draft-irtf-cfrg-vrf.txt

CFRG                                                         S. Goldberg
Internet-Draft                                         Boston University
Intended status: Informational                                 L. Reyzin
Expires: 10 February 2023                 Boston University and Algorand
                                                         D. Papadopoulos
                          Hong Kong University of Science and Technology
                                                               J. Vcelak
                                                                     NS1
                                                           9 August 2022


                   Verifiable Random Functions (VRFs)
                         draft-irtf-cfrg-vrf-15

Abstract

   A Verifiable Random Function (VRF) is the public-key version of a
   keyed cryptographic hash.  Only the holder of the secret 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 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
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   This Internet-Draft will expire on 10 February 2023.

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 . . . . . . . . . . . . . . . . . . . . .   5
     3.2.  Full Collison Resistance  . . . . . . . . . . . . . . . .   6
     3.3.  Trusted Uniqueness and Trusted Collision Resistance . . .   6
     3.4.  Full Pseudorandomness or Selective Pseudorandomness . . .   7
     3.5.  Unpredictability Under Malicious Key Generation . . . . .   8
   4.  RSA Full Domain Hash VRF (RSA-FDH-VRF)  . . . . . . . . . . .   8
     4.1.  RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . .  10
     4.2.  RSA-FDH-VRF Proof to Hash . . . . . . . . . . . . . . . .  10
     4.3.  RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . .  11
     4.4.  RSA-FDH-VRF Ciphersuites  . . . . . . . . . . . . . . . .  12
   5.  Elliptic Curve VRF (ECVRF)  . . . . . . . . . . . . . . . . .  12
     5.1.  ECVRF Proving . . . . . . . . . . . . . . . . . . . . . .  15
     5.2.  ECVRF Proof to Hash . . . . . . . . . . . . . . . . . . .  16
     5.3.  ECVRF Verifying . . . . . . . . . . . . . . . . . . . . .  16
     5.4.  ECVRF Auxiliary Functions . . . . . . . . . . . . . . . .  18
       5.4.1.  ECVRF Encode to Curve . . . . . . . . . . . . . . . .  18
       5.4.2.  ECVRF Nonce Generation  . . . . . . . . . . . . . . .  20
       5.4.3.  ECVRF Challenge Generation  . . . . . . . . . . . . .  22
       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  . . . . . . . . . . . . . . . . . . . . .  29
       7.1.1.  Uniqueness and collision resistance under malicious key
               generation  . . . . . . . . . . . . . . . . . . . . .  29
       7.1.2.  Pseudorandomness under malicious key generation . . .  29
       7.1.3.  Unpredictability under malicious key generation . . .  30
     7.2.  Security Levels . . . . . . . . . . . . . . . . . . . . .  30
     7.3.  Selective vs. Full Pseudorandomness . . . . . . . . . . .  31
     7.4.  Proper pseudorandom nonce for ECVRF . . . . . . . . . . .  31
     7.5.  Side-channel attacks  . . . . . . . . . . . . . . . . . .  32



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     7.6.  Proofs provide no secrecy for the VRF input . . . . . . .  32
     7.7.  Prehashing  . . . . . . . . . . . . . . . . . . . . . . .  33
     7.8.  Hash function domain separation . . . . . . . . . . . . .  33
     7.9.  Hash function salting . . . . . . . . . . . . . . . . . .  34
     7.10. Futureproofing  . . . . . . . . . . . . . . . . . . . . .  34
   8.  Change Log  . . . . . . . . . . . . . . . . . . . . . . . . .  34
   9.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  36
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  36
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  37
     10.2.  Informative References . . . . . . . . . . . . . . . . .  38
   Appendix A.  Test Vectors for the RSA-FDH-VRF ciphersuites  . . .  39
     A.1.  RSA-FDH-VRF-SHA256  . . . . . . . . . . . . . . . . . . .  41
     A.2.  RSA-FDH-VRF-SHA384  . . . . . . . . . . . . . . . . . . .  43
     A.3.  RSA-FDH-VRF-SHA512  . . . . . . . . . . . . . . . . . . .  45
   Appendix B.  Test Vectors for the ECVRF ciphersuites  . . . . . .  47
     B.1.  ECVRF-P256-SHA256-TAI . . . . . . . . . . . . . . . . . .  47
     B.2.  ECVRF-P256-SHA256-SSWU  . . . . . . . . . . . . . . . . .  49
     B.3.  ECVRF-EDWARDS25519-SHA512-TAI . . . . . . . . . . . . . .  51
     B.4.  ECVRF-EDWARDS25519-SHA512-ELL2  . . . . . . . . . . . . .  53
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  55

1.  Introduction

   A Verifiable Random Function (VRF) [MRV99] is the public-key version
   of a keyed cryptographic hash.  Only the holder of the VRF secret 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 secret key and uses the VRF hashing to construct a hash-based
   data structure on the input data.

   Due to the nature of the VRF, only the Prover can answer queries
   about whether or not some data is stored in the data structure.
   Anyone who knows the VRF public 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 VRFs based on RSA and elliptic curves.  The
   choices of VRFs for inclusion into this document were based, in part,
   on synergy with existing RFCs and commonly available implementations
   of individual components that are used within the VRFs.






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   The particular choice of the VRF for a given application 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).

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 secret key for the VRF.  (Note: the secret key is also
      sometimes called "private key".)

   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 VRF secret key SK and public key PK.

   Verifier:  The Verifier holds the VRF public key PK.

   Adversary:  Potential attacker; often used to define a security
      property.

   Malicious (or adversarial):  Performed by an adversary.

2.  VRF Algorithms

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

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



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      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 secret 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)

   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:
   uniqueness (full or trusted), collision resistance (full or trusted),
   and pseudorandomness (full or selective).  Some are designed to also
   ensure unpredictability under malicious key generation.  We now
   describe these properties.

3.1.  Full Uniqueness

   Uniqueness means that, for any fixed VRF public key and for any input
   alpha, it is infeasible to find proofs for more than one VRF output
   beta.

   More precisely, "full uniqueness" means that an adversary cannot find



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   *  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.

3.2.  Full Collison Resistance

   Like cryptographic hash functions, VRFs are collision resistant.
   Collison resistance means that it is infeasible to find two different
   inputs alpha1 and alpha2 with the same output beta.

   More precisely, "full collision resistance" means that an adversary
   cannot find

   *  a VRF public key PK,

   *  two VRF inputs alpha1 and alpha2 that are not equal to each other,

   *  and two proofs pi1 and pi2

   such that

   *  VRF_verify(PK, alpha1, pi1) outputs (VALID, beta1),

   *  VRF_verify(PK, alpha2, pi2) outputs (VALID, beta2),

   *  and beta1 is equal to beta2.

