Movatterモバイル変換

[0]ホーム
URL:

RFC 9381VRFsAugust 2023
Goldberg, et al.Informational[Page]
Stream:
Internet Research Task Force (IRTF)
RFC:
9381
Category:
Informational
Published:
ISSN:
2070-1721
Authors:
S. Goldberg
Boston University
L. Reyzin
Boston University and Algorand
D. Papadopoulos
Hong Kong University of Science and Technology
J. Včelák
NS1

RFC 9381

Verifiable Random Functions (VRFs)

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 document is not an Internet Standards Track specification; it is published for informational purposes.

This document is a product of the Internet Research Task Force (IRTF). The IRTF publishes the results of Internet-related research and development activities. These results might not be suitable for deployment. This RFC represents the consensus of the Crypto Forum Research Group of the Internet Research Task Force (IRTF). Documents approved for publication by the IRSG are not candidates for any level of Internet Standard; see Section 2 of RFC 7841.

Information about the current status of this document, any errata, and how to provide feedback on it may be obtained athttps://www.rfc-editor.org/info/rfc9381.

Copyright Notice

Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved.

This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.

▲

Table of Contents

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 in this document were based, in part, on synergy with existing RFCs and commonly available implementations of individual components that are used within the VRFs.

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 the case of elliptic curves). Differences in the security properties provided by the different options are discussed in Sections 3 and7.

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", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14[RFC2119][RFC8174] when, and only when, they appear in all capitals, as shown here.

1.2.Terminology

The following terminology is used throughout this document:

SK:
The secret key for the VRF. (Note: The secret key is also sometimes called a "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:
Holds the VRF secret key SK and public key PK.
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:

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 the 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

  • 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 Collision Resistance

Like cryptographic hash functions, VRFs are collision resistant. Collision 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. These properties are called "trusted uniqueness" and "trusted collision resistance"; they 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 and then given to the adversary. In other words, they are guaranteed to hold even if the adversary has knowledge of SK and PK but are not guaranteed to hold if the adversary has the ability to choose SK and PK.

As further discussed inSection 7.1.1, some of the 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 resistanceMUST 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, beta is indistinguishable from a random value.

More precisely, suppose that 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).

The "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 inSection 7.3, the 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). In particular, if the key is generated maliciously, even parties other than the Prover may know SK, and thus pseudorandomness cannot be guaranteed.

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 to see 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 inSection 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 inSection 7.1.3, only some of the 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 inSection 3, as further discussed inSection 7. Its security follows from the standard RSA assumption in the random oracle model. Formal security proofs are provided in[PWHVNRG17].

The VRF computes the proof pi as a deterministic RSA signature on input alpha using the RSA Full Domain Hashing algorithm[RFC8017] parameterized with the selected hash algorithm. RSA signature verification is used to verify the correctness of the proof. The VRF hash output beta is simply obtained by hashing the proof pi with the selected hash algorithm.

The key pair for the RSA-FDH-VRFMUST satisfy the conditions specified inSection 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 inSection 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 non-negative integer to an octet string as defined inSection 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 non-negative integer as defined inSection 4.2 of [RFC8017] (given a big-endian encoding of an integer, produces the integer)
RSASP1:
RSA signature primitive as defined inSection 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 inSection 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 inAppendix B.2.1 of [RFC8017] (given an input, produces a random-oracle-like output of desired length)
||:
octet string concatenation

4.1.RSA-FDH-VRF Proving

RSAFDHVRF_prove(K, alpha_string[, MGF_salt])

Input:


K:
RSA private key
alpha_string:
VRF hash input, an octet string
Optional input:


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


pi_string:
proof, an octet string of length k

Steps:

  1. mgf_domain_separator = 0x01
  2. EM = MGF1(suite_string || mgf_domain_separator || MGF_salt || alpha_string, k - 1)
  3. m = OS2IP(EM)
  4. s = RSASP1(K, m)
  5. pi_string = I2OSP(s, k)
  6. Output pi_string

4.2.RSA-FDH-VRF Proof to Hash

RSAFDHVRF_proof_to_hash(pi_string)

Input:


pi_string:
proof, an octet string of length k
Output:


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

Important note:

RSAFDHVRF_proof_to_hash should be run only on a pi_string value that is known to have been produced by RSAFDHVRF_prove, or from within RSAFDHVRF_verify as specified inSection 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:


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

"INVALID"

Steps:

  1. s = OS2IP(pi_string)
  2. m = RSAVP1((n, e), s); if RSAVP1 returns "signature representative out of range", output "INVALID" and stop
  3. mgf_domain_separator = 0x01
  4. EM' = MGF1(suite_string || mgf_domain_separator || MGF_salt || alpha_string, k - 1)
  5. m' = OS2IP(EM')
  6. If m and m' are equal, output ("VALID", RSAFDHVRF_proof_to_hash(pi_string)); else output "INVALID"

4.4.RSA-FDH-VRF Ciphersuites

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

  • suite_string = 0x01.
  • The hash function Hash is SHA-256 as specified in[RFC6234], with hLen = 32.
  • MGF_salt = I2OSP(k, 4) || I2OSP(n, k).

