Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-10
The information below is for an old version of the document.
| Document | Type |
This is an older version of an Internet-Draft that was ultimately published as RFC 9381.
|
|
|---|---|---|---|
| Authors | Sharon Goldberg , Leonid Reyzin , Dimitrios Papadopoulos , Jan Včelák | ||
| Last updated | 2021-11-17 | ||
| Replaces | draft-goldbe-vrf | ||
| RFC stream | Internet Research Task Force (IRTF) | ||
| Formats | |||
| IETF conflict review | conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf, conflict-review-irtf-cfrg-vrf | ||
| Additional resources | Mailing list discussion | ||
| Stream | IRTF state | In RG Last Call | |
| Consensus boilerplate | Unknown | ||
| Document shepherd | (None) | ||
| IESG | IESG state | Became RFC 9381 (Informational) | |
| Telechat date | (None) | ||
| Responsible AD | (None) | ||
| Send notices to | (None) |
draft-irtf-cfrg-vrf-10
CFRG S. Goldberg
Internet-Draft Boston University
Intended status: Standards Track L. Reyzin
Expires: 21 May 2022 Boston University and Algorand
D. Papadopoulos
Hong Kong University of Science and Technology
J. Vcelak
NS1
17 November 2021
Verifiable Random Functions (VRFs)
draft-irtf-cfrg-vrf-10
Abstract
A Verifiable Random Function (VRF) is the public-key version of a
keyed cryptographic hash. Only the holder of the private key can
compute the hash, but anyone with public key can verify the
correctness of the hash. VRFs are useful for preventing enumeration
of hash-based data structures. This document specifies several VRF
constructions that are secure in the cryptographic random oracle
model. One VRF uses RSA and the other VRF uses Elliptic Curves (EC).
Status of This Memo
This Internet-Draft is submitted in full conformance with the
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This Internet-Draft will expire on 21 May 2022.
Copyright Notice
Copyright (c) 2021 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
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provided without warranty as described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Rationale . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2. Requirements . . . . . . . . . . . . . . . . . . . . . . 3
1.3. Terminology . . . . . . . . . . . . . . . . . . . . . . . 3
2. VRF Algorithms . . . . . . . . . . . . . . . . . . . . . . . 4
3. VRF Security Properties . . . . . . . . . . . . . . . . . . . 5
3.1. Full Uniqueness or Trusted Uniqueness . . . . . . . . . . 5
3.2. Full Collison Resistance or Trusted Collision
Resistance . . . . . . . . . . . . . . . . . . . . . . . 5
3.3. Full Pseudorandomness or Selective Pseudorandomness . . . 6
3.4. A random-oracle-like unpredictability property . . . . . 7
4. RSA Full Domain Hash VRF (RSA-FDH-VRF) . . . . . . . . . . . 7
4.1. RSA-FDH-VRF Proving . . . . . . . . . . . . . . . . . . . 8
4.2. RSA-FDH-VRF Proof to Hash . . . . . . . . . . . . . . . . 9
4.3. RSA-FDH-VRF Verifying . . . . . . . . . . . . . . . . . . 9
5. Elliptic Curve VRF (ECVRF) . . . . . . . . . . . . . . . . . 10
5.1. ECVRF Proving . . . . . . . . . . . . . . . . . . . . . . 12
5.2. ECVRF Proof to Hash . . . . . . . . . . . . . . . . . . . 13
5.3. ECVRF Verifying . . . . . . . . . . . . . . . . . . . . . 14
5.4. ECVRF Auxiliary Functions . . . . . . . . . . . . . . . . 15
5.4.1. ECVRF Hash to Curve . . . . . . . . . . . . . . . . . 15
5.4.2. ECVRF Nonce Generation . . . . . . . . . . . . . . . 17
5.4.3. ECVRF Hash Points . . . . . . . . . . . . . . . . . . 19
5.4.4. ECVRF Decode Proof . . . . . . . . . . . . . . . . . 19
5.5. ECVRF Ciphersuites . . . . . . . . . . . . . . . . . . . 20
5.6. When the ECVRF Keys are Untrusted . . . . . . . . . . . . 23
5.6.1. ECVRF Validate Key . . . . . . . . . . . . . . . . . 23
6. Implementation Status . . . . . . . . . . . . . . . . . . . . 25
7. Security Considerations . . . . . . . . . . . . . . . . . . . 26
7.1. Key Generation . . . . . . . . . . . . . . . . . . . . . 26
7.1.1. Uniqueness and collision resistance with untrusted
keys . . . . . . . . . . . . . . . . . . . . . . . . 26
7.1.2. Pseudorandomness with untrusted keys . . . . . . . . 27
7.2. Selective vs. Full Pseudorandomness . . . . . . . . . . . 27
7.3. Proper pseudorandom nonce for ECVRF . . . . . . . . . . . 28
7.4. Side-channel attacks . . . . . . . . . . . . . . . . . . 28
7.5. Proofs provide no secrecy for the VRF input . . . . . . . 29
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7.6. Prehashing . . . . . . . . . . . . . . . . . . . . . . . 29
7.7. Hash function domain separation and futureproofing . . . 29
8. Change Log . . . . . . . . . . . . . . . . . . . . . . . . . 30
9. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 32
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 32
10.1. Normative References . . . . . . . . . . . . . . . . . . 32
10.2. Informative References . . . . . . . . . . . . . . . . . 33
Appendix A. Test Vectors for the ECVRFs . . . . . . . . . . . . 34
A.1. ECVRF-P256-SHA256-TAI . . . . . . . . . . . . . . . . . . 34
A.2. ECVRF-P256-SHA256-SSWU . . . . . . . . . . . . . . . . . 35
A.3. ECVRF-EDWARDS25519-SHA512-TAI . . . . . . . . . . . . . . 37
A.4. ECVRF-EDWARDS25519-SHA512-ELL2 . . . . . . . . . . . . . 39
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 41
1. Introduction
1.1. Rationale
A Verifiable Random Function (VRF) [MRV99] is the public-key version
of a keyed cryptographic hash. Only the holder of the private VRF
key can compute the hash, but anyone with corresponding public key
can verify the correctness of the hash.
A key application of the VRF is to provide privacy against offline
enumeration (e.g. dictionary attacks) on data stored in a hash-based
data structure. In this application, a Prover holds the VRF private
key and uses the VRF hashing to construct a hash-based data structure
on the input data. Due to the nature of the VRF, only the Prover can
answer queries about whether or not some data is stored in the data
structure. Anyone who knows the public VRF key can verify that the
Prover has answered the queries correctly. However, no offline
inferences (i.e. inferences without querying the Prover) can be made
about the data stored in the data structure.
1.2. Requirements
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
document are to be interpreted as described in [RFC2119].
1.3. Terminology
The following terminology is used through this document:
SK: The private key for the VRF.
PK: The public key for the VRF.
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alpha or alpha_string: The input to be hashed by the VRF.
beta or beta_string: The VRF hash output.
pi or pi_string: The VRF proof.
Prover: The Prover holds the private VRF key SK and public VRF key
PK.
Verifier: The Verifier holds the public VRF key PK.
2. VRF Algorithms
A VRF comes with a key generation algorithm that generates a public
VRF key PK and private VRF key SK.
The prover hashes an input alpha using the private VRF key SK to
obtain a VRF hash output beta
beta = VRF_hash(SK, alpha)
The VRF_hash algorithm is deterministic, in the sense that it always
produces the same output beta given the same pair of inputs (SK,
alpha). The prover also uses the private key SK to construct a proof
pi that beta is the correct hash output
pi = VRF_prove(SK, alpha)
The VRFs defined in this document allow anyone to deterministically
obtain the VRF hash output beta directly from the proof value pi by
using the function VRF_proof_to_hash:
beta = VRF_proof_to_hash(pi)
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 VRF also comes with an algorithm
VRF_verify(PK, alpha, pi)
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that outputs (VALID, beta = VRF_proof_to_hash(pi)) if pi is valid,
and INVALID otherwise.
3. VRF Security Properties
VRFs are designed to ensure the following security properties.
3.1. Full Uniqueness or Trusted Uniqueness
Uniqueness means that, for any fixed public VRF key and for any input
alpha, there is a unique VRF output beta that can be proved to be
valid. Uniqueness must hold even for an adversarial Prover that
knows the VRF private key SK.
