode_to_curve_salt || alpha_string

   2.  H = encode(string_to_be_hashed)

       (the encode function is discussed below)

   3.  Output H






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   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).

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 depends on the ciphersuite, as specified 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 nonce 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 (in bits)
      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






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      qlen is the binary length of q, i.e., the smallest integer such
      that 2^qlen > q (this qlen is not to be confused with qLen in this
      document, which is the length of q in octets)

      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].  To
   use this algorithm, hLen MUST be at least 64.

   ECVRF_nonce_generation_RFC8032(SK, h_string)

   Input:

      SK - an ECVRF secret key

      h_string - an octet string

   Output:

      k - an integer nonce 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 Challenge Generation

   ECVRF_challenge_generation(P1, P2, P3, P4, P5)

   Input:

      P1, P2, P3, P4, P5 - EC points

   Output:

      c - challenge value, integer between 0 and 2^(8*cLen)-1

   Steps:




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   1.  challenge_generation_domain_separator_front = 0x02

   2.  Initialize str = suite_string ||
       challenge_generation_domain_separator_front

   3.  for PJ in [P1, P2, P3, P4, P5]:

       str = str || point_to_string(PJ)

   4.  challenge_generation_domain_separator_back = 0x00

   5.  str = str || challenge_generation_domain_separator_back

   6.  c_string = Hash(str)

   7.  truncated_c_string = c_string[0]...c_string[cLen-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+cLen+qLen octets)

   Output:

      "INVALID", or

      Gamma - a point on E

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

      s - integer between 0 and q-1

   Steps:

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

   2.  c_string = pi_string[ptLen]...pi_string[ptLen+cLen-1]

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

   4.  Gamma = string_to_point(gamma_string)



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

   6.  c = string_to_int(c_string)

   7.  s = string_to_int(s_string)

   8.  if s >= q output "INVALID" and stop

   9.  Output Gamma, c, and s

5.4.5.  ECVRF Validate Key

   ECVRF_validate_key(Y)

   Input:

      Y - public key, a point on E

   Output:

      "VALID" or "INVALID"

   Important note: the public key Y given to this procedure MUST be a
   valid point on E.

   Steps:

   1.  Let Y' = cofactor*Y

   2.  If Y' is the identity element of the elliptic curve group, output
       "INVALID" and stop

   3.  Output "VALID"

   Note that if the cofactor = 1, then Step 1 simply sets Y'=Y.  In
   particular, for the P-256 curve, ECVRF_validate_key simply ensures
   that Y is not the point at infinity.

   Any algorithm with identical input-output behavior MAY be used in
   place of the above steps.  For example, if the total number of Y
   values that could cause Step 2 to output "INVALID" is small, it may
   be more efficient to simply check Y against a fixed list of such
   values.  For example, the following algorithm MAY be used for the
   edwards25519 curve:

   1.   PK_string = point_to_string(Y)

   2.   oneTwentySeven_string = 0x7F



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   3.   y_string[31] = y_string[31] & oneTwentySeven_string

        (this step clears the high-order bit of octet 31)

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

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

   6.   bad_y2 = 2707385501144840649318225287225658788936804267575313519
        463743609750303402022

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

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

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

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

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

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

   13.  Output "VALID"

   (This algorithm works for the following reason.  Note that there are
   8 bad points -- namely, the points whose order is 1, 2, 4, or 8 -- on
   the edwards25519 curve.  Their y coordinates happen to be 0 (two
   points of order 4), 1 (one point of order 1), bad_y2 (two points of
   order 8), p-bad_y2 (two points of order 8), and p-1 (one point of
   order 2).  They can obtained by converting the points specified in
   [X25519] to Edwards coordinates.  Thus, bad_pk[0] (of order 4),
   bad_pk[2] (of order 8), and bad_pk[3] (of order 8) each match two bad
   points, depending on the sign of the x-coordinate, which was cleared
   in step 3, in order to make sure that it does not affect the
   comparison. bad_pk[1] (of order 1) and bad_pk[4] (of order 2) each
   match one bad point, because x-coordinate is 0 for these two points.
   Note that the first 5 list elements cover the 8 bad points.  However,
   in case the y-coordinate of the public key Y had not been modular
   reduced by p, the list also includes bad_pk[5] and bad_pk[6], which
   are simply bad_pk[0] and bad_pk[1] shifted by p.  There is no need to
   shift the other bad_pk values by p (or any bad_pk values by a larger
   multiple of p), because their y coordinate would exceed 2^255; and we
   ensure that y_string corresponds to an integer less than 2^255 in
   step 3.)