3.3.  Trusted Uniqueness and Trusted Collision Resistance

   Full uniqueness and full collision resistance hold even if the VRF
   keys are generated maliciously.  For some applications, it is
   sufficient for a VRF to possess weaker security properties than full
   uniqueness and full collision resistance, called "trusted uniqueness"
   and "trusted collision resistance".  These properties are the same as
   full uniqueness and full collision resistance, respectively, but are
   not guaranteed to hold if the adversary gets to choose the VRF public
   key PK.  Instead, they are guaranteed to hold only if the VRF keys PK
   and SK are generated as specified by the VRF key generation algorithm



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   and then given to the adversary.  In other words, they are guaranteed
   to hold even if the adversary has the knowledge of SK and PK, but not
   guaranteed to hold if the adversary has the ability to choose SK and
   PK.

   As further discussed in Section 7.1.1, some VRFs specified in this
   document satisfy only trusted uniqueness and trusted collision
   resistance.  VRFs in this document that satisfy only trusted
   uniqueness and trusted collision resistance MUST NOT be used in
   applications that need protection against adversarial VRF key
   generation.

3.4.  Full Pseudorandomness or Selective Pseudorandomness

   Pseudorandomness ensures that when someone who does not know SK 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 secret VRF keys (PK, SK) were
   generated correctly.  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 adversary who does not know the VRF secret key
   SK.  This holds even if the adversary sees VRF hash outputs beta' and
   proofs pi' for multiple other inputs alpha' (and even if those other
   inputs alpha' are chosen by the adversary).

   "Full pseudorandomness" security property holds even against an
   adversary who 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 that
   suffices in many applications.  This security property holds against
   an adversary who chooses the target VRF input alpha first, before it
   learns the VRF public key PK and obtains VRF outputs beta' and proofs
   pi' on other inputs alpha' of its choice.

   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 is always
   distinguishable from random by the Prover, or by any other party that
   knows the VRF secret key SK.  Such a party can easily distinguish
   beta from a random value by comparing beta to the result of
   VRF_hash(SK, alpha).



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   Similarly, the VRF output beta is always distinguishable from random
   by any party that knows a valid VRF proof pi corresponding to the VRF
   input alpha, even if this party does not know the VRF secret key SK.
   Such a party can easily distinguish beta from a random value by
   checking whether VRF_verify(PK, alpha, pi) returns (VALID, beta).

   Additionally, the VRF output beta may be distinguishable from random
   if VRF key generation was not done correctly.  (For example, if VRF
   keys were generated with bad randomness.)

3.5.  Unpredictability Under Malicious Key Generation

   As explained in Section 3.4, pseudorandomness cannot hold against
   malicious key generation.  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 uniformly random, no matter how the VRF keys are generated.

   A formal definition of this property appears in Section 3.2 of
   [DGKR18].  As further discussed in Section 7.1.3, only some VRFs
   specified in this document satisfy this property.

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.





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   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)

      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



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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)

   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:



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      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

   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



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   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:

   *  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].




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   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

      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)





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      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.

      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 secret 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



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5.1.  ECVRF Proving

   ECVRF_prove(SK, alpha_string[, encode_to_curve_salt])

   Input:

      SK - VRF secret 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

       (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



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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)

   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



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      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:

   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"



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   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

   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):



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      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)

       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.




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   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

   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



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   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

      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




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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:

   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



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      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)

   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"




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   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

   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),



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   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.)

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 secret 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



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      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.

   *  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 secret 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.)




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   *  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.)

   *  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.



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   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.

   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







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7.1.  Key Generation

   Implementations of VRFs defined in this document MUST ensure that
   they generate VRF keys correctly and using good randomness.  However,
   in some applications keys may be generated by an adversary who does
   not necessarily implement this document.  We now discuss the
   implications of this possibility.

7.1.1.  Uniqueness and collision resistance under malicious key
        generation

   See Section 3 for definitions of uniqueness and collision resistance
   properties.

   The RSA-FDH-VRF satisfies only the "trusted" variants of uniqueness
   and collision resistance.  Thus, for RSA-FDH-VRF, 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
   are chosen not in compliance with [RFC8017]); thus, RSA-FDH-VRF
   defined in this document does not have "full uniqueness" and "full
   collision resistance".  Therefore, if malicious key generation is a
   concern, the RSA-FDH-VRF has to be enhanced by additional
   cryptographic checks (such as zero-knowledge proofs) that its public
   key has the right form.  These enhancements 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", ensuring uniqueness even under malicious key generation.
   The ECVRF also satisfies "trusted collision resistance".  It
   additionally satisfies "full collision resistance" if validate_key
   parameter given to the ECVRF_verify is TRUE.  This setting of
   ECVRF_verify ensures collision resistance under malicious key
   generation.

7.1.2.  Pseudorandomness under malicious key generation

   Without good randomness, the "pseudorandomness" properties of the VRF
   (defined in Section 3.4) 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.






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7.1.3.  Unpredictability under malicious key generation

   Unpredictability under malicious key generation (defined in
   Section 3.5) does not hold for the RSA-FDH-VRF.  (Specifically, the
   VRF output may be predictable if the RSA function specified in the
   public key is far from bijective because the modulus n or the
   exponent e are chosen not in compliance with [RFC8017].)  If
   unpredictability under malicious key generation is desired, the RSA-
   FDH-VRF has to be enhanced by additional cryptographic checks (such
   as zero-knowledge proofs) that its public key has the right form.
   These enhancements are left for future specification.

   Unpredictability under malicious key generation holds for the ECVRF
   if validate_key parameter given to the ECVRF_verify is TRUE.

7.2.  Security Levels

   As shown in [PWHVNRG17], RSA-FDH-VRF satisfies 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 uniqueness (both trusted and full): 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).

   *  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.





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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 3.4 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 to ensure specific levels of provable security.  Such
   applications may consider the following three options:

   *  They may limit the number of proofs produced for a given secret
      key, to reduce the loss in the provable security level.

   *  They may work to ensure that selective pseudorandomness is
      sufficient for the application.  That is, they may design the
      application in such a way 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 VRF
   secret scalar x (and thus break pseudorandomness of the VRF) after
   observing several valid VRF proofs pi, using, for example, techniques
   described in [BreHen19].  The nonce generation methods specified in
   the ECVRF ciphersuites of Section 5.5 are designed with this
   requirement in mind.




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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 secret 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.
   Specifically, in the RSA-FDH-VRF, hashing is used in MGF1 and in
   proof_to_hash; in the ECVRF, hashing is used 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 inputs
   given to the hash function for a given SK and alpha are
   overwhelmingly unlikely to equal each other or to any inputs given 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.










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   *  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].

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.



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      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 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.



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      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.

      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 addressed IRTF
      Chair comments

      14 - Addressed IRSG review comments, which resulted in a
      substantial reworking of section 3.

      15 - Added RSA-FDH-VRF test vectors.

9.  Contributors

   This document 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, Mallory
   Knodel, David C.  Lawerence, Derek Ting-Haye Leung, Antonio
   Marcedone, Piotr Nojszewski, Chris Peikert, Colin Perkins, Trevor
   Perrin, Sam Scott, Stanislav Smyshlyaev, Adam Suhl, Nick Sullivan,
   Christopher Wood, Jiayu Xu, and Annie Yousar provided valuable input
   to this draft.  Christopher Wood, Malte Thomsen, Marcus Rasmussen,
   and Tobias Vestergaard provided independent verification of the test
   vectors.  Riad Wahby helped this document align with draft-irtf-cfrg-
   hash-to-curve.

