CFRG D. Connolly
InternetDraft Zcash Foundation
Intended status: Informational C. Komlo
Expires: 12 January 2023 University of Waterloo, Zcash Foundation
I. Goldberg
University of Waterloo
C. A. Wood
Cloudflare
11 July 2022
TwoRound Threshold Schnorr Signatures with FROST
draftirtfcfrgfrost07
Abstract
In this draft, we present the tworound signing variant of FROST, a
Flexible RoundOptimized 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
keyrecovery attack that is possible when even only one 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/draftirtfcfrgfrost.
Status of This Memo
This InternetDraft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as InternetDrafts. The list of current Internet
Drafts is at https://datatracker.ietf.org/drafts/current/.
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InternetDrafts are draft documents valid for a maximum of six months
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This InternetDraft will expire on 12 January 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/
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Please review these documents carefully, as they describe your rights
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 7
3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 8
3.1. PrimeOrder Group . . . . . . . . . . . . . . . . . . . . 8
3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 9
4. Helper functions . . . . . . . . . . . . . . . . . . . . . . 9
4.1. Nonce generation . . . . . . . . . . . . . . . . . . . . 10
4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 10
4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 10
4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 11
4.2.3. Deriving the constant term of a polynomial . . . . . 12
4.3. List Operations . . . . . . . . . . . . . . . . . . . . . 13
4.4. Binding Factors Computation . . . . . . . . . . . . . . . 14
4.5. Group Commitment Computation . . . . . . . . . . . . . . 15
4.6. Signature Challenge Computation . . . . . . . . . . . . . 16
5. TwoRound FROST Signing Protocol . . . . . . . . . . . . . . 16
5.1. Round One  Commitment . . . . . . . . . . . . . . . . . 19
5.2. Round Two  Signature Share Generation . . . . . . . . . 19
5.3. Signature Share Verification and Aggregation . . . . . . 22
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 24
6.1. FROST(Ed25519, SHA512) . . . . . . . . . . . . . . . . . 25
6.2. FROST(ristretto255, SHA512) . . . . . . . . . . . . . . 26
6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 27
6.4. FROST(P256, SHA256) . . . . . . . . . . . . . . . . . . 28
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7. Security Considerations . . . . . . . . . . . . . . . . . . . 30
7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 31
7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 31
7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 31
7.4. Input Message Validation . . . . . . . . . . . . . . . . 32
8. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 32
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 32
9.1. Normative References . . . . . . . . . . . . . . . . . . 32
9.2. Informative References . . . . . . . . . . . . . . . . . 33
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 34
Appendix B. Schnorr Signature Verification for PrimeOrder
Groups . . . . . . . . . . . . . . . . . . . . . . . . . 34
Appendix C. Trusted Dealer Key Generation . . . . . . . . . . . 35
C.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 36
C.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 37
Appendix D. Test Vectors . . . . . . . . . . . . . . . . . . . . 39
D.1. FROST(Ed25519, SHA512) . . . . . . . . . . . . . . . . . 40
D.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 41
D.3. FROST(ristretto255, SHA512) . . . . . . . . . . . . . . 43
D.4. FROST(P256, SHA256) . . . . . . . . . . . . . . . . . . 44
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 46
1. Introduction
DISCLAIMER: This is a workinprogress 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/draftirtfcfrgfrost.
Instructions are on that page as well.
Unlike signatures in a singleparty setting, threshold signatures
require cooperation among a threshold number of signers each holding
a share of a common private key. The security of threshold schemes
in general assume that an adversary can corrupt strictly fewer than a
threshold number of participants.
This document presents a variant of a Flexible RoundOptimized
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 Schnorrbased threshold signature
constructions. The variant of FROST presented in this document
requires two rounds to compute a signature. Singleround signing
with FROST is out of scope.
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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 keyrecovery attack in multi
party discrete logarithmbased signatures, such as FROST.
While an optimization to FROST was shown in [Schnorr21] that reduces
scalar multiplications from linear in the number of signers 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 signers that started round one of the
protocol is the same set of signers 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
draft06
* Make verification a perciphersuite functionality (#219)
* Use persigner values of rho to mitigate protocol malleability
(#217)
* Correct primeorder subgroup checks (#215, #211)
* Fix bug in ed25519 ciphersuite description (#205)
* Various editorial improvements (#208, #209, #210, #218)
draft05
* Update test vectors to include version string (#202, #203)
* Rename THRESHOLD_LIMIT to MIN_SIGNERS (#192)
* Use noncontiguous signers for the test vectors (#187)
* Add more reasoning why the coordinator MUST abort (#183)
* Add a function to generate nonces (#182)
* Add MUST that all participants have the same view of VSS
commitment (#174)
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* Use THRESHOLD_LIMIT instead of t and MAX_SIGNERS instead of n
(#171)
* Specify what the dealer is trusted to do (#166)
* Clarify types of NUM_SIGNERS 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 singleparty 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)
draft04
* 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).
* Define cofactor checks for verification (#118)
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* Various editorial improvements and add contributors (#124, #123,
#119, #116, #113, #109)
draft03
* 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.
draft02
* 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.
draft01
* Specify operations, notation and cryptographic dependencies.
draft00
* Outline CFRG draft based on draftkomlofrost.
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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 terminology is used throughout this document.
* A participant is an entity that is trusted to hold and use a
signing key share.
* MAX_SIGNERS denotes the number of participants, and the number of
shares that s is split into. This value MUST NOT exceed 2^161.