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5.5.  ECVRF Ciphersuites

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

   *  suite_string = 0x01.

   *  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, fLen = qLen = 32 and
      cofactor = 1.

   *  cLen = 16.

   *  The key pair generation primitive is specified in Section 3.2.1 of
      [SECG1] (q, B, SK, and Y in this document correspond to 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.

   *  encode_to_curve_salt = PK_string

   *  The ECVRF_nonce_generation function is as specified in
      Section 5.4.2.1.

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

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

   *  The point_to_string function converts a point on E to an octet
      string according to the encoding specified in Section 2.3.3 of
      [SECG1] with point compression on.  This implies ptLen = fLen + 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.)

   *  The string_to_point function converts an octet string to an a
      point on E 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 a point on the curve E.

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






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   *  The ECVRF_encode_to_curve function is as specified in
      Section 5.4.1.1, with interpret_hash_value_as_a_point(s) =
      string_to_point(0x02 || s).

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

   *  suite_string = 0x02.

   *  the ECVRF_encode_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 8.2)

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

   *  suite_string = 0x03.

   *  The EC group G is the edwards25519 elliptic curve with parameters
      defined in Table 1 of [RFC8032].  For this group, fLen = qLen = 32
      and cofactor = 8.

   *  cLen = 16.

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

   *  encode_to_curve_salt = PK_string

   *  The ECVRF_nonce_generation function is as specified in
      Section 5.4.2.2.

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

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

   *  The point_to_string function converts an point on E to an octet
      string according to the encoding specified in Section 5.1.2 of
      [RFC8032].  This implies ptLen = fLen = 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.)




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   *  The string_to_point function converts an octet string to a point
      on E 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 a point on the curve E.

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

   *  The ECVRF_encode_to_curve function is as specified in
      Section 5.4.1.1, with interpret_hash_value_as_a_point(s) =
      string_to_point(s[0]...s[31]).

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

   *  suite_string = 0x04.

   *  the ECVRF_encode_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 8.5).

6.  Implementation Status

   Note to RFC editor: Remove before publication

   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
   efficient, but can be used to generate test vectors.

   A Python implementation of an older version of ECVRF-
   EDWARDS25519-SHA512-ELL2 from the -05 version of this draft is
   available at https://github.com/integritychain/draft-irtf-cfrg-vrf-
   05.

   A C implementation of an older version of ECVRF-
   EDWARDS25519-SHA512-ELL2 from the -03 version of this draft is
   available at https://github.com/algorand/libsodium/tree/draft-irtf-
   cfrg-vrf-03/src/libsodium/crypto_vrf/ietfdraft03.

   A Rust implementation of an older version of ECVRF-P256-SHA256-TAI
   from the -05 version of this draft, as well as variants for the
   sect163k1 and secp256k1 curves, is available at
   https://crates.io/crates/vrf.





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   A C implementation of a variant of ECVRF-P256-SHA256-TAI from the -05
   version of this draft adapted for the secp256k1 curve is available at
   https://github.com/aergoio/secp256k1-vrf.

   An implementation of an earlier 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 implementation that
   is similar to the ECVRF-P256-SHA256-TAI, with a few changes including
   the use of SHA-512 instead of SHA-256.  Its implementation is
   available at
   https://github.com/google/keytransparency/blob/master/core/crypto/
   vrf/

   An implementation by Ryuji Ishiguro following an older version of
   ECVRF-EDWARDS25519-SHA512-TAI from the -00 version of this draft is
   available at https://github.com/r2ishiguro/vrf.

   An implementation similar to ECVRF-EDWARDS25519-SHA512-ELL2 (with
   some changes, including the use of SHA-3) 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 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 the VRF key is generated correctly, using good randomness.