10.  References




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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>.

   [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>.




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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.

   [BreHen19] Breitner, J. and N. Heninger, "Biased Nonce Sense: Lattice
              Attacks against Weak ECDSA Signatures in
              Cryptocurrencies", in Financial Cryptography, 2019,
              <https://eprint.iacr.org/2019/023>.

   [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>.

   [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>.





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   [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 RSA-FDH-VRF ciphersuites

   The test vectors in this section were generated using code at
   https://github.com/reyzin/rsa-fdh-vrf.

   There are three keys used in the nine examples below.  First, we
   provide the keys.  They are shown in hexadecimal big-endian notation.

   2048-bit key:

      p = efb52a568fa3038fffb853e2183791c6bc81ceee86d20e8f9b6401dc79a8f1
      f6248d3a25fdb3f99245fce41667da038f59745b87cc1aed8b4a9c1d74e7d5c16c
      f7343f2b12f1b5055337369bf018fa07adc0d16f2164a516e80d2b4734f0c6563d
      6ee6d4a9e1a54e300cfe9ee679afc3d14a152dfb49b6cfb208bbf921f764af
      q = ecbca5ee88bbc635d8263aaba84f6502fdb2b4998a40f7c149133d840b6b1b
      d9a972fe2a981c770272b78fda213f76a062dd865dd116d4c8980975ee9347fe0f
      500567e51d78dbee4a34e626051cf018d7feb72f19189525d4f70b6467d0cef514
      633ab08a9e7a9ec632064b7b5e3e82128fe563757a614092fc5cf624d10e1b
      n = ddaba77202bafb796b85bcec98958aa58ae2d117cbc66a6e75c4c2af983985
      a3064eaef93e2b03393256d94d75d6a6656b2956524ed8711898a0c3abae84371d
      a0283bc5f433fc384d810a3c118ed302c0b03da16bee70b80ba3480e7acc1eb358
      b3f20fbe90cc4c8a7e2ba9e28b2a3800a5efbaa3c264f79b231f7cdc9577818df1
      bac60ef7a3f78a44f046fd29b0689556da7a7f61eefe67427f3f691aee0a4b1efe
      2ee2e0e6091143ebb7d69254c9d8ab01ff5e0ad7329f566082f9251e64f436c547
      e68de75351ea3a09746ceb7efed2d234121088aaed01696583c172ec88bc173a0d
      4d8ec43f4dcc18ff8379317e83ef9685536283368c9c6deb783075
      e = 010001
      d = d5c5ceab929a841e2a654536de4788f7f0a2a086d44bbb245f8aab3df00db9
      24e8d644c3b502820f4cce98adacf09e73bc0e9762b50ae2b697aaa24914fa08b5
      1758f59c07cf827341bb2a0597e126f9c69db031d60692c9cadf62842444696f08
      223154a1b0be752a325725748644e6d12935b1c66f983379773bcc8c65d06262e9
      3b5bb774dd2784265c23e9a7fc5e8871eb6bcc9968a6bc360a98874b623ec59f41
      af0a9ecec6af095cb7e5aca11472363950dcbbfcf678fe003358b4ff0060a391da
      a45a1bd81c166b6221fb07e4f5da75e27d8d5fdbbf87ecbd7f5a4d804597070faa
      ed22f197511b218788816689375245ddf7fa12337f3e7e898fb9d9

   3072-bit key:

      p = ee5adea28491084e6635bd73fd95649915a11da410d3f361c8eccc90a4b834
      25146da7b9e9d3994fd37d5fad7fb759ae451eb99b1102d4671ead23a2925133d1
      9df49cf9d7e9dcb69fd7555ca095338d0d2a84abb6825050eaf5fffaeff17ccb08
      33c6079081dfcbd98ced36a593557d29d64b0e0253ce2ee4e07fe2a06269dfe5ca
      230fad221a593a69d9534b2521c1b41d116cafdee02106228ff41433605453e237
      777626953e79b46a84f50069e25b4f50496a928708abce30559eb183cf




Goldberg, et al.        Expires 10 February 2023               [Page 39]

Internet-Draft                     VRF                       August 2022


      q = fb585bbc12f5695951f70a25e27682dc568acf56115ad749709b2a6e915cdd
      66dfa06db09b390c00b7c7ebeea00845f73c999d8ea9352b1128bdf10113c7500b
      76a03f6b38d0920b5589961549be3d841ccc306f3edd600a53b4b9d4fa1249af87
      af58dfb3ed694289477e853f7d062f58911f7bdb98033b001ee90f11b78f031cff
      ac2b5a07e11b01a2a6c1cda059a728f8253a5ff87267623253fc022d3993b27e2f
      344b99eb6072ff7c7ee160724f8fbca562be49247ffae42b55ea79dad5
      n = ea055cef495dec2d8fb3aef519ca87bd1575fa0ae15dd433f4a5f6c40d34ed
      6ba2388172ab7d2183ed970a669d427dc2774ced66a3f082b8e23e94e7de7532f4
      f30bb4a5bbf2e1db2cba0752858a7c7a9bb892c5d6af7e90a7cee8f0097d14498c
      8b482f86348640af61b66640538e834f23ba8f906048db0e57b6fdc162ba2a8a0e
      aedd5423f23d8f89413223d89f473029cba11a211eb59e41fb8f0b8ddc651d115d
      9f07ac30296485a9adbd71cc5d9e4a448bd6d70785e838a978b2e66513eb897c96
      2e85f00a36cc0a3a613183d8bd1572f895901eb8155af9797dbd4aa14726f41571
      2bf0eb29fa0a9e938cf5325def05d3af7e686227456d903233e316c8cc50341615
      e59b665f0a4a2c32cfccbf9469bdf89564481fb7afc27a7127741f79424e0a35cd
      c466dd33ef5a2067f75c86e06af9c03c68c6e78be5f1a4f49ea03569cd9f74c3a0
      ff290ca4ce2c2fa5b770ef8032b26a517c257b7b1c424622c5c04cf20f2290a268
      939e0cc79dfbac71842f94727b07bfafaded7db6c7f13b
      e = 010001
      d = 6e68e957dbfd7c1862dc1b87780b9dcf0ff9016770bc9c09873b66194941d7
      6218bf2013c1e4df9326dd4402f5df110656d2ec8ea87a28b2a1cb74e590872aeb
      765fe772ea21c57d6ab4ba0fad019189273f05c061719afd14af02277dd28d67c5
      ef50b75b521ca51819b9bcb44cb7c82be66776a45f490050dc0171e77374f1ed00
      d06f8beb09b711a9682107d8840d4a23edf6ac25441fdbf2b584dfa6a67cee21eb
      51c484f09416e11914e774713f1a17600fb9e4e99fbbd83fdcba4b09145dd98094
      49a1713777161c912d5d595362314b0ea9d1199e97780e8b3293a39af4019fcc74
      6aaf78dbb7db06852c3358a9ed02ab1d15831a148b27b932c117445a4a6f5114ed
      fa3ccc9a9862df714b78a5362aab5e30501b4a729af73e3cdcabe19aac4928b668
      969780ad33d9df206d904b978a055f4abbc64987744526856e16ef55962453e3ed
      7a8055b0d79d051c50c94584ec7501dbd4856d7a21e43f25d8749e683cca2f53f5
      75af1d80f39d8e6932ffdf201d179cbf98314c4048c6c1

