Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-11
The information below is for an old version of the document.
| Document | Type |
This is an older version of an Internet-Draft whose latest revision state is "Active".
|
|
|---|---|---|---|
| Authors | Deirdre Connolly , Chelsea Komlo , Ian Goldberg , Christopher A. Wood | ||
| Last updated | 2022-11-14 (Latest revision 2022-10-07) | ||
| RFC stream | Internet Research Task Force (IRTF) | ||
| Formats | |||
| IETF conflict review | conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost | ||
| Additional resources | Mailing list discussion | ||
| Stream | IRTF state | In RG Last Call | |
| Consensus boilerplate | Unknown | ||
| Document shepherd | Stanislav V. Smyshlyaev | ||
| IESG | IESG state | I-D Exists | |
| Telechat date | (None) | ||
| Responsible AD | (None) | ||
| Send notices to | smyshsv@gmail.com |
draft-irtf-cfrg-frost-11
CFRG D. Connolly
Internet-Draft Zcash Foundation
Intended status: Informational C. Komlo
Expires: 10 April 2023 University of Waterloo, Zcash Foundation
I. Goldberg
University of Waterloo
C. A. Wood
Cloudflare
7 October 2022
Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-11
Abstract
In this draft, we present the two-round signing variant of FROST, a
Flexible Round-Optimized Schnorr Threshold signature scheme. FROST
signatures can be issued after a threshold number of entities
cooperate to issue a signature, allowing for improved distribution of
trust and redundancy with respect to a secret key. Further, this
draft specifies signatures that are compatible with [RFC8032].
However, unlike [RFC8032], the protocol for producing signatures in
this draft is not deterministic, so as to ensure protection against a
key-recovery attack that is possible when even only one signer
participant is malicious.
Discussion Venues
This note is to be removed before publishing as an RFC.
Discussion of this document takes place on the Crypto Forum Research
Group mailing list (cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg.
Source for this draft and an issue tracker can be found at
https://github.com/cfrg/draft-irtf-cfrg-frost.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
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Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on 10 April 2023.
Copyright Notice
Copyright (c) 2022 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. Code Components
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described in Section 4.e of the Trust Legal Provisions and are
provided without warranty as described in the Revised BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 8
3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 8
3.1. Prime-Order Group . . . . . . . . . . . . . . . . . . . . 9
3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 10
4. Helper Functions . . . . . . . . . . . . . . . . . . . . . . 10
4.1. Nonce generation . . . . . . . . . . . . . . . . . . . . 11
4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 11
4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 11
4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 12
4.3. List Operations . . . . . . . . . . . . . . . . . . . . . 13
4.4. Binding Factors Computation . . . . . . . . . . . . . . . 15
4.5. Group Commitment Computation . . . . . . . . . . . . . . 15
4.6. Signature Challenge Computation . . . . . . . . . . . . . 16
5. Two-Round FROST Signing Protocol . . . . . . . . . . . . . . 17
5.1. Round One - Commitment . . . . . . . . . . . . . . . . . 20
5.2. Round Two - Signature Share Generation . . . . . . . . . 21
5.3. Signature Share Verification and Aggregation . . . . . . 23
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 25
6.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 26
6.2. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 27
6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 29
6.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 30
6.5. FROST(secp256k1, SHA-256) . . . . . . . . . . . . . . . . 32
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6.6. Ciphersuite Requirements . . . . . . . . . . . . . . . . 33
7. Security Considerations . . . . . . . . . . . . . . . . . . . 33
7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 34
7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 35
7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 35
7.4. Input Message Hashing . . . . . . . . . . . . . . . . . . 35
7.5. Input Message Validation . . . . . . . . . . . . . . . . 36
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 36
8.1. Normative References . . . . . . . . . . . . . . . . . . 36
8.2. Informative References . . . . . . . . . . . . . . . . . 37
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 38
Appendix B. Schnorr Signature Generation and Verification for
Prime-Order Groups . . . . . . . . . . . . . . . . . . . 38
Appendix C. Trusted Dealer Key Generation . . . . . . . . . . . 40
C.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 41
C.1.1. Deriving the constant term of a polynomial . . . . . 43
C.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 44
Appendix D. Random Scalar Generation . . . . . . . . . . . . . . 46
D.1. Rejection Sampling . . . . . . . . . . . . . . . . . . . 46
D.2. Wide Reduction . . . . . . . . . . . . . . . . . . . . . 46
Appendix E. Test Vectors . . . . . . . . . . . . . . . . . . . . 46
E.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 47
E.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 49
E.3. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 51
E.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 52
E.5. FROST(secp256k1, SHA-256) . . . . . . . . . . . . . . . . 54
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 56
1. Introduction
DISCLAIMER: This is a work-in-progress draft of FROST.
RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this
draft is maintained in GitHub. Suggested changes should be submitted
as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost.
Instructions are on that page as well.
Unlike signatures in a single-party setting, threshold signatures
require cooperation among a threshold number of signing participants
each holding a share of a common private key. The security of
threshold schemes in general assumes that an adversary can corrupt
strictly fewer than a threshold number of signer participants.
This document presents a variant of a Flexible Round-Optimized
Schnorr Threshold (FROST) signature scheme originally defined in
[FROST20]. FROST reduces network overhead during threshold signing
operations while employing a novel technique to protect against
forgery attacks applicable to prior Schnorr-based threshold signature
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constructions. The variant of FROST presented in this document
requires two rounds to compute a signature. Single-round signing
with FROST is out of scope.
For select ciphersuites, the signatures produced by this draft are
compatible with [RFC8032]. However, unlike [RFC8032], signatures
produced by FROST are not deterministic, since deriving nonces
deterministically allows for a complete key-recovery attack in multi-
party discrete logarithm-based signatures, such as FROST.
While an optimization to FROST was shown in [Schnorr21] that reduces
scalar multiplications from linear in the number of signing
participants to constant, this draft does not specify that
optimization due to the malleability that this optimization
introduces, as shown in [StrongerSec22]. Specifically, this
optimization removes the guarantee that the set of signer
participants that started round one of the protocol is the same set
of signing participants that produced the signature output by round
two.
Key generation for FROST signing is out of scope for this document.
However, for completeness, key generation with a trusted dealer is
specified in Appendix C.
1.1. Change Log
draft-11
* Update version string constant (#307)
* Make SerializeElement reject the identity element (#306)
* Make ciphersuite requirements explicit (#302)
* Fix various editorial issues (#303, #301, #299, #297)
draft-10
* Update version string constant (#296)
* Fix some editorial issues from Ian Goldberg (#295)
draft-09
* Add single-signer signature generation to complement RFC8032
functions (#293)
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* Address Thomas Pornin review comments from
https://mailarchive.ietf.org/arch/msg/crypto-panel/
bPyYzwtHlCj00g8YF1tjj-iYP2c/ (#292, #291, #290, #289, #287, #286,
#285, #282, #281, #280, #279, #278, #277, #276, #275, #273, #272,
#267)
* Correct Ed448 ciphersuite (#246)
* Various editorial changes (#241, #240)
draft-08
* Add notation for Scalar multiplication (#237)
* Add secp2561k1 ciphersuite (#223)
* Remove RandomScalar implementation details (#231)
* Add domain separation for message and commitment digests (#228)
draft-07
* Fix bug in per-rho signer computation (#222)
draft-06
* Make verification a per-ciphersuite functionality (#219)
* Use per-signer values of rho to mitigate protocol malleability
(#217)
* Correct prime-order subgroup checks (#215, #211)
* Fix bug in ed25519 ciphersuite description (#205)
* Various editorial improvements (#208, #209, #210, #218)
draft-05
* Update test vectors to include version string (#202, #203)
* Rename THRESHOLD_LIMIT to MIN_PARTICIPANTS (#192)
* Use non-contiguous signers for the test vectors (#187)
* Add more reasoning why the coordinator MUST abort (#183)
* Add a function to generate nonces (#182)
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* Add MUST that all participants have the same view of VSS
commitment (#174)
* Use THRESHOLD_LIMIT instead of t and MAX_PARTICIPANTS instead of n
(#171)
* Specify what the dealer is trusted to do (#166)
* Clarify types of NUM_PARTICIPANTS and THRESHOLD_LIMIT (#165)
* Assert that the network channel used for signing should be
authenticated (#163)
* Remove wire format section (#156)
* Update group commitment derivation to have a single scalarmul
(#150)
* Use RandomNonzeroScalar for single-party Schnorr example (#148)
* Fix group notation and clarify member functions (#145)
* Update existing implementations table (#136)
* Various editorial improvements (#135, #143, #147, #149, #153,
#158, #162, #167, #168, #169, #170, #175, #176, #177, #178, #184,
#186, #193, #198, #199)
draft-04
* Added methods to verify VSS commitments and derive group info
(#126, #132).
* Changed check for participants to consider only nonnegative
numbers (#133).
* Changed sampling for secrets and coefficients to allow the zero
element (#130).
* Split test vectors into separate files (#129)
* Update wire structs to remove commitment shares where not
necessary (#128)
* Add failure checks (#127)
* Update group info to include each participant's key and clarify
how public key material is obtained (#120, #121).
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* Define cofactor checks for verification (#118)
* Various editorial improvements and add contributors (#124, #123,
#119, #116, #113, #109)
draft-03
* Refactor the second round to use state from the first round (#94).
* Ensure that verification of signature shares from the second round
uses commitments from the first round (#94).
* Clarify RFC8032 interoperability based on PureEdDSA (#86).
* Specify signature serialization based on element and scalar
serialization (#85).
* Fix hash function domain separation formatting (#83).
* Make trusted dealer key generation deterministic (#104).
* Add additional constraints on participant indexes and nonce usage
(#105, #103, #98, #97).
* Apply various editorial improvements.
draft-02
* Fully specify both rounds of FROST, as well as trusted dealer key
generation.
* Add ciphersuites and corresponding test vectors, including suites
for RFC8032 compatibility.
* Refactor document for editorial clarity.
draft-01
* Specify operations, notation and cryptographic dependencies.
draft-00
* Outline CFRG draft based on draft-komlo-frost.
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2. Conventions and Definitions
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.
The following notation is used throughout the document.
* random_bytes(n): Outputs n bytes, sampled uniformly at random
using a cryptographically secure pseudorandom number generator
(CSPRNG).
* count(i, L): Outputs the number of times the element i is
represented in the list L.
* len(l): Outputs the length of input list l, e.g., len([1,2,3]) =
3).
* reverse(l): Outputs the list l in reverse order, e.g.,
reverse([1,2,3]) = [3,2,1].
* range(a, b): Outputs a list of integers from a to b-1 in ascending
order, e.g., range(1, 4) = [1,2,3].
* pow(a, b): Outputs the integer result of a to the power of b,
e.g., pow(2, 3) = 8.
* || denotes concatenation of byte strings, i.e., x || y denotes the
byte string x, immediately followed by the byte string y, with no
extra separator, yielding xy.
* nil denotes an empty byte string.
Unless otherwise stated, we assume that secrets are sampled uniformly
at random using a cryptographically secure pseudorandom number
generator (CSPRNG); see [RFC4086] for additional guidance on the
generation of random numbers.
3. Cryptographic Dependencies
FROST signing depends on the following cryptographic constructs:
* Prime-order Group, Section 3.1;
* Cryptographic hash function, Section 3.2;
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These are described in the following sections.