* MIN_SIGNERS denotes the threshold number of participants required
to issue a signature, where MIN_SIGNERS <= MAX_SIGNERS.
* NUM_SIGNERS denotes the number of signers that participate in an
invocation of FROST signing, where MIN_SIGNERS <= NUM_SIGNERS <=
MAX_SIGNERS.
* An identifier is an integer value associated with a participant,
or signer, and is a value in the range [1, MAX_SIGNERS].
Additionally, the following notation is used throughout the document.
* encode_uint16(x): Convert two byte unsigned integer (uint16) x to
a 2byte, bigendian byte string. For example, encode_uint16(310)
= [0x01, 0x36].
* random_bytes(n): Outputs n bytes, sampled uniformly at random
using a cryptographically secure pseudorandom number generator
(CSPRNG).
* 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 b1 in ascending
order, e.g., range(1, 4) = [1,2,3].
* pow(a, b): Output the integer result of a to the power of b, e.g.,
pow(2, 3) = 8.
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*  denotes concatenation, i.e., x  y = 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:
* Primeorder Group, Section 3.1;
* Cryptographic hash function, Section 3.2;
These are described in the following sections.
3.1. PrimeOrder 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). Scalar
multiplication is equivalent to the repeated application of the group
operation on an element A with itself r1 times, this is denoted as
r*A = A + ... + A. For any element A, p * A = 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 r1 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 equality
comparison as == and assignment of values by =.
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).
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* SerializeElement(A): Maps an Element A to a unique byte array buf
of fixed length Ne.
* DeserializeElement(buf): Attempts to map a byte array buf to an
Element A, and fails if the input is not a valid byte
representation of an element of the group. This function can
raise a DeserializeError if deserialization fails or A is the
identity element of the group; see Section 6 for groupspecific
input validation steps.
* SerializeScalar(s): Maps a Scalar s to a unique byte array buf of
fixed length Ns.
* DeserializeScalar(buf): Attempts to map a byte array buf to a
Scalar s. This function can raise a DeserializeError if
deserialization fails; see Section 6 for groupspecific 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 four
separate domainseparated hashes, H1, H2, H3, and H4, where H1, H2,
and H4 map arbitrary byte strings to Scalar elements of the prime
order group scalar field, and H3 is an alias for H with a domain
separator. The details of H1, H2, H3, and H4 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
These sections describes these operations in more detail.
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4.1. Nonce generation
To hedge against a bad RNG that outputs predictable values, we
generate nonces by sourcing fresh randomness and combine with the
secret key, to create a domainseparated hash function from the
ciphersuite hash function H, H4:
nonce_generate(secret):
Inputs:
 secret, a Scalar
Outputs: nonce, a Scalar
def nonce_generate(secret):
k_enc = random_bytes(32)
secret_enc = G.SerializeScalar(secret)
return H4(k_enc  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 is represented as a list of t
coefficients, where the constant term of the polynomial is in the
first position and the highestdegree 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 xcoordinate and
ycoordinate 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 xcoordinates as input. Note that
derive_lagrange_coefficient does not permit any xcoordinate to equal
0. Lagrange coefficients are used in FROST to evaluate a polynomial
f at xcoordinate 0, i.e., f(0), given a list of t other
xcoordinates.
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derive_lagrange_coefficient(x_i, L):
Inputs:
 x_i, an xcoordinate contained in L, a Scalar
 L, the set of xcoordinates, each a Scalar
Outputs: L_i, the ith Lagrange coefficient
Errors:
 "invalid parameters", if any xcoordinate is equal to 0 or if x_i
is not 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"
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
4.2.3. 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 t1. Recovering the
constant term occurs with a set of t points using polynomial
interpolation, defined as follows.
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Inputs:
 points, a set of t 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):
L = []
for point in points:
L.append(point.x)
f_zero = Scalar(0)
for point in points:
delta = point.y * derive_lagrange_coefficient(point.x, L)
f_zero = f_zero + delta
return f_zero
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 signer commitments into a bytestring for use in the FROST
protocol.
Inputs:
 commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each signer, where each element in the list
indicates the signer 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 signer 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 = encode_uint16(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.
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Inputs:
 commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each signer, where each element in the list
indicates the signer 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 signer identifier.
Outputs: A list of signer 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.
Inputs:
 binding_factor_list = [(i, binding_factor), ...],
a list of binding factors for each signer, where each element in the list
indicates the signer identifier i and their binding factor. This list MUST be sorted
in ascending order by signer identifier.
 identifier, Identifier i of the signer.
Outputs: A Scalar value.
Errors: "invalid participant", when the designated participant is not known
def binding_factor_for_participant(binding_factor_list, identifier):
binding_factors = []
for (i, binding_factor) in commitment_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 signer commitment list and message to be signed.
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Inputs:
 commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each signer, where each element in the list
indicates the signer 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 signer 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 = H3(msg)
encoded_commitment_hash = H3(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  encode_uint16(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.
Inputs:
 commitment_list =
[(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...], a list
of commitments issued by each signer, where each element in the list
indicates the signer 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 signer 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 + (binding_factor * binding_nonce_commitment))
return group_commitment
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4.6. Signature Challenge Computation
This section describes the subroutine for creating the permessage
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
5. TwoRound FROST Signing Protocol
We now present the tworound variant of the FROST threshold signature
protocol for producing Schnorr signatures. It involves signer
participants and a Coordinator. Signer participants are entities
with signing key shares that participate in the threshold signing
protocol. The Coordinator is an entity with the following
responsibilities:
1. Determining which signers will participate (at least MIN_SIGNERS
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 the selection of all participants, including
Coordinator and set of signers, are all chosen external to the
protocol. Note that it is possible to deploy the protocol without a
distinguished Coordinator; see Section 7.3 for more information.