   4096-bit key:

      p = ac803464c8b2082153e15d5a0698d0a2990397fa01c1edd6171a5315e743c9
      9feb7acd31c37529d4f83405e657c390488d19f7da9ef9d9f9cff4b460d2a26eb1
      0f90cf4aaf55a19e21dc3bb697723a673e12bbc6580adc7bb72adaddf4682d656f
      f5b992e62379bc7b0ac977f2bfbcfac634e04ed597ef302684be72c6bf7db10b80
      f452d412d09e63e017acba378ccc6ea58e683e5641d1e72248f3201a5632f4af75
      25e91f9e0733731d264fe36802f416cb3e182b21e67a12e3bfba9a9cf40a45ff32
      addfae78063933120238ac61fbb995300a8602aa84f993bed375d6ccba86ad0c8e
      fa5f0950aa2c92779febce9d05fa7a1f0d6e5c0d785de93c108297









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      q = feb39bb6ee78adfa524e9c0821f60c20d3cff74f8b49731d67ea27d218bcb2
      0c87498d30dfd398bc23daff7b33dc330db93e6c0e5e6196e035446c6db7cfdb98
      68b9518d94670b31f9c4d2109cf32c9cc8ac2fc4a6c2e1078510522c81610a81a7
      07997933ee24030b572a76ee51aa683312ecaa51d8558b3b19cccf65fc867354ae
      193fd5c4f5d5a7180c5ca1e90fcc42f6915dff69a3d1e49046f6c3ef841b262ba8
      9ddcfde2ed3caeb5bd594181a76f6f1ce01fc65c6f925f6d5b77037c2cbf7b6047
      e19f7b9c846c80238f1c8284c33bfd90c79de91381bb883b0de568aaf4b4a3c3f9
      c98f92e9f6a51f010bcc1dacfd72bfdfda29f527d7f4913153bef7
      n = aba03a8d8527bfc0cbea1cb9a100f4ee7870aedd74a6406f108f7a07f37433
      6025357e256d655b342d73369102d03c7dcf3c14ed70aac7ebb62498c570068f71
      f1f165e14527f96d946ba839412252eacea604e7d6fd47a0bb9de776679fa9ad64
      85a076fda04a2015322626dcd2eb91d6b6248802e6d453eb4cbf5e1bfebed02d6a
      b36cfe3dd1e8b9749d4853a029940a0bed3aa3128fd8e2e6cd1115db15405bb383
      7012f56bdc5a6895ec5cc6bca52f7952cfe3c7d5d81d4d3d1c9a29a429eeedfbf5
      8da0a5b17480875b8071f49eb568fc8d8c023c83b3ed870c3775aaf0578485d757
      b4ab18d8e5fdb30c2b5586047e6203ab1636e376f1c7031f171e2807a2058ece89
      0cc8fae29ba819df76b45ddb514caee63db1c5e7a3af7468febff82bfe2eb79e3c
      5d1383b7ebee86f02e9cc1853f0f4486f7eb8fee23a2f794317ffd1c39471086df
      bfc0e3c0f412f917225f5c551557f38c11f172eca257e4b5908a571e4daa7c7434
      903701f21937df87d10de9b50ada97e65855d5e786db8f3f86248b55d999ec3153
      8bd1a409f3e13de46dccc05325774e89016708f8a96240ae1c16641e8b12ab0725
      7e88aa50d3546e7a91073d85ed601775a3c08e9b7c242d20664dfd4e70a05218d9
      f2c7d760fab3cd772d9362527917cf5b51817e8c2aef51cb3b0dd8cb838097e513
      537f1d9c3c4708f44ed270db963c7d72cf11b1
      e = 010001
      d = 1efd8dd524282b4deb04592f83cd226d353e53b5156d37d15652321ce16f28
      1fc258487105b1f9a81054ef937bc89243bd7a01e56624d078d5a9021514c77a7b
      7eceb230dd45fc9a36e4c1b9a4f347b9b29af3e3d14466fcb5242c398b389f70f9
      e7cf33ed54564e38c597720909e513ae8bb149060d1c6612e506e13d78e087c2cb
      b39e88c22cf73315c598dbd0ddf1276743ed04a943644c84949ef32d5e4702c805
      81e54a7fb18879be28b21008dc63182b45f2c190f1b748cd322efc39f2807c64b4
      d06023cb49583418e7b6ac0f447eb2abf48e2ad335583cbc8dff2760c2cce14623
      46326708336f7e374253ed213e990044927c52d29591f414571e509afc2396a6af
      9843303a19673bcdec1e3fc7c0d6c3f43b4bf88ce83e2bdcfb5e39069fe32800cf
      3f6f6d9917b8083a66ce23a9ab5b0c95bbcc6dfc21d38dadecc20725b13ce2954b
      a1bd45ec151a8877fed317cac60b2afaa96c826df6d1c48e7c10649dccc75bdf90
      5c362c6934da06c3ce30f5befc1cf776d7fda673625147b1108ecb5473f7f58827
      9533eb184d748230443694b9761b01532ba707563ffa4962321e44fdb710025e8a
      6e00d29bf01ea040618ee111b5d79ac860083f91aa614777cc99d739458f7c53d6
      3cea7155b118068e0b30b35ed6d0cfc75672f18d075157a3ed31bfa1ce2cea2343
      57ec76117cc687c274636077abc437cb70a029

A.1.  RSA-FDH-VRF-SHA256

   Example 1, using the 2048-bit key above:

      alpha = (the empty string)




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      EM = 092ea69ca4f5630d4bd1012805ad23528a5f44c040829b4a0208491913ee3
      9711889bce5347765072efb0b7f8ad9798c830085d9babe10c29f1a649dbb9a64c
      93a8cdaa325d37814faa15a1071ba81c39275f3cd66ce70fd21ee3acc7ac127c5d
      e8f2a816b05aff19e4e63451cfe51fef059b2547302387449b4df1ab8eaa5bfc84
      dbbc5edf3b07eb8fe3fe2a93858bd0d55d6f0686f2eb449ed4c609b3083de04b49
      d409a425509d89d282de806a6ce66892edc30337f780b15c7695b26383516f1fc1
      8f7eab52557c654467600e2e272ef41e7e4a060b42f7533bae603a7fa50f497a64
      a1508b93826d99643a2001d1c958a7a06da0370668634d678a5de
      pi = 14234ff8a9487e1b36a23086e258135b8a8a7ff2e23f19c0dfeca0c0a943f
      119ebd336fdc292ef67b56e32ba06f9941893754a8b97c82f68974b2b34c17f6d4
      3bfd55eb110cd7ea3452d59a24e4ddb8d4cdf040c814e22e3537ca09c2e2dc5dd8
      ea281e6492ad335378f9f437eed30c51eeeee66ef14efb4000c75c802e9c5a6bb8
      039c0258d4347981159d0ef6990b5e9c8ac2fb03915d7ff1ffa0626e2e11714a63
      342e59124c1fcea8e2816c1d9a7751feaaa66cf6c82cd3c58ffde66460d98246ab
      358cc33baefae4dfb0d191e9b6d6c0e3f92c35200408925dc8bef39b78d1259f81
      63a5003a693555f05290ef2e68345f27c6e2a8847c5c919d92e7505
      beta =
      79f0615d4677fb72571889453644013f1a31b08d222e3cee349d64ce1c41045a