7.1.1.  Uniqueness and collision resistance with untrusted keys

   The RSA-FDH-VRF satisfies the "trusted uniqueness" (see Section 3.1)
   and "trusted collision resistance" (see Section 3.2) properties as
   long as the VRF keys are generated correctly.  Uniqueness and
   collision resistance may not hold if the keys are generated
   adversarially (specifically, if the RSA function specified in the
   public key is not bijective because the modulus n or the exponent e



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   are chosen not in compliance with the stadnard); thus, RSA-FDH-VRF
   defined in this document does not have "full uniqueness" and "full
   collision resistance".  Therefore, if adversarial key generation is a
   concern, the RSA-FDH-VRF has to be enhanced by additional
   cryptographic checks that its public key has the right form.  These
   enhacements are left for future specification.

   For the ECVRF, the Verifier MUST obtain E and B from a trusted
   source, such as a ciphersuite specification, rather than from the
   prover.  If the verifier does so, then the ECVRF satisfies the "full
   uniqueness" (see Section 3.1) and "trusted collision resistance" (see
   Section 3.2) properties.  It additonally satisfies "full collision
   resistance" if validate_key parameter given to the ECVRF_verify is
   TRUE.

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.  Security Levels

   As shown in [PWHVNRG17], RSA-FDH-VRF satifies the trusted uniqueness
   property unconditionally.  The security level of the RSA-FDH-VRF,
   measured in bits, for the other two properties is as follows (in the
   random oracle model for the functions MGF1 and Hash):

   *  For trusted collision resistance: approximately 8*min(k/2, hLen/2)
      (as shown in [PWHVNRG17]).

   *  For selective pseudorandomness: approximately as strong as the
      security, in bits, of the RSA problem for the key (n, e) (as shown
      in [GNPRVZ15]).

   As shown in [PWHVNRG17], the security level of the ECVRF, measured in
   bits, is as follows (in the random oracle model for the functions
   Hash and ECVRF_encode_to_curve):

   *  For trusted uniqueness: approximately 8*min(qLen, cLen).

   *  For collision resistance (trusted or full, depending on whether
      validation is performed as explained in Section 7.1.1):
      approximately 8*min(qLen/2, hLen/2).




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   *  For the selective pseudorandomness property: approximately as
      strong as the security, in bits, of the decisional Diffie-Hellman
      problem in the group G (which is at most 8*qLen/2).

   See Section 3 for the definitions of these security properties.  See
   Section 7.3 for the discussion of full pseudorandomness.

7.3.  Selective vs. Full Pseudorandomness

   [PWHVNRG17] presents cryptographic reductions to an underlying hard
   problem (namely, the RSA problem for RSA-FDH-VRF and the Decisional
   Diffie-Hellman problem for the ECVRF) to prove that the VRFs
   specified in this document possess not only selective
   pseudorandomness, but also full pseudorandomness (see Section 3.3 for
   an explanation of these notions).  However, the cryptographic
   reductions are tighter for selective pseudorandomness than for full
   pseudorandomness.  Specifically, the approximate provable security
   level, measured in bits, for full pseudorandomness may be obtained
   from the provable security level for selective pseudorandomness
   (given in Section 7.2) by subtracting the binary logarithm of the
   number of proofs produced for a given secret key.  This holds for
   both the RSA-FDH-VRF and the ECVRF.

   While no known attacks against full pseudorandomness are stronger
   than similar attacks against selective pseudorandomness, some
   applications may be concerned about tightness of cryptographic
   reductions.  Such applications may consider the following two
   options:

   *  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.

   *  They 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 by specifying a new
      ciphersuite.

7.4.  Proper pseudorandom nonce for ECVRF

   The security of the ECVRF defined in this document relies on the fact
   that the nonce k used in the ECVRF_prove algorithm is chosen
   uniformly and pseudorandomly modulo q, and is unknown to the
   adversary.  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



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   specified in the ECVRF ciphersuites of Section 5.5 are designed with
   this requirement in mind.

7.5.  Side-channel attacks

   Side channel attacks on cryptographic primitives are an important
   issue.  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), which is used in VRF_prove.  In most
   applications, VRF_proof_to_hash and VRF_verify algorithms take only
   inputs that are public, and thus side channel attacks are typically
   not a concern for these algorithms.