3.1. Prime-Order Group
FROST depends on an abelian group of prime order p. We represent
this group as the object G that additionally defines helper functions
described below. The group operation for G is addition + with
identity element I. For any elements A and B of the group G, A + B =
B + A is also a member of G. Also, for any A in G, there exists an
element -A such that A + (-A) = (-A) + A = I. For convenience, we
use - to denote subtraction, e.g., A - B = A + (-B). Integers, taken
modulo the group order p, are called scalars; arithmetic operations
on scalars are implicitly performed modulo p. Since p is prime,
scalars form a finite field. Scalar multiplication is equivalent to
the repeated application of the group operation on an element A with
itself r-1 times, denoted as ScalarMult(A, r). We denote the sum,
difference, and product of two scalars using the +, -, and *
operators, respectively. (Note that this means + may refer to group
element addition or scalar addition, depending on types of the
operands.) For any element A, ScalarMult(A, p) = I. We denote B as
a fixed generator of the group. Scalar base multiplication is
equivalent to the repeated application of the group operation B with
itself r-1 times, this is denoted as ScalarBaseMult(r). The set of
scalars corresponds to GF(p), which we refer to as the scalar field.
This document uses types Element and Scalar to denote elements of the
group G and its set of scalars, respectively. We denote Scalar(x) as
the conversion of integer input x to the corresponding Scalar value
with the same numeric value. For example, Scalar(1) yields a Scalar
representing the value 1. We denote equality comparison as == and
assignment of values by =. Finally, it is assumed that group element
addition, negation, and equality comparisons can be efficiently
computed for arbitrary group elements.
We now detail a number of member functions that can be invoked on G.
* Order(): Outputs the order of G (i.e. p).
* Identity(): Outputs the identity Element of the group (i.e. I).
* RandomScalar(): Outputs a random Scalar element in GF(p), i.e., a
random scalar in [0, p - 1].
* ScalarMult(A, k): Output the scalar multiplication between Element
A and Scalar k.
* ScalarBaseMult(k): Output the scalar multiplication between Scalar
k and the group generator B.
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* SerializeElement(A): Maps an Element A to a canonical byte array
buf of fixed length Ne. This function can raise an error if A is
the identity element of the group.
* DeserializeElement(buf): Attempts to map a byte array buf to an
Element A, and fails if the input is not the valid canonical byte
representation of an element of the group. This function can
raise an error if deserialization fails or A is the identity
element of the group; see Section 6 for group-specific input
validation steps.
* SerializeScalar(s): Maps a Scalar s to a canonical byte array buf
of fixed length Ns.
* DeserializeScalar(buf): Attempts to map a byte array buf to a
Scalar s. This function can raise an error if deserialization
fails; see Section 6 for group-specific input validation steps.
3.2. Cryptographic Hash Function
FROST requires the use of a cryptographically secure hash function,
generically written as H, which functions effectively as a random
oracle. For concrete recommendations on hash functions which SHOULD
be used in practice, see Section 6. Using H, we introduce separate
domain-separated hashes, H1, H2, H3, H4, and H5:
* H1, H2, and H3 map arbitrary byte strings to Scalar elements of
the prime-order group scalar field.
* H4 and H5 are aliases for H with distinct domain separators.
The details of H1, H2, H3, H4, and H5 vary based on ciphersuite. See
Section 6 for more details about each.
4. Helper Functions
Beyond the core dependencies, the protocol in this document depends
on the following helper operations:
* Nonce generation, Section 4.1;
* Polynomial operations, Section 4.2;
* Encoding operations, Section 4.3;
* Signature binding Section 4.4, group commitment Section 4.5, and
challenge computation Section 4.6.
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These sections describes these operations in more detail.
4.1. Nonce generation
To hedge against a bad RNG that outputs predictable values, nonces
are generated with the nonce_generate function by combining fresh
randomness with the secret key as input to a domain-separated hash
function built from the ciphersuite hash function H. This domain-
separated hash function is denoted H3. This function always samples
32 bytes of fresh randomness to ensure that the probability of nonce
reuse is at most 2^-128 as long as no more than 2^64 signatures are
computed by a given signing participant.
nonce_generate(secret):
Inputs:
- secret, a Scalar
Outputs: nonce, a Scalar
def nonce_generate(secret):
random_bytes = random_bytes(32)
secret_enc = G.SerializeScalar(secret)
return H3(random_bytes || secret_enc)
4.2. Polynomial Operations
This section describes operations on and associated with polynomials
over Scalars that are used in the main signing protocol. A
polynomial of maximum degree t+1 is represented as a list of t
coefficients, where the constant term of the polynomial is in the
first position and the highest-degree coefficient is in the last
position. A point on the polynomial is a tuple (x, y), where y =
f(x). For notational convenience, we refer to the x-coordinate and
y-coordinate of a point p as p.x and p.y, respectively.
4.2.1. Evaluation of a polynomial
This section describes a method for evaluating a polynomial f at a
particular input x, i.e., y = f(x) using Horner's method.
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polynomial_evaluate(x, coeffs):
Inputs:
- x, input at which to evaluate the polynomial, a Scalar
- coeffs, the polynomial coefficients, a list of Scalars
Outputs: Scalar result of the polynomial evaluated at input x
def polynomial_evaluate(x, coeffs):
value = 0
for coeff in reverse(coeffs):
value *= x
value += coeff
return value
4.2.2. Lagrange coefficients
The function derive_lagrange_coefficient derives a Lagrange
coefficient to later perform polynomial interpolation, and is
provided a list of x-coordinates as input. Note that
derive_lagrange_coefficient does not permit any x-coordinate to equal
0. Lagrange coefficients are used in FROST to evaluate a polynomial
f at x-coordinate 0, i.e., f(0), given a list of t other
x-coordinates.
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derive_lagrange_coefficient(x_i, L):
Inputs:
- x_i, an x-coordinate contained in L, a Scalar
- L, the set of x-coordinates, each a Scalar
Outputs: L_i, the i-th Lagrange coefficient
Errors:
- "invalid parameters", if 1) any x-coordinate is equal to 0, 2) if x_i
is not in L, or if 3) any x-coordinate is represented more than once in L.
def derive_lagrange_coefficient(x_i, L):
if x_i == 0:
raise "invalid parameters"
for x_j in L:
if x_j == 0:
raise "invalid parameters"
if x_i not in L:
raise "invalid parameters"
for x_j in L:
if count(x_j, L) > 1:
raise "invalid parameters"
numerator = Scalar(1)
denominator = Scalar(1)
for x_j in L:
if x_j == x_i: continue
numerator *= x_j
denominator *= x_j - x_i
L_i = numerator / denominator
return L_i
4.3. List Operations
This section describes helper functions that work on lists of values
produced during the FROST protocol. The following function encodes a
list of participant commitments into a bytestring for use in the
FROST protocol.
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Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates the participant identifier i and their two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by participant identifier.
Outputs: A byte string containing the serialized representation of commitment_list
def encode_group_commitment_list(commitment_list):
encoded_group_commitment = nil
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
encoded_commitment = G.SerializeScalar(identifier) ||
G.SerializeElement(hiding_nonce_commitment) ||
G.SerializeElement(binding_nonce_commitment)
encoded_group_commitment = encoded_group_commitment || encoded_commitment
return encoded_group_commitment
The following function is used to extract participant identifiers
from a commitment list.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates the participant identifier i and their two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by participant identifier.
Outputs: A list of participant identifiers
def participants_from_commitment_list(commitment_list):
identifiers = []
for (identifier, _, _) in commitment_list:
identifiers.append(identifier)
return identifiers
The following function is used to extract a binding factor from a
list of binding factors.
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Inputs:
- binding_factor_list = [(i, binding_factor), ...],
a list of binding factors for each participant, where each element in the list
indicates the participant identifier i and their binding factor. This list MUST be sorted
in ascending order by participant identifier.
- identifier, participant identifier, a Scalar.
Outputs: A Scalar value.
Errors: "invalid participant", when the designated participant is not known
def binding_factor_for_participant(binding_factor_list, identifier):
for (i, binding_factor) in binding_factor_list:
if identifier == i:
return binding_factor
raise "invalid participant"
4.4. Binding Factors Computation
This section describes the subroutine for computing binding factors
based on the participant commitment list and message to be signed.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates the participant identifier i and their two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by participant identifier.
- msg, the message to be signed.
Outputs: A list of (identifier, Scalar) tuples representing the binding factors.
def compute_binding_factors(commitment_list, msg):
msg_hash = H4(msg)
encoded_commitment_hash = H5(encode_group_commitment_list(commitment_list))
rho_input_prefix = msg_hash || encoded_commitment_hash
binding_factor_list = []
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
rho_input = rho_input_prefix || G.SerializeScalar(identifier)
binding_factor = H1(rho_input)
binding_factor_list.append((identifier, binding_factor))
return binding_factor_list
4.5. Group Commitment Computation
This section describes the subroutine for creating the group
commitment from a commitment list.
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Inputs:
- commitment_list =
[(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...], a list
of commitments issued by each participant, where each element in the list
indicates the participant identifier i and their two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be
sorted in ascending order by participant identifier.
- binding_factor_list = [(i, binding_factor), ...],
a list of (identifier, Scalar) tuples representing the binding factor Scalar
for the given identifier. This list MUST be sorted in ascending order by identifier.
Outputs: An Element in G representing the group commitment
def compute_group_commitment(commitment_list, binding_factor_list):
group_commitment = G.Identity()
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
binding_factor = binding_factor_for_participant(binding_factors, identifier)
group_commitment = group_commitment +
hiding_nonce_commitment + G.ScalarMult(binding_nonce_commitment, binding_factor)
return group_commitment
4.6. Signature Challenge Computation
This section describes the subroutine for creating the per-message
challenge.
Inputs:
- group_commitment, an Element in G representing the group commitment
- group_public_key, public key corresponding to the group signing key, an
Element in G.
- msg, the message to be signed.
Outputs: A Scalar representing the challenge
def compute_challenge(group_commitment, group_public_key, msg):
group_comm_enc = G.SerializeElement(group_commitment)
group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg
challenge = H2(challenge_input)
return challenge
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5. Two-Round FROST Signing Protocol
This section describes the two-round variant of the FROST threshold
signature protocol for producing Schnorr signatures. The protocol is
configured to run with a selection of NUM_PARTICIPANTS signer
participants and a Coordinator. NUM_PARTICIPANTS is a positive
integer at least MIN_PARTICIPANTS but no larger than
MAX_PARTICIPANTS, where MIN_PARTICIPANTS <= MAX_PARTICIPANTS,
MIN_PARTICIPANTS is a positive integer and MAX_PARTICIPANTS is a
positive integer less than the group order. A signer participant, or
simply participant, is an entity that is trusted to hold and use a
signing key share. The Coordinator is an entity with the following
responsibilities:
1. Determining which participants will participate (at least
MIN_PARTICIPANTS in number);
2. Coordinating rounds (receiving and forwarding inputs among
participants); and
3. Aggregating signature shares output by each participant, and
publishing the resulting signature.
FROST assumes that the Coordinator and the set of signer
participants, are chosen externally to the protocol. Note that it is
possible to deploy the protocol without a distinguished Coordinator;
see Section 7.3 for more information.