   Example 2, using the 3072-bit key above:

      alpha = 74657374 (4 bytes; ASCII "test")
      EM = 20a059b7f7034d0d7696c63328cbbd4b40f7c656a632b4129915018fe6c5d
      ee8b5bde68ec2a5a78b1ca8483386e3a1a0fa07b4d329ea55facc3145c663ca90d
      f5ae46c903211a21bf908dc9a33bd09410cc09c7b4de5fbc79de3413bc80bccf2d
      3aca2fc9c60c776619849ed3e704057ac3d5deacff845d5bc8084ac730c19a1466
      8e53b5b8b90446b2272eaf59cdf985a7804c7b91cea1ce2582099b7b0f20163b11
      d23110939dd62081b5aa46c62db76b2ac28473d2488970d480bdd8bef8cae9e812
      74fe3f9925b012c1b55cba8c4291ec7433223cb872e422bb9e0d3775670d587e40
      3660ff440a9c11a18a488abc716ae36840b2ef5b0db4a90d88f91d79536cef378b
      f8e76d173288e26241df522a3cf6bece49c960e43a2d93e7bed10b90580c5b3aff
      056507b4ef27368579832cb4aeecc99c2d8ba402117457df5ae0ed28068ef8b2e0
      d4582f8edacfcad02c83bfab778460b979e9e984827bbefe2b544c0f3ed715dde6
      dc1d7fc7c0f1f87d78aed8e148004b9f62e0321214c7c
      pi = 69f6042d400dfad4bdb9974fb73d12ec7823c6632df6b0a97ebc14d8a443f
      74e1eb1a99b37204ba5c7e53bdaf7e3e3fae9efe47cc01d0b061585c8d757ecf00
      663b3e1bd447d55b6ebd066b814a8d9c4434b224e9cb053a1fd038a58f3bf6b0c7
      5b6f48f3c8d1ca398a730c133f86f244655f24c445324fdacd291d6d907f93efb2
      4b59e509f2f370392f5e262fc106292792352d93800f0a1e3a389786619a622f60
      05cab78ea5f0b5b7ca91ad2a9c6c34fc4a3f9b0332b99e907ffa7f750cdc8342e1
      2da78f13ad49953bae1751c983ce3cd3335288ac856f85057a7f05acba6465a1c6
      901ba30bc65b79fb7a847c42a5b4942d600ef316030f2ccafbc6f2e1ff0b46fb5c
      8517cd98c93f81acf370cfdab559bb4270d07db5466e2342d56c476089f4738404
      34cbcdbd1853b487a6df346208d12c17a48fe50b73b96f640a9761f570a516f615
      7432b83dd18a1d05cc27b6f283a02fcfda147cf1471772e469961004bde7fa1585
      7e7bf97b5a83c33fddbd9f4b2e2488f4ed5f7463c93f30b
      beta =
      bfe966f3fabde6f38a2792ad59bc836bbca39de6eff64f15a42886deff6dfcc5



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Internet-Draft                     VRF                       August 2022


   Example 3, using the 4096-bit key above:

      alpha = 73616d706c65 (6 bytes; ASCII "sample")
      EM = 17524fa1710b2f8a04e55da403b9b287b99a47afe9b81d3421482e3959b73
      b4d4d4f4b52243ff2bfd2d29b1f030b521d0699065faa2b8903cca2b24cff42956
      1234fcbd7bcccdac61b7dcb7bc61cd857287b4b42357adbd2fc83ecfc0d5bc199e
      1f6e298b5e470bfc540bc85e933b02035792d65d861096dc03f048cae51c9adc6c
      1ec09e7f5e595681b3d3976d94ba1a65c83c7e82503db5478d3d91b2e00a0f24e7
      ffee1faed68aa62ad7ba4b2912ceb636064766f0535d3ca1369760d8edebc3c8d7
      f5b4de784b644b59e44e24e436298cc33a3cd0f676d6fa0b76ca3b9b11aa68e078
      9e83bd27b3af08518b9a5eb5f34f4953a79dc25c1285b20fa73e558dd99638eb51
      bf89c80d7989f6e925d8ca5ed1d3f29cc1e1065400e4abbdbcf898791be12c5ae2
      5661bf7de58a4cb6608c9a4dcc18150638068bb6452b25589ae0a943a67f024dd4
      b5d9e7940c01886f798316156e5771c19457f9104618e271ae7863b65fd07f87fc
      d7862690115ce2d963eeac60f78b47c037d6ed3000b43d8149cee08df10a97158e
      e1daaf0a3963d23fb6ab0615891734e3039417d8ce03bfc18920c832a40385de95
      d99b546b4bd24ecbfb2e75e9158ae1769bcff444990f54aa40e6a14e0aca52df00
      062afbf81f6ce8193c53f8d26ac71324fc1db878379178abd695cf04a0fae3432d
      1efffa73bba15b4e81fbaf598146a0c3edafc
      pi = 745cc4b6cb75b925194374cdf91b498e8d687c5d9cae1eb5352446c554c2c
      43ac4aa3e2db5cf5e366df635ce156a277ebdbe78c5598588c98257069253127e5
      7c9735b498f2939f14e1d019795cbd74cee2693acda2666624f174e8f666494aa1
      2641bce0677acd20e5552d2690117bddb38678a18acdc380bd9d93f3b10960f9be
      0c141fc14f5f30da324ff14020cb5b8aed9fbca3fc44b4973d8e5527bd81f5ae5d
      a67e5cc995abd1f7f9cdd3fa89b243fd4d5d5086ddb4eed77a2851fda1d4463f5e
      e037a4015aa40c420c2e609d5d0da4ef4a1622131022bdd9c9dc26d177b392663e
      a42050ef485fe9d53a8d28d84b82a21101bed5b213c82b578ce7c9c6f7c1bf9eca
      3c248ace9f8835f3850158749111ce1a3bdf5766add72a95a47c8866a4817c42c5
      cbd85d7bef52afab567e564f6625be9e04be6f7da012af68e6623ce4f29c692ba0
      b5f7665bb435a2168bd3b88aae0c6168bb87ea6977f35bb5ad833d96dd14d340f2
      a67b241b01fd8caf415842fd0a9dd5f4ccf4e70f15efdb85332e1df2bb186be15f
      7195176435e01bfd00592710023c3a88ac0eea7189b32296f865a310375111a5f1
      1b74d0c74b98dfe4c41ccbe695ea801ba47f37b878c1ed0fff8302705b63c89120
      9ea63defa892969e015a86d97945189444524e5fb660f2b9d1dce337a12e0d003e
      a6262ca3194515cc3aa10b1a03ac9dd6995b54d
      beta =
      b663c5f90da1c12cd5d0e6d049679459e6f79f9fe16bc8b8e7e4d64d66500bd9