   The VRF input alpha may be also a sensitive input to VRF_prove and
   may need to be protected against side channel attacks.  Below we
   discuss one particular class of such attacks: timing attacks that can
   be used to leak information about the VRF input alpha.

   The ECVRF_encode_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].


7.6.  Proofs provide no secrecy for the 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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7.7.  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 Initialize-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_encode_to_curve.  The curve
   point H becomes the representative of alpha thereafter.

7.8.  Hash function domain separation

   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
   encode_to_curve, nonce_generation, challenge_generation, and
   proof_to_hash).  The theoretical analysis treats each of these
   functions as a separate hash function, modeled as a 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 second 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:

   *  inputs to the hash function used during nonce_generation are
      unlikely to equal inputs used in encode_to_curve, proof_to_hash,
      and challenge_generation.  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.

   *  the second octets of the inputs to the hash function used in
      proof_to_hash, challenge_generation, and
      encode_to_curve_try_and_increment are all different.

   *  the last octet of the input to the hash function used in
      proof_to_hash, challenge_generation, and
      encode_to_curve_try_and_increment is always zero, and therefore
      different from the last octet of the input to the hash function
      used in ECVRF_encode_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].



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7.9.  Hash function salting

   In case a hash collision is found, in order to make it more difficult
   for the adversary to exploit such a collision, the MGF1 function for
   the RSA-FDH-VRF and ECVRF_encode_to_curve function for the ECVRF use
   a public value in addition to alpha (as a so-called salt).  This
   value is determined by the ciphersuite.  For the ciphersuites defined
   in this document, it is set equal to the string representation of the
   RSA modulus and EC public key, respectively.  Implementations that do
   not use one of the ciphersuites (see Section 7.10) MAY use a
   different salt.  For example, if a group of public keys to share the
   same salt, then the hash of the VRF input alpha will be the same for
   the entire group of public keys, which may aid in some protocol that
   uses the VRF.

7.10.  Futureproofing

   if future designs need to specify variants (e.g., additional
   ciphersuites) of the RSA-FDH-VRF or the ECVRF 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.  The
   suite_string constants in this document are all single octets; if a
   future suite_string constant is longer than one octet, then it should
   start with a different octet than the suite_string constants in this
   document.  Then, for the RSA-FDH-VRF, the inputs to the hash function
   used in MGF1 and proof_to_hash will be different from other
   ciphersuites.  For the ECVRF, the inputs ECVRF_encode_to_curve hash
   function used in producing H are then guaranteed to be different from
   other ciphersuites; since all the other hashing done by the prover
   depends on H, inputs to all the hash functions used by the prover
   will also be different from other ciphersuites as long as
   ECVRF_encode_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



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      that nonces are deterministic.  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 futureproofing.  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 a 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.

      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
      parameter; 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 - Incorporated suggestions from crypto panel review by Chloe
      Martindale.  Changed Reyzin's affiliation.  Updated references.

      09 - Added a note to remove the implementation page before
      publication.




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      10 - Added a check in ECVRF_decode_proof to ensure that s is
      reduced mod q.  Connected security properties (Section 3) and
      security considerations (Section 7) with more cross-references.

      11 - Processed last call comments.  Clarified various notation,
      including lengths of various parameters for ECVRF; added error
      handling to RSA-FDH-VRF; added security levels section; clarified
      full vs trusted uniqueness and full vs selective pseudorandomness;
      added RSA ciphersuites; made key validation clearer; renamed
      hash_to_curve to encode_to_curve to be consistent with the
      hash_to_curve draft; allowed a more general salt in hashing, added
      the public key as input to ECVRF_challenge_generation, and added
      an explanation about the salt.

      12 - Added k_string to edwards25519 test vectors

      13 - Clarified key validation for edwards25519 and adddressed IRTF
      Chair comments

9.  Contributors

   This document also would not be possible without the work of Moni
   Naor, Sachin Vasant, and Asaf Ziv. Chloe Martindale provided a
   thorough cryptographer's review.  Liliya Akhmetzyanova, Tony Arcieri,
   Gary Belvin, Mario Cao Cueto, Brian Chen, Sergey Gorbunov, Shumon
   Huque, Gorka Irazoqui Apecechea, Marek Jankowski, Burt Kaliski, David
   C.  Lawerence, Derek Ting-Haye Leung, Antonio Marcedone, Piotr
   Nojszewski, Chris Peikert, Trevor Perrin, Sam Scott, Stanislav
   Smyshlyaev, Adam Suhl, Nick Sullivan, Christopher Wood, Jiayu Xu, and
   Annie Yousar provided valuable input to this draft.  Riad Wahby
   helped this document align with draft-irtf-cfrg-hash-to-curve.