FROST produces signatures that are indistinguishable from those
produced with a single participant using a signing key s with
corresponding public key PK, where s is a Scalar value and PK =
G.ScalarBaseMult(s). As a threshold signing protocol, the group
signing key s is secret-shared amongst each participant and used to
produce signatures. In particular, FROST assumes each participant is
configured with the following information:
* An identifier, which is a Scalar value denoted i in the range [1,
MAX_PARTICIPANTS] and MUST be distinct from the identifier of
every other participant.
* A signing key share sk_i, which is a Scalar value representing the
i-th secret share of the group signing key s. The public key
corresponding to this signing key share is PK_i =
G.ScalarBaseMult(sk_i).
The Coordinator and each participant are additionally configured with
common group information, denoted "group info," which consists of the
following:
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* Group public key, which is an Element in G denoted PK.
* Public keys PK_i for each participant, which are Element values in
G denoted PK_i for each i in [1, MAX_PARTICIPANTS].
This document does not specify how this information, including the
signing key shares, are configured and distributed to participants.
In general, two possible configuration mechanisms are possible: one
that requires a single, trusted dealer, and the other which requires
performing a distributed key generation protocol. We highlight key
generation mechanism by a trusted dealer in Appendix C for reference.
The signing variant of FROST in this document requires participants
to perform two network rounds: 1) generating and publishing
commitments, and 2) signature share generation and publication. The
first round serves for each participant to issue a commitment to a
nonce. The second round receives commitments for all participants as
well as the message, and issues a signature share with respect to
that message. The Coordinator performs the coordination of each of
these rounds. At the end of the second round, the Coordinator then
performs an aggregation step and outputs the final signature. This
complete interaction is shown in Figure 1.
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(group info) (group info, (group info,
| signing key share) signing key share)
| | |
v v v
Coordinator Signer-1 ... Signer-n
------------------------------------------------------------
message
------------>
|
== Round 1 (Commitment) ==
| participant commitment | |
|<-----------------------+ |
| ... |
| participant commitment (commit state) ==\
|<-----------------------------------------+ |
|
== Round 2 (Signature Share Generation) == |
| |
| participant input | | |
+------------------------> | |
| signature share | | |
|<-----------------------+ | |
| ... | |
| participant input | |
+------------------------------------------> /
| signature share |<=======/
<------------------------------------------+
|
== Aggregation ==
|
signature |
<-----------+
Figure 1: FROST signature overview
Details for round one are described in Section 5.1, and details for
round two are described in Section 5.2. Note that each participant
persists some state between the two rounds, and this state is deleted
as described in Section 5.2. The final Aggregation step is described
in Section 5.3.
FROST assumes that all inputs to each round, especially those of
which are received over the network, are validated before use. In
particular, this means that any value of type Element or Scalar is
deserialized using DeserializeElement and DeserializeScalar,
respectively, as these functions perform the necessary input
validation steps.
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FROST assumes reliable message delivery between the Coordinator and
participants in order for the protocol to complete. An attacker
masquerading as another participant will result only in an invalid
signature; see Section 7. However, in order to identify any
participant which has misbehaved (resulting in the protocol aborting)
to take actions such as excluding them from future signing
operations, we assume that the network channel is additionally
authenticated; confidentiality is not required.
5.1. Round One - Commitment
Round one involves each participant generating nonces and their
corresponding public commitments. A nonce is a pair of Scalar
values, and a commitment is a pair of Element values. Each
participant's behavior in this round is described by the commit
function below. Note that this function invokes nonce_generate
twice, once for each type of nonce produced. The output of this
function is a pair of secret nonces (hiding_nonce, binding_nonce) and
their corresponding public commitments (hiding_nonce_commitment,
binding_nonce_commitment).
Inputs: sk_i, the secret key share, a Scalar
Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs,
where each value in the nonce pair is a Scalar and each value in
the nonce commitment pair is an Element
def commit(sk_i):
hiding_nonce = nonce_generate(sk_i)
binding_nonce = nonce_generate(sk_i)
hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
nonce = (hiding_nonce, binding_nonce)
comm = (hiding_nonce_commitment, binding_nonce_commitment)
return (nonce, comm)
The outputs nonce and comm from participant P_i should both be stored
locally and kept for use in the second round. The nonce value is
secret and MUST NOT be shared, whereas the public output comm is sent
to the Coordinator. The nonce values produced by this function MUST
NOT be reused in more than one invocation of FROST, and it MUST be
generated from a source of secure randomness.
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5.2. Round Two - Signature Share Generation
In round two, the Coordinator is responsible for sending the message
to be signed, and for choosing which participants will participate
(of number at least MIN_PARTICIPANTS). Signers additionally require
locally held data; specifically, their private key and the nonces
corresponding to their commitment issued in round one.
The Coordinator begins by sending each participant the message to be
signed along with the set of signing commitments for all participants
in the participant list. Each participant MUST validate the inputs
before processing the Coordinator's request. In particular, the
Signer MUST validate commitment_list, deserializing each group
Element in the list using DeserializeElement from Section 3.1. If
deserialization fails, the Signer MUST abort the protocol. Moreover,
each participant MUST ensure that their identifier appears in
commitment_list along with their commitment from the first round.
Applications which require that participants not process arbitrary
input messages are also required to also perform relevant
application-layer input validation checks; see Section 7.5 for more
details.
Upon receipt and successful input validation, each Signer then runs
the following procedure to produce its own signature share.
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Inputs:
- identifier, Identifier i of the participant. Note identifier will never equal 0.
- sk_i, Signer secret key share, a Scalar.
- group_public_key, public key corresponding to the group signing key,
an Element in G.
- nonce_i, pair of Scalar values (hiding_nonce, binding_nonce) generated in
round one.
- msg, the message to be signed (sent by the Coordinator).
- commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each participant and sent by the Coordinator.
Each element in the list indicates the participant identifier j and their two commitment
Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by participant identifier.
Outputs: a Scalar value representing the signature share
def sign(identifier, sk_i, group_public_key, nonce_i, msg, commitment_list):
# Compute the binding factor(s)
binding_factor_list = compute_binding_factors(commitment_list, msg)
binding_factor = binding_factor_for_participant(binding_factor_list, identifier)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor_list)
# Compute Lagrange coefficient
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_lagrange_coefficient(identifier, participant_list)
# Compute the per-message challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge)
return sig_share
The output of this procedure is a signature share. Each participant
then sends these shares back to the Coordinator. Each participant
MUST delete the nonce and corresponding commitment after this round
completes, and MUST use the nonce to generate at most one signature
share.
Note that the lambda_i value derived during this procedure does not
change across FROST signing operations for the same signing group.
As such, participants can compute it once and store it for reuse
across signing sessions.
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Upon receipt from each Signer, the Coordinator MUST validate the
input signature share using DeserializeElement. If validation fails,
the Coordinator MUST abort the protocol. If validation succeeds, the
Coordinator then verifies the set of signature shares using the
following procedure.
5.3. Signature Share Verification and Aggregation
After participants perform round two and send their signature shares
to the Coordinator, the Coordinator verifies each signature share for
correctness. In particular, for each participant, the Coordinator
uses commitment pairs generated during round one and the signature
share generated during round two, along with other group parameters,
to check that the signature share is valid using the following
procedure.
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Inputs:
- identifier, Identifier i of the participant. Note: identifier MUST never equal 0.
- PK_i, the public key for the ith participant, where PK_i = G.ScalarBaseMult(sk_i),
an Element in G
- comm_i, pair of Element values in G (hiding_nonce_commitment, binding_nonce_commitment)
generated in round one from the ith participant.
- sig_share_i, a Scalar value indicating the signature share as produced in
round two from the ith participant.
- commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each participant, where each element
in the list indicates the participant identifier j and their two commitment
Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by participant identifier.
- group_public_key, public key corresponding to the group signing key,
an Element in G.
- msg, the message to be signed.
Outputs: True if the signature share is valid, and False otherwise.
def verify_signature_share(identifier, PK_i, comm_i, sig_share_i, commitment_list,
group_public_key, msg):
# Compute the binding factors
binding_factor_list = compute_binding_factors(commitment_list, msg)
binding_factor = binding_factor_for_participant(binding_factor_list, identifier)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor_list)
# Compute the commitment share
(hiding_nonce_commitment, binding_nonce_commitment) = comm_i
comm_share = hiding_nonce_commitment + G.ScalarMult(binding_nonce_commitment, binding_factor)
# Compute the challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute Lagrange coefficient
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_lagrange_coefficient(identifier, participant_list)
# Compute relation values
l = G.ScalarBaseMult(sig_share_i)
r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i)
return l == r
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If any signature share fails to verify, i.e., if
verify_signature_share returns False for any participant share, the
Coordinator MUST abort the protocol for correctness reasons (this is
true regardless of the size or makeup of the signing set selected by
the Coordinator). Excluding one participant means that their nonce
will not be included in the joint response z and consequently the
output signature will not verify. This is because the group
commitment will be with respect to a different signing set than the
the aggregated response.
Otherwise, if all shares from participants that participated in
Rounds 1 and 2 are valid, the Coordinator performs the aggregate
operation and publishes the resulting signature.
Inputs:
- group_commitment, the group commitment returned by compute_group_commitment,
an Element in G.
- sig_shares, a set of signature shares z_i, Scalar values, for each participant,
of length NUM_PARTICIPANTS, where MIN_PARTICIPANTS <= NUM_PARTICIPANTS <= MAX_PARTICIPANTS.
Outputs: (R, z), a Schnorr signature consisting of an Element R and Scalar z.
def aggregate(group_commitment, sig_shares):
z = 0
for z_i in sig_shares:
z = z + z_i
return (group_commitment, z)
The output signature (R, z) from the aggregation step MUST be encoded
as follows (using notation from Section 3 of [TLS]):
struct {
opaque R_encoded[Ne];
opaque z_encoded[Ns];
} Signature;
Where Signature.R_encoded is G.SerializeElement(R) and
Signature.z_encoded is G.SerializeScalar(z).
6. Ciphersuites
A FROST ciphersuite must specify the underlying prime-order group
details and cryptographic hash function. Each ciphersuite is denoted
as (Group, Hash), e.g., (ristretto255, SHA-512). This section
contains some ciphersuites. Each ciphersuite also includes a context
string, denoted contextString, which is an ASCII string literal (with
no NULL terminating character).
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The RECOMMENDED ciphersuite is (ristretto255, SHA-512) Section 6.2.
The (Ed25519, SHA-512) ciphersuite is included for backwards
compatibility with [RFC8032].
The DeserializeElement and DeserializeScalar functions instantiated
for a particular prime-order group corresponding to a ciphersuite
MUST adhere to the description in Section 3.1. Validation steps for
these functions are described for each the ciphersuites below.
Future ciphersuites MUST describe how input validation is done for
DeserializeElement and DeserializeScalar.
Each ciphersuite includes explicit instructions for verifying
signatures produced by FROST. Note that these instructions are
equivalent to those produced by a single participant.
Each ciphersuite adheres to the requirements in Section 6.6. Future
ciphersuites MUST also adhere to these requirements.
6.1. FROST(Ed25519, SHA-512)
This ciphersuite uses edwards25519 for the Group and SHA-512 for the
Hash function H meant to produce signatures indistinguishable from
Ed25519 as specified in [RFC8032]. The value of the contextString
parameter is "FROST-ED25519-SHA512-v11".