More precisely, "full uniqueness" states that a computationally-
bounded adversary cannot choose a VRF public key PK, a VRF input
alpha, and two proofs pi1 and pi2 such that VRF_verify(PK, alpha,
pi1) outputs (VALID, beta1), VRF_verify(PK, alpha, pi2) outputs
(VALID, beta2), and beta1 is not equal to beta2.
For many applications, a slightly weaker security property called
"trusted uniqueness" suffices. Trusted uniqueness is the same as
full uniqueness, but it is guaranteed to hold only if the VRF keys PK
and SK were generated in a trustworthy manner.
As further discussed in Section 7.1.1, some VRFs specified in this
document satisfy only trusted uniqueness, while others satisfy full
uniqueness. VRFs in this document that satisfy only trusted
uniqueness but not full uniqueness MUST NOT be used if the key
generation process cannot be trusted.
3.2. Full Collison Resistance or Trusted Collision Resistance
Like any cryptographic hash function, VRFs need to be collision
resistant. Collison resistance must hold even for an adversarial
Prover that knows the VRF private key SK.
More precisely, "full collision resistance" states that it should be
computationally infeasible for an adversary to find two distinct VRF
inputs alpha1 and alpha2 that have the same VRF hash beta, even if
that adversary knows the private VRF key SK.
For many applications, a slightly weaker security property called
"trusted collision resistance" suffices. Trusted collision
resistance is the same as collision resistance, but it is guaranteed
to hold only if the VRF keys PK and SK were generated in a
trustworthy manner.
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As further discussed in Section 7.1.1, some VRFs specified in this
document satisfy only trusted collision resistance, while others
satisfy full collision resistance. VRFs in this document that
satisfy only trusted collision resistance but not full collision
resistance MUST NOT be used if the key generation process cannot be
trusted.
3.3. Full Pseudorandomness or Selective Pseudorandomness
Pseudorandomness ensures that when an adversarial Verifier sees a VRF
hash output beta without its corresponding VRF proof pi, then beta is
indistinguishable from a random value.
More precisely, suppose the public and private VRF keys (PK, SK) were
generated in a trustworthy manner. Pseudorandomness ensures that the
VRF hash output beta (without its corresponding VRF proof pi) on any
adversarially-chosen "target" VRF input alpha looks indistinguishable
from random for any computationally bounded adversary who does not
know the private VRF key SK. This holds even if the adversary also
gets to choose other VRF inputs alpha' and observe their
corresponding VRF hash outputs beta' and proofs pi'.
With "full pseudorandomness", the adversary is allowed to choose the
"target" VRF input alpha at any time, even after it observes VRF
outputs beta' and proofs pi' on a variety of chosen inputs alpha'.
"Selective pseudorandomness" is a weaker security property which
suffices in many applications. Here, the adversary must choose the
target VRF input alpha independently of the public VRF key PK, and
before it observes VRF outputs beta' and proofs pi' on inputs alpha'
of its choice.
As further discussed in Section 7.2, VRFs specified in this document
satisfy both full pseudorandomness and selective pseudorandomness,
but their quantitative security against the selective
pseudorandomness attack is stronger.
It is important to remember that the VRF output beta does not look
random to the Prover, or to any other party that knows the private
VRF key SK! Such a party can easily distinguish beta from a random
value by comparing beta to the result of VRF_hash(SK, alpha).
Also, the VRF output beta does not look random to any party that
knows the valid VRF proof pi corresponding to the VRF input alpha,
even if this party does not know the private VRF key SK. Such a
party can easily distinguish beta from a random value by checking
whether VRF_verify(PK, alpha, pi) returns (VALID, beta).
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Also, the VRF output beta may not look random if VRF key generation
was not done in a trustworthy fashion. (For example, if VRF keys
were generated with bad randomness.)
3.4. A random-oracle-like unpredictability property
As explained in Section 3.3, pseudorandomness is guaranteed only if
the VRF keys were generated in a trustworthy fashion. For instance,
if an adversary outputs VRF keys that are deterministically generated
(or hard-coded and publicly known), then the outputs are easily
derived by anyone and are therefore not pseudorandom.
There is, however, a different type of unpredictability that is
desirable in certain VRF applications (such as [GHMVZ17] and
[DGKR18]). This property is similar to the unpredictability achieved
by an (ordinary, unkeyed) cryptographic hash function: if the input
has enough entropy (i.e., cannot be predicted), then the correct
output is indistinguishable from uniform.
A formal definition of this property appears in Section 3.2 of
[DGKR18]. The VRF schemes presented in this specification are
believed to satisfy this property if the public key was generated in
a trustworthy manner. Additionally, the ECVRF is believed to also
satisfy this property even if the public key was not generated in a
trustworthy manner, as long as the public key satisfies the key
validation procedure in Section 5.6.
4. RSA Full Domain Hash VRF (RSA-FDH-VRF)
The RSA Full Domain Hash VRF (RSA-FDH-VRF) is a VRF that satisfies
the "trusted uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. Its security
follows from the standard RSA assumption in the random oracle model.
Formal security proofs are in [PWHVNRG17].
The VRF computes the proof pi as a deterministic RSA signature on
input alpha using the RSA Full Domain Hash Algorithm [RFC8017]
parametrized with the selected hash algorithm. RSA signature
verification is used to verify the correctness of the proof. The VRF
hash output beta is simply obtained by hashing the proof pi with the
selected hash algorithm.
The key pair for RSA-FDH-VRF MUST be generated in a way that it
satisfies the conditions specified in Section 3 of [RFC8017].
In this section, the notation from [RFC8017] is used.
Parameters used:
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(n, e) - RSA public key
K - RSA private key
k - length in octets of the RSA modulus n (k must be less than
2^32)
Fixed options:
Hash - cryptographic hash function
hLen - output length in octets of hash function Hash
Primitives used:
I2OSP - Conversion of a nonnegative integer to an octet string as
defined in Section 4.1 of [RFC8017] (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 secret key and an input, raises the input to
the secret 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]
|| - octet string concatenation
4.1. RSA-FDH-VRF Proving
RSAFDHVRF_prove(K, alpha_string)
Input:
K - RSA private key
alpha_string - VRF hash input, an octet string
Output:
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pi_string - proof, an octet string of length k
Steps:
1. one_string = 0x01
2. EM = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) ||
alpha_string, k - 1)
3. m = OS2IP(EM)
4. s = RSASP1(K, m)
5. pi_string = I2OSP(s, k)
6. Output pi_string
4.2. RSA-FDH-VRF Proof to Hash
RSAFDHVRF_proof_to_hash(pi_string)
Input:
pi_string - proof, an octet string of length k
Output:
beta_string - VRF hash output, an octet string of length hLen
Important note:
RSAFDHVRF_proof_to_hash should be run only on pi_string that is
known to have been produced by RSAFDHVRF_prove, or from within
RSAFDHVRF_verify as specified in Section 4.3.
Steps:
1. two_string = 0x02
2. beta_string = Hash(two_string || pi_string)
3. Output beta_string
4.3. RSA-FDH-VRF Verifying
RSAFDHVRF_verify((n, e), alpha_string, pi_string)
Input:
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(n, e) - RSA public key
alpha_string - VRF hash input, an octet string
pi_string - proof to be verified, an octet string of length n
Output:
("VALID", beta_string), where beta_string is the VRF hash output,
an octet string of length hLen; or
"INVALID"
Steps:
1. s = OS2IP(pi_string)
2. m = RSAVP1((n, e), s)
3. EM = I2OSP(m, k - 1)
4. one_string = 0x01
5. EM' = MGF1(one_string || I2OSP(k, 4) || I2OSP(n, k) ||
alpha_string, k - 1)
6. If EM and EM' are equal, output ("VALID",
RSAFDHVRF_proof_to_hash(pi_string)); else output "INVALID".
5. Elliptic Curve VRF (ECVRF)
The Elliptic Curve Verifiable Random Function (ECVRF) is a VRF that
satisfies the trusted uniqueness, trusted collision resistance, and
full pseudorandomness properties defined in Section 3. The security
of this VRF follows from the decisional Diffie-Hellman (DDH)
assumption in the random oracle model. Formal security proofs are in
[PWHVNRG17].