Because key generation is not specified, all signers are assumed to
have the (public) group state that we refer to as "group info" below,
and their corresponding signing key shares.
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In particular, it is assumed that the Coordinator and each signer
participant P_i knows the following group info:
* Group public key, an Element in G, denoted PK, corresponding to
the group secret key s, which is a Scalar. PK is an output from
the group's key generation protocol, such as trusted_dealer_keygen
or a DKG.
* Public keys for each signer, denoted PK_i, which are similarly
outputs from the group's key generation protocol, Element values
in G.
And that each participant with identifier i additionally knows the
following:
* Participant is signing key share sk_i, which is the ith secret
share of s, a Scalar.
By construction, PK = G.ScalarBaseMult(s) and PK_i =
G.ScalarMultBase(sk_i) for each participant i.
The exact key generation mechanism is out of scope for this
specification. In general, key generation is a protocol that outputs
(1) a shared, group public key PK owned by each Signer, and (2)
individual shares of the signing key owned by each Signer. In
general, two possible key generation 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.
This signing variant of FROST requires signers 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 signers 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 Signer1 ... Signern

message
>

== Round 1 (Commitment) ==
 signer commitment  
<+ 
 ... 
 signer commitment 
<+
== Round 2 (Signature Share Generation) ==

 signer input  
+> 
 signature share  
<+ 
 ... 
 signer 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. 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
signer 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 signer 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 signer in round one generates a nonce nonce = (hiding_nonce,
binding_nonce) and commitment comm = (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 private output nonce from Participant P_i is stored locally and
kept private for use in the second round. This nonce MUST NOT be
reused in more than one invocation of FROST, and it MUST be generated
from a source of secure randomness. The public output comm from
Participant P_i is sent to the Coordinator.
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 signers will participate (of
number at least MIN_SIGNERS). Signers additionally require locally
held data; specifically, their private key and the nonces
corresponding to their commitment issued in round one.
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The Coordinator begins by sending each signer the message to be
signed along with the set of signing commitments for all signers in
the participant list. Each signer 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 signer
MUST ensure that their identifier as well as their commitment as from
the first round appears in commitment_list. Applications which
require that signers not process arbitrary input messages are also
required to also perform relevant applicationlayer input validation
checks; see Section 7.4 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 signer. 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 signer and sent by the Coordinator.
Each element in the list indicates the signer 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 signer 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 permessage 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 signer then
sends these shares back to the Coordinator. Each signer 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, signers 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 signers perform round two and send their signature shares to
the Coordinator, the Coordinator verifies each signature share for
correctness. In particular, for each signer, 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 signer. Note identifier will never equal 0.
 PK_i, the public key for the ith signer, 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 signer.
 sig_share_i, a Scalar value indicating the signature share as produced in
round two from the ith signer.
 commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each signer, where each element
in the list indicates the signer 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 signer 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 + (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 + ((challenge * lambda_i) * PK_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 signer share, the
Coordinator MUST abort the protocol for correctness reasons.
Excluding one signer means that their nonce will not be included in
the joint response z and consequently the output signature will not
verify.
Otherwise, if all signer shares 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 signer,
of length NUM_SIGNERS, where MIN_SIGNERS <= NUM_SIGNERS <= MAX_SIGNERS.
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 primeorder group
details and cryptographic hash function. Each ciphersuite is denoted
as (Group, Hash), e.g., (ristretto255, SHA512). This section
contains some ciphersuites.
The RECOMMENDED ciphersuite is (ristretto255, SHA512) Section 6.2.
The (Ed25519, SHA512) ciphersuite is included for backwards
compatibility with [RFC8032].
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The DeserializeElement and DeserializeScalar functions instantiated
for a particular primeorder 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 signer.
6.1. FROST(Ed25519, SHA512)
This ciphersuite uses edwards25519 for the Group and SHA512 for the
Hash function H meant to produce signatures indistinguishable from
Ed25519 as specified in [RFC8032]. The value of the contextString
parameter is empty.
* Group: edwards25519 [RFC8032]
 Order: 2^252 + 27742317777372353535851937790883648493 (see
[RFC7748])
 Identity: As defined in [RFC7748].
 RandomScalar: Implemented by repeatedly generating a random
32byte string and invoking DeserializeScalar on the result
until success.
 SerializeElement: Implemented as specified in [RFC8032],
Section 5.1.2.
 DeserializeElement: 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 primeorder 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.
 SerializeScalar: Implemented by outputting the littleendian
32byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHA512, and Nh = 64.
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 H1(m): Implemented by computing H("rho"  m), interpreting the
64byte digest as a littleendian integer, and reducing the
resulting integer modulo L =
2^252+27742317777372353535851937790883648493.
 H2(m): Implemented by computing H(m), interpreting the 64byte
digest as a littleendian integer, and reducing the resulting
integer modulo L =
2^252+27742317777372353535851937790883648493.
 H3(m): Implemented as an alias for H, i.e., H(m).
 H4(m): Implemented by computing H("nonce"  m), interpreting
the 64byte digest as a littleendian integer, and reducing the
resulting integer modulo L =
2^252+27742317777372353535851937790883648493.
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 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, SHA512)
This ciphersuite uses ristretto255 for the Group and SHA512 for the
Hash function H. The value of the contextString parameter is "FROST
RISTRETTO255SHA512v5".