* Group: edwards25519 [RFC8032]
- Order(): Return 2^252 + 27742317777372353535851937790883648493
(see [RFC7748])
- Identity(): As defined in [RFC7748].
- RandomScalar(): Implemented by returning a uniformly random
Scalar in the range [0, G.Order() - 1]. Refer to Appendix D
for implementation guidance.
- SerializeElement(A): Implemented as specified in [RFC8032],
Section 5.1.2. Additionally, this function validates that the
input element is not the group identity element.
- DeserializeElement(buf): Implemented as specified in [RFC8032],
Section 5.1.3. Additionally, this function validates that the
resulting element is not the group identity element and is in
the prime-order subgroup. The latter check can be implemented
by multiplying the resulting point by the order of the group
and checking that the result is the identity element. Note
that optimizations for this check exist; see [Pornin22].
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- SerializeScalar(s): Implemented by outputting the little-endian
32-byte encoding of the Scalar value with the top three bits
set to zero.
- DeserializeScalar(buf): Implemented by attempting to
deserialize a Scalar from a little-endian 32-byte string. This
function can fail if the input does not represent a Scalar in
the range [0, G.Order() - 1]. Note that this means the top
three bits of the input MUST be zero.
* Hash (H): SHA-512
- H1(m): Implemented by computing H(contextString || "rho" || m),
interpreting the 64-byte digest as a little-endian integer, and
reducing the resulting integer modulo
2^252+27742317777372353535851937790883648493.
- H2(m): Implemented by computing H(m), interpreting the 64-byte
digest as a little-endian integer, and reducing the resulting
integer modulo 2^252+27742317777372353535851937790883648493.
- H3(m): Implemented by computing H(contextString || "nonce" ||
m), interpreting the 64-byte digest as a little-endian integer,
and reducing the resulting integer modulo
2^252+27742317777372353535851937790883648493.
- H4(m): Implemented by computing H(contextString || "msg" || m).
- H5(m): Implemented by computing H(contextString || "com" || m).
Normally H2 would also include a domain separator, but for backwards
compatibility with [RFC8032], it is omitted.
Signature verification is as specified in Section 5.1.7 of [RFC8032]
with the constraint that implementations MUST check the group
equation [8][S]B = [8]R + [8][k]A'. The alternative check [S]B = R +
[k]A' is not safe or interoperable in practice.
6.2. FROST(ristretto255, SHA-512)
This ciphersuite uses ristretto255 for the Group and SHA-512 for the
Hash function H. The value of the contextString parameter is "FROST-
RISTRETTO255-SHA512-v11".
* Group: ristretto255 [RISTRETTO]
- Order(): Return 2^252 + 27742317777372353535851937790883648493
(see [RISTRETTO])
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- Identity(): As defined in [RISTRETTO].
- RandomScalar(): Implemented by returning a uniformly random
Scalar in the range [0, G.Order() - 1]. Refer to Appendix D
for implementation guidance.
- SerializeElement(A): Implemented using the 'Encode' function
from [RISTRETTO]. Additionally, this function validates that
the input element is not the group identity element.
- DeserializeElement(buf): Implemented using the 'Decode'
function from [RISTRETTO]. Additionally, this function
validates that the resulting element is not the group identity
element.
- SerializeScalar(s): Implemented by outputting the little-endian
32-byte encoding of the Scalar value with the top three bits
set to zero.
- DeserializeScalar(buf): Implemented by attempting to
deserialize a Scalar from a little-endian 32-byte string. This
function can fail if the input does not represent a Scalar in
the range [0, G.Order() - 1]. Note that this means the top
three bits of the input MUST be zero.
* Hash (H): SHA-512
- H1(m): Implemented by computing H(contextString || "rho" || m)
and mapping the output to a Scalar as described in [RISTRETTO],
Section 4.4.
- H2(m): Implemented by computing H(contextString || "chal" || m)
and mapping the output to a Scalar as described in [RISTRETTO],
Section 4.4.
- H3(m): Implemented by computing H(contextString || "nonce" ||
m) and mapping the output to a Scalar as described in
[RISTRETTO], Section 4.4.
- H4(m): Implemented by computing H(contextString || "msg" || m).
- H5(m): Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in Appendix B.
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6.3. FROST(Ed448, SHAKE256)
This ciphersuite uses edwards448 for the Group and SHAKE256 for the
Hash function H meant to produce signatures indistinguishable from
Ed448 as specified in [RFC8032]. The value of the contextString
parameter is "FROST-ED448-SHAKE256-v11".
* Group: edwards448 [RFC8032]
- Order(): Return 2^446 - 138180668098951153520073867485154268803
36692474882178609894547503885
- Identity(): As defined in [RFC7748].
- RandomScalar(): Implemented by returning a uniformly random
Scalar in the range [0, G.Order() - 1]. Refer to Appendix D
for implementation guidance.
- SerializeElement(A): Implemented as specified in [RFC8032],
Section 5.2.2. Additionally, this function validates that the
input element is not the group identity element.
- DeserializeElement(buf): Implemented as specified in [RFC8032],
Section 5.2.3. Additionally, this function validates that the
resulting element is not the group identity element and is in
the prime-order subgroup. The latter check can be implemented
by multiplying the resulting point by the order of the group
and checking that the result is the identity element. Note
that optimizations for this check exist; see [Pornin22].
- SerializeScalar(s): Implemented by outputting the little-endian
48-byte encoding of the Scalar value.
- DeserializeScalar(buf): Implemented by attempting to
deserialize a Scalar from a little-endian 48-byte string. This
function can fail if the input does not represent a Scalar in
the range [0, G.Order() - 1].
* Hash (H): SHAKE256
- H1(m): Implemented by computing H(contextString || "rho" || m),
interpreting the 114-byte digest as a little-endian integer,
and reducing the resulting integer modulo 2^446 - 1381806680989
5115352007386748515426880336692474882178609894547503885.
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- H2(m): Implemented by computing H("SigEd448" || 0 || 0 || m),
interpreting the 114-byte digest as a little-endian integer,
and reducing the resulting integer modulo 2^446 - 1381806680989
5115352007386748515426880336692474882178609894547503885.
- H3(m): Implemented by computing H(contextString || "nonce" ||
m), interpreting the 114-byte digest as a little-endian
integer, and reducing the resulting integer modulo 2^446 - 1381
806680989511535200738674851542688033669247488217860989454750388
5.
- H4(m): Implemented by computing H(contextString || "msg" || m).
- H5(m): Implemented by computing H(contextString || "com" || m).
Normally H2 would also include a domain separator, but for backwards
compatibility with [RFC8032], it is omitted.
Signature verification is as specified in Section 5.2.7 of [RFC8032]
with the constraint that implementations MUST check the group
equation [4][S]B = [4]R + [4][k]A'. The alternative check [S]B = R +
[k]A' is not safe or interoperable in practice.
6.4. FROST(P-256, SHA-256)
This ciphersuite uses P-256 for the Group and SHA-256 for the Hash
function H. The value of the contextString parameter is "FROST-
P256-SHA256-v11".
* Group: P-256 (secp256r1) [x9.62]
- Order(): Return 0xffffffff00000000ffffffffffffffffbce6faada7179
e84f3b9cac2fc632551
- Identity(): As defined in [x9.62].
- RandomScalar(): Implemented by returning a uniformly random
Scalar in the range [0, G.Order() - 1]. Refer to Appendix D
for implementation guidance.
- SerializeElement(A): Implemented using the compressed Elliptic-
Curve-Point-to-Octet-String method according to [SEC1],
yielding a 33 byte output. Additionally, this function
validates that the input element is not the group identity
element.
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- DeserializeElement(buf): Implemented by attempting to
deserialize a 33 byte input string to a public key using the
compressed Octet-String-to-Elliptic-Curve-Point method
according to [SEC1], and then performs partial public-key
validation as defined in section 5.6.2.3.4 of [KEYAGREEMENT].
This includes checking that the coordinates of the resulting
point are in the correct range, that the point is on the curve,
and that the point is not the point at infinity. Additionally,
this function validates that the resulting element is not the
group identity element. If these checks fail, deserialization
returns an error.
- SerializeScalar(s): Implemented using the Field-Element-to-
Octet-String conversion according to [SEC1].
- DeserializeScalar(buf): Implemented by attempting to
deserialize a Scalar from a 32-byte string using Octet-String-
to-Field-Element from [SEC1]. This function can fail if the
input does not represent a Scalar in the range [0, G.Order() -
1].
* Hash (H): SHA-256
- H1(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "rho", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
- H2(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "chal", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
- H3(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "nonce", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
- H4(m): Implemented by computing H(contextString || "msg" || m).
- H5(m): Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in Appendix B.
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6.5. FROST(secp256k1, SHA-256)
This ciphersuite uses secp256k1 for the Group and SHA-256 for the
Hash function H. The value of the contextString parameter is "FROST-
secp256k1-SHA256-v11".
* Group: secp256k1 [SEC2]
- Order(): Return 0xffffffff00000000ffffffffffffffffbce6faada7179
e84f3b9cac2fc632551
- Identity(): As defined in [SEC2].
- RandomScalar(): Implemented by returning a uniformly random
Scalar in the range [0, G.Order() - 1]. Refer to Appendix D
for implementation guidance.
- SerializeElement(A): Implemented using the compressed Elliptic-
Curve-Point-to-Octet-String method according to [SEC1].
Additionally, this function validates that the input element is
not the group identity element.
- DeserializeElement(buf): Implemented by attempting to
deserialize a public key using the compressed Octet-String-to-
Elliptic-Curve-Point method according to [SEC1], and then
performs partial public-key validation as defined in section
3.2.2.1 of [SEC1]. This includes checking that the coordinates
of the resulting point are in the correct range, that the point
is on the curve, and that the point is not the point at
infinity. Additionally, this function validates that the
resulting element is not the group identity element. If these
checks fail, deserialization returns an error.
- SerializeScalar(s): Implemented using the Field-Element-to-
Octet-String conversion according to [SEC1].
- DeserializeScalar(buf): Implemented by attempting to
deserialize a Scalar from a 32-byte string using Octet-String-
to-Field-Element from [SEC1]. This function can fail if the
input does not represent a Scalar in the range [0, G.Order() -
1].
* Hash (H): SHA-256
- H1(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "rho", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
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- H2(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "chal", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
- H3(m): Implemented as hash_to_field(m, 1) from [HASH-TO-CURVE],
Section 5.2 using expand_message_xmd with SHA-256 with
parameters DST = contextString || "nonce", F set to the scalar
field, p set to G.Order(), m = 1, and L = 48.
- H4(m): Implemented by computing H(contextString || "msg" || m).
- H5(m): Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in Appendix B.
6.6. Ciphersuite Requirements
Future documents that introduce new ciphersuites MUST adhere to the
following requirements.
1. H1, H2, and H3 all have output distributions that are close to
(indistinguishable from) the uniform distribution.
2. All hash functions MUST be domain separated with a per-suite
context string. Note that the FROST(Ed25519, SHA-512)
ciphersuite does not adhere to this requirement for backwards
compatibility with [RFC8032].
3. The group MUST be of prime-order, and all deserialization
functions MUST output elements that belong to to their respective
sets of Elements or Scalars, or failure when deserialization
fails.
7. Security Considerations
A security analysis of FROST exists in [FROST20] and [Schnorr21].
The protocol as specified in this document assumes the following
threat model.