To additionally satisfy "full uniqueness" and "full collision
resistance", the Verifier MUST additionally perform the validation
procedure specified in Section 5.6 upon receipt of the public VRF
key.
Notation used:
Elliptic curve operations are written in additive notation, with
P+Q denoting point addition and x*P denoting scalar multiplication
of a point P by a scalar x
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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
2n - length, in octets, of a field element in F, rounded up to the
nearest even integer
E - elliptic curve (EC) defined over F
ptLen - length, in octets, of an EC point encoded as an octet
string
G - subgroup of E of large prime order
q - prime order of group G
qLen - length of q in octets, i.e., smallest integer such that
2^(8qLen)>q (note that in the typical case, qLen equals 2n or is
close to 2n)
cofactor - number of points on E divided by q
B - generator of group G
Hash - cryptographic hash function
hLen - output length in octets of Hash; must be at least 2n
ECVRF_hash_to_curve - a function that hashes strings to an EC
point.
ECVRF_nonce_generation - a function that derives a pseudorandom
nonce from SK and the input as part of ECVRF proving.
suite_string - a single nonzero octet specifying the ECVRF
ciphersuite, which determines the above options as well as type
conversions and parameter generation
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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 EC point to an ptLen-octet string
string_to_point - conversion of an ptLen-octet string to EC point.
string_to_point returns INVALID if the octet string does not
convert to a valid EC point.
Note that with certain software libraries (for big integer and
elliptic curve arithmetic), the int_to_string and point_to_string
conversions are not needed, when the libraries encode integers and
EC points in the same way as required by the ciphersuites. For
example, in some implementations, EC point operations will take
octet strings as inputs and produce octet strings as outputs,
without introducing a separate elliptic curve point type.
Parameters used (the generation of these parameters is specified in
Section 5.5):
SK - VRF private key
x - VRF secret scalar, an integer
- Note: depending on the ciphersuite used, the VRF secret scalar
may be equal to SK; else, it is derived from SK
Y = x*B - VRF public key, an EC point
5.1. ECVRF Proving
ECVRF_prove(SK, alpha_string)
Input:
SK - VRF private key
alpha_string = input alpha, an octet string
Output:
pi_string - VRF proof, octet string of length ptLen+n+qLen
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Steps:
1. Use SK to derive the VRF secret scalar x and the VRF public key Y
= x*B
(this derivation depends on the ciphersuite, as per Section 5.5;
these values can be cached, for example, after key generation,
and need not be rederived each time)
2. H = ECVRF_hash_to_curve(Y, alpha_string)
3. h_string = point_to_string(H)
4. Gamma = x*H
5. k = ECVRF_nonce_generation(SK, h_string)
6. c = ECVRF_hash_points(H, Gamma, k*B, k*H) (see Section 5.4.3)
7. s = (k + c*x) mod q
8. pi_string = point_to_string(Gamma) || int_to_string(c, n) ||
int_to_string(s, qLen)
9. Output pi_string
5.2. ECVRF Proof to Hash
ECVRF_proof_to_hash(pi_string)
Input:
pi_string - VRF proof, octet string of length ptLen+n+qLen
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:
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1. D = ECVRF_decode_proof(pi_string) (see Section 5.4.4)
2. If D is "INVALID", output "INVALID" and stop
3. (Gamma, c, s) = D
4. three_string = 0x03
5. zero_string = 0x00
6. beta_string = Hash(suite_string || three_string ||
point_to_string(cofactor * Gamma) || zero_string)
7. Output beta_string
5.3. ECVRF Verifying
ECVRF_verify(Y, pi_string, alpha_string)
Input:
Y - public key, an EC point
pi_string - VRF proof, octet string of length ptLen+n+qLen
alpha_string - VRF input, octet string
Output:
("VALID", beta_string), where beta_string is the VRF hash output,
octet string of length hLen; or
"INVALID"
Steps:
1. D = ECVRF_decode_proof(pi_string) (see Section 5.4.4)
2. If D is "INVALID", output "INVALID" and stop
3. (Gamma, c, s) = D
4. H = ECVRF_hash_to_curve(Y, alpha_string)
5. U = s*B - c*Y
6. V = s*H - c*Gamma
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7. c' = ECVRF_hash_points(H, Gamma, U, V) (see Section 5.4.3)
8. If c and c' are equal, output ("VALID",
ECVRF_proof_to_hash(pi_string)); else output "INVALID"
5.4. ECVRF Auxiliary Functions
5.4.1. ECVRF Hash to Curve
The ECVRF_hash_to_curve algorithm takes in the VRF input alpha and
converts it to H, an EC point in G. This algorithm is the only place
the VRF input alpha is used for proving and verifying. See
Section 7.6 for further discussion.
This section specifies a number of such algorithms, which are not
compatible with each other. The choice of a particular algorithm
from the options specified in this section is made in Section 5.5.
5.4.1.1. ECVRF_hash_to_curve_try_and_increment
The following ECVRF_hash_to_curve_try_and_increment(Y, alpha_string)
algorithm implements ECVRF_hash_to_curve in a simple and generic way
that works for any elliptic curve.
The running time of this algorithm depends on alpha_string. For the
ciphersuites specified in Section 5.5, this algorithm is expected to
find a valid curve point after approximately two attempts (i.e., when
ctr=1) on average.
However, because the running time of algorithm depends on
alpha_string, this algorithm SHOULD be avoided in applications where
it is important that the VRF input alpha remain secret.
ECVRF_hash_to_try_and_increment(Y, alpha_string)
Input:
Y - public key, an EC point
alpha_string - value to be hashed, an octet string
Output:
H - hashed value, a finite EC point in G
Fixed option (specified in Section 5.5):
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arbitrary_string_to_point - conversion of an arbitrary octet
string to an EC point.
Steps:
1. ctr = 0
2. PK_string = point_to_string(Y)
3. one_string = 0x01
4. zero_string = 0x00
5. H = "INVALID"
6. 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 || one_string || PK_string ||
alpha_string || ctr_string || zero_string)
c. H = arbitrary_string_to_point(hash_string)
d. If H is not "INVALID" and cofactor > 1, set H = cofactor * H
e. ctr = ctr + 1
7. Output H
5.4.1.2. ECVRF_hash_to_curve_h2c_suite
The ECVRF_hash_to_curve_h2c_suite(Y, alpha_string) algorithm
implements ECVRF_hash_to_curve using one of the several hash-to-curve
options defined in [I-D.irtf-cfrg-hash-to-curve]. The specific
choice of the hash-to-curve option (called Suite ID in
[I-D.irtf-cfrg-hash-to-curve]) is given by the h2c_suite_ID_string
parameter.
ECVRF_hash_to_curve_h2c_suite(Y, alpha_string)
Input:
alpha_string - value to be hashed, an octet string
Y - public key, an EC point
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Output:
H - hashed value, a finite EC point in G
Fixed option (specified in Section 5.5):
h2c_suite_ID_string - a hash-to-curve suite ID, encoded in ASCII
(see discussion below)
Steps
1. PK_string = point_to_string(Y)
2. string_to_hash = PK_string || alpha_string
3. H = encode(string_to_hash)
(the encode function is discussed below)
4. Output H
The encode function is provided by the hash-to-curve suite whose ID
is h2c_suite_ID_string, as specified in
[I-D.irtf-cfrg-hash-to-curve], Section 8. The domain separation tag
DST, a parameter to the hash-to-curve suite, SHALL be set to
"ECVRF_" || h2c_suite_ID_string || suite_string
where "ECVRF_" is represented as a 6-byte ASCII encoding (in
hexadecimal, octets 45 43 56 52 46 5F).
5.4.2. ECVRF Nonce Generation
The following algorithms generate the nonce value k in a
deterministic pseudorandom fashion. This section specifies a number
of such algorithms, which are not compatible with each other. The
choice of a particular algorithm from the options specified in this
section is made in Section 5.5.