A.2.  RSA-FDH-VRF-SHA384

   Example 4, using the 2048-bit key above:

      alpha = (the empty string)
      EM = 1fa5c0079423d46edb63a833abb2e6ecfd5f39d1f2bd68fc666274d9e8ed8
      ea8a13411126861167a4ba1d014d5ae213372de6bb4227b12e68e16e13ce108536
      acb25f7219c49388f757219716fcb74eb0245b826c7e47ca793864885684b7673e
      2f8579f26e78d63a940eacb23bf7619290cb5cd20859482c410fbd6d83a61f8940
      866f512be7ac041fc23c3ee71d918ec994f3efa62f4f1f44eaa29f5b37a1e93e24



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Internet-Draft                     VRF                       August 2022


      73d8677fcbec312838379a3e05899ce44227c0c428fbd7d4f2d0b46cfde7254e39
      67b220f8661f5dfbce7a3bf19364f522914478cead3eff0f0e02d166c251319bcf
      86701af1c48436f49ceac990f52940f7da6ac6f5fdafa5c55dc77
      pi = cffe6067bd9a1285dc1e8e543e8582c1250407cbfbcb2d01c4ddbc0d4ecb5
      edeb721fb33147cf95f3084f7ce611f9877814770b14b8a671abc7ff085cf5cbe9
      1e72d17f076d62db478d4758412a4e4b77a5591dc32b764a501d27e34e56189ba7
      347a96f141ed1290f8ef7c4ce4009a9aba0715cbd0148721ea72bce00a22e59460
      421a21e4d121fc0b4eda62479d93724afae7556abe66326487be38cfb795ac1968
      c33a3890f2d8c0f7dfbe88bc76f16cbfd2b0f7ee8663abfd7b789caa5f6c77dd1c
      a991c9a9cc532f7550ad6184c8ece12ca4bea7e67f32405416a1f83245b09d06e7
      b4214157fb444be12a2eddc4381678f2b862fb240fcedd2da7ffcb3
      beta = dc37e83f8de0e990abada5096a05ca74754cfe7fe8e46b831e241009194
      15415dcd5a305f5fb8195713cebc78649c8d1

   Example 5, using the 3072-bit key above:

      alpha = 74657374 (4 bytes; ASCII "test")
      EM = fa2fd7c735c961b43b01c005faefc4e39505ede3914076d4dee40d52acf72
      7de1782386ad6e9e07faf7666c8f45fde93b024d97c40651b957cfcccc42b8596a
      68a5495c02313ed9ecbb705ab0689c38b9e57af035189e377ad50b4704004c2a97
      3d9f7554204b03e8b925a973d41a9c3432246eb2eab2f729f03d3a63c9c38c0cc2
      baa440ed5e2d61644405e4b5c1acaac85d8de75a4de00419a478e6c44a97b3e898
      75c318400ce8d75b84c416ffd501ba78dd3203f21c6610fcaa4d8fc94f45e80dc6
      5b7e48967199e7acdb18d82413b7018192a6fa2da5d6838adb8e6139f8d12abcce
      7d5fd20cfa031c4971e563d4863d498591dc652a937db5e0bfd68535e3c9db9611
      8874287c2291a5d3b29aa142795e60f1ade2c8c4d627ee678b652f5fded61f9a60
      d2fa9cf5fb5e6a7fd63d81c91ea2269388f0a96fae77da0957695779385c757489
      56972ab1cb5e19ad3cc6a357b9ffd368ca985dd9c0e53dd42aff5985f7a234af96
      ad9e34e459a958b808e858f6d7be2e964c33cefad9660
      pi = 22c9278e7171183cf6a3ce108f0400e308a9177c39a171f77777c106c966e
      b041824ce43fa56c5c77576646dd110e0b5d7f838bd5b1d1bf2c1feb1520397dd5
      2d3cea6dbb49d786aa3bf3f5235e7692e583d290c7192102a6e0cb64f5229a326d
      4d00267fd75aae9687167ea0d3d450b2d63519ad605e64c77438728a190a129b11
      63939a5b7b0721b8d81efbf99a96944f63bf80ecc932fe40402d67c3e099a317cd
      1d13ac6947096308050ea6dad18fdb0958ae565d07d29e619673798f52b8d1dfdb
      f29b4641324ea6db5b9f35870acde7bf68e0829534d1c1f43ca9a16861efd82fb8
      83e35d581f613d2dfbc89d01a84fdf081a3a850f2e865188cd995857222160c547
      80dc310a6ec100b9bac30f3af92e641360cad8dc255b56fa28e88ffcbef8ebe6ba
      8557e4ec44a7d0ebef882ade36db0d89be71ecaa2b35026c9d328d2384b54ae68d
      e2ea70160ddde9aced5a8d896590fc185b408732cc04a249eff27501594902bf3a
      f4a3743c4da50c5d62a74746007dedb8358ecfef78c75ab
      beta = 5bdf742667ad10080f4ca573ec66f751e82e4077d0db1b281df421af68d
      39412e70362dc5101b4b46e1e453eea7e0989

   Example 6, using the 4096-bit key above:

      alpha = 73616d706c65 (6 bytes; ASCII "sample")