* Secure key distribution. The signer key shares are generated and
distributed securely, i.e., via a trusted dealer that performs key
generation (see Appendix C.2) or through a distributed key
generation protocol.
* Honest-but-curious coordinator. We assume an honest-but-curious
Coordinator which, at the time of signing, does not perform a
denial of service attack. A denial of service would include any
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action which either prevents the protocol from completing or
causing the resulting signature to be invalid. Such actions for
the latter include sending inconsistent values to participants,
such as messages or the set of individual commitments. Note that
the Coordinator is _not_ trusted with any private information and
communication at the time of signing can be performed over a
public but reliable channel.
Under this threat model, FROST aims to achieve signature
unforgeability assuming at most (MIN_PARTICIPANTS-1) corrupted
participants. In particular, so long as an adversary corrupts fewer
than MIN_PARTICIPANTS participants, the scheme remains secure against
Existential Unforgeability Under Chosen Message Attack (EUF-CMA)
attacks, as defined in [BonehShoup], Definition 13.2. Satisfying
this requirement requires the ciphersuite to adhere to the
requirements in Section 6.6.
FROST does not aim to achieve the following goals:
* Post quantum security. FROST, like plain Schnorr signatures,
requires the hardness of the Discrete Logarithm Problem.
* Robustness. In the case of failure, FROST requires aborting the
protocol.
* Downgrade prevention. All participants in the protocol are
assumed to agree on what algorithms to use.
* Metadata protection. If protection for metadata is desired, a
higher-level communication channel can be used to facilitate key
generation and signing.
The rest of this section documents issues particular to
implementations or deployments.
7.1. Nonce Reuse Attacks
Section 4.1 describes the procedure that participants use to produce
nonces during the first round of signing. The randomness produced in
this procedure MUST be sampled uniformly at random. The resulting
nonces produced via nonce_generate are indistinguishable from values
sampled uniformly at random. This requirement is necessary to avoid
replay attacks initiated by other participants, which allow for a
complete key-recovery attack. The Coordinator MAY further hedge
against nonce reuse attacks by tracking participant nonce commitments
used for a given group key, at the cost of additional state.
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7.2. Protocol Failures
We do not specify what implementations should do when the protocol
fails, other than requiring that the protocol abort. Examples of
viable failure include when a verification check returns invalid or
if the underlying transport failed to deliver the required messages.
7.3. Removing the Coordinator Role
In some settings, it may be desirable to omit the role of the
Coordinator entirely. Doing so does not change the security
implications of FROST, but instead simply requires each participant
to communicate with all other participants. We loosely describe how
to perform FROST signing among participants without this coordinator
role. We assume that every participant receives as input from an
external source the message to be signed prior to performing the
protocol.
Every participant begins by performing commit() as is done in the
setting where a Coordinator is used. However, instead of sending the
commitment to the Coordinator, every participant instead will publish
this commitment to every other participant. Then, in the second
round, participants will already have sufficient information to
perform signing. They will directly perform sign(). All
participants will then publish their signature shares to one another.
After having received all signature shares from all other
participants, each participant will then perform
verify_signature_share and then aggregate directly.
The requirements for the underlying network channel remain the same
in the setting where all participants play the role of the
Coordinator, in that all messages that are exchanged are public and
so the channel simply must be reliable. However, in the setting that
a player attempts to split the view of all other players by sending
disjoint values to a subset of players, the signing operation will
output an invalid signature. To avoid this denial of service,
implementations may wish to define a mechanism where messages are
authenticated, so that cheating players can be identified and
excluded.
7.4. Input Message Hashing
FROST signatures do not pre-hash message inputs. This means that the
entire message must be known in advance of invoking the signing
protocol. Applications can apply pre-hashing in settings where
storing the full message is prohibitively expensive. In such cases,
pre-hashing MUST use a collision-resistant hash function with a
security level commensurate with the security in inherent to the
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ciphersuite chosen. It is RECOMMENDED that applications which choose
to apply pre-hashing use the hash function (H) associated with the
chosen ciphersuite in a manner similar to how H4 is defined. In
particular, a different prefix SHOULD be used to differentiate this
pre-hash from H4. One possible example is to construct this pre-hash
over message m as H(contextString \|\| "pre-hash" \|\| m).
7.5. Input Message Validation
Some applications may require that participants only process messages
of a certain structure. For example, in digital currency
applications wherein multiple participants may collectively sign a
transaction, it is reasonable to require that each participant check
the input message to be a syntactically valid transaction.
As another example, use of threshold signatures in [TLS] to produce
signatures of transcript hashes might require the participants
receive the source handshake messages themselves, and recompute the
transcript hash which is used as input message to the signature
generation process, so that they can verify that they are signing a
proper TLS transcript hash and not some other data.
In general, input message validation is an application-specific
consideration that varies based on the use case and threat model.
However, it is RECOMMENDED that applications take additional
precautions and validate inputs so that participants do not operate
as signing oracles for arbitrary messages.
8. References
8.1. Normative References
[HASH-TO-CURVE]
Faz-Hernández, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
16, 15 June 2022, <https://datatracker.ietf.org/doc/html/
draft-irtf-cfrg-hash-to-curve-16>.
[KEYAGREEMENT]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
Davis, "Recommendation for pair-wise key-establishment
schemes using discrete logarithm cryptography", National
Institute of Standards and Technology report,
DOI 10.6028/nist.sp.800-56ar3, April 2018,
<https://doi.org/10.6028/nist.sp.800-56ar3>.
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[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/rfc/rfc2119>.
[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/rfc/rfc8032>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[RISTRETTO]
de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I.,
Tankersley, G., and F. Valsorda, "The ristretto255 and
decaf448 Groups", Work in Progress, Internet-Draft, draft-
irtf-cfrg-ristretto255-decaf448-03, 25 February 2022,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
ristretto255-decaf448-03>.
[SEC1] "Elliptic Curve Cryptography, Standards for Efficient
Cryptography Group, ver. 2", 2009,
<https://secg.org/sec1-v2.pdf>.
[SEC2] "Recommended Elliptic Curve Domain Parameters, Standards
for Efficient Cryptography Group, ver. 2", 2010,
<https://secg.org/sec2-v2.pdf>.
[x9.62] ANS, "Public Key Cryptography for the Financial Services
Industry: the Elliptic Curve Digital Signature Algorithm
(ECDSA)", ANS X9.62-2005, November 2005.
8.2. Informative References
[BonehShoup]
Boneh, D. and V. Shoup, "A Graduate Course in Applied
Cryptography", January 2020,
<http://toc.cryptobook.us/book.pdf>.
[FROST20] Komlo, C. and I. Goldberg, "Two-Round Threshold Signatures
with FROST", 22 December 2020,
<https://eprint.iacr.org/2020/852.pdf>.
[Pornin22] Pornin, T., "Point-Halving and Subgroup Membership in
Twisted Edwards Curves", 6 September 2022,
<https://eprint.iacr.org/2022/1164.pdf>.
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[RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106, RFC 4086,
DOI 10.17487/RFC4086, June 2005,
<https://www.rfc-editor.org/rfc/rfc4086>.
[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, <https://www.rfc-editor.org/rfc/rfc7748>.
[Schnorr21]
Crites, E., Komlo, C., and M. Maller, "How to Prove
Schnorr Assuming Schnorr", 11 October 2021,
<https://eprint.iacr.org/2021/1375>.
[StrongerSec22]
Bellare, M., Tessaro, S., and C. Zhu, "Stronger Security
for Non-Interactive Threshold Signatures: BLS and FROST",
1 June 2022, <https://eprint.iacr.org/2022/833>.
[TLS] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
<https://www.rfc-editor.org/rfc/rfc8446>.
Appendix A. Acknowledgments
This document was improved based on input and contributions by the
Zcash Foundation engineering team. In addition, the authors of this
document would like to thank Isis Lovecruft, Alden Torres, T.
Wilson-Brown, and Conrado Gouvea for their inputs and contributions.
Appendix B. Schnorr Signature Generation and Verification for Prime-
Order Groups
This section contains descriptions of functions for generating and
verifying Schnorr signatures. It is included to complement the
routines present in [RFC8032] for prime-order groups, including
ristretto255, P-256, and secp256k1. The functions for generating and
verifying signatures are prime_order_sign and prime_order_verify,
respectively.
The function prime_order_sign produces a Schnorr signature over a
message given a full secret signing key as input (as opposed to a key
share.)
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prime_order_sign(msg, sk):
``
Inputs:
- msg, message to sign, a byte string
- sk, secret key, a Scalar
Outputs: (R, z), a Schnorr signature consisting of an Element R and Scalar z.
def prime_order_sign(msg, sk):
r = G.RandomScalar()
R = G.ScalarBaseMult(r)
PK = G.ScalarBaseMult(sk)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
z = r + (c * sk) // Scalar addition and multiplication
return (R, z)
The function prime_order_verify verifies Schnorr signatures with
validated inputs. Specifically, it assumes that signature R
component and public key belong to the prime-order group.
prime_order_verify(msg, sig, PK):
Inputs:
- msg, signed message, a byte string
- sig, a tuple (R, z) output from signature generation
- PK, public key, an Element
Outputs: 1 if signature is valid, and 0 otherwise
def prime_order_verify(msg, sig = (R, z), PK):
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
l = G.ScalarBaseMult(z)
r = R + G.ScalarMult(PK, c)
return l == r
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Appendix C. Trusted Dealer Key Generation
One possible key generation mechanism is to depend on a trusted
dealer, wherein the dealer generates a group secret s uniformly at
random and uses Shamir and Verifiable Secret Sharing as described in
Appendix C.1 and Appendix C.2 to create secret shares of s, denoted
s_i for i = 0, ..., MAX_PARTICIPANTS, to be sent to all
MAX_PARTICIPANTS participants. This operation is specified in the
trusted_dealer_keygen algorithm. The mathematical relation between
the secret key s and the MAX_SIGNER secret shares is formalized in
the secret_share_combine(shares) algorithm, defined in Appendix C.1.
The dealer that performs trusted_dealer_keygen is trusted to 1)
generate good randomness, and 2) delete secret values after
distributing shares to each participant, and 3) keep secret values
confidential.
Inputs:
- secret_key, a group secret, a Scalar, that MUST be derived from at least Ns bytes of entropy
- MAX_PARTICIPANTS, the number of shares to generate, an integer
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer
Outputs:
- participant_private_keys, MAX_PARTICIPANTS shares of the secret key s, each a tuple
consisting of the participant identifier and the key share (a Scalar).
- group_public_key, public key corresponding to the group signing key, an
Element in G.
- vss_commitment, a vector commitment of Elements in G, to each of the coefficients
in the polynomial defined by secret_key_shares and whose first element is
G.ScalarBaseMult(s).
def trusted_dealer_keygen(secret_key, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
# Generate random coefficients for the polynomial
coefficients = []
for i in range(0, MIN_PARTICIPANTS - 1):
coefficients.append(G.RandomScalar())
participant_private_keys, coefficients = secret_share_shard(secret_key, coefficients, MAX_PARTICIPANTS, MIN_PARTICIPANTS)
vss_commitment = vss_commit(coefficients):
return participant_private_keys, vss_commitment[0], vss_commitment
It is assumed the dealer then sends one secret key share to each of
the NUM_PARTICIPANTS participants, along with vss_commitment. After
receiving their secret key share and vss_commitment, participants
MUST abort if they do not have the same view of vss_commitment.