5.4.2.1. ECVRF Nonce Generation from RFC 6979
ECVRF_nonce_generation_RFC6979(SK, h_string)
Input:
SK - an ECVRF secret key
h_string - an octet string
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Output:
k - an integer between 1 and q-1
The ECVRF_nonce_generation function is as specified in [RFC6979]
Section 3.2 where
Input m is set equal to h_string
The "suitable for DSA or ECDSA" check in step h.3 is omitted
The hash function H is Hash and its output length hlen is set as
hLen*8
The secret key x is set equal to the VRF secret scalar x
The prime q is the same as in this specification
qlen is the binary length of q, i.e., the smallest integer such
that 2^qlen > q
All the other values and primitives as defined in [RFC6979]
5.4.2.2. ECVRF Nonce Generation from RFC 8032
The following is from Steps 2-3 of Section 5.1.6 in [RFC8032].
ECVRF_nonce_generation_RFC8032(SK, h_string)
Input:
SK - an ECVRF secret key
h_string - an octet string
Output:
k - an integer between 0 and q-1
Steps:
1. hashed_sk_string = Hash(SK)
2. truncated_hashed_sk_string =
hashed_sk_string[32]...hashed_sk_string[63]
3. k_string = Hash(truncated_hashed_sk_string || h_string)
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4. k = string_to_int(k_string) mod q
5.4.3. ECVRF Hash Points
ECVRF_hash_points(P1, P2, ..., PM)
Input:
P1...PM - EC points in G
Output:
c - hash value, integer between 0 and 2^(8n)-1
Steps:
1. two_string = 0x02
2. Initialize str = suite_string || two_string
3. for PJ in [P1, P2, ... PM]:
str = str || point_to_string(PJ)
4. zero_string = 0x00
5. str = str || zero_string
6. c_string = Hash(str)
7. truncated_c_string = c_string[0]...c_string[n-1]
8. c = string_to_int(truncated_c_string)
9. Output c
5.4.4. ECVRF Decode Proof
ECVRF_decode_proof(pi_string)
Input:
pi_string - VRF proof, octet string (ptLen+n+qLen octets)
Output:
"INVALID", or
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Gamma - EC point
c - integer between 0 and 2^(8n)-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+n-1]
3. s_string = pi_string[ptLen+n]...pi_string[ptLen+n+qLen-1]
4. Gamma = string_to_point(gamma_string)
5. if Gamma = "INVALID" output "INVALID" and stop
6. c = string_to_int(c_string)
7. s = string_to_int(s_string)
8. if s >= q output "INVALID" and stop
9. Output Gamma, c, and s
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, 2n = qLen = 32 and
cofactor = 1.
* The key pair generation primitive is specified in Section 3.2.1 of
[SECG1] (q, B, SK, and Y in this document correspond to n, G, d,
and Q in Section 3.2.1 of [SECG1]). In this ciphersuite, the
secret scalar x is equal to the private key SK.
* 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.)
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* 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 an EC point to an octet
string according to the encoding specified in Section 2.3.3 of
[SECG1] with point compression on. This implies ptLen = 2n + 1 =
33. (Note that certain software implementations do not introduce
a separate elliptic curve point type and instead directly treat
the EC point as an octet string per above encoding. When using
such an implementation, the point_to_string function can be
treated as the identity function.)
* The string_to_point function converts an octet string to an EC
point according to the encoding specified in Section 2.3.4 of
[SECG1]. This function MUST output INVALID if the octet string
does not decode to an EC point.
* The hash function Hash is SHA-256 as specified in [RFC6234], with
hLen = 32.
* The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1, with arbitrary_string_to_point(s) =
string_to_point(0x02 || s). The input s to
arbitrary_string_to_point is a 32-octet string and the output is
either an EC point or "INVALID".
This document defines ECVRF-P256-SHA256-SSWU as identical to ECVRF-
P256-SHA256-TAI, except that:
* suite_string = 0x02
* the ECVRF_hash_to_curve function is as specified in
Section 5.4.1.2 with h2c_suite_ID_string = P256_XMD:SHA-
256_SSWU_NU_ (the suite is defined in
[I-D.irtf-cfrg-hash-to-curve] Section 8.2)
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, 2n = qLen = 32
and cofactor = 8.
* The private key and generation of the secret scalar and the public
key are specified in Section 5.1.5 of [RFC8032].
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* The ECVRF_nonce_generation function is as specified in
Section 5.4.2.2.
* The int_to_string function as specified in the first paragraph of
Section 5.1.2 of [RFC8032]. (This is little-endian
representation.)
* The string_to_int function interprets the string as an integer in
little-endian representation.
* The point_to_string function converts an EC point to an octet
string according to the encoding specified in Section 5.1.2 of
[RFC8032]. This implies ptLen = 2n = 32. (Note that certain
software implementations do not introduce a separate elliptic
curve point type and instead directly treat the EC point as an
octet string per above encoding. When using such and
implementation, the point_to_string function can be treated as the
identity function.)
* The string_to_point function converts an octet string to an EC
point according to the encoding specified in Section 5.1.3 of
[RFC8032]. This function MUST output INVALID if the octet string
does not decode to an EC point.
* The hash function Hash is SHA-512 as specified in [RFC6234], with
hLen = 64.
* The ECVRF_hash_to_curve function is as specified in
Section 5.4.1.1, with arbitrary_string_to_point(s) =
string_to_point(s[0]...s[31]).
This document defines ECVRF-EDWARDS25519-SHA512-ELL2 as identical to
ECVRF-EDWARDS25519-SHA512-TAI, except:
* suite_string = 0x04;
* the ECVRF_hash_to_curve function is as specified in
Section 5.4.1.2 with h2c_suite_ID_string = edwards25519_XMD:SHA-
512_ELL2_NU_ (the suite is defined in
[I-D.irtf-cfrg-hash-to-curve] Section 8.5).
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5.6. When the ECVRF Keys are Untrusted
The ECVRF as specified above is a VRF that satisfies the "trusted
uniqueness", "trusted collision resistance", and "full
pseudorandomness" properties defined in Section 3. In order to
obtain "full uniqueness" and "full collision resistance" (which
provide protection against a malicious VRF public key), the Verifier
MUST perform the following additional validation procedure upon
receipt of the public VRF key. The public VRF key MUST NOT be used
if this procedure returns "INVALID".
Note that this procedure is not sufficient if the elliptic curve E or
the point B, the generator of group G, is untrusted. If the prover
is untrusted, the Verifier MUST obtain E and B from a trusted source,
such as a ciphersuite specification, rather than from the prover.
Ciphersuites in this document specify E and B.
This procedure supposes that the public key provided to the Verifier
is an octet string. The procedure returns "INVALID" if the public
key in invalid. Otherwise, it returns Y, the public key as an EC
point.
5.6.1. ECVRF Validate Key
ECVRF_validate_key(PK_string)
Input:
PK_string - public key, an octet string
Output:
"INVALID", or
Y - public key, an EC point
Steps:
1. Y = string_to_point(PK_string)
2. If Y is "INVALID", output "INVALID" and stop
3. If cofactor*Y is the identity element of the elliptic curve
group, output "INVALID" and stop
4. Output Y
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Note that if the cofactor = 1, then Step 3 need not multiply Y by the
cofactor; instead, it suffices to output "INVALID" if Y is the
identity element of the elliptic curve group. Moreover, when
cofactor>1, it is not necessary to verify that Y is in the subgroup
G; Step 3 suffices. Therefore, if the cofactor is small, the total
number of points that could cause Step 3 to output "INVALID" may be
small, and it may be more efficient to simply check Y against a fixed
list of such points. For example, the following algorithm can be
used for the edwards25519 curve:
1. Y = string_to_point(PK_string)
2. If Y is "INVALID", output "INVALID" and stop
3. y_string = PK_string
4. oneTwentySeven_string = 0x7F
5. y_string[31] = y_string[31] & oneTwentySeven_string
(this step clears the high-order bit of octet 31)
6. bad_pk[0] = int_to_string(0, 32)
7. bad_pk[1] = int_to_string(1, 32)
8. bad_y2 = 2707385501144840649318225287225658788936804267575313519
463743609750303402022
9. bad_pk[2] = int_to_string(bad_y2, 32)
10. bad_pk[3] = int_to_string(p-bad_y2, 32)
11. bad_pk[4] = int_to_string(p-1, 32)
12. bad_pk[5] = int_to_string(p, 32)
13. bad_pk[6] = int_to_string(p+1, 32)
14. If y_string is in bad_pk[0]...bad_pk[6], output "INVALID" and
stop
15. Output Y
(bad_pk[0], bad_pk[2], bad_pk[3] each match two bad public keys,
depending on the sign of the x-coordinate, which was cleared in step
5, in order to make sure that it does not affect the comparison.
bad_pk[1] and bad_pk[4] each match one bad public key, because
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x-coordinate is 0 for these two public keys. bad_pk[5] and bad_pk[6]
are simply bad_pk[0] and bad_pk[1] shifted by p, in case the
y-coordinate had not been modular reduced by p. There is no need to
shift the other bad_pk values by p, because they will exceed 2^255.