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      EM = 1b1d2f330ee20b9b1754f5e6ee4126cf03ea2c7f4e8c52d96111da7f99509
      042428ec2f2eafdf41716c04a9976a26df77b3d4cea8b10b216e7786fb49e923d9
      84a2ee13ad82b95783b68fcf3444b65d1353619602ae06e392dc030be105d4cebc
      6ff8a647b79115357833bd5312b9d3f0df1a307e782ff4db8de0eb16259d6bff2f
      57b3dd60a57693d607c42013cbcfc140a77d4a651492854afbacca377ed6729d1c
      be72999a62a96190fb630e5abc54d5cbe93254426df4e2315dbc777360ffb2401b
      3dedbed1acacf4b3a63b5ff8e5ab6c0f8ffd9e2a34fffd68a8a593c64de2660dce
      daaab13cd42ebf5720d49f3120b01f45f29d1f465e995b148c9266aa97793a9da2
      f38831d00f95f9688b1c50b52a4cbcc14f8287db822381cdddf609c9c178286b1b
      c2f94d7ef4d5ceb1293dd7b0fac16d1b3a8b2a7fcc454e52efd2de5a799397fd55
      a909641fa775463f4808b520c3ebe0f94e2765f8538d91a4f53bb746e7d5eaf55b
      3876503760f5c015f9e52bc54bdfc9632028db5e88b7dc0b1e9f1661d0a9b3574e
      46311de8ef6278c4c14f68375763e5df0d4cf221a4c3e84493ed0c36984c172d87
      b513857af4b6c10174dea9db6464e2bab210aa492987f0255d2c5588b1c79769da
      03b62f691d5c4e5fac65505c317bf96b4f70e97c002aa0a032b02e48ee3aead570
      3bde3186ce138f29ba36219fd3558af417945
      pi = 89d801e364fd48c3b8672e7d7abd8a2a1e5bd36bb1e38af5aaefa2f01cde6
      86fa2e33f88fdcc8eb3babcf1c66cbbf7dcddb614041813990787be5feabe86bbe
      c373d2cbf7c080caa0e37a339d5de1d1455de28f9bef76cd72500c669e9cab4599
      b55dc155d9dd5810174c170f646d3b0b459347c17347c0281eecf5055cf887d6bd
      0a2c962c77d5ff9355a53cea64c34ea0888110ec4eb32da69022e293a8843d4c06
      c9d6e020c594335720467a8337c6a939fb2c5d710f7bdab48a52f4e7483dae062c
      1b9f66f7c9038ba9ceef3d61cb4cc004319c94a267a2425b5f042cd7f1a17922d6
      596a88a6fefaef41fc87742f2badee7d7613179589b4d02611ac8fd7895d926f48
      4f79542cdf7f034dd536c9596da2f588ac9840f6bb05875bd17107e7458cc5ea36
      8a7699fd60c35b54253a718c26cf518712be9d86213b2c6bddd0b7dd169f9240e7
      7bfc44223675454f9c5596ad2e6e607ea65011a721ecbfa993172ae372ae874377
      9b33278d25e11ced77b14bc481fce60e4fc10a8a211d8b359906509d6830c653d9
      1c1a86865219db43f62c70ac6780644d2bd73c5c256527a3eaefaebaf1f2207324
      17e17dbf598636616f70f2088969ac796a853dc8a5f270a1c505797e83d1675e4f
      40b59c150ca06c49bb0967a2e0c7e74eff9e182d0f7bb6f54f68fe788b89d2191c
      87bbf7f3927978449c2174baa581dc64a9c58ed
      beta = 8ec4d150788513c85eea3490d1a1ee1b7a397602d3f9c8b467527f09fab
      5252e539f82e8002825608295ebbba19644dd

A.3.  RSA-FDH-VRF-SHA512

   Example 7, using the 2048-bit key above:

      alpha = (the empty string)
      EM = 7b08a7fff4e5d8fd4978ac5a0ddf48537a2bb3f952dcdc00affb25d747b40
      85c29c68dddaa87378db32396219ce784acebe70699286318f42794927f546de5d
      85bbefd80a02c3aa714fc17090baa0d0f7fb504e1af0b79ea02d41dc0bf576b8f2
      1472dd4c55f96bd64772d3ebd0347abe74b9fdf35b754d0405e42ceb0e290fdd91
      ef766a3e27ff59cd86572d15274f6fd49400ec4d126145f3cae200d67d5d108999
      61658ece7dcbf41f1cca63f8b50399955416a1f55e0af116fac2a9fd1f2dc0085e
      6ad6c1c4bc12d9308d9a030c3e2ea7f037d1c98beb23d43d67a97e5bf52382b8e8
      90c5967ab42f2010cac985d3a52fe726045746d4ffef901127646



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      pi = a280db108df5ad6ac1bed67efbc5c6fc6da0d301b9c0b41d26e379cd223c6
      13c59d52c987e4baaa6de4de2103284ddd56aa0b662dfe8faa8f6a503b83b7c81f
      481e23a08761d49a151ada1d9daa132138bbd6f80204c7fa87716b120df957224f
      92b32a3a0f96c3b209080c408618a92382ab5575f10a57c24ee0ffd01d6b822dc3
      6b27600bf36aafadf0a01e65aa6a0f2fc1a9dcd207d9bf5181a9ca69120e154108
      00a26efd3ce619349592eeff7b1851737bd033a83f88744ddd3d3e782efb6d2438
      ffda22ddcaa32c821c6730a05d5bdab88c354809d615884744ff10276496bee70b
      62feb6ed07a3948823e9ee2a453dbcd4450192c9de0128adfc7e147
      beta = 808ca1f8f66a48118aacb011394bd4e5f0011c89ca913943d467b81cc5c
      43086e588abdde061c3ee30f4c15b2a6b51ad0ada42c0737fd7b2206fb43d35c8e
      d22

   Example 8, using the 3072-bit key above:

      alpha = 74657374 (4 bytes; ASCII "test")
      EM = 803b6618f0ad47da2db309b1f57807a286500020c9e2b1427ebd9fff1104e
      3aa8a69210441cd58344bd810c4900825c84b1e5e36825f1e397df54c4419f8525
      d9a09a49e7fedc18b8d906cbd9ea831c55f2aaa0461e19ddd6ec9d14dafa1fcf49
      b77458a65427b7f060bc7425538e5d3af1813752cb452d0b098514110399734d1f
      55870c65ea3e799e6d9024a9e2fb95883e580578811a8c7d34b18f8fabc6c05fb9
      697335fcf2cb1b7576ee7a39dcff129e1f142106c45f30a8ae62370f576d1d1d8c
      6307fccfe25cb431f348dea81b6b7e6307bbefda2a0b23036653a612226392a573
      b7d62e28f9fecc7f4be0bf0a3049ce8ed276b34130faa943aeeedf962b42a3fc6c
      881bbf9a62039e9c0850f1393a2a02c6848d06c3520e086541d8af99ea3ef9f9da
      2e3b2bd3172682a47e5965899bc576b66e29a0b8dcf06871202a1a4e7f2ff19bdd
      9eac2241129a73d7d01303b80372ac62a0d5b6bfd1d7119e561ace229cd53d2c99
      63d6127b9ade16dce4b07d1cd89247ffc438811dc8b3c
      pi = 1aa828e0a751074fed2fa776fd29336a84987c064eeebcd3a8129fb688b47
      eb7109987d01db0c3624ba7cc75e2f1ad60f5e204a250a329048bc34df34d41bfe
      ea6651774d249ff9fc29aeabdf524400527aa1c4100b1af86b2dcc2e7aecc77f38
      6b80f29ccd807cb705b5057431832dafe56733a1e7bfcba1d052a26d1a8512f297
      b5abad5afd64fbcf21b57531a9b2c8217c0d9f1c875c196d998f61e8017f6b6ebe
      7317545ed390e18305bc96abb1514ec271963d02bed91ccf029d022189f84bac8c
      fa216da54e39919118348dfea6f4f6532b49da7820ee2a21f42b762e107722ad0a
      bf62271e0640d6b1c4d1a39b94ebd74b4283de2d6550cbdb1f29cac51671e9c8fc
      0ea0fdbb082a14a221e0531615f2bcfba0d70e99e4997cb00f81fcab2b95566322
      0234a5e90f29bd08e6fa50dd92770d9e514e0f9eb27aee634877bcea681ffd7da2
      b5be2f80c1dde1243b17ac726401cf961c5ce06640eb93352402c1ebc59c92188c
      511b375d63124846b46017fe36dc13fc2d34dbd80b312e0
      beta = 9202b6715b7921c5eb35572ed9ebb85848d3345efadd665049ce889be46
      322586d4177864c9179468473518c6b6ac2e9c85ae5ee5fcd3c0d8e6d4d8f18be6
      238