Otherwise, each participant MUST perform
vss_verify(secret_key_share_i, vss_commitment), and abort if the
check fails. The trusted dealer MUST delete the secret_key and
secret_key_shares upon completion.
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Use of this method for key generation requires a mutually
authenticated secure channel between the dealer and participants to
send secret key shares, wherein the channel provides confidentiality
and integrity. Mutually authenticated TLS is one possible deployment
option.
C.1. Shamir Secret Sharing
In Shamir secret sharing, a dealer distributes a secret Scalar s to n
participants in such a way that any cooperating subset of
MIN_PARTICIPANTS participants can recover the secret. There are two
basic steps in this scheme: (1) splitting a secret into multiple
shares, and (2) combining shares to reveal the resulting secret.
This secret sharing scheme works over any field F. In this
specification, F is the scalar field of the prime-order group G.
The procedure for splitting a secret into shares is as follows.
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secret_share_shard(s, coefficients, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
Inputs:
- s, secret value to be shared, a Scalar
- coefficients, an array of size MIN_PARTICIPANTS - 1 with randomly generated
Scalars, not including the 0th coefficient of the polynomial
- MAX_PARTICIPANTS, the number of shares to generate, an integer less than 2^16
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer greater than 0
Outputs:
- secret_key_shares, A list of MAX_PARTICIPANTS number of secret shares, each a tuple
consisting of the participant identifier and the key share (a Scalar)
- coefficients, a vector of MIN_PARTICIPANTS coefficients which uniquely determine a polynomial f.
Errors:
- "invalid parameters", if MIN_PARTICIPANTS > MAX_PARTICIPANTS or if MIN_PARTICIPANTS is less than 2
def secret_share_shard(s, coefficients, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
if MIN_PARTICIPANTS > MAX_PARTICIPANTS:
raise "invalid parameters"
if MIN_PARTICIPANTS < 2:
raise "invalid parameters"
# Prepend the secret to the coefficients
coefficients = [s] + coefficients
# Evaluate the polynomial for each point x=1,...,n
secret_key_shares = []
for x_i in range(1, MAX_PARTICIPANTS + 1):
y_i = polynomial_evaluate(Scalar(x_i), coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_share.append(secret_key_share_i)
return secret_key_shares, coefficients
Let points be the output of this function. The i-th element in
points is the share for the i-th participant, which is the randomly
generated polynomial evaluated at coordinate i. We denote a secret
share as the tuple (i, points[i]), and the list of these shares as
shares. i MUST never equal 0; recall that f(0) = s, where f is the
polynomial defined in a Shamir secret sharing operation.
The procedure for combining a shares list of length MIN_PARTICIPANTS
to recover the secret s is as follows; the algorithm
polynomial_interpolation is defined in {{dep-polynomial-
interpolate}}.
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secret_share_combine(shares):
Inputs:
- shares, a list of at minimum MIN_PARTICIPANTS secret shares, each a tuple (i, f(i))
where i and f(i) are Scalars
Outputs: The resulting secret s, a Scalar, that was previously split into shares
Errors:
- "invalid parameters", if fewer than MIN_PARTICIPANTS input shares are provided
def secret_share_combine(shares):
if len(shares) < MIN_PARTICIPANTS:
raise "invalid parameters"
s = polynomial_interpolation(shares)
return s
C.1.1. Deriving the constant term of a polynomial
Secret sharing requires "splitting" a secret, which is represented as
a constant term of some polynomial f of degree t-1. Recovering the
constant term occurs with a set of t points using polynomial
interpolation, defined as follows.
Inputs:
- points, a set of t distinct points on a polynomial f, each a tuple of two
Scalar values representing the x and y coordinates
Outputs: The constant term of f, i.e., f(0)
def polynomial_interpolation(points):
x_coords = []
for point in points:
x_coords.append(point.x)
f_zero = Scalar(0)
for point in points:
delta = point.y * derive_lagrange_coefficient(point.x, x_coords)
f_zero = f_zero + delta
return f_zero
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C.2. Verifiable Secret Sharing
Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret
sharing, adding a verification step to demonstrate the consistency of
a participant's share with a public commitment to the polynomial f
for which the secret s is the constant term. This check ensures that
all participants have a point (their share) on the same polynomial,
ensuring that they can later reconstruct the correct secret.
The procedure for committing to a polynomial f of degree at most
MIN_PARTICIPANTS-1 is as follows.
vss_commit(coeffs):
Inputs:
- coeffs, a vector of the MIN_PARTICIPANTS coefficients which uniquely determine
a polynomial f.
Outputs: a commitment vss_commitment, which is a vector commitment to each of the
coefficients in coeffs, where each element of the vector commitment is an Element in G.
def vss_commit(coeffs):
vss_commitment = []
for coeff in coeffs:
A_i = G.ScalarBaseMult(coeff)
vss_commitment.append(A_i)
return vss_commitment
The procedure for verification of a participant's share is as
follows. If vss_verify fails, the participant MUST abort the
protocol, and failure should be investigated out of band.
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vss_verify(share_i, vss_commitment):
Inputs:
- share_i: A tuple of the form (i, sk_i), where i indicates the participant
identifier, and sk_i the participant's secret key, a secret share of the
constant term of f, where sk_i is a Scalar.
- vss_commitment: A VSS commitment to a secret polynomial f, a vector commitment
to each of the coefficients in coeffs, where each element of the vector commitment
is an Element
Outputs: 1 if sk_i is valid, and 0 otherwise
vss_verify(share_i, vss_commitment)
(i, sk_i) = share_i
S_i = ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
S_i' += G.ScalarMult(vss_commitment[j], pow(i, j))
if S_i == S_i':
return 1
return 0
We now define how the Coordinator and participants can derive group
info, which is an input into the FROST signing protocol.
derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS, vss_commitment):
Inputs:
- MAX_PARTICIPANTS, the number of shares to generate, an integer
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer
- vss_commitment: A VSS commitment to a secret polynomial f, a vector commitment to each of the
coefficients in coeffs, where each element of the vector commitment is an Element in G.
Outputs:
- PK, the public key representing the group, an Element.
- participant_public_keys, a list of MAX_PARTICIPANTS public keys PK_i for i=1,...,MAX_PARTICIPANTS,
where each PK_i is the public key, an Element, for participant i.
derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS, vss_commitment)
PK = vss_commitment[0]
participant_public_keys = []
for i in range(1, MAX_PARTICIPANTS+1):
PK_i = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
PK_i += G.ScalarMult(vss_commitment[j], pow(i, j))
participant_public_keys.append(PK_i)
return PK, participant_public_keys
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Appendix D. Random Scalar Generation
Two popular algorithms for generating a random integer uniformly
distributed in the range [0, G.Order() -1] are as follows:
D.1. Rejection Sampling
Generate a random byte array with Ns bytes, and attempt to map to a
Scalar by calling DeserializeScalar in constant time. If it
succeeds, return the result. If it fails, try again with another
random byte array, until the procedure succeeds. Failure to
implement DeserializeScalar in constant time can leak information
about the underlying corresponding Scalar.
As an optimization, if the group order is very close to a power of 2,
it is acceptable to omit the rejection test completely. In
particular, if the group order is p, and there is an integer b such
that p - 2<sup>b</sup>| < 2<sup>(b/2)</sup>, then RandomScalar can
simply return a uniformly random integer of at most b bits.
D.2. Wide Reduction
Generate a random byte array with l = ceil(((3 *
ceil(log2(G.Order()))) / 2) / 8) bytes, and interpret it as an
integer; reduce the integer modulo G.Order() and return the result.
See Section 5 of [HASH-TO-CURVE] for the underlying derivation of l.
Appendix E. Test Vectors
This section contains test vectors for all ciphersuites listed in
Section 6. All Element and Scalar values are represented in
serialized form and encoded in hexadecimal strings. Signatures are
represented as the concatenation of their constituent parts. The
input message to be signed is also encoded as a hexadecimal string.
Each test vector consists of the following information.
* Configuration. This lists the fixed parameters for the particular
instantiation of FROST, including MAX_PARTICIPANTS,
MIN_PARTICIPANTS, and NUM_PARTICIPANTS.
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* Group input parameters. This lists the group secret key and
shared public key, generated by a trusted dealer as described in
Appendix C, as well as the input message to be signed. The
randomly generated coefficients produced by the trusted dealer to
share the group signing secret are also listed. Each coefficient
is identified by its index, e.g., share_polynomial_coefficients[1]
is the coefficient of the first term in the polynomial. Note that
the 0-th coefficient is omitted as this is equal to the group
secret key. All values are encoded as hexadecimal strings.
* Signer input parameters. This lists the signing key share for
each of the NUM_PARTICIPANTS participants.
* Round one parameters and outputs. This lists the NUM_PARTICIPANTS
participants engaged in the protocol, identified by their integer
identifier, and for each participant: the hiding and binding
commitment values produced in Section 5.1; the randomness values
used to derive the commitment nonces in nonce_generate; the
resulting group binding factor input computed in part from the
group commitment list encoded as described in Section 4.3; and
group binding factor as computed in Section 5.2).
* Round two parameters and outputs. This lists the NUM_PARTICIPANTS
participants engaged in the protocol, identified by their integer
identifier, along with their corresponding output signature share
as produced in Section 5.2.
* Final output. This lists the aggregate signature as produced in
Section 5.3.