10.  References

10.1.  Normative References

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

   [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>.







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   [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>.

   [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>.

   [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>.

   [I-D.irtf-cfrg-hash-to-curve]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
              and C. A. Wood, "Hashing to Elliptic Curves", Work in
              Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
              16, 15 June 2022, <https://datatracker.ietf.org/doc/html/
              draft-irtf-cfrg-hash-to-curve-16>.

   [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>.

   [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

   [ANSI.X9-62-2005]
              "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>.



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   [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>.

   [GNPRVZ15] Goldberg, S., Naor, M., Papadopoulos, D., Reyzin, L.,
              Vasant, S., and A. Ziv, "NSEC5: Provably Preventing DNSSEC
              Zone Enumeration", in NDSS, 2015,
              <https://eprint.iacr.org/2014/582.pdf>.

   [I-D.vcelak-nsec5]
              Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
              D. C. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
              Existence", Work in Progress, Internet-Draft, draft-
              vcelak-nsec5-08, 29 December 2018,
              <https://datatracker.ietf.org/doc/html/draft-vcelak-
              nsec5-08>.

   [MRV99]    Micali, 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>.

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

Appendix A.  Test Vectors for the ECVRFs

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

A.1.  ECVRF-P256-SHA256-TAI

   The example secret keys and messages in Examples 1 and 2 are taken
   from Appendix A.2.5 of [RFC6979].

   Example 1:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721




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      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 73616d706c65 (ASCII "sample")
      try_and_increment succeeded on ctr = 1
      H =
      0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4
      k =
      0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77
      U = k*B =
      02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672
      V = k*H =
      02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394
      pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5
      071b4a53f0a46f018bc2c56e58d383f2305e0975972c26feea0eb122fe7893c15a
      f376b33edf7de17c6ea056d4d82de6bc02f
      beta =
      a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e

   Example 2:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 74657374 (ASCII "test")
      try_and_increment succeeded on ctr = 3
      H =
      02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974
      k =
      5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961
      U = k*B =
      022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48e
      V = k*H =
      033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119
      pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080
      ddb56c19e067b15a8a8174905b13617804534214f935b94c2287f797e393eb0816
      969d864f37625b443f30f1a5a33f2b3c854
      beta =
      a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d

   The example secret key in Example 3 is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   Example 3:

      SK = x =
      2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8




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      PK =
      03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
      alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
      417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
      (ASCII "Example using ECDSA key from Appendix L.4.2 of
      ANSI.X9-62-2005")
      try_and_increment succeeded on ctr = 1
      H =
      0258055c26c4b01d01c00fb57567955f7d39cd6f6e85fd37c58f696cc6b7aa761d
      k =
      5689e2e08e1110b4dda293ac21667eac6db5de4a46a519c73d533f69be2f4da3
      U = k*B =
      020f465cd0ec74d2e23af0abde4c07e866ae4e5138bded5dd1196b8843f380db84
      V = k*H =
      036cb6f811428fc4904370b86c488f60c280fa5b496d2f34ff8772f60ed24b2d1d
      pi = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b
      8c667d091c00b0f5c3619d10ecea44363b5a599cadc5b2957e223fec62e81f7b48
      25fc799a771a3d7334b9186bdbee87316b1
      beta =
      90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19

A.2.  ECVRF-P256-SHA256-SSWU

   The example secret keys and messages in Examples 4 and 5 are taken
   from Appendix A.2.5 of [RFC6979].