* Group: ristretto255 [RISTRETTO]
 Order: 2^252 + 27742317777372353535851937790883648493 (see
[RISTRETTO])
 Identity: As defined in [RISTRETTO].
 RandomScalar: Implemented by repeatedly generating a random
32byte string and invoking DeserializeScalar on the result
until success.
 SerializeElement: Implemented using the 'Encode' function from
[RISTRETTO].
 DeserializeElement: Implemented using the 'Decode' function
from [RISTRETTO].
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 SerializeScalar: Implemented by outputting the littleendian
32byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHA512, and Nh = 64.
 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  "digest" 
m).
 H4(m): Implemented by computing H(contextString  "nonce" 
m) and mapping the output to a Scalar as described in
[RISTRETTO], Section 4.4.
Signature verification is as specified in Appendix B.
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 empty.
* Group: edwards448 [RFC8032]
 Order: 2^446  138180668098951153520073867485154268803366924748
82178609894547503885
 Identity: As defined in [RFC7748].
 RandomScalar: Implemented by repeatedly generating a random
48byte string and invoking DeserializeScalar on the result
until success.
 SerializeElement: Implemented as specified in [RFC8032],
Section 5.2.2.
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 DeserializeElement: Implemented as specified in [RFC8032],
Section 5.2.3. Additionally, this function validates that the
resulting element is not the group identity element.
 SerializeScalar: Implemented by outputting the littleendian
48byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 48byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHAKE256, and Nh = 114.
 H1(m): Implemented by computing H("rho"  m), interpreting the
lower 57 bytes as a littleendian integer, and reducing the
resulting integer modulo L = 2^446  13818066809895115352007386
748515426880336692474882178609894547503885.
 H2(m): Implemented by computing H(m), interpreting the lower 57
bytes as a littleendian integer, and reducing the resulting
integer modulo L = 2^446  138180668098951153520073867485154268
80336692474882178609894547503885.
 H3(m): Implemented as an alias for H, i.e., H(m).
 H4(m): Implemented by computing H("nonce"  m), interpreting
the lower 57 bytes as a littleendian integer, and reducing the
resulting integer modulo L = 2^446  13818066809895115352007386
748515426880336692474882178609894547503885.
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 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(P256, SHA256)
This ciphersuite uses P256 for the Group and SHA256 for the Hash
function H. The value of the contextString parameter is "FROST
P256SHA256v5".
* Group: P256 (secp256r1) [x9.62]
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 Order: 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9ca
c2fc632551
 Identity: As defined in [x9.62].
 RandomScalar: Implemented by repeatedly generating a random
32byte string and invoking DeserializeScalar on the result
until success.
 SerializeElement: Implemented using the compressed Elliptic
CurvePointtoOctetString method according to [SECG].
 DeserializeElement: Implemented by attempting to deserialize a
public key using the compressed OctetStringtoEllipticCurve
Point method according to [SECG], and then performs partial
publickey 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: Implemented using the FieldElementtoOctet
String conversion according to [SECG].
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string using OctetStringtoField
Element from [SECG]. This function can fail if the input does
not represent a Scalar between the value 0 and G.Order()  1.
* Hash (H): SHA256, and Nh = 32.
 H1(m): Implemented using hash_to_field from [HASHTOCURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA256, DST
= contextString  "rho", and prime modulus equal to Order().
 H2(m): Implemented using hash_to_field from [HASHTOCURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA256, DST
= contextString  "chal", and prime modulus equal to Order().
 H3(m): Implemented by computing H(contextString  "digest" 
m).
 H4(m): Implemented using hash_to_field from [HASHTOCURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA256, DST
= contextString  "nonce", and prime modulus equal to Order().
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Signature verification is as specified in Appendix B.
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.
* Trusted dealer. The dealer that performs key generation is
trusted to follow the protocol, although participants still are
able to verify the consistency of their shares via a VSS
(verifiable secret sharing) step; see Appendix C.2.
* Unforgeability assuming at most (MIN_SIGNERS1) corrupted signers.
So long as an adversary corrupts fewer than MIN_SIGNERS
participants, the scheme remains secure against Existential
Unforgeability Under Chosen Message Attack (EUFCMA) attacks, as
defined in [BonehShoup], Definition 13.2.
* Coordinator. We assume the Coordinator at the time of signing
does not perform a denial of service attack. A denial of service
would include any 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
signer 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.
The protocol as specified in this document does not target 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
higherlevel communication channel can be used to facilitate key
generation and signing.
The rest of this section documents issues particular to
implementations or deployments.
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7.1. Nonce Reuse Attacks
Nonces generated by each participant in the first round of signing
must be sampled uniformly at random and cannot be derived from some
deterministic function. This is to avoid replay attacks initiated by
other signers, which allows for a complete keyrecovery attack. The
Coordinator MAY further hedge against nonce reuse attacks by tracking
signer nonce commitments used for a given group key, at the cost of
additional state.
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 signers 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, signers 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 signers, each signer
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.
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7.4. Input Message Validation
Some applications may require that signers only process messages of a
certain structure. For example, in digital currency applications
wherein multiple signers may collectively sign a transaction, it is
reasonable to require that each signer check the input message to be
a syntactically valid transaction. As another example, use of
threshold signatures in TLS [TLS] to produce signatures of transcript
hashes might require that signers check that the input message is a
valid TLS transcript from which the corresponding transcript hash can
be derived.
In general, input message validation is an applicationspecific
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 signers do not operate as
signing oracles for arbitrary messages.
8. Contributors
* Isis Lovecruft
* T. WilsonBrown
* Alden Torres
9. References
9.1. Normative References
[HASHTOCURVE]
FazHernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, InternetDraft, draftirtfcfrghashtocurve
16, 15 June 2022, .
[KEYAGREEMENT]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
Davis, "Recommendation for pairwise keyestablishment
schemes using discrete logarithm cryptography", National
Institute of Standards and Technology report,
DOI 10.6028/nist.sp.80056ar3, April 2018,
.
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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,
.
[RFC8032] Josefsson, S. and I. Liusvaara, "EdwardsCurve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[RISTRETTO]
Valence, H. D., Grigg, J., Hamburg, M., Lovecruft, I.,
Tankersley, G., and F. Valsorda, "The ristretto255 and
decaf448 Groups", Work in Progress, InternetDraft, draft
irtfcfrgristretto255decaf44803, 25 February 2022,
.
[SECG] "Elliptic Curve Cryptography, Standards for Efficient
Cryptography Group, ver. 2", 2009,
.
[x9.62] ANSI, "Public Key Cryptography for the Financial Services
Industry: the Elliptic Curve Digital Signature Algorithm
(ECDSA)", ANSI X9.621998, September 1998.
9.2. Informative References
[BonehShoup]
Boneh, D. and V. Shoup, "A Graduate Course in Applied
Cryptography", January 2020,
.
[FROST20] Komlo, C. and I. Goldberg, "TwoRound Threshold Signatures
with FROST", 22 December 2020,
.
[RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106, RFC 4086,
DOI 10.17487/RFC4086, June 2005,
.
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[RFC7748] Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
for Security", RFC 7748, DOI 10.17487/RFC7748, January
2016, .
[Schnorr21]
Crites, E., Komlo, C., and M. Maller, "How to Prove
Schnorr Assuming Schnorr", 11 October 2021,
.
[StrongerSec22]
Bellare, M., Tessaro, S., and C. Zhu, "Stronger Security
for NonInteractive Threshold Signatures: BLS and FROST",
1 June 2022, .
[TLS] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
.
Appendix A. Acknowledgments
This document was improved based on input and contributions by the
Zcash Foundation engineering team.
Appendix B. Schnorr Signature Verification for PrimeOrder Groups
This section contains a routine for verifying Schnorr signatures with
validated inputs. Specifically, it assumes that signature R
component and public key belong to the primeorder 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 + (c * PK)
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 to be sent
to all other participants. We highlight at a high level how this
operation can be performed.
The dealer 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:
 s, a group secret, Scalar, that MUST be derived from at least Ns bytes of entropy
 MAX_SIGNERS, the number of shares to generate, an integer
 MIN_SIGNERS, the threshold of the secret sharing scheme, an integer
Outputs:
 signer_private_keys, MAX_SIGNERS shares of the secret key s, each a Scalar value.
 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(s, MAX_SIGNERS, MIN_SIGNERS):
signer_private_keys, coefficients = secret_share_shard(secret_key, MAX_SIGNERS, MIN_SIGNERS)
vss_commitment = vss_commit(coefficients):
PK = G.ScalarBaseMult(secret_key)
return signer_private_keys, vss_commitment
It is assumed the dealer then sends one secret key share to each of
the NUM_SIGNERS 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.
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.
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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_SIGNERS
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 primeorder group G.
The procedure for splitting a secret into shares is as follows.
secret_share_shard(s, MAX_SIGNERS, MIN_SIGNERS):
Inputs:
 s, secret value to be shared, a Scalar
 MAX_SIGNERS, the number of shares to generate, an integer
 MIN_SIGNERS, the threshold of the secret sharing scheme, an integer
Outputs:
 secret_key_shares, A list of MAX_SIGNERS number of secret shares, which is a tuple
consisting of the participant identifier and the key share, each of which is a Scalar
 coefficients, a vector of the t coefficients which uniquely determine
a polynomial f.
Errors:
 "invalid parameters", if MIN_SIGNERS > MAX_SIGNERS or if MIN_SIGNERS is less than 2
def secret_share_shard(s, MAX_SIGNERS, MIN_SIGNERS):
if MIN_SIGNERS > MAX_SIGNERS:
raise "invalid parameters"
if MIN_SIGNERS < 2:
raise "invalid parameters"
# Generate random coefficients for the polynomial, yielding
# a polynomial of degree (MIN_SIGNERS  1)
coefficients = [s]
for i in range(1, MIN_SIGNERS):
coefficients.append(G.RandomScalar())
# Evaluate the polynomial for each point x=1,...,n
secret_key_shares = []
for x_i in range(1, MAX_SIGNERS + 1):
y_i = polynomial_evaluate(x_i, coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_share.append(secret_key_share_i)
return secret_key_shares, coefficients
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Let points be the output of this function. The ith element in
points is the share for the ith 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_SIGNERS to
recover the secret s is as follows.
secret_share_combine(shares):
Inputs:
 shares, a list of at minimum MIN_SIGNERS secret shares, each a tuple (i, f(i))
Outputs: The resulting secret s, a Scalar, that was previously split into shares
Errors:
 "invalid parameters", if less than MIN_SIGNERS input shares are provided
def secret_share_combine(shares):
if len(shares) < MIN_SIGNERS:
raise "invalid parameters"
s = polynomial_interpolation(shares)
return s
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 ensure 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
MIN_SIGNERS1 is as follows.
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vss_commit(coeffs):
Inputs:
 coeffs, a vector of the MIN_SIGNERS 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.
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, commitment)
(i, sk_i) = share_i
S_i = ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, MIN_SIGNERS):
S_i' += pow(i, j) * vss_commitment[j]
if S_i == S_i':
return 1
return 0
We now define how the Coordinator and signer participants can derive
group info, which is an input into the FROST signing protocol.
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derive_group_info(MAX_SIGNERS, MIN_SIGNERS, vss_commitment):
Inputs:
 MAX_SIGNERS, the number of shares to generate, an integer
 MIN_SIGNERS, 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.
 signer_public_keys, a list of MAX_SIGNERS public keys PK_i for i=1,...,MAX_SIGNERS,
where each PK_i is the public key, an Element, for participant i.
derive_group_info(MAX_SIGNERS, MIN_SIGNERS, vss_commitment)
PK = vss_commitment[0]
signer_public_keys = []
for i in range(1, MAX_SIGNERS+1):
PK_i = G.Identity()
for j in range(0, MIN_SIGNERS):
PK_i += pow(i, j) * vss_commitment[j]
signer_public_keys.append(PK_i)
return PK, signer_public_keys
Appendix D. 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_SIGNERS, MIN_SIGNERS, and
NUM_SIGNERS.
* 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. All values
are encoded as hexadecimal strings.
* Signer input parameters: This lists the signing key share for each
of the NUM_SIGNERS signers.
* Round one parameters and outputs: This lists the NUM_SIGNERS
participants engaged in the protocol, identified by their integer
identifier, the hiding and binding commitment values produced in
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Section 5.1, as well as 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_SIGNERS
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.
D.1. FROST(Ed25519, SHA512)
// Configuration information
MAX_SIGNERS: 3
MIN_SIGNERS: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
90c6e13a98304
group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040
d380fb9738673
message: 74657374
// Signer input parameters
S1 signer_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033
f2ec83d93509
S2 signer_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee
d76bf409e80d
S3 signer_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8
bdea643a9a02
// Round one parameters
participants: 1,3
// Signer round one outputs
S1 hiding_nonce: d9aad97e1a1127bb87702ce8d81d8c07c7cbca89e784868d8e38
76ff6b459700
S1 binding_nonce: 5063be2774520d08a5ccd7f1213fb1179a5fa292bf13bc91cb2
8e7bd4d4a690c
S1 hiding_nonce_commitment: 33fc1eacb8d168e54ab811320196b7715a9918461
e1e00c3688e503ace1628d5
S1 binding_nonce_commitment: b32d41ce5a230b459de8c49b0619cb5fbde46690
752ec94ef05aa1f8647301df
S1 binding_factor_input: ee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e
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473f8819a5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8
e6f57f50028a8ff7b5e7b0a8efad964ba83310b56607920b6c3c979e38e583aa9d02a
df541c58ea92a259b5c8a184d0a7c5ea978e42a8ff84608c38cbb22475bd54858c4ff
d524e0001
S1 binding_factor: a523eba59830d1b5dbe6914e954862c5396b979bcd258fe323
e324335db81101
S3 hiding_nonce: 86961f3a429ac0c5696f49e6d796817ff653f83c07f34e9e1f4d
4c8c515b7900
S3 binding_nonce: 72225ec11c1315d9f1ea0e78b1160ed95800fadd0191d23fd2f
2c90ac96cb307
S3 hiding_nonce_commitment: 493651312d26af93d2bc5b92eeecc12f1d6da9e18
4911e0943ebeb5ec59d3926
S3 binding_nonce_commitment: 8dae85381a582288c934741defbcaeba7a1b944e
3a2df0aa0ac96aec4431c690
S3 binding_factor_input: ee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e
473f8819a5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8
e6f57f50028a8ff7b5e7b0a8efad964ba83310b56607920b6c3c979e38e583aa9d02a
df541c58ea92a259b5c8a184d0a7c5ea978e42a8ff84608c38cbb22475bd54858c4ff
d524e0003
S3 binding_factor: c900ec81622c4b4b756139607357c1bf531df1a3b055304af2
15278aadb84b02
// Round two parameters
participants: 1,3
// Signer round two outputs
S1 sig_share: caae171b83bff0c2c6f56a1276892918ba228146f6344b85d2ec6ef
eb6f16d0d
S3 sig_share: ea6fdbf61683cf5f1f742e1b91583f0f667f0369efd2e33399b96d5
a3ff0300d
sig: 5da10008c13c04dd72328ba8e0f72b63cad43c3bf4b7eaada1c78225afbd977e
c74afdb47fdfadca0fcda18a28e8891220a284afe5072fb96ba6dc58f6e19e0a
D.2. FROST(Ed448, SHAKE256)
// Configuration information
MAX_SIGNERS: 3
MIN_SIGNERS: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: 6298e1eef3c379392caaed061ed8a31033c9e9e3420726f23b4
04158a401cd9df24632adfe6b418dc942d8a091817dd8bd70e1c72ba52f3c00
group_public_key: 1588564c56a8edb53b55399df5b65fd2abe777717baa2ef440b
13fe13b7ce077347f5e4346ab4475f9258fb947978b0123884832a46c6be800
message: 74657374
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// Signer input parameters
S1 signer_share: 4a2b2f5858a932ad3d3b18bd16e76ced3070d72fd79ae4402df2
01f525e754716a1bc1b87a502297f2a99d89ea054e0018eb55d39562fd0100
S2 signer_share: 2503d56c4f516444a45b080182b8a2ebbe4d9b2ab509f25308c8
8c0ea7ccdc44e2ef4fc4f63403a11b116372438a1e287265cadeff1fcb0700
S3 signer_share: 00db7a8146f995db0a7cf844ed89d8e94c2b5f259378ff66e39d
172828b264185ac4decf7219e4aa4478285b9c0eef4fccdf3eea69dd980d00
// Round one parameters
participants: 1,3
// Signer round one outputs
S1 hiding_nonce: a7fa56e3dc9935845e58275131eeda30d648432cba7ec3e3c522
dea613962439cdbd016cd78d54eb72ba8ec4e1b4e6cb41d3afb55a28f40300
S1 binding_nonce: 66480d4125faf4033babeee514f0b8d26118618ad05d6e3f8e6
4ea7082249b460c9eee5259f6ca6d1036db968923a89534b679c6ec96181b00
S1 hiding_nonce_commitment: b4b449e692a233b7661da0dbe4c337dd1c8c8369f
c0786d6d1537ab371bb8afc4e59812de18300aef79b26920696c180e2f78f96bfb0d9
1100
S1 binding_nonce_commitment: 705e18f4f601754c700ef93591fb24af5d3ca0c8
052a890de5aa2dc9231903b5d0d8a56c0dfe5b3c66e94b8615f705e7a5086fe93c020
b5600
S1 binding_factor_input: b54ff7255705a71ee2925e4a3e30e41aed489a579d55
95e0df13e32e1e4dd202a7c7f68b31d6418d9845eb4d757adda6ab189e1bb340db818
e5b3bc725d992faf63e9b0500db10517fe09d3f566fba3a80e46a403e0c7d41548fbf
75cf2662b00225b502961f98d8c9ff937de0b24c231845c16de964d8b11ffe861c657
afc6656a15d98dc9e6df3d2371d0fd2e0d990ad977470d0a371c1510accd90bb9fe51
4da13c4c2d97488a7980cb7ea47ac5124ec710faa8692c009794b7c7a9e29b8cc5ea4
cd9418c853676e55971349c313f84b902c1a112a0ecdbecb5fb6030ad874161ff7c00
01
S1 binding_factor: 63e240eeaa6d10b99561d7eb813fd4164f3cde8eeffcf2c973
c9de583ea075e471efbeb949af4fb11e7659bfacbd67eba4d9aa58c653190f00
S3 hiding_nonce: 6341f043b08f518d5f12ce4d699e3827e0ad7a8f2a4bcdcee64f
afe99dfbbe4187a01ebdf967a3503bbd84af24e0af93b078ab8d1cda533c00
S3 binding_nonce: 1716d9dc1e4c97553708f2ebc65039a50d00919a68940afd660
f31d1939e6e5f4a88631693f1acb2e737feff2bef7b0cdb1d3baae603272900
S3 hiding_nonce_commitment: 8dd1e8cf1e0330bbcdeced3e8e325e48bba2b0caf
34185a53bd8227f1c96be778681164417a582d39f1bea23a8dfe5a9e0a96d3dbbf8ee
6180
S3 binding_nonce_commitment: 58df1966884f46af333e26b6c1cace2720e2bd70
39a21b1b8483e28974237bcea8c5649cfe460e821afc94021d0b686029681a1148cd3
f0e00
S3 binding_factor_input: b54ff7255705a71ee2925e4a3e30e41aed489a579d55
95e0df13e32e1e4dd202a7c7f68b31d6418d9845eb4d757adda6ab189e1bb340db818
e5b3bc725d992faf63e9b0500db10517fe09d3f566fba3a80e46a403e0c7d41548fbf
75cf2662b00225b502961f98d8c9ff937de0b24c231845c16de964d8b11ffe861c657
afc6656a15d98dc9e6df3d2371d0fd2e0d990ad977470d0a371c1510accd90bb9fe51
4da13c4c2d97488a7980cb7ea47ac5124ec710faa8692c009794b7c7a9e29b8cc5ea4
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cd9418c853676e55971349c313f84b902c1a112a0ecdbecb5fb6030ad874161ff7c00
03
S3 binding_factor: bb8b3d669199e180628a91097a03422c12103b2f34c7931f98
0accb20574a506d8cf966c444fcf5fd5bbfbc6943440aa981ef6fb070fad0600
// Round two parameters
participants: 1,3
// Signer round two outputs
S1 sig_share: a2697f5e866a4b61651f16df4105b80a47365294522dbfa26ed9f8b
cb66954dec45326f5645590f2e0a8664e8870c053ec8ba5a58526a42f00
S3 sig_share: 1bce211bc3a8ccd27721c091bc426f422314f70b0bde3f4c45bfad2
48e57643f68983bcba53e6c500bbb4d19de4b5320e44a757c8997042c00
sig: 60cf90055083501d04f38c133c01f121444a6c6889745363555cea964285d5eb
bdb25690cdff9ca96a28b10bab68aa721b0fca9288a7efbe80caf248ceb6509f1088b
110e38b85ba2bda1373f11330b02aca74dc6445c1b81d2dec61c00a94fc42ec63b467
66bc1374d0d61a220fbea81b00
D.3. FROST(ristretto255, SHA512)
// Configuration information
MAX_SIGNERS: 3
MIN_SIGNERS: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: 1b25a55e463cfd15cf14a5d3acc3d15053f08da49c8afcf3ab2
65f2ebc4f970b
group_public_key: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3dd9c3c0d
05083c7254f57
message: 74657374
// Signer input parameters
S1 signer_share: 5c3430d391552f6e60ecdc093ff9f6f4488756aa6cebdbad75a7
68010b8f830e
S2 signer_share: b06fc5eac20b4f6e1b271d9df2343d843e1e1fb03c4cbb673f28
72d459ce6f01
S3 signer_share: f17e505f0e2581c6acfe54d3846a622834b5e7b50cad9a2109a9
7ba7a80d5c04
// Round one parameters
participants: 1,3
// Signer round one outputs
S1 hiding_nonce: eb0dc12ae7b746d36e3f2de46ce3833a05b9d4af5434eeb8cafa
efda76906d00
S1 binding_nonce: 491e91aa9df514ef598d5e0c7c5cdd088fbde4965b96069d546
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c0f04f1822b03
S1 hiding_nonce_commitment: c6fe28df6a13f2ea80a911dd7a284e4b185bc8d3e
3102adaf88807a5e3d3813c
S1 binding_nonce_commitment: a413722bcfc71ba044bb2846b814401e60fed6b2
fc5bfb25f5a49e63474b7011
S1 binding_factor_input: 678630bf982c566949d7f22d2aefb94f252c664216d3
32f34e2c8fdcd7045f207f854504d0daa534a5b31dbdf4183be30eb4fdba4f962d8a6
b69cf20c2734043c229faa47541463641bcc7c23a4576d74e536dea0d7f7ae6e2c846
1a63f4fe97599d8d83005d520a104f937ce3b8181281348fad246e1c0d89ed4cca7d5
22e750001
S1 binding_factor: 2e81f15e28874f517b6d2023291e49000f71f998852b484aae
f945000478ea05
S3 hiding_nonce: abd12b8e6f255ee1e540eab029003a6e956567617720f61115f0
941615892209
S3 binding_nonce: 218e22625f93f262f025bd2d13c46ba722aa29fe585ceed66ff
442d98fe4e509
S3 hiding_nonce_commitment: 5450c4c98c3fc6bb579bded17fcdc23073d2ecfb7
61e3f9433cbc991e1496068
S3 binding_nonce_commitment: 0ae0cf608fcba285ec1f6c84c955572c91a4fafc
c1f1120f4f30b25e40fbcc0a
S3 binding_factor_input: 678630bf982c566949d7f22d2aefb94f252c664216d3
32f34e2c8fdcd7045f207f854504d0daa534a5b31dbdf4183be30eb4fdba4f962d8a6
b69cf20c2734043c229faa47541463641bcc7c23a4576d74e536dea0d7f7ae6e2c846
1a63f4fe97599d8d83005d520a104f937ce3b8181281348fad246e1c0d89ed4cca7d5
22e750003
S3 binding_factor: 240d5257c68e377c1994481081a8a4c4362b9e82e523088c30
d91f8c2811890e
// Round two parameters
participants: 1,3
// Signer round two outputs
S1 sig_share: efae3a83437fa8cd96194aacc56a7eb841630c280da99e7764a81d1
340323306
S3 sig_share: 96ddc4582e45eabce46f07b9e9375f8b49d35d1510fd34ac02b1e79
d6100a602
sig: 7ec584cef9a383afb43883b73bcaa6313afe878bd5fe75a608311b866a76ec67
858cffdb71c4928a7b895165afa2dd438b366a3d1da6d323675905b1a132d908
D.4. FROST(P256, SHA256)
// Configuration information
MAX_SIGNERS: 3
MIN_SIGNERS: 2
NUM_SIGNERS: 2
// Group input parameters
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group_secret_key: 8ba9bba2e0fd8c4767154d35a0b7562244a4aaf6f36c8fb8735
fa48b301bd8de
group_public_key: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99cfbd02b4
dc00585ec45ab70
message: 74657374
// Signer input parameters
S1 signer_share: 0c9c1a0fe806c184add50bbdcac913dda73e482daf95dcb9f35d
bb0d8a9f7731
S2 signer_share: 8d8e787bef0ff6c2f494ca45f4dad198c6bee01212d6c8406715
9c52e1863ad5
S3 signer_share: 0e80d6e8f6192c003b5488ce1eec8f5429587d48cf001541e713
b2d53c09d928
// Round one parameters
participants: 1,3
// Signer round one outputs
S1 hiding_nonce: 33a519cf070a166f9ef41a798d03423743f3e7d0b0efd5d0d963
773c4c53205e
S1 binding_nonce: 307d208d0c5728f323ae374f1ebd7f14a1a49b77d9d4bc1eab2
22218a17765ff
S1 hiding_nonce_commitment: 021e5c8b286dc859314eb1c0a2024a2077ad49b60
3112dd7bfaf326591d3fab332
S1 binding_nonce_commitment: 039431f230cf2bd90ad556a7f3d6b5a5686efd19
4c863356628d7296c2a3fa5900
S1 binding_factor_input: 7a753fed12531fbcd151e1d84702927c39063e780e91
c01f02bd11b60d7632bf44df5a9e0d49f359549018a13a586b5ede02cadef80472f75
d195b82160f43ea0001
S1 binding_factor: 71f09a2c4a1fc2f7a1379102809b4ac3247837c532cc5cf091
3782496c515655
S3 hiding_nonce: a614eadb972dc37b88aeceb6e899903f3104742d13f379a0e014
541decbea4a4
S3 binding_nonce: e509791018504c5bb87edaf0f44761cc840888507c4cd802379
71d78e65f70f2
S3 hiding_nonce_commitment: 0