   Example 9, using the 4096-bit key above:

      alpha = 73616d706c65 (6 bytes; ASCII "sample")





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      EM = 289607894786ccf223b1e758232f3402aa50cd48bca3d2bd64b2ea9b4a69d
      c91756a42b2c1feaaa777763b9b7c91888c580433ac85f5fbb360f129ecee739f6
      9b560657687d38f7d43c84f6605005c38f56c91310eff27bae49b14d8a36542d69
      b70efca8637b845be0f029c085a7b6aad6ca0eb65fffdf8d55d9538d3b54044ebb
      c26e092b2f3ddaece7aa5b4b234ec848bfdc72a4ecdc10c66fb845cfc5ad39756e
      7f26007cea0ebf1e878636f4e39308fe7a317a9b7e90051536ac028bc1a2ec200a
      5dad0e3b74717bd9e7ea620919e315799e0fea7b0895fcbc0b95686b2495dc23e3
      cd56a16652b0df0dbd3ca8a6c96b13973a0c31a5541e211229da8a56e588a616c7
      21baab8e2d30313008c2374887f147598468b378bf8949ab1165b9348245d0a6a5
      f795918fce05f0d072f81f78c7224e7f1c4684877d714f231d5775c88759086121
      2eae2d174761158ac7d653b18f0d4b71362c0eb8a67bed1a48a4e7dc739b2b4469
      4514cae7f192d236afb1ab2409f24dfa94a2d705d0087860d844ba04564bb6733c
      a20089417d74eaed86d7e68ced681e9d88a9c3d7e6a33927592820cb9a38d45393
      32e509296489e54cd6b8495f100c36debe1f719578b15e8a99cc8febc3212e8147
      8aeca616a5230ae84e7079f52aefb2ec2a97157fb5d60e1ddcf03b134be2c93ffb
      a41d5d068750adc8df07e5a264640f7e586bd
      pi = 17d7635cac33b0b72ea1c0afb1f681d1a96c5073ed9f88ed8bb54eb428d7b
      2db4ee3355eee512ddc7af50694b37fd389f990278e22095b2582c78c4ed6070b0
      c7382b0308b6d546141a9b0d6ebb3af97abd93c16a5d34a2d805d8aa444fed2297
      d017571a693d221fda094d40500ab9b203d397a7543e72b26b06e561d49696e01d
      eebfed58b46611dd5a346e227d7519f8ffd1dc76a172c9f7f355c3e7e5ee7773ed
      ab00a22af5c39367f3779da68ce6da9f8a594f5f6149012501181653572fe5549a
      9c2bf36148b3bdc94feaedd600727fe5c11b7dcbfd73002ae08061cb4b84ba47f1
      bf8c5d46bc2acb7cb4964a6dca7eedc396e663a64121d93dade8b83cea09d76653
      cca1a8d20d6b7323a890651dc575025ba1be02d08c5946f50cde438339b06e8633
      198da0d467d2cac7d98ae62dd71353f6fb19aa9daac851d0ce237b21db93b91e51
      8d5c1ac36cdf874975deb7aab3942acc3980f221f33ad1254eb8ac3138e087d045
      c4746e0b7eedcaf2a1a173559783eba8691555c1b0e468f8efe6501679b760038e
      d6fc9ce6aa5ae24b3f1178713793c8e5ee96035a2f0ee02e2d10ac098613358d3c
      ff10f4dff3437f2a48252c5d6805288fbd7ee05356f80db12aaeabf6638677abf5
      b8eb2376fb76861cf1b817d5a0b878dae6beac44f078f37d982d941a77582a7778
      4fabd632e28d664d9f705f31e24d1ca623dfac7
      beta = 6026f6defaf534cc79ce7c1b0370fb53e4825d2d44f549f696e06d693c3
      9e852e21a5e3b6ff093618dd277b40678957e1b90e8e6ca742efed30dc309b3b24
      2b8

Appendix B.  Test Vectors for the ECVRF ciphersuites

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

B.1.  ECVRF-P256-SHA256-TAI

   The example secret keys and messages in Examples 10 and 11 are taken
   from Appendix A.2.5 of [RFC6979].

   Example 10:




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      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 73616d706c65 (6 bytes; 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 11:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 74657374 (4 bytes; 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 12 is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   Example 12:





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      SK = x =
      2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
      PK =
      03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
      alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
      417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
      (62 bytes; 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

B.2.  ECVRF-P256-SHA256-SSWU

   The example secret keys and messages in Examples 13 and 14 are taken
   from Appendix A.2.5 of [RFC6979].

   Example 13:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 73616d706c65 (6 bytes; 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



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      V = k*H =
      03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3
      pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394
      c7888782fffde7b842c38c20c08de6ec6c2e7027a97000f2c9fa4425d5c03e639f
      b48fde58114d755985498d7eb234cf4aed9
      beta =
      21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b

   Example 14:

      SK = x =
      c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
      PK =
      0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
      alpha = 74657374 (4 bytes; 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 15 is taken from Appendix L.4.2 of
   [ANSI.X9-62-2005].

   Example 15:

      SK = x =
      2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
      PK =
      03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d






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      alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
      417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
      (62 bytes; ASCII "Example using ECDSA key from Appendix L.4.2 of
      ANSI.X9-62-2005")
      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

B.3.  ECVRF-EDWARDS25519-SHA512-TAI

   The example secret keys and messages in Examples 16, 17, and 18 are
   taken from Section 7.1 of [RFC8032].

   Example 16:

      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




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      U = k*B =
      aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071
      V = k*H =
      5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f
      pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f7
      23f26f8a57ccaed74ee1b190bed1f479d9727d2d0f9b005a6e456a35d4fb0daab1
      268a1b0db10836d9826a528ca76567805
      beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757
      876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd
      8ae

   Example 17:

      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 18:

      SK =
      c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
      PK =
      fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
      alpha = af82 (2 bytes)
      x =
      909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c



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      try_and_increment succeeded on ctr = 0
      H =
      bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a
      k_string = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59f
      c31a0febc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6b
      b6d6c3a
      k =
      5ffdbc72135d936014e8ab708585fda379405542b07e3bd2c0bd48437fbac60a
      U = k*B =
      2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5
      V = k*H =
      88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506
      pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccb
      f8096bb474e53895c362d8628ee9f9ea3c0e52c7a5c691b6c18c9979866568add7
      a2d41b00b05081ed0f58ee5e31b3a970e
      beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c
      61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df
      19f

B.4.  ECVRF-EDWARDS25519-SHA512-ELL2

   The example secret keys and messages in Examples 19, 20, and 21 are
   taken from Section 7.1 of [RFC8032].

   Example 19:

      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




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      U = k*B =
      762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9
      V = k*H =
      4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af
      pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab
      46f14adf9a3cd8b8412d9038531e865c341cafa73589b023d14311c331a9ad15ff
      2fb37831e00f0acaa6d73bc9997b06501
      beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80
      c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6
      c54

   Example 20:

      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 21:





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










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   Leonid Reyzin
   Boston University and Algorand
   111 Cummington Mall
   Boston, MA 02215
   United States of America

   Email: reyzin@bu.edu


   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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