   Example 4:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 73616d706c65 (ASCII "sample")
      In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c
      2a4b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7e
      In SSWU: u =
      df565615a2372e8b31b8771f7503bafc144e48b05688b97958cc27ce29a8d810
      In SSWU: x1 =
      e7e39eb8a4c982426fcff629e55a3e13516cfeb62c02c369b1e750316f5e94eb
      In SSWU: gx1 is a nonsquare
      H =
      02b31973e872d4a097e2cfae9f37af9f9d73428fde74ac537dda93b5f18dbc5842
      k =
      e92820035a0a8afe132826c6312662b6ea733fc1a0d33737945016de54d02dd8
      U = k*B =
      031490f49d0355ffcdf66e40df788bee93861917ee713acff79be40d20cc91a30a
      V = k*H =
      03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3



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      pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394
      c7888782fffde7b842c38c20c08de6ec6c2e7027a97000f2c9fa4425d5c03e639f
      b48fde58114d755985498d7eb234cf4aed9
      beta =
      21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b

   Example 5:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 74657374 (ASCII "test")
      In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109af
      b243b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3
      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 = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa
      8db2743aad348a3006dc1caad7da28687320740bf7dd78fe13c298867321ce3b36
      b79ec3093b7083ac5e4daf3465f9f43c627
      beta =
      8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd

   The example secret key in Example 6 is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   Example 6:

      SK = x =
      2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
      PK =
      03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
      alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
      417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
      (ASCII "Example using ECDSA key from Appendix L.4.2 of
      ANSI.X9-62-2005")




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      In SSWU: uniform_bytes = 9b81d55a242d3e8438d3bcfb1bee985a87fd14480
      2c9268cf9adeee160e6e9ff765569797a0f701cb4316018de2e7dd4
      In SSWU: u =
      e43c98c2ae06d13839fedb0303e5ee815896beda39be83fb11325b97976efdce
      In SSWU: x1 =
      be9e195a50f175d3563aed8dc2d9f513a5536c1e9aee1757d86c08d32d582a86
      In SSWU: gx1 is a nonsquare
      H =
      022dd5150e5a2a24c66feab2f68532be1486e28e07f1b9a055cf38ccc16f6595ff
      k =
      8e29221f33564f3f66f858ba2b0c14766e1057adbd422c3e7d0d99d5e142b613
      U = k*B =
      03a8823ff9fd16bf879261c740b9c7792b77fee0830f21314117e441784667958d
      V = k*H =
      02d48fbb45921c755b73b25be2f23379e3ce69294f6cee9279815f57f4b422659d
      pi = 039f8d9cdc162c89be2871cbcb1435144739431db7fab437ab7bc4e2651a9
      e99d5488405a11a6c7fc8defddd9e1573a563b7333aab4effe73ae9803274174c6
      59269fd39b53e133dcd9e0d24f01288de9a
      beta =
      4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a

A.3.  ECVRF-EDWARDS25519-SHA512-TAI

   The example secret keys and messages in Examples 7, 8, and 9 are
   taken from Section 7.1 of [RFC8032].

   Example 7:

      SK =
      9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
      PK =
      d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
      alpha = (the empty string)
      x =
      307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
      try_and_increment succeeded on ctr = 0
      H =
      91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318
      k_string = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106
      f345b8594b6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2
      a6afae4
      k =
      8a49edbd1492a8ee09766befe50a7d563051bf3406cbffc20a88def030730f0f
      U = k*B =
      aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071
      V = k*H =
      5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f




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      pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f7
      23f26f8a57ccaed74ee1b190bed1f479d9727d2d0f9b005a6e456a35d4fb0daab1
      268a1b0db10836d9826a528ca76567805
      beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757
      876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd
      8ae

   Example 8:

      SK =
      4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
      PK =
      3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
      alpha = 72 (1 byte)
      x =
      68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
      try_and_increment succeeded on ctr = 1
      H =
      5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551
      k_string = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b234
      2acb9c167cab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc
      16cc5ce
      k =
      d8c3a66921444cb3427d5d989f9b315aa8ca3375e9ec4d52207711a1fdb44107
      U = k*B =
      1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3
      V = k*H =
      fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423
      pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed
      5933bf0864a62558b3ed7f2fea45c92a465301b3bbf5e3e54ddf2d935be3b67926
      da3ef39226bbc355bdc9850112c8f4b02
      beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496
      310eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5
      031

   Example 9:

      SK =
      c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
      PK =
      fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
      alpha = af82 (2 bytes)
      x =
      909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
      try_and_increment succeeded on ctr = 0
      H =
      bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a