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

  • suite_string = 0x02.
  • The hash function Hash is SHA-384 as specified in[RFC6234], with hLen = 48.
  • MGF_salt = I2OSP(k, 4) || I2OSP(n, k).

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

  • 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 inSection 3. If the validate_key parameter given to ECVRF_verify is TRUE, then the ECVRF additionally satisfies "full collision resistance" and "unpredictability under malicious key generation". SeeSection 7 for further discussion. Formal security proofs are provided in[PWHVNRG17].

Notation used:


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

x^y:
x raised to the power y
x*y:
x multiplied by y
s || t:
concatenation of octet strings s and t
0xMN (where M and N are hexadecimal digits):
a single octet with value M*16+N; equivalently, int_to_string(M*16+N, 1), where int_to_string is as defined below
Fixed options (specified inSection 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., the smallest integer such that 2^(8qLen) > q
cLen:
length, in octets, of a challenge value used by the VRF (note that in the typical case, cLen is qLen/2 or close to it)
cofactor:
number of points on E divided by q
B:
generator of group G
Hash:
cryptographic hash function
hLen:
output length, in octets, of Hash (hLen must be at least cLen; in the typical case, it is at least qLen)
ECVRF_encode_to_curve:
a function that hashes strings to points on E
ECVRF_nonce_generation:
a function that derives a pseudorandom nonce from SK and the input as part of ECVRF proving
suite_string:
an octet string specifying the ECVRF ciphersuite, which determines the above options as well as type conversions and parameter generation
Type conversions (specified inSection 5.5):


int_to_string(a, len):
conversion of non-negative integer a to octet string of length len
string_to_int(a_string):
conversion of an octet string a_string to a non-negative integer
point_to_string:
conversion of a point on E to a ptLen-octet string
string_to_point:
conversion of a 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 inSection 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, a point on E
PK_string = point_to_string(Y):
VRF public key represented as an octet string
encode_to_curve_salt:
a public value used as a hash function salt

5.1.ECVRF Proving

ECVRF_prove(SK, alpha_string[, encode_to_curve_salt])

Input:


SK:
VRF 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, an 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 perSection 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) (seeSection 5.4.1)
  3. h_string = point_to_string(H)
  4. Gamma = x*H
  5. k = ECVRF_nonce_generation(SK, h_string) (seeSection 5.4.2)
  6. c = ECVRF_challenge_generation(Y, H, Gamma, k*B, k*H) (seeSection 5.4.3)
  7. s = (k + c*x) mod q
  8. pi_string = point_to_string(Gamma) || int_to_string(c, cLen) || int_to_string(s, qLen)
  9. Output pi_string

5.2.ECVRF Proof to Hash

ECVRF_proof_to_hash(pi_string)

Input:


pi_string:
VRF proof, an octet string of length ptLen+cLen+qLen
Output:


"INVALID", or
beta_string:
VRF hash output, an octet string of length hLen

Important note:

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

Steps:

  1. D = ECVRF_decode_proof(pi_string) (seeSection 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, an octet string
pi_string:
VRF proof, an 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 implementationMAY 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, itMUST specify which option it supports
Output:


("VALID", beta_string), where beta_string is the VRF hash output, an 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) (seeSection 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) (seeSection 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) (seeSection 5.4.3)
  11. If c and c' are equal, output ("VALID", ECVRF_proof_to_hash(pi_string)); else output "INVALID"

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

5.4.ECVRF Auxiliary Functions

5.4.1.ECVRF Encode to Curve

The ECVRF_encode_to_curve algorithm takes a public salt (seeSection 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. SeeSection 7.7 for further discussion.

This section specifies a number of such algorithms; these algorithms are not compatible with each other and are intended for use with the various ciphersuites specified inSection 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 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, hLenMUST be at least fLen.

The running time of this algorithm depends on alpha_string. For the ciphersuites specified inSection 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 algorithm's running time depends on alpha_string, this algorithmSHOULD 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 inSection 5.5):


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

Steps:

  1. ctr = 0
  2. encode_to_curve_domain_separator_front = 0x01
  3. encode_to_curve_domain_separator_back = 0x00
  4. H = "INVALID"

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

    1. ctr_string = int_to_string(ctr, 1)
    2. 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)
    3. H = interpret_hash_value_as_a_point(hash_string)
    4. If H is not "INVALID" and cofactor > 1, set H = cofactor * H
    5. ctr = ctr + 1
  6. Output H

Note that even though the loop is infinite as written and int_to_string(ctr, 1) may fail when ctr reaches 256, each of the options for the interpret_hash_value_as_a_point function specified inSection 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[RFC9380]. The specific choice of the hash-to-curve option (called the Suite ID in[RFC9380]) is given by the h2c_suite_ID_string parameter.