These bad keys, which represent all points of order 1, 2, 4, and 8,
have been obtained by converting the points specified in [X25519] to
Edwards coordinates.)
6. Implementation Status
Note to RFC editor: Remove before publication
A reference C++ implementation of ECVRF-P256-SHA256-TAI, ECVRF-
P256-SHA256-SSWU, ECVRF-EDWARDS25519-SHA512-TAI, and ECVRF-
EDWARDS25519-SHA512-ELL2 is available at https://github.com/reyzin/
ecvrf. This implementation is neither secure nor especially
efficient, but can be used to generate test vectors.
A Python implementation of an older version of ECVRF-
EDWARDS25519-SHA512-ELL2 from the -05 version of this draft is
available at https://github.com/integritychain/draft-irtf-cfrg-vrf-
05.
A C implementation of an older version of ECVRF-
EDWARDS25519-SHA512-ELL2 from the -03 version of this draft is
available at https://github.com/algorand/libsodium/tree/draft-irtf-
cfrg-vrf-03/src/libsodium/crypto_vrf/ietfdraft03.
A Rust implementation of an older version of ECVRF-P256-SHA256-TAI
from the -05 version of this draft, as well as variants for the
sect163k1 and secp256k1 curves, is available at
https://crates.io/crates/vrf.
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/
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An implementation by Ryuji Ishiguro following an older version of
ECVRF-EDWARDS25519-SHA512-TAI from the -00 version of this draft is
available at https://github.com/r2ishiguro/vrf.
An implementation similar to ECVRF-EDWARDS25519-SHA512-ELL2 (with
some changes, including the use of SHA-3) is available as part of the
CONIKS implementation in Golang at https://github.com/coniks-sys/
coniks-go/tree/master/crypto/vrf.
Open Whisper Systems also uses a VRF similar to ECVRF-
EDWARDS25519-SHA512-ELL2, called VXEdDSA, and specified here
https://whispersystems.org/docs/specifications/xeddsa/ and here
https://moderncrypto.org/mail-archive/curves/2017/000925.html.
Implementations in C and Java are available at
https://github.com/signalapp/curve25519-java and
https://github.com/wavesplatform/curve25519-java.
7. Security Considerations
7.1. Key Generation
Applications that use the VRFs defined in this document MUST ensure
that the VRF key is generated correctly, using good randomness.
7.1.1. Uniqueness and collision resistance with untrusted keys
The ECVRF as specified in Section 5.1-Section 5.5 satisfies the
"trusted uniqueness" (see Section 3.1) and "trusted collision
resistance" (see Section 3.2) properties as long as the VRF keys are
generated correctly, with good randomness. If the Verifier trusts
the VRF keys are generated correctly, it MAY use the public key Y as
is.
However, if the ECVRF uses keys that could be generated
adversarially, then the Verifier MUST first perform the validation
procedure ECVRF_validate_key(PK_string) (specified in Section 5.6)
upon receipt of the public key as an octet string. If the validation
procedure outputs "INVALID", then the public key MUST not be used.
Otherwise, the procedure will output a valid public key Y, and the
ECVRF with public key Y satisfies the "full uniqueness" and "full
collision resistance" properties.
The RSA-FDH-VRF satisfies the "trusted uniqueness" and "trusted
collision resistance" properties as long as the VRF keys are
generated correctly, with good randomness. These properties may not
hold if the keys are generated adversarially (e.g., if the RSA
function specified in the public key is not bijective). Meanwhile,
the "full uniqueness" and "full collision resistance" are properties
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that hold even if VRF keys are generated by an adversary. The RSA-
FDH-VRF defined in this document does not have these properties.
However, if adversarial key generation is a concern, the RSA-FDH-VRF
may be modified to have these properties by adding additional
cryptographic checks that its public key has the right form. These
modifications are left for future specification.
7.1.2. Pseudorandomness with untrusted keys
Without good randomness, the "pseudorandomness" properties of the VRF
may not hold. Note that it is not possible to guarantee
pseudorandomness in the face of adversarially generated VRF keys.
This is because an adversary can always use bad randomness to
generate the VRF keys, and thus, the VRF output may not be
pseudorandom.
7.2. Selective vs. Full Pseudorandomness
[PWHVNRG17] presents cryptographic reductions to an underlying hard
problem (e.g. Decisional Diffie-Hellman for the ECVRF, or the
standard RSA assumption for RSA-FDH-VRF) that prove the VRFs
specified in this document possess full pseudorandomness as well as
selective pseudorandomness (see Section 3.3 for an explanation of
these notions). However, the cryptographic reductions are tighter
for selective pseudorandomness than for full pseudorandomness. This
means that the VRFs have quantitively stronger security guarantees
for selective pseudorandomness.
Applications that are concerned about tightness of cryptographic
reductions therefore have two options.
* They may choose to ensure that selective pseudorandomness is
sufficient for the application. That is, that pseudorandomness of
outputs matters only for inputs that are chosen independently of
the VRF key.
* If full pseudorandomness is required for the application, the
application may increase security parameters to make up for the
loose security reduction. For RSA-FDH-VRF, this means increasing
the RSA key length. For ECVRF, this means increasing the
cryptographic strength of the EC group G. For both RSA-FDH-VRF
and ECVRF, the cryptographic strength of the hash function Hash
may also potentially need to be increased.
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7.3. Proper pseudorandom nonce for ECVRF
The security of the ECVRF defined in this document relies on the fact
that the nonce k used in the ECVRF_prove algorithm is chosen
uniformly and pseudorandomly modulo q, and is unknown to the
adversary. Otherwise, an adversary may be able to recover the
private VRF key x (and thus break pseudorandomness of the VRF) after
observing several valid VRF proofs pi. The nonce generation methods
specified in the ECVRF ciphersuites of Section 5.5 are designed with
this requirement in mind.
7.4. Side-channel attacks
Side channel attacks on cryptographic primitives are an important
issue. Implementers should take care to avoid side-channel attacks
that leak information about the VRF private key SK (and the nonce k
used in the ECVRF), which is used in VRF_prove. In most
applications, VRF_proof_to_hash and VRF_verify algorithms take only
inputs that are public, and thus side channel attacks are typically
not a concern for these algorithms.
The VRF input alpha may be also a sensitive input to VRF_prove and
may need to be protected against side channel attacks. Below we
discuss one particular class of such attacks: timing attacks that can
be used to leak information about the VRF input alpha.
The ECVRF_hash_to_curve_try_and_increment algorithm defined in
Section 5.4.1.1 SHOULD NOT be used in applications where the VRF
input alpha is secret and is hashed by the VRF on-the-fly. This is
because the algorithm's running time depends on the VRF input alpha,
and thus creates a timing channel that can be used to learn
information about alpha. That said, for most inputs the amount of
information obtained from such a timing attack is likely to be small
(1 bit, on average), since the algorithm is expected to find a valid
curve point after only two attempts. However, there might be inputs
which cause the algorithm to make many attempts before it finds a
valid curve point; for such inputs, the information leaked in a
timing attack will be more than 1 bit.
ECVRF-P256-SHA256-SSWU and ECVRF-EDWARDS25519-SHA512-ELL2 can be made
to run in time independent of alpha, following recommendations in
[I-D.irtf-cfrg-hash-to-curve].
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7.5. 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.
7.6. Prehashing
The VRFs specified in this document allow for read-once access to the
input alpha for both signing and verifying. Thus, additional
prehashing of alpha (as specified, for example, in [RFC8032] for
EdDSA signatures) is not needed, even for applications that need to
handle long alpha or to support the Initialized-Update-Finalize (IUF)
interface (in such an interface, alpha is not supplied all at once,
but rather in pieces by a sequence of calls to Update). The ECVRF,
in particular, uses alpha only in ECVRF_hash_to_curve. The curve
point H becomes the representative of alpha thereafter. Note that
the suite_string octet and the public key are hashed together with
alpha in ECVRF_hash_to_curve, which ensures that the curve (including
the generator B) and the public key are included indirectly into
subsequent hashes.