E.1. FROST(Ed25519, SHA-512)
// Configuration information
MAX_PARTICIPANTS: 3
MIN_PARTICIPANTS: 2
NUM_PARTICIPANTS: 2
// Group input parameters
group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
90c6e13a98304
group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040
d380fb9738673
message: 74657374
share_polynomial_coefficients[1]: 178199860edd8c62f5212ee91eff1295d0d
670ab4ed4506866bae57e7030b204
// Signer input parameters
P1 participant_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f
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4a033f2ec83d93509
P2 participant_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685
c07eed76bf409e80d
P3 participant_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c
46da8bdea643a9a02
// Round one parameters
participant_list: 1,3
// Signer round one outputs
P1 hiding_nonce_randomness: 9d06a6381c7a4493929761a73692776772b274236
fb5cfcc7d1b48ac3a9c249f
P1 binding_nonce_randomness: db184d7bc01a3417fe1f2eb3cf5479bb027145e6
369a5f879f32d334ab256b23
P1 hiding_nonce: 70652da3e8d7533a0e4b9e9104f01b48c396b5b553717784ed8d
05c6a36b9609
P1 binding_nonce: 4f9e1ad260b5c0e4fe0e0719c6324f89fecd053758f77c957f5
6967e634a710e
P1 hiding_nonce_commitment: 44105304351ceddc58e15ddea35b2cb48e60ced54
ceb22c3b0e5d42d098aa1d8
P1 binding_nonce_commitment: b8274b18a12f2cef74ae42f876cec1e31daab5cb
162f95a56cd2487409c9d1dd
P1 binding_factor_input: c5b95020cba31a9035835f074f718d0c3af02a318d6b
4723bbd1c088f4889dd7b9ff8e79f9a67a9d27605144259a7af18b7cca2539ffa5c4f
1366a98645da8f4e077d604fff64f20e2377a37e5a10ce152194d62fe856ef4cd935d
4f1cb0088c2083a2722ad3f5a84d778e257da0df2a7cadb004b1f5528352af778b94e
e1c2a0100000000000000000000000000000000000000000000000000000000000000
P1 binding_factor: 2d5630c36d33258b1208c4205fa759b762d09bfa06b29cf792
cf98758c0b3305
P3 hiding_nonce_randomness: 31ca9b07936d6b342a43d97f23b7bec5a5f5a0957
5a075393868dd8df5c05a54
P3 binding_nonce_randomness: c1db96a85d8b593e14fdb869c0955625478afa6a
987ad217e7f2261dcab26819
P3 hiding_nonce: 233adcb0ec0eddba5f1cc5268f3f4e6fc1dd97fb1e4a1754e6dd
c92ed834ca0b
P3 binding_nonce: b59fc8a32fe02ec0a44c4671f3d1f82ea3924b7c7c0179398fc
9137e82757803
P3 hiding_nonce_commitment: d31bd81ce216b1c83912803a574a0285796275cb8
b14f6dc92c8b09a6951f0a2
P3 binding_nonce_commitment: e1c863cfd08df775b6747ef2456e9bf9a03cc281
a479a95261dc39137fcf0967
P3 binding_factor_input: c5b95020cba31a9035835f074f718d0c3af02a318d6b
4723bbd1c088f4889dd7b9ff8e79f9a67a9d27605144259a7af18b7cca2539ffa5c4f
1366a98645da8f4e077d604fff64f20e2377a37e5a10ce152194d62fe856ef4cd935d
4f1cb0088c2083a2722ad3f5a84d778e257da0df2a7cadb004b1f5528352af778b94e
e1c2a0300000000000000000000000000000000000000000000000000000000000000
P3 binding_factor: 1137be5cdf3d18e44367acee8485e9a66c3164077af80619b6
291e3943bbef04
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// Round two parameters
participant_list: 1,3
// Signer round two outputs
P1 sig_share: c4b26af1e91fbc8440a0dad253e72620da624553c5b625fd51e6ea1
79fc09f05
P3 sig_share: 9369640967d0cb98f4dedfde58a845e0e18e0a7164396358439060e
d282b4e08
sig: ae11c539fdc709b78fef5ee1f5a2250297e3e1b62a86a86c26d93c389934ba0e
571ccffa50f0871d357fbab1ac8f6c00bcf14fc429f0885595764b05c8ebed0d
E.2. FROST(Ed448, SHAKE256)
// Configuration information
MAX_PARTICIPANTS: 3
MIN_PARTICIPANTS: 2
NUM_PARTICIPANTS: 2
// Group input parameters
group_secret_key: 6298e1eef3c379392caaed061ed8a31033c9e9e3420726f23b4
04158a401cd9df24632adfe6b418dc942d8a091817dd8bd70e1c72ba52f3c00
group_public_key: 3832f82fda00ff5365b0376df705675b63d2a93c24c6e81d408
01ba265632be10f443f95968fadb70d10786827f30dc001c8d0f9b7c1d1b000
message: 74657374
share_polynomial_coefficients[1]: dbd7a514f7a731976620f0436bd135fe8dd
dc3fadd6e0d13dbd58a1981e587d377d48e0b7ce4e0092967c5e85884d0275a7a740b
6abdcd0500
// Signer input parameters
P1 participant_share: 4a2b2f5858a932ad3d3b18bd16e76ced3070d72fd79ae44
02df201f525e754716a1bc1b87a502297f2a99d89ea054e0018eb55d39562fd0100
P2 participant_share: 2503d56c4f516444a45b080182b8a2ebbe4d9b2ab509f25
308c88c0ea7ccdc44e2ef4fc4f63403a11b116372438a1e287265cadeff1fcb0700
P3 participant_share: 00db7a8146f995db0a7cf844ed89d8e94c2b5f259378ff6
6e39d172828b264185ac4decf7219e4aa4478285b9c0eef4fccdf3eea69dd980d00
// Round one parameters
participant_list: 1,3
// Signer round one outputs
P1 hiding_nonce_randomness: 89bf16040081ff2990336b200613787937ebe1f02
4b8cdff90eb6f1c741d91c1
P1 binding_nonce_randomness: cd646348bb98fd2a4b2f27fb7d6da18201c16184
7352576b4bf125190e965483
P1 hiding_nonce: 67a6f023e77361707c6e894c625e809e80f33fdb310810053ae2
9e28e7011f3193b9020e73c183a98cc3a519160ed759376dd92c9483162200
P1 binding_nonce: 4812e8d7c8b7a50ced80b507902d074ef8647bc1146979683da
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8d0fecd93fa3c8230cade2fb4344600aa04bd4b7a21d046c5b63ee865b12a00
P1 hiding_nonce_commitment: 649c6a53b109897d962d033f23d01fd4e1053dddf
3746d2ddce9bd66aea38ccfc3df061df03ca399eb806312ab3037c0c31523142956ad
a780
P1 binding_nonce_commitment: 0064cc729a8e2fcf417e43788ecec37b10e9e1dc
b3ae90854efbfaad00a0ef3cdd52e18d56f073c8ff0947cb71ff0bb17c3d45d096409
ddb00
P1 binding_factor_input: 106dadce87ca867018702d69a02effd165e1ac1a511c
957cff1897ceff2e34ca212fe798d84f0bde6054bf0fa77fd4cd4bc4853d6dc8dbd19
d340923f0ebbbb35172df4ab865a45d55af31fa0e6606ea97cf8513022b2b133d0f9f
6b8d3be184221fc4592bf12bd7fb4127bb67e51a6dc9e5f1ed5243362fb46a6da5524
18ca967d43d9bc811a21917a3018de58f11c25f6b9ad8bec3699e06b87dd3ab67a732
6c30878c7c55ec1a45802af65da193ce99634158539e38c232a627895c5f14e2e20d4
87382ccc9c99cd0a0df266a292f283bb9b6854e344ecc32d5e1852fdde5fde7779801
000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000
P1 binding_factor: 3412ac894a91a6bc0e3e7c790f3e8ef5d1288e54de780aba38
4cbb3081b602dd188010e5b0c9ac2b5dca0aae54cfd0f5c391cece8092131d00
P3 hiding_nonce_randomness: 3718dabb4fd3d7dd9adad4878c6de8b33c8841cfe
7cc95a85592952a2c9c554d
P3 binding_nonce_randomness: 3becbc90798211a0f52543dd1f24869a143fdf74
3409581af4db30f045773d64
P3 hiding_nonce: 4f2666770317d14ec9f7fd6690c075c34b4cde7f6d9bceda9e94
33ec8c0f2dc983ff1622c3a54916ce7c161381d263fad62539cddab2101600
P3 binding_nonce: 88f66df8bb66389932721a40de4aa5754f632cac114abc10526
88104d19f3b1a010880ebcd0c4c0f8cf567d887e5b0c3c0dc78821166550f00
P3 hiding_nonce_commitment: 8dcf049167e28d5f53fa7ebbbd136abcaf2be9f2c
02448c8979002f92577b22027640def7ddd5b98f9540c2280f36a92d4747bbade0b0c
4280
P3 binding_nonce_commitment: 12e837b89a2c085481fcf0ca640a17a24b6fc96b
032d40e4301c78e7232a9f49ffdcad2c21acbc992e79dfc3c6c07cb94e4680b3dcc99
35580
P3 binding_factor_input: 106dadce87ca867018702d69a02effd165e1ac1a511c
957cff1897ceff2e34ca212fe798d84f0bde6054bf0fa77fd4cd4bc4853d6dc8dbd19
d340923f0ebbbb35172df4ab865a45d55af31fa0e6606ea97cf8513022b2b133d0f9f
6b8d3be184221fc4592bf12bd7fb4127bb67e51a6dc9e5f1ed5243362fb46a6da5524
18ca967d43d9bc811a21917a3018de58f11c25f6b9ad8bec3699e06b87dd3ab67a732
6c30878c7c55ec1a45802af65da193ce99634158539e38c232a627895c5f14e2e20d4
87382ccc9c99cd0a0df266a292f283bb9b6854e344ecc32d5e1852fdde5fde7779803
000000000000000000000000000000000000000000000000000000000000000000000
0000000000000000000000000000000000000000000
P3 binding_factor: 6aa48a3635d7b962489283ee1ccda8ea66e5677b1e17f2f475
eb565e3ae8ea73360f24c04e3775dadd1f2923adcda3d105536ad28c3c561100
// Round two parameters
participant_list: 1,3
// Signer round two outputs
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P1 sig_share: c5057c80d13e565545dac6f3aa333065c809a14a94fea3c8e4e87e3
86a9cb89602de7355c5d19ebb09d553b100ef1858104fc7c43992d83400
P3 sig_share: 2b490ea08411f78c620c668fff8ba70b25b7c89436f20cc45419213
de70f93fb6c9094c79293697d72e741b68d2e493446005145d0b7fc3500
sig: 83ac141d289a5171bc894b058aee2890316280719a870fc5c1608b7740302315
5d7a9dc15a2b7920bb5826dd540bf76336be99536cebe36280fd093275c38dd4be525
767f537fd6a4f5d8a9330811562c84fded5f851ac4b926f6e081d586508397cbc9567
8e1d628c564f180a0a4ad52a00
E.3. FROST(ristretto255, SHA-512)
// Configuration information
MAX_PARTICIPANTS: 3
MIN_PARTICIPANTS: 2
NUM_PARTICIPANTS: 2
// Group input parameters
group_secret_key: 1b25a55e463cfd15cf14a5d3acc3d15053f08da49c8afcf3ab2
65f2ebc4f970b
group_public_key: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3dd9c3c0d
05083c7254f57
message: 74657374
share_polynomial_coefficients[1]: 410f8b744b19325891d73736923525a4f59
6c805d060dfb9c98009d34e3fec02
// Signer input parameters
P1 participant_share: 5c3430d391552f6e60ecdc093ff9f6f4488756aa6cebdba
d75a768010b8f830e
P2 participant_share: b06fc5eac20b4f6e1b271d9df2343d843e1e1fb03c4cbb6
73f2872d459ce6f01
P3 participant_share: f17e505f0e2581c6acfe54d3846a622834b5e7b50cad9a2
109a97ba7a80d5c04
// Round one parameters
participant_list: 1,3
// Signer round one outputs