ECVRF_encode_to_curve_h2c_suite(encode_to_curve_salt, alpha_string)

Fixed option (specified inSection 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 (as specified inSection 8 of [RFC9380]) whose ID is h2c_suite_ID_string. The domain separation tag DST, a parameter in 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; these algorithms 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 inSection 5.5.

5.4.2.1.ECVRF Nonce Generation from RFC 6979

ECVRF_nonce_generation_RFC6979(SK, h_string)

Input:


SK:
an ECVRF secret key
h_string:
an octet string
Output:


k:
an integer nonce between 1 and q-1

The ECVRF_nonce_generation function is implemented according to the process specified inSection 3.2 of [RFC6979], 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 (note that hlen is not to be confused with hLen, which is used in this document to represent the length of the output of Hash in octets).
  • 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, which is used in this document to represent the length of q in octets).
  • All the other values and primitives are as defined in[RFC6979].
5.4.2.2.ECVRF Nonce Generation from RFC 8032

The following is derived from Steps 2 and 3 inSection 5.1.6 of [RFC8032]. To use this algorithm, hLenMUST be at least 64.

ECVRF_nonce_generation_RFC8032(SK, h_string)

Input:


SK:
an ECVRF secret key
h_string:
an octet string
Output:


k:
an integer nonce between 0 and q-1

Steps:

  1. hashed_sk_string = Hash(SK)
  2. truncated_hashed_sk_string = hashed_sk_string[32]...hashed_sk_string[63]
  3. k_string = Hash(truncated_hashed_sk_string || h_string)
  4. k = string_to_int(k_string) mod q

5.4.3.ECVRF Challenge Generation

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

Input:


P1, P2, P3, P4, P5:
EC points
Output:


c:
challenge value, an 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, an octet string (ptLen+cLen+qLen octets)
Output:


"INVALID", or
Gamma:
a point on E
c:
an integer between 0 and 2^(8*cLen)-1
s:
an 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 provided as input to this procedureMUST be a valid point on E.

Steps:

  1. Let Y' = cofactor*Y
  2. If Y' is the identity element of the elliptic curve group, output "INVALID" and stop
  3. Output "VALID"

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

Any algorithm with identical input-output behaviorMAY 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 algorithmMAY 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 = 2707385501144840649318225287225658788936804267575313519463743609750303402022
  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 eight 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 be obtained by converting the points specified in[X25519] to Edwards coordinates. Thus, bad_pk[0] (of order 4), bad_pk[2] (of order 8), and bad_pk[3] (of order 8) each match two bad points, depending on the sign of the x-coordinate. This sign is 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 the x-coordinate is 0 for these two points. Note that the first five list elements cover the eight bad points. However, to cover the case when the y-coordinate of the public key Y has 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-coordinates would exceed 2^255, and the algorithm ensures 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 the finite field and curve parameters as specified in Section 3.2.1.3 of[SP-800-186] andSection 2.6 of [RFC5114]. 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 inSection 5.4.2.1.
  • The int_to_string function is the I2OSP function specified inSection 4.1 of [RFC8017]. (This is big-endian representation.)
  • The string_to_int function is the OS2IP function specified inSection 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 that ptLen = fLen + 1 = 33. (Note that certain software implementations do not introduce a separate elliptic curve point type and instead directly treat the EC point as an octet string per the 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 a point on E according to the encoding specified in Section 2.3.4 of[SECG1]. This functionMUST 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 inSection 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 inSection 5.4.1.2, with h2c_suite_ID_string = P256_XMD:SHA-256_SSWU_NU_ (the suite is defined inSection 8.2 of [RFC9380]).

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

  • suite_string = 0x03.
  • The EC group G is the edwards25519 elliptic curve, with the finite field and curve parameters as defined in Table 1 inSection 5.1 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 inSection 5.1.5 of [RFC8032].
  • encode_to_curve_salt = PK_string.
  • The ECVRF_nonce_generation function is as specified inSection 5.4.2.2.
  • The int_to_string function is implemented as specified in the first paragraph ofSection 5.1.2 of [RFC8032]. (This is little-endian representation.)
  • The string_to_int function interprets the string as an integer in little-endian representation.
  • The point_to_string function converts a point on E to an octet string according to the encoding specified inSection 5.1.2 of [RFC8032]. This implies that 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 the 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 a point on E according to the encoding specified inSection 5.1.3 of [RFC8032]. This functionMUST 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 inSection 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 that

  • suite_string = 0x04.
  • The ECVRF_encode_to_curve function is as specified inSection 5.4.1.2, with h2c_suite_ID_string = edwards25519_XMD:SHA-512_ELL2_NU_ (the suite is defined inSection 8.5 of [RFC9380]).

6.IANA Considerations

This document has no IANA actions.

7.Security Considerations

7.1.Key Generation

Implementations of the VRFs defined in this documentMUST ensure that they generate VRF keys correctly and use 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

SeeSection 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 the 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 without complying with[RFC8017]); thus, the RSA-FDH-VRF as 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) to ensure that its public key has the right form. These enhancements are left for future specifications.

For the ECVRF, the VerifierMUST 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 "full uniqueness", ensuring uniqueness even under malicious key generation. The ECVRF also satisfies "trusted collision resistance". It additionally satisfies "full collision resistance" if the validate_key parameter given to 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 inSection 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.

7.1.3.Unpredictability under Malicious Key Generation

Unpredictability under malicious key generation (defined inSection 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 without complying 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) to ensure that its public key has the right form. These enhancements are left for future specifications.

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

7.2.Security Levels

As shown in[PWHVNRG17], the 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 inSection 7.1.1):

approximately 8*min(qLen/2, hLen/2).
For selective pseudorandomness:
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).

SeeSection 3 for the definitions of these security properties andSection 7.3 for the discussion of full pseudorandomness.

7.3.Selective vs. Full Pseudorandomness

[PWHVNRG17] presents cryptographic reductions to an underlying hard problem (namely, the RSA problem for the 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 (seeSection 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 inSection 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 such 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 lossy security reductions. For the RSA-FDH-VRF, this means increasing the RSA key length. For the ECVRF, this means increasing the cryptographic strength of the EC group G by specifying a new ciphersuite.

7.4.Proper Pseudorandom Nonce for the 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 ofSection 5.5 are designed with this requirement in mind.

7.5.Side-Channel Attacks

Side-channel attacks on cryptographic primitives are an important issue. Implementers should take care to avoid side-channel attacks that leak information about the VRF secret key SK (and the nonce k used in the ECVRF), which is used in VRF_prove. In most applications, the 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 also be 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 inSection 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 that 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 that is independent of alpha, following recommendations in[RFC9380].

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.

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 Edwards-curve Digital Signature Algorithm (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 inputs given to the hash function for a given SK and alpha are overwhelmingly unlikely to be equal to 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 inputs 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 in nonce_generation are unlikely to equal inputs used in encode_to_curve, proof_to_hash, and challenge_generation. This follows, since nonce_generation inputs a secret to the hash function that is not used by honest parties as input to any other hash function and is not available to the adversary.
  • The second octets of the inputs to the hash function used in proof_to_hash, challenge_generation, and encode_to_curve_try_and_increment are all different.
  • The last octet of the inputs to the hash function used in proof_to_hash, challenge_generation, and encode_to_curve_try_and_increment is always zero and is 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 per[RFC9380].

7.9.Hash Function Salting

If 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 the 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 (seeSection 7.10)MAY use a different salt. For example, if a group of public keys shares the same salt, then the hash of the VRF input alpha will be the same for the entire group of public keys; this can be helpful for some protocols that use the VRF.

7.10.Futureproofing

If future designs need to specify variants (e.g., additional ciphersuites) of the RSA-FDH-VRF or the ECVRF as defined 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 discussed 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 discussed 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 to the 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.References

8.1.Normative References

[RFC2119]
Bradner, S.,"Key words for use in RFCs to Indicate Requirement Levels",BCP 14,RFC 2119,DOI 10.17487/RFC2119,,<https://www.rfc-editor.org/info/rfc2119>.
[RFC5114]
Lepinski, M. andS. Kent,"Additional Diffie-Hellman Groups for Use with IETF Standards",RFC 5114,DOI 10.17487/RFC5114,,<https://www.rfc-editor.org/info/rfc5114>.
[RFC6234]
Eastlake 3rd, D. andT. Hansen,"US Secure Hash Algorithms (SHA and SHA-based HMAC and HKDF)",RFC 6234,DOI 10.17487/RFC6234,,<https://www.rfc-editor.org/info/rfc6234>.
[RFC6979]
Pornin, T.,"Deterministic Usage of the Digital Signature Algorithm (DSA) and Elliptic Curve Digital Signature Algorithm (ECDSA)",RFC 6979,DOI 10.17487/RFC6979,,<https://www.rfc-editor.org/info/rfc6979>.
[RFC8017]
Moriarty, K., Ed.,Kaliski, B.,Jonsson, J., andA. Rusch,"PKCS #1: RSA Cryptography Specifications Version 2.2",RFC 8017,DOI 10.17487/RFC8017,,<https://www.rfc-editor.org/info/rfc8017>.
[RFC8032]
Josefsson, S. andI. Liusvaara,"Edwards-Curve Digital Signature Algorithm (EdDSA)",RFC 8032,DOI 10.17487/RFC8032,,<https://www.rfc-editor.org/info/rfc8032>.
[RFC8174]
Leiba, B.,"Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words",BCP 14,RFC 8174,DOI 10.17487/RFC8174,,<https://www.rfc-editor.org/info/rfc8174>.
[RFC9380]
Faz-Hernandez, A.,Scott, S.,Sullivan, N.,Wahby, R. S., andC. A. Wood,"Hashing to Elliptic Curves",RFC 9380,DOI 10.17487/RFC9380,,<https://www.rfc-editor.org/info/rfc9380>.
[SECG1]
Standards for Efficient Cryptography Group (SECG),"SEC 1: Elliptic Curve Cryptography",Version 2.0,,<https://www.secg.org/sec1-v2.pdf>.
[SP-800-186]
National Institute for Standards and Technology (NIST),"Recommendations for Discrete Logarithm-based Cryptography: Elliptic Curve Domain Parameters",NIST SP 800-186,DOI 10.6028/NIST.SP.800-186,,<https://nvlpubs.nist.gov/nistpubs/SpecialPublications/NIST.SP.800-186.pdf>.

8.2.Informative References

[ANSI.X9-62-2005]
American National Standards Institute (ANSI),"Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)",ANSI X9.62,,<https://standards.globalspec.com/std/1955141/ANSI%20X9.62>.
[BreHen19]
Breitner, J. andN. Heninger,"Biased Nonce Sense: Lattice Attacks against Weak ECDSA Signatures in Cryptocurrencies",Cryptology ePrint Archive, Paper 2019/023,,<https://eprint.iacr.org/2019/023>.
[DGKR18]
David, B.,Gaži, P.,Kiayias, A., andA. Russell,"Ouroboros Praos: An adaptively-secure, semi-synchronous proof-of-stake blockchain",Cryptology ePrint Archive, Paper 2017/573,,<https://eprint.iacr.org/2017/573>.
[GHMVZ17]
Gilad, Y.,Hemo, R.,Micali, S.,Vlachos, G., andN. Zeldovich,"Algorand: Scaling Byzantine Agreements for Cryptocurrencies",Cryptology ePrint Archive, Paper 2017/454,,<https://eprint.iacr.org/2017/454>.
[GNPRVZ15]
Goldberg, S.,Naor, M.,Papadopoulos, D.,Reyzin, L.,Vasant, S., andA. Ziv,"NSEC5: Provably Preventing DNSSEC Zone Enumeration",Cryptology ePrint Archive, Paper 2014/582,,<https://eprint.iacr.org/2014/582>.
[MRV99]
Micali, S.,Rabin, M., andS. Vadhan,"Verifiable Random Functions",FOCS '99: Proceedings of the 40th Annual Symposium on Foundations of Computer Science, pp. 120-130,DOI 10.1109/SFFCS.1999.814584,,<https://dash.harvard.edu/handle/1/5028196>.
[PWHVNRG17]
Papadopoulos, D.,Wessels, D.,Huque, S.,Naor, M.,Včelák, J.,Reyzin, L., andS. Goldberg,"Making NSEC5 Practical for DNSSEC",Cryptology ePrint Archive, Paper 2017/099,,<https://eprint.iacr.org/2017/099>.
[X25519]
Bernstein, D.J.,"How do I validate Curve25519 public keys?",,<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 provided 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 = efb52a568fa3038fffb853e2183791c6bc81ceee86d20e8f9b6401dc79a8f1f6248d3a25fdb3f99245fce41667da038f59745b87cc1aed8b4a9c1d74e7d5c16cf7343f2b12f1b5055337369bf018fa07adc0d16f2164a516e80d2b4734f0c6563d6ee6d4a9e1a54e300cfe9ee679afc3d14a152dfb49b6cfb208bbf921f764afq = ecbca5ee88bbc635d8263aaba84f6502fdb2b4998a40f7c149133d840b6b1bd9a972fe2a981c770272b78fda213f76a062dd865dd116d4c8980975ee9347fe0f500567e51d78dbee4a34e626051cf018d7feb72f19189525d4f70b6467d0cef514633ab08a9e7a9ec632064b7b5e3e82128fe563757a614092fc5cf624d10e1bn = 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 = 010001d = 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

3072-bit key:

p = ee5adea28491084e6635bd73fd95649915a11da410d3f361c8eccc90a4b83425146da7b9e9d3994fd37d5fad7fb759ae451eb99b1102d4671ead23a2925133d19df49cf9d7e9dcb69fd7555ca095338d0d2a84abb6825050eaf5fffaeff17ccb0833c6079081dfcbd98ced36a593557d29d64b0e0253ce2ee4e07fe2a06269dfe5ca230fad221a593a69d9534b2521c1b41d116cafdee02106228ff41433605453e237777626953e79b46a84f50069e25b4f50496a928708abce30559eb183cfq = fb585bbc12f5695951f70a25e27682dc568acf56115ad749709b2a6e915cdd66dfa06db09b390c00b7c7ebeea00845f73c999d8ea9352b1128bdf10113c7500b76a03f6b38d0920b5589961549be3d841ccc306f3edd600a53b4b9d4fa1249af87af58dfb3ed694289477e853f7d062f58911f7bdb98033b001ee90f11b78f031cffac2b5a07e11b01a2a6c1cda059a728f8253a5ff87267623253fc022d3993b27e2f344b99eb6072ff7c7ee160724f8fbca562be49247ffae42b55ea79dad5n = 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 = 010001d = 6e68e957dbfd7c1862dc1b87780b9dcf0ff9016770bc9c09873b66194941d76218bf2013c1e4df9326dd4402f5df110656d2ec8ea87a28b2a1cb74e590872aeb765fe772ea21c57d6ab4ba0fad019189273f05c061719afd14af02277dd28d67c5ef50b75b521ca51819b9bcb44cb7c82be66776a45f490050dc0171e77374f1ed00d06f8beb09b711a9682107d8840d4a23edf6ac25441fdbf2b584dfa6a67cee21eb51c484f09416e11914e774713f1a17600fb9e4e99fbbd83fdcba4b09145dd9809449a1713777161c912d5d595362314b0ea9d1199e97780e8b3293a39af4019fcc746aaf78dbb7db06852c3358a9ed02ab1d15831a148b27b932c117445a4a6f5114edfa3ccc9a9862df714b78a5362aab5e30501b4a729af73e3cdcabe19aac4928b668969780ad33d9df206d904b978a055f4abbc64987744526856e16ef55962453e3ed7a8055b0d79d051c50c94584ec7501dbd4856d7a21e43f25d8749e683cca2f53f575af1d80f39d8e6932ffdf201d179cbf98314c4048c6c1

4096-bit key:

p = 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 = 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 = 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 = 010001d = 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

A.1.RSA-FDH-VRF-SHA256

Example 1, using the 2048-bit key above:

alpha =  (the empty string)EM = 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 = 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 =79f0615d4677fb72571889453644013f1a31b08d222e3cee349d64ce1c41045a

Example 2, using the 3072-bit key above:

alpha = 74657374 (4 bytes; ASCII "test")EM = 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 = 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 =bfe966f3fabde6f38a2792ad59bc836bbca39de6eff64f15a42886deff6dfcc5

Example 3, using the 4096-bit key above:

alpha = 73616d706c65 (6 bytes; ASCII "sample")EM = 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 = 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 =b663c5f90da1c12cd5d0e6d049679459e6f79f9fe16bc8b8e7e4d64d66500bd9

A.2.RSA-FDH-VRF-SHA384

Example 4, using the 2048-bit key above:

alpha =  (the empty string)EM = 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 = 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 = dc37e83f8de0e990abada5096a05ca74754cfe7fe8e46b831e24100919415415dcd5a305f5fb8195713cebc78649c8d1

Example 5, using the 3072-bit key above:

alpha = 74657374 (4 bytes; ASCII "test")EM = 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 = 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 = 5bdf742667ad10080f4ca573ec66f751e82e4077d0db1b281df421af68d39412e70362dc5101b4b46e1e453eea7e0989

Example 6, using the 4096-bit key above:

alpha = 73616d706c65 (6 bytes; ASCII "sample")EM = 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 = 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 = 8ec4d150788513c85eea3490d1a1ee1b7a397602d3f9c8b467527f09fab5252e539f82e8002825608295ebbba19644dd

A.3.RSA-FDH-VRF-SHA512

Example 7, using the 2048-bit key above:

alpha =  (the empty string)EM = 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 = 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 = 808ca1f8f66a48118aacb011394bd4e5f0011c89ca913943d467b81cc5c43086e588abdde061c3ee30f4c15b2a6b51ad0ada42c0737fd7b2206fb43d35c8ed22

Example 8, using the 3072-bit key above:

alpha = 74657374 (4 bytes; ASCII "test")EM = 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 = 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 = 9202b6715b7921c5eb35572ed9ebb85848d3345efadd665049ce889be46322586d4177864c9179468473518c6b6ac2e9c85ae5ee5fcd3c0d8e6d4d8f18be6238

Example 9, using the 4096-bit key above:

alpha = 73616d706c65 (6 bytes; ASCII "sample")EM = 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 = 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 = 6026f6defaf534cc79ce7c1b0370fb53e4825d2d44f549f696e06d693c39e852e21a5e3b6ff093618dd277b40678957e1b90e8e6ca742efed30dc309b3b242b8

Appendix B.Test Vectors for the ECVRF Ciphersuites

The test vectors in this section were generated using code provided 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 fromAppendix A.2.5 of [RFC6979].

Example 10:

SK = x =c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721PK =0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6alpha = 73616d706c65 (6 bytes; ASCII "sample")try_and_increment succeeded on ctr = 1H =0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4k =0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77U = k*B =02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672V = k*H =02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5071b4a53f0a46f018bc2c56e58d383f2305e0975972c26feea0eb122fe7893c15af376b33edf7de17c6ea056d4d82de6bc02fbeta =a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e

Example 11:

SK = x =c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721PK =0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6alpha = 74657374 (4 bytes; ASCII "test")try_and_increment succeeded on ctr = 3H =02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974k =5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961U = k*B =022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48eV = k*H =033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080ddb56c19e067b15a8a8174905b13617804534214f935b94c2287f797e393eb0816969d864f37625b443f30f1a5a33f2b3c854beta =a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d

The example secret key in Example 12 is taken from Appendix L.4.2 of[ANSI.X9-62-2005].

Example 12:

SK = x =2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8PK =03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274dalpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035(62 bytes; ASCII "Example using ECDSA key from Appendix L.4.2 ofANSI.X9-62-2005")try_and_increment succeeded on ctr = 1H =0258055c26c4b01d01c00fb57567955f7d39cd6f6e85fd37c58f696cc6b7aa761dk =5689e2e08e1110b4dda293ac21667eac6db5de4a46a519c73d533f69be2f4da3U = k*B =020f465cd0ec74d2e23af0abde4c07e866ae4e5138bded5dd1196b8843f380db84V = k*H =036cb6f811428fc4904370b86c488f60c280fa5b496d2f34ff8772f60ed24b2d1dpi = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b8c667d091c00b0f5c3619d10ecea44363b5a599cadc5b2957e223fec62e81f7b4825fc799a771a3d7334b9186bdbee87316b1beta =90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19

B.2.ECVRF-P256-SHA256-SSWU

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

Example 13:

SK = x =c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721PK =0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6alpha = 73616d706c65 (6 bytes; ASCII "sample")In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c2a4b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7eIn SSWU: u =df565615a2372e8b31b8771f7503bafc144e48b05688b97958cc27ce29a8d810In SSWU: x1 =e7e39eb8a4c982426fcff629e55a3e13516cfeb62c02c369b1e750316f5e94ebIn SSWU: gx1 is a nonsquareH =02b31973e872d4a097e2cfae9f37af9f9d73428fde74ac537dda93b5f18dbc5842k =e92820035a0a8afe132826c6312662b6ea733fc1a0d33737945016de54d02dd8U = k*B =031490f49d0355ffcdf66e40df788bee93861917ee713acff79be40d20cc91a30aV = k*H =03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394c7888782fffde7b842c38c20c08de6ec6c2e7027a97000f2c9fa4425d5c03e639fb48fde58114d755985498d7eb234cf4aed9beta =21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b

Example 14:

SK = x =c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721PK =0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6alpha = 74657374 (4 bytes; ASCII "test")In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109afb243b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3In SSWU: u =d8b0107f7e7aa36390240d834852f8703a6dc407019d6196bda5861b8fc00181In SSWU: x1 =ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7In SSWU: gx1 is a squareH =03ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7k =febc3451ea7639fde2cf41ffd03f463124ecb3b5a79913db1ed069147c8a7deaU = k*B =031200f9900e96f811d1247d353573f47e0d9da601fc992566234fc1a5b37749aeV = k*H =02d3715dcfee136c7ae50e95ffca76f4ca6c29ddfb92a39c31a0d48e75c6605cd1pi = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa8db2743aad348a3006dc1caad7da28687320740bf7dd78fe13c298867321ce3b36b79ec3093b7083ac5e4daf3465f9f43c627beta =8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd

The example secret key in Example 15 is taken from Appendix L.4.2 of[ANSI.X9-62-2005].

Example 15:

SK = x =2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8PK =03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274dalpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035(62 bytes; ASCII "Example using ECDSA key from Appendix L.4.2 ofANSI.X9-62-2005")In SSWU: uniform_bytes = 9b81d55a242d3e8438d3bcfb1bee985a87fd144802c9268cf9adeee160e6e9ff765569797a0f701cb4316018de2e7dd4In SSWU: u =e43c98c2ae06d13839fedb0303e5ee815896beda39be83fb11325b97976efdceIn SSWU: x1 =be9e195a50f175d3563aed8dc2d9f513a5536c1e9aee1757d86c08d32d582a86In SSWU: gx1 is a nonsquareH =022dd5150e5a2a24c66feab2f68532be1486e28e07f1b9a055cf38ccc16f6595ffk =8e29221f33564f3f66f858ba2b0c14766e1057adbd422c3e7d0d99d5e142b613U = k*B =03a8823ff9fd16bf879261c740b9c7792b77fee0830f21314117e441784667958dV = k*H =02d48fbb45921c755b73b25be2f23379e3ce69294f6cee9279815f57f4b422659dpi = 039f8d9cdc162c89be2871cbcb1435144739431db7fab437ab7bc4e2651a9e99d5488405a11a6c7fc8defddd9e1573a563b7333aab4effe73ae9803274174c659269fd39b53e133dcd9e0d24f01288de9abeta =4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a

B.3.ECVRF-EDWARDS25519-SHA512-TAI

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

Example 16:

SK =9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60PK =d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511aalpha =  (the empty string)x =307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94ftry_and_increment succeeded on ctr = 0H =91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318k_string = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106f345b8594b6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2a6afae4k =8a49edbd1492a8ee09766befe50a7d563051bf3406cbffc20a88def030730f0fU = k*B =aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071V = k*H =5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954fpi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f723f26f8a57ccaed74ee1b190bed1f479d9727d2d0f9b005a6e456a35d4fb0daab1268a1b0db10836d9826a528ca76567805beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd8ae

Example 17:

SK =4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fbPK =3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660calpha = 72 (1 byte)x =68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51try_and_increment succeeded on ctr = 1H =5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551k_string = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b2342acb9c167cab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc16cc5cek =d8c3a66921444cb3427d5d989f9b315aa8ca3375e9ec4d52207711a1fdb44107U = k*B =1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3V = k*H =fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed5933bf0864a62558b3ed7f2fea45c92a465301b3bbf5e3e54ddf2d935be3b67926da3ef39226bbc355bdc9850112c8f4b02beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496310eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5031

Example 18:

SK =c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7PK =fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025alpha = af82 (2 bytes)x =909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95ctry_and_increment succeeded on ctr = 0H =bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576ak_string = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59fc31a0febc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6bb6d6c3ak =5ffdbc72135d936014e8ab708585fda379405542b07e3bd2c0bd48437fbac60aU = k*B =2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5V = k*H =88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccbf8096bb474e53895c362d8628ee9f9ea3c0e52c7a5c691b6c18c9979866568add7a2d41b00b05081ed0f58ee5e31b3a970ebeta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df19f

B.4.ECVRF-EDWARDS25519-SHA512-ELL2

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

Example 19:

SK =9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60PK =d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511aalpha =  (the empty string)x =307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94fIn Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca5298b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269aeIn Elligator2: u =30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55cIn Elligator2: gx1 =8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08In Elligator2: gx1 is a squareH =b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9k_string = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a42458f817bed7a386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88dab8d41ck =55cbb247af9b8372259a97b2cfec656d78868deb33b203d51b9961c364522400U = k*B =762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9V = k*H =4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860afpi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab46f14adf9a3cd8b8412d9038531e865c341cafa73589b023d14311c331a9ad15ff2fb37831e00f0acaa6d73bc9997b06501beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6c54

Example 20:

SK =4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fbPK =3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660calpha = 72 (1 byte)x =68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd77bb99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6dIn Elligator2: u =3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773dIn Elligator2: gx1 =25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464dIn Elligator2: gx1 is a nonsquareH =76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57k_string = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37c4c37f1d41e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab7544e444dk =9565956daeedf376cad61b829b2a4d21ba1b52e9b3e2457477a64630a9711003U = k*B =8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061V = k*H =fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8bpi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c509072ef055b48372bb82efbdce8e10c8cb9a2f9d60e93908f93df1623ad78a86a028d6bc064dbfc75a6a57379ef855dc6733801beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4635987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc735

Example 21:

SK =c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7PK =fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025alpha = af82 (2 bytes)x =909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95cIn Elligator2: uniform_bytes = be0aed556e36cdfddf8f1eeddbb7356a24fad64cf95a922a098038f215588b216beabbfe6acf20256188e883292b7a3aIn Elligator2: u =f6675dc6d17fc790d4b3f1c6acf689a13d8b5815f23880092a925af94cd6fa24In Elligator2: gx1 =a63d48e3247c903e22fdfb88fd9295e396712a5fe576af335dbe16f99f0af26cIn Elligator2: gx1 is a squareH = 13d2a8b5ca32db7e98094a61f656a08c6c964344e058879a386a947a4e189ed1k_string = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5aaf79cf576087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861b3d683bk =1fda4077f737098b3f361c33a36cccafd7e9e9b720e1f84011254e25f37eed02U = k*B =a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60V = k*H =fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25api = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57fdce35b46edfc655bc828d44ad09d1150f31374e7ef73027e14760d42e77341fe05467bb286cc2c9d7fde29120a0b2320d04beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5cd090a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314a58

Contributors

This document would not be possible without the work ofMoni Naor,Sachin Vasant, andAsaf 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. Lawrence,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, andAnnie Yousar provided valuable input to this document.Christopher Wood,Malte Thomsen,Marcus Rasmussen, andTobias Vestergaard provided independent verification of the test vectors.Riad Wahby helped this document align with[RFC9380].

Authors' Addresses

Sharon Goldberg
Boston University
665 Commonwealth Avenue
Boston,MA02215
United States of America
Email:goldbe@cs.bu.edu
Leonid Reyzin
Boston University and Algorand
665 Commonwealth Avenue
Boston,MA02215
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 Včelák
NS1
Email:jvcelak@ns1.com
[8]ページ先頭
©2009-2025 Movatter.jp