7.7. Hash function domain separation and futureproofing
Hashing is used for different purposes in the two VRFs (namely, in
the RSA-FDH-VRF, in MGF1 and in proof_to_hash; in the ECVRF, in
hash_to_curve, nonce_generation, hash_points, and proof_to_hash).
The theoretical analysis assumes each of these functions is a
separate random oracle. This analysis still holds even if the same
hash function is used, as long as the four queries made to the hash
function for a given SK and alpha are overwhelmingly unlikely to
equal each other or to any queries made to the hash function for the
same SK and different alpha. This is indeed the case for the RSA-
FDH-VRF defined in this document, because the first octets of the
input to the hash function used in MGF1 and in proof_to_hash are
different.
This is also the case for the ECVRF ciphersuites defined in this
document, because:
* inputs to the hash function used during nonce_generation are
unlikely to equal inputs used in hash_to_curve, proof_to_hash, and
hash_points. This follows since nonce_generation inputs a secret
to the hash function that is not used by honest parties as input
to any other hash function, and is not available to the adversary.
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* the second octets of the inputs to the hash function used in
proof_to_hash, hash_points, and
ECVRF_hash_to_curve_try_and_increment are all different.
* the last octet of the input to the hash function used in
proof_to_hash, hash_points, and
ECVRF_hash_to_curve_try_and_increment is always zero, and
therefore different from the last octet of the input to the hash
function used in ECVRF_hash_to_curve_h2c_suite, which is set equal
to the nonzero length of the domain separation tag by
[I-D.irtf-cfrg-hash-to-curve].
For the RSA VRF, if future designs need to specify variants of the
design in this document, such variants should use different first
octets in inputs to MGF1 and to the hash function used in
proof_to_hash, in order to avoid the possibility that an adversary
can obtain a VRF output under one variant, and then claim it was
obtained under another variant
For the elliptic curve VRF, if future designs need to specify
variants (e.g., additional ciphersuites) of the design in this
document, then, to avoid the possibility that an adversary can obtain
a VRF output under one variant, and then claim it was obtained under
another variant, they should specify a different suite_string
constant. This way, the inputs to the hash_to_curve hash function
used in producing H are guaranteed to be different; since all the
other hashing done by the prover depends on H, inputs all the hash
functions used by the prover will also be different as long as
hash_to_curve is collision resistant.
8. Change Log
Note to RFC Editor: if this document does not obsolete an existing
RFC, please remove this appendix before publication as an RFC.
00 - Forked this document from draft-goldbe-vrf-01.
01 - Minor updates, mostly highlighting TODO items.
02 - Added specification of elligator2 for Curve25519, along with
ciphersuites for ECVRF-ED25519-SHA512-Elligator. Changed ECVRF-
ED25519-SHA256 suite_string to ECVRF-ED25519-SHA512. (This change
made because Ed25519 in [RFC8032] signatures use SHA512 and not
SHA256.) Made ECVRF nonce generation a separate component, so
that nonces are 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
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to ECVRF for futureproofing. Ensured domain separation for hash
functions by modifying hash_points and added discussion about
domain separation. Updated todos in the "additional
pseudorandomness property" section. Added a discussion of secrecy
into security considerations. Removed B and PK=Y from
ECVRF_hash_points because they are already present via H, which is
computed via hash_to_curve using the suite_string (which
identifies B) and Y.
03 - Changed Ed25519 conversions to little-endian, to match RFC
8032; added simple key validation for Ed25519; added Simple SWU
cipher suite; clarified Elligator and removed the extra x0 bit, to
make Montgomery and Edwards Elligator the same; added domain
separation for RSA VRF; improved notation throughout; added nonce
generation as a section; changed counter in try-and-increment from
four bytes to one, to avoid endian issues; renamed try-and-
increment ciphersuites to -TAI; added qLen as a separate
parameter; changed output length to hLen for ECVRF, to match
RSAVRF; made Verify return beta so unverified proofs don't end up
in proof_to_hash; added test vectors.
04 - Clarified handling of optional arguments x and PK in
ECVRF_prove. Edited implementation status to bring it up to date.
05 - Renamed ed25519 into the more commonly used edwards25519.
Corrected ECVRF_nonce_generation_RFC6979 (thanks to Gorka Irazoqui
Apecechea and Mario Cao Cueto for finding the problem) and
corresponding test vectors for the P256 suites. Added a reference
to the Rust implementation.
06 - Made some variable names more descriptive. Added a few
implementation references.
07 - Incorporated hash-to-curve draft by reference to replace our
own Elligator2 and Simple SWU. Clarified discussion of EC
parameters and functions. Added a 0 octet to all hashing to
enforce domain separation from hashing done inside hash-to-curve.
08 - Incorporated suggestions from crypto panel review by Chloe
Martindale. Changed Reyzin's affiliation. Updated references.
09 - Added a note to remove the implementation page before
publication.
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.
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9. Contributors
This document also would not be possible without the work of Moni
Naor (Weizmann Institute), Sachin Vasant (Cisco Systems), and Asaf
Ziv (Facebook). Shumon Huque, David C. Lawerence, Trevor Perrin,
Annie Yousar, Stanislav Smyshlyaev, Liliya Akhmetzyanova, Tony
Arcieri, Sergey Gorbunov, Sam Scott, Nick Sullivan, Christopher Wood,
Marek Jankowski, Derek Ting-Haye Leung, Adam Suhl, Gary Belvinm,
Piotr Nojszewski, Gorka Irazoqui Apecechea, and Mario Cao Cueto
provided valuable input to this draft. Riad Wahby was very helpful
with the integration of the hash-to-curve draft.
10. References
10.1. Normative References
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/info/rfc2119>.
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
<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>.
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[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-
13, 10 November 2021,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
hash-to-curve-13>.
[FIPS-186-4]
National Institute for Standards and Technology, "Digital
Signature Standard (DSS)", FIPS PUB 186-4, July 2013,
<https://csrc.nist.gov/publications/detail/fips/186/4/
final>.
[SECG1] Standards for Efficient Cryptography Group (SECG), "SEC 1:
Elliptic Curve Cryptography", Version 2.0, May 2009,
<http://www.secg.org/sec1-v2.pdf>.
10.2. Informative References
[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>.
[MRV99] Micali, S., Rabin, M., and S. Vadhan, "Verifiable Random
Functions", in FOCS, 1999,
<https://dash.harvard.edu/handle/1/5028196>.
[X25519] Bernstein, D.J., "How do I validate Curve25519 public
keys?", 2006, <https://cr.yp.to/ecdh.html#validate>.
[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>.
[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>.
[I-D.vcelak-nsec5]
Vcelak, J., Goldberg, S., Papadopoulos, D., Huque, S., and
D. C. Lawrence, "NSEC5, DNSSEC Authenticated Denial of
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Existence", Work in Progress, Internet-Draft, draft-
vcelak-nsec5-08, 29 December 2018,
<https://datatracker.ietf.org/doc/html/draft-vcelak-
nsec5-08>.
[ANSI.X9-62-2005]
"Public Key Cryptography for the Financial Services
Industry: The Elliptic Curve Digital Signature Algorithm
(ECDSA)", ANSI X9.62, 2005.
Appendix A. Test Vectors for the ECVRFs
The test vectors in this section were generated using the reference
implementation at https://github.com/reyzin/ecvrf.
A.1. ECVRF-P256-SHA256-TAI
The example secret keys and messages in Examples 1 and 2 are taken
from Appendix A.2.5 of [RFC6979].