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      k_string = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59f
      c31a0febc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6b
      b6d6c3a
      k =
      5ffdbc72135d936014e8ab708585fda379405542b07e3bd2c0bd48437fbac60a
      U = k*B =
      2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5
      V = k*H =
      88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506
      pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccb
      f8096bb474e53895c362d8628ee9f9ea3c0e52c7a5c691b6c18c9979866568add7
      a2d41b00b05081ed0f58ee5e31b3a970e
      beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c
      61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df
      19f

A.4.  ECVRF-EDWARDS25519-SHA512-ELL2

   The example secret keys and messages in Examples 10, 11, and 12 are
   taken from Section 7.1 of [RFC8032].

   Example 10:

      SK =
      9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
      PK =
      d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
      alpha = (the empty string)
      x =
      307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
      In Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca
      5298b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269ae
      In Elligator2: u =
      30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55c
      In Elligator2: gx1 =
      8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08
      In Elligator2: gx1 is a square
      H =
      b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9
      k_string = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a4245
      8f817bed7a386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88d
      ab8d41c
      k =
      55cbb247af9b8372259a97b2cfec656d78868deb33b203d51b9961c364522400
      U = k*B =
      762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9
      V = k*H =
      4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af



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      pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab
      46f14adf9a3cd8b8412d9038531e865c341cafa73589b023d14311c331a9ad15ff
      2fb37831e00f0acaa6d73bc9997b06501
      beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80
      c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6
      c54

   Example 11:

      SK =
      4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
      PK =
      3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
      alpha = 72 (1 byte)
      x =
      68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
      In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd7
      7bb99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6d
      In Elligator2: u =
      3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773d
      In Elligator2: gx1 =
      25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464d
      In Elligator2: gx1 is a nonsquare
      H =
      76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57
      k_string = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37
      c4c37f1d41e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab75
      44e444d
      k =
      9565956daeedf376cad61b829b2a4d21ba1b52e9b3e2457477a64630a9711003
      U = k*B =
      8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061
      V = k*H =
      fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8b
      pi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c50907
      2ef055b48372bb82efbdce8e10c8cb9a2f9d60e93908f93df1623ad78a86a028d6
      bc064dbfc75a6a57379ef855dc6733801
      beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4
      635987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc
      735

   Example 12:

      SK =
      c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
      PK =
      fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
      alpha = af82 (2 bytes)



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      x =
      909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
      In Elligator2: uniform_bytes = be0aed556e36cdfddf8f1eeddbb7356a24f
      ad64cf95a922a098038f215588b216beabbfe6acf20256188e883292b7a3a
      In Elligator2: u =
      f6675dc6d17fc790d4b3f1c6acf689a13d8b5815f23880092a925af94cd6fa24
      In Elligator2: gx1 =
      a63d48e3247c903e22fdfb88fd9295e396712a5fe576af335dbe16f99f0af26c
      In Elligator2: gx1 is a square
      H =
      13d2a8b5ca32db7e98094a61f656a08c6c964344e058879a386a947a4e189ed1
      k_string = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5
      aaf79cf576087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861
      b3d683b
      k =
      1fda4077f737098b3f361c33a36cccafd7e9e9b720e1f84011254e25f37eed02
      U = k*B =
      a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60
      V = k*H =
      fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25a
      pi = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57f
      dce35b46edfc655bc828d44ad09d1150f31374e7ef73027e14760d42e77341fe05
      467bb286cc2c9d7fde29120a0b2320d04
      beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5c
      d090a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314
      a58

Authors' Addresses

   Sharon Goldberg
   Boston University
   111 Cummington Mall
   Boston, MA 02215
   United States of America
   Email: goldbe@cs.bu.edu


   Leonid Reyzin
   Boston University and Algorand
   111 Cummington Mall
   Boston, MA 02215
   United States of America
   Email: reyzin@bu.edu








Goldberg, et al.        Expires 17 December 2022               [Page 45]


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   Dimitrios Papadopoulos
   Hong Kong University of Science and Technology
   Clearwater Bay
   Hong Kong
   Email: dipapado@cse.ust.hk


   Jan Vcelak
   NS1
   16 Beaver St
   New York, NY 10004
   United States of America
   Email: jvcelak@ns1.com






































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