P1 hiding_nonce_randomness: 81800157bb554f299fe0b6bd658e4c4591d74168b
5177bf55e8dceed59dc80c7
P1 binding_nonce_randomness: e9b37de02fde28f601f09051ed9a277b02ac81c8
03a5c72492d58635001fe355
P1 hiding_nonce: 40f58e8df202b21c94f826e76e4647efdb0ea3ca7ae7e3689bc0
cbe2e2f6660c
P1 binding_nonce: 373dd42b5fe80e88edddf82e03744b6a12d59256f546de612d4
bbd91a6b1df06
P1 hiding_nonce_commitment: b8c7319a56b296537436e5a6f509a871a3c74eff1
534ec1e2f539ccd8b322411
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P1 binding_nonce_commitment: 7af5d4bece8763ce3630370adbd978699402f624
fd3a7d2c71ea5839efc3cf54
P1 binding_factor_input: 9c245d5fc2e451c5c5a617cc6f2a20629fb317d9b1c1
915ab4bfa319d4ebf922c54dd1a5b3b754550c72734ac9255db8107a2b01f361754d9
f13f428c2f6de9e4f609ae0dbe8bd1f95bee9f9ea219154d567ef174390bac737bb67
ee1787c8a34279728d4aa99a6de2d5ce6deb86afe6bc68178f01223bb5eb934c8a23b
6354e0100000000000000000000000000000000000000000000000000000000000000
P1 binding_factor: 607df5e2e3a8b5e2704716693e18f548100a32b86a5685d393
2a774c3f107e06
P3 hiding_nonce_randomness: daeb223c4a913943cff2fb0b0e638dfcc281e1e89
36ee6c3fef4d49ad9cbfaa0
P3 binding_nonce_randomness: c425768d952ab8f18b9720c54b93e612ba2cca17
0bb7518cac080896efa7429b
P3 hiding_nonce: 491477c9dbe8717c77c6c1e2c5f4cec636c7c154313a44c91fea
63e309f3e100
P3 binding_nonce: 3ae1bba7d6f2076f81596912dd916efae5b3c2ef89695632119
4fdd2e52ebc0f
P3 hiding_nonce_commitment: e4466b7670ac4f9d9b7b67655860dd1ab341be18a
654bb1966df53c76c85d511
P3 binding_nonce_commitment: ce47cd595d25d7effc3c095efa2a687a1728a5ec
ab402b39e0c0ad9a525ea54f
P3 binding_factor_input: 9c245d5fc2e451c5c5a617cc6f2a20629fb317d9b1c1
915ab4bfa319d4ebf922c54dd1a5b3b754550c72734ac9255db8107a2b01f361754d9
f13f428c2f6de9e4f609ae0dbe8bd1f95bee9f9ea219154d567ef174390bac737bb67
ee1787c8a34279728d4aa99a6de2d5ce6deb86afe6bc68178f01223bb5eb934c8a23b
6354e0300000000000000000000000000000000000000000000000000000000000000
P3 binding_factor: 2bd27271c28746eb93e2114d6778c12b44c9287d84b85dc780
eb08da6f689900
// Round two parameters
participant_list: 1,3
// Signer round two outputs
P1 sig_share: c38f438c325ce6bfa4272b37e7707caaeb57fa8c7ddcc05e0725acb
8a7d9cd0c
P3 sig_share: 4cb9917be3bd53f1d60f1c3d1a3ff563565fa15a391133e7f980e55
d3aeb7904
sig: 204d5d93aa486192ecf2f64ce7dbc1db76948fb1077d1a719ae1ecca6143501e
2275dfaafbb62759a59a4fd122b692f941b79be7b6edf34501a69116e2c44701
E.4. FROST(P-256, SHA-256)
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// Configuration information
MAX_PARTICIPANTS: 3
MIN_PARTICIPANTS: 2
NUM_PARTICIPANTS: 2
// Group input parameters
group_secret_key: 8ba9bba2e0fd8c4767154d35a0b7562244a4aaf6f36c8fb8735
fa48b301bd8de
group_public_key: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99cfbd02b4
dc00585ec45ab70
message: 74657374
share_polynomial_coefficients[1]: 80f25e6c0709353e46bfbe882a11bdbb1f8
097e46340eb8673b7e14556e6c3a4
// Signer input parameters
P1 participant_share: 0c9c1a0fe806c184add50bbdcac913dda73e482daf95dcb
9f35dbb0d8a9f7731
P2 participant_share: 8d8e787bef0ff6c2f494ca45f4dad198c6bee01212d6c84
067159c52e1863ad5
P3 participant_share: 0e80d6e8f6192c003b5488ce1eec8f5429587d48cf00154
1e713b2d53c09d928
// Round one parameters
participant_list: 1,3
// Signer round one outputs
P1 hiding_nonce_randomness: f4e8cf80aec3f888d997900ac7e3e349944b5a6b4
7649fc32186d2f1238103c6
P1 binding_nonce_randomness: a7f220770b6f10ff54ec6afa55f99bd08cc92fa1
a488c86e9bf493e9cb894cdf
P1 hiding_nonce: f871dfcf6bcd199342651adc361b92c941cb6a0d8c8c1a3b91d7
9e2c1bf3722d
P1 binding_nonce: bd3ece3634a1b303dea0586ed67a91fe68510f11ebe66e88683
09b1551ef2388
P1 hiding_nonce_commitment: 03987febbc67a8ed735affdff4d3a5adf22c05c80
f97f311ab7437a3027372deb3
P1 binding_nonce_commitment: 02a1960477d139035b986d6adcb06491378beb92
ccd097ad94e76291c52343849d
P1 binding_factor_input: 350c8b523feea9bb35720e9fbe0405ed48d78caa4fb6
0869f34367e144c68bb0fc77bf512409ad8b91e2ace4909229891a446c45683f5eb2f
843dbec224527dc000000000000000000000000000000000000000000000000000000
0000000001
P1 binding_factor: cb415dd1d866493ee7d2db7cb33929d7e430e84d80c58070e2
bbb1fdbf76a9c8
P3 hiding_nonce_randomness: 1b6149d252a0a0a6618b8d22a1c49897f9b0d23a4
8f19598e191e05dc7b7ae33
P3 binding_nonce_randomness: e13994bb75aafe337c32afdbfd08ae60dd108fc7
68845edaa871992044cabf1b
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P3 hiding_nonce: 802e9321f9f63688c6c1a9681a4a4661f71770e0cef92b8a5997
155d18fb82ef
P3 binding_nonce: 8b6b692ae634a24536f45dda95b2398af71cd605fb7a0bbdd94
08d211ab99eba
P3 hiding_nonce_commitment: 0212cac45ebd4100c97506939391f9be4ffc3ca29
60e2ef95aeaa38abdede204ca
P3 binding_nonce_commitment: 03017ce754d310eabda0f5681e61ce3d713cdd33
7070faa6a68471af49694a4e7e
P3 binding_factor_input: 350c8b523feea9bb35720e9fbe0405ed48d78caa4fb6
0869f34367e144c68bb0fc77bf512409ad8b91e2ace4909229891a446c45683f5eb2f
843dbec224527dc000000000000000000000000000000000000000000000000000000
0000000003
P3 binding_factor: dfd82467569334e952edecb10d92adf85b8e299db0b40be313
1a12efdfa3e796
// Round two parameters
participant_list: 1,3
// Signer round two outputs
P1 sig_share: c5acd980310aaf87cb7a9a90428698ef3e6b1e5860f7fb06329bc0e
fe3f14ca5
P3 sig_share: 1e064fbd35467377eb3fe161ff975e9ec3ed8e2e0d4c73f3a6b0a02
3777e1264
sig: 029e07d4171dbf9a730ed95e9d95bda06fa4db76c88c519f7f3ca5483019f46c
b0e3b3293d665122ffb6ba7bf2421df78e0258ac866e446ef9d94c61135b6f5f09
E.5. FROST(secp256k1, SHA-256)
// Configuration information
MAX_PARTICIPANTS: 3
MIN_PARTICIPANTS: 2
NUM_PARTICIPANTS: 2
// Group input parameters
group_secret_key: 0d004150d27c3bf2a42f312683d35fac7394b1e9e318249c1bf
e7f0795a83114
group_public_key: 02f37c34b66ced1fb51c34a90bdae006901f10625cc06c4f646
63b0eae87d87b4f
message: 74657374
share_polynomial_coefficients[1]: fbf85eadae3058ea14f19148bb72b45e439
9c0b16028acaf0395c9b03c823579
// Signer input parameters
P1 participant_share: 08f89ffe80ac94dcb920c26f3f46140bfc7f95b493f8310
f5fc1ea2b01f4254c
P2 participant_share: 04f0feac2edcedc6ce1253b7fab8c86b856a797f44d83d8
2a385554e6e401984
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P3 participant_share: 00e95d59dd0d46b0e303e500b62b7ccb0e555d49f5b849f
5e748c071da8c0dbc
// Round one parameters
participant_list: 1,3
// Signer round one outputs
P1 hiding_nonce_randomness: 80cbea5e405d169999d8c4b30b755fedb26ab07ec
8198cda4873ed8ce5e16773
P1 binding_nonce_randomness: f6d5b38197843046b68903048c1feba433e35001
45281fa8bb1e26fdfeef5e7f
P1 hiding_nonce: acc83278035223c1ba464e2d11bfacfc872b2b23e1041cf5f613
0da21e4d8068
P1 binding_nonce: c3ef169995bc3d2c2d48f30b83d0c63751e67ceb057695bcb2a
6aa40ed5d926b
P1 hiding_nonce_commitment: 036673d68a928793c33ae07776908eae8ea15dd94
7ed81284e939aaba118573a5e
P1 binding_nonce_commitment: 03d2a96dd4ec1ee29dc22067109d1290dabd8016
cb41856ee8ff9281c3fa1baffd
P1 binding_factor_input: a645d8249457bbcac34fa7b740f66bcce08fc39506b8
bbf1a1c81092f6272eda82ae39234d714f87a7b91dd67d124a06561a36817c1ecaa25
5c3527d694fc4f1000000000000000000000000000000000000000000000000000000
0000000001
P1 binding_factor: d7bcbd29408dedc9e138262d99b09d8b5705d76eb5de2369d9
103e4423f8ac79
P3 hiding_nonce_randomness: b9794047604beda0c5c0529ac9dfd83c0a80399a7
bdf4c3e23cef2faf69cdcc3
P3 binding_nonce_randomness: c28ce6252631620b84c2702b34774fab365e286e
bc77030a112ebccccbffa78b
P3 hiding_nonce: cb3387defef07fc9010c0564ba6495ed41876626ed86b886ca26
cbbd3566ffbc
P3 binding_nonce: 4559459735eb68e8c16319a9fd9a14016053957cb8cea273a24
b7c7bc1ee26f6
P3 hiding_nonce_commitment: 030278e6e6055fb963b40e0c3c37099f803f3f389
30fc89092517f8ce1b47e8d6b
P3 binding_nonce_commitment: 028eb6d238c6c0fc6216906706ad0ff9943c6c1d
6079cdf74f674481ebb2485db3
P3 binding_factor_input: a645d8249457bbcac34fa7b740f66bcce08fc39506b8
bbf1a1c81092f6272eda82ae39234d714f87a7b91dd67d124a06561a36817c1ecaa25
5c3527d694fc4f1000000000000000000000000000000000000000000000000000000
0000000003
P3 binding_factor: ecc057259f3c8b195308c9b73aaaf840660a37eb264ebce342
412c58102ee437
// Round two parameters
participant_list: 1,3
// Signer round two outputs
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P1 sig_share: 1750b2a314a81b66fd81366583617aaafcffa68f14495204795aa04
34b907aa3
P3 sig_share: e4dbbbbbcb035eb3512918b0368c4ab2c836a92dccff3251efa7a4a
acc7d3790
sig: 0259696aac722558e8638485d252bb2556f6241a7adfdf284c8c87a3428d4644
8dfc2c6e5edfab7a1a4eaa4f15b9edc55dc5364fbce1488456690244ee180db233
Authors' Addresses
Deirdre Connolly
Zcash Foundation
Email: durumcrustulum@gmail.com
Chelsea Komlo
University of Waterloo, Zcash Foundation
Email: ckomlo@uwaterloo.ca
Ian Goldberg
University of Waterloo
Email: iang@uwaterloo.ca
Christopher A. Wood
Cloudflare
Email: caw@heapingbits.net
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