Example 1:
SK = x =
c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK =
0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
try_and_increment succeeded on ctr = 1
H =
0272a877532e9ac193aff4401234266f59900a4a9e3fc3cfc6a4b7e467a15d06d4
k =
0d90591273453d2dc67312d39914e3a93e194ab47a58cd598886897076986f77
U = k*B =
02bb6a034f67643c6183c10f8b41dc4babf88bff154b674e377d90bde009c21672
V = k*H =
02893ebee7af9a0faa6da810da8a91f9d50e1dc071240c9706726820ff919e8394
pi = 035b5c726e8c0e2c488a107c600578ee75cb702343c153cb1eb8dec77f4b5
071b498e7c291a16dafb9ccff8c2ae1f039fa92a328d5f7e0d483ee18353067a13
f699944a78892ff24939bcd044827eef884
beta =
a3ad7b0ef73d8fc6655053ea22f9bede8c743f08bbed3d38821f0e16474b505e
Example 2:
SK = x =
c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK =
0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
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alpha = 74657374 (ASCII "test")
try_and_increment succeeded on ctr = 3
H =
02173119b4fff5e6f8afed4868a29fe8920f1b54c2cf89cc7b301d0d473de6b974
k =
5852353a868bdce26938cde1826723e58bf8cb06dd2fed475213ea6f3b12e961
U = k*B =
022779a2cafcb65414c4a04a4b4d2adf4c50395f57995e89e6de823250d91bc48e
V = k*H =
033b4a14731672e82339f03b45ff6b5b13dee7ada38c9bf1d6f8f61e2ce5921119
pi = 034dac60aba508ba0c01aa9be80377ebd7562c4a52d74722e0abae7dc3080
ddb56c874cc95b7d29a6a65cb518fe6f4418256385f12b1eccbad023c901bb983f
f707b109b3a3b526ca3a1e8661f7b8481a2
beta =
a284f94ceec2ff4b3794629da7cbafa49121972671b466cab4ce170aa365f26d
The example secret key in Example 3 is taken from Appendix L.4.2 of
[ANSI.X9-62-2005].
Example 3:
SK = x =
2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK =
03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
(ASCII "Example using ECDSA key from Appendix L.4.2 of
ANSI.X9-62-2005")
try_and_increment succeeded on ctr = 1
H =
0258055c26c4b01d01c00fb57567955f7d39cd6f6e85fd37c58f696cc6b7aa761d
k =
5689e2e08e1110b4dda293ac21667eac6db5de4a46a519c73d533f69be2f4da3
U = k*B =
020f465cd0ec74d2e23af0abde4c07e866ae4e5138bded5dd1196b8843f380db84
V = k*H =
036cb6f811428fc4904370b86c488f60c280fa5b496d2f34ff8772f60ed24b2d1d
pi = 03d03398bf53aa23831d7d1b2937e005fb0062cbefa06796579f2a1fc7e7b
8c6679d92353c8a4fdfddb2a8540094b686cb5fb50f730d833a098a0399ccad32f
3fec4da2299891fc75ebda42baeb65e8c11
beta =
90871e06da5caa39a3c61578ebb844de8635e27ac0b13e829997d0d95dd98c19
A.2. ECVRF-P256-SHA256-SSWU
The example secret keys and messages in Examples 4 and 5 are taken
from Appendix A.2.5 of [RFC6979].
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Example 4:
SK = x =
c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK =
0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 73616d706c65 (ASCII "sample")
In SSWU: uniform_bytes = 5024e98d6067dec313af09ff0cbe78218324a645c
2a4b0aae2453f6fe91aa3bd9471f7b4a5fbf128e4b53f0c59603f7e
In SSWU: u =
df565615a2372e8b31b8771f7503bafc144e48b05688b97958cc27ce29a8d810
In SSWU: x1 =
e7e39eb8a4c982426fcff629e55a3e13516cfeb62c02c369b1e750316f5e94eb
In SSWU: gx1 is a nonsquare
H =
02b31973e872d4a097e2cfae9f37af9f9d73428fde74ac537dda93b5f18dbc5842
k =
e92820035a0a8afe132826c6312662b6ea733fc1a0d33737945016de54d02dd8
U = k*B =
031490f49d0355ffcdf66e40df788bee93861917ee713acff79be40d20cc91a30a
V = k*H =
03701df0228138fa3d16612c0d720389326b3265151bc7ac696ea4d0591cd053e3
pi = 0331d984ca8fece9cbb9a144c0d53df3c4c7a33080c1e02ddb1a96a365394
c7888a39dfe7432f119228473f37db3f87ca470c63b0237432a791f18f823c1215
e276b7ac0962725ba8daec2bf90c0ccc91a
beta =
21e66dc9747430f17ed9efeda054cf4a264b097b9e8956a1787526ed00dc664b
Example 5:
SK = x =
c9afa9d845ba75166b5c215767b1d6934e50c3db36e89b127b8a622b120f6721
PK =
0360fed4ba255a9d31c961eb74c6356d68c049b8923b61fa6ce669622e60f29fb6
alpha = 74657374 (ASCII "test")
In SSWU: uniform_bytes = 910cc66d84a57985a1d15843dad83fd9138a109af
b243b7fa5d64d766ec9ca3894fdcf46ebeb21a3972eb452a4232fd3
In SSWU: u =
d8b0107f7e7aa36390240d834852f8703a6dc407019d6196bda5861b8fc00181
In SSWU: x1 =
ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7
In SSWU: gx1 is a square
H =
03ccc747fa7318b9486ce4044adbbecaa084c27be6eda88eb7b7f3d688fd0968c7
k =
febc3451ea7639fde2cf41ffd03f463124ecb3b5a79913db1ed069147c8a7dea
U = k*B =
031200f9900e96f811d1247d353573f47e0d9da601fc992566234fc1a5b37749ae
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V = k*H =
02d3715dcfee136c7ae50e95ffca76f4ca6c29ddfb92a39c31a0d48e75c6605cd1
pi = 03f814c0455d32dbc75ad3aea08c7e2db31748e12802db23640203aebf1fa
8db2721e0499b7cecd68027a82f6095da076625a5f2f62908f1c283d5ee9b9e852
d85bedf64f2452a4e5094729e101824443e
beta =
8e7185d2b420e4f4681f44ce313a26d05613323837da09a69f00491a83ad25dd
The example secret key in Example 6 is taken from Appendix L.4.2 of
[ANSI.X9-62-2005].
Example 6:
SK = x =
2ca1411a41b17b24cc8c3b089cfd033f1920202a6c0de8abb97df1498d50d2c8
PK =
03596375e6ce57e0f20294fc46bdfcfd19a39f8161b58695b3ec5b3d16427c274d
alpha = 4578616d706c65207573696e67204543445341206b65792066726f6d20
417070656e646978204c2e342e32206f6620414e53492e58392d36322d32303035
(ASCII "Example using ECDSA key from Appendix L.4.2 of
ANSI.X9-62-2005")
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
e99d5288aac70a5e4bd07df303c1d460eb6336bb5fa95436a07c2f6b7aec6fef7c
c4846ea901ee1e238dee12bf752029b0b2e
beta =
4fbadf33b42a5f42f23a6f89952d2e634a6e3810f15878b46ef1bb85a04fe95a
A.3. ECVRF-EDWARDS25519-SHA512-TAI
The example secret keys and messages in Examples 7, 8, and 9 are
taken from Section 7.1 of [RFC8032].
Example 7:
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SK =
9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK =
d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x =
307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
try_and_increment succeeded on ctr = 0
H =
91bbed02a99461df1ad4c6564a5f5d829d0b90cfc7903e7a5797bd658abf3318
k = 7100f3d9eadb6dc4743b029736ff283f5be494128df128df2817106f345b85
94b6d6da2d6fb0b4c0257eb337675d96eab49cf39e66cc2c9547c2bf8b2a6afae4
U = k*B =
aef27c725be964c6a9bf4c45ca8e35df258c1878b838f37d9975523f09034071
V = k*H =
5016572f71466c646c119443455d6cb9b952f07d060ec8286d678615d55f954f
pi = 8657106690b5526245a92b003bb079ccd1a92130477671f6fc01ad16f26f7
23f5e8bd1839b414219e8626d393787a192241fc442e6569e96c462f62b8079b9e
d83ff2ee21c90c7c398802fdeebea4001
beta = 90cf1df3b703cce59e2a35b925d411164068269d7b2d29f3301c03dd757
876ff66b71dda49d2de59d03450451af026798e8f81cd2e333de5cdf4f3e140fdd
8ae
Example 8:
SK =
4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK =
3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x =
68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
try_and_increment succeeded on ctr = 1
H =
5b659fc3d4e9263fd9a4ed1d022d75eaacc20df5e09f9ea937502396598dc551
k = 42589bbf0c485c3c91c1621bb4bfe04aed7be76ee48f9b00793b2342acb9c1
67cab856f9f9d4febc311330c20b0a8afd3743d05433e8be8d32522ecdc16cc5ce
U = k*B =
1dcb0a4821a2c48bf53548228b7f170962988f6d12f5439f31987ef41f034ab3
V = k*H =
fd03c0bf498c752161bae4719105a074630a2aa5f200ff7b3995f7bfb1513423
pi = f3141cd382dc42909d19ec5110469e4feae18300e94f304590abdced48aed
593f7eaf3eb2f1a968cba3f6e23b386aeeaab7b1ea44a256e811892e13eeae7c9f
6ea8992557453eac11c4d5476b1f35a08
beta = eb4440665d3891d668e7e0fcaf587f1b4bd7fbfe99d0eb2211ccec90496
310eb5e33821bc613efb94db5e5b54c70a848a0bef4553a41befc57663b56373a5
031
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Example 9:
SK =
c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
PK =
fc51cd8e6218a1a38da47ed00230f0580816ed13ba3303ac5deb911548908025
alpha = af82 (2 bytes)
x =
909a8b755ed902849023a55b15c23d11ba4d7f4ec5c2f51b1325a181991ea95c
try_and_increment succeeded on ctr = 0
H =
bf4339376f5542811de615e3313d2b36f6f53c0acfebb482159711201192576a
k = 38b868c335ccda94a088428cbf3ec8bc7955bfaffe1f3bd2aa2c59fc31a0fe
bc59d0e1af3715773ce11b3bbdd7aba8e3505d4b9de6f7e4a96e67e0d6bb6d6c3a
U = k*B =
2bae73e15a64042fcebf062abe7e432b2eca6744f3e8265bc38e009cd577ecd5
V = k*H =
88cba1cb0d4f9b649d9a86026b69de076724a93a65c349c988954f0961c5d506
pi = 9bc0f79119cc5604bf02d23b4caede71393cedfbb191434dd016d30177ccb
f80e29dc513c01c3a980e0e545bcd848222d08a6c3e3665ff5a4cab13a643bef81
2e284c6b2ee063a2cb4f456794723ad0a
beta = 645427e5d00c62a23fb703732fa5d892940935942101e456ecca7bb217c
61c452118fec1219202a0edcf038bb6373241578be7217ba85a2687f7a0310b2df
19f
A.4. ECVRF-EDWARDS25519-SHA512-ELL2
The example secret keys and messages in Examples 10, 11, and 12 are
taken from Section 7.1 of [RFC8032].
Example 10:
SK =
9d61b19deffd5a60ba844af492ec2cc44449c5697b326919703bac031cae7f60
PK =
d75a980182b10ab7d54bfed3c964073a0ee172f3daa62325af021a68f707511a
alpha = (the empty string)
x =
307c83864f2833cb427a2ef1c00a013cfdff2768d980c0a3a520f006904de94f
In Elligator2: uniform_bytes = d620782a206d9de584b74e23ae5ee1db5ca
5298b3fc527c4867f049dee6dd419b3674967bd614890f621c128d72269ae
In Elligator2: u =
30f037b9745a57a9a2b8a68da81f397c39d46dee9d047f86c427c53f8b29a55c
In Elligator2: gx1 =
8cb66318fb2cea01672d6c27a5ab662ae33220961607f69276080a56477b4a08
In Elligator2: gx1 is a square
H =
b8066ebbb706c72b64390324e4a3276f129569eab100c26b9f05011200c1bad9
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k = b5682049fee54fe2d519c9afff73bbfad724e69a82d5051496a42458f817be
d7a386f96b1a78e5736756192aeb1818a20efb336a205ffede351cfe88dab8d41c
U = k*B =
762f5c178b68f0cddcc1157918edf45ec334ac8e8286601a3256c3bbf858edd9
V = k*H =
4652eba1c4612e6fce762977a59420b451e12964adbe4fbecd58a7aeff5860af
pi = 7d9c633ffeee27349264cf5c667579fc583b4bda63ab71d001f89c10003ab
46f25898f6bd7d4ed4c75f0282b0f7bb9d0e61b387b76db60b3cbf34bf09109ccb
33fab742a8bddc0c8ba3caf5c0b75bb04
beta = 9d574bf9b8302ec0fc1e21c3ec5368269527b87b462ce36dab2d14ccf80
c53cccf6758f058c5b1c856b116388152bbe509ee3b9ecfe63d93c3b4346c1fbc6
c54
Example 11:
SK =
4ccd089b28ff96da9db6c346ec114e0f5b8a319f35aba624da8cf6ed4fb8a6fb
PK =
3d4017c3e843895a92b70aa74d1b7ebc9c982ccf2ec4968cc0cd55f12af4660c
alpha = 72 (1 byte)
x =
68bd9ed75882d52815a97585caf4790a7f6c6b3b7f821c5e259a24b02e502e51
In Elligator2: uniform_bytes = 04ae20a9ad2a2330fb33318e376a2448bd7
7bb99e81d126f47952b156590444a9225b84128b66a2f15b41294fa2f2f6d
In Elligator2: u =
3092f033b16d4d5f74a3f7dc7091fe434b449065152b95476f121de899bb773d
In Elligator2: gx1 =
25d7fe7f82456e7078e99fdb24ef2582b4608357cdba9c39a8d535a3fd98464d
In Elligator2: gx1 is a nonsquare
H =
76ac3ccb86158a9104dff819b1ca293426d305fd76b39b13c9356d9b58c08e57
k = 88bf479281fd29a6cbdffd67e2c5ec0024d92f14eaed58f43f22f37c4c37f1
d41e65c036fbf01f9fba11d554c07494d0c02e7e5c9d64be88ef78cab7544e444d
U = k*B =
8ec26e77b8cb3114dd2265fe1564a4efb40d109aa3312536d93dfe3d8d80a061
V = k*H =
fe799eb5770b4e3a5a27d22518bb631db183c8316bb552155f442c62a47d1c8b
pi = 47b327393ff2dd81336f8a2ef10339112401253b3c714eeda879f12c50907
2ef9bf1a234f833f72d8fff36075fd9b836da28b5569e74caa418bae7ef521f2dd
d35f5727d271ecc70b4a83c1fc8ebc40c
beta = 38561d6b77b71d30eb97a062168ae12b667ce5c28caccdf76bc88e093e4
635987cd96814ce55b4689b3dd2947f80e59aac7b7675f8083865b46c89b2ce9cc
735
Example 12:
SK =
c5aa8df43f9f837bedb7442f31dcb7b166d38535076f094b85ce3a2e0b4458f7
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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 = a7ddd74a3a7d165d511b02fa268710ddbb3b939282d276fa2efcfa5aaf79cf
576087299ca9234aacd7cd674d912deba00f4e291733ef189a51e36c861b3d683b
U = k*B =
a012f35433df219a88ab0f9481f4e0065d00422c3285f3d34a8b0202f20bac60
V = k*H =
fb613986d171b3e98319c7ca4dc44c5dd8314a6e5616c1a4f16ce72bd7a0c25a
pi = 926e895d308f5e328e7aa159c06eddbe56d06846abf5d98c2512235eaa57f
dce6187befa109606682503b3a1424f0f729ca0418099fbd86a48093e6a8de2630
7b8d93e02da927e6dd5b73c8f119aee0f
beta = 121b7f9b9aaaa29099fc04a94ba52784d44eac976dd1a3cca458733be5c
d090a7b5fbd148444f17f8daf1fb55cb04b1ae85a626e30a54b4b0f8abf4a43314
a58
Authors' Addresses
Sharon Goldberg
Boston University
111 Cummington Mall
Boston, MA 02215
United States of America
Email: goldbe@cs.bu.edu
Leonid Reyzin
Boston University and Algorand
111 Cummington Mall
Boston, MA 02215
United States of America
Email: reyzin@bu.edu
Goldberg, et al. Expires 21 May 2022 [Page 41]
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Dimitrios Papadopoulos
Hong Kong University of Science and Technology
Clearwater Bay
Hong Kong
Email: dipapado@cse.ust.hk
Jan Vcelak
NS1
16 Beaver St
New York, NY 10004
United States of America
Email: jvcelak@ns1.com
Goldberg, et al. Expires 21 May 2022 [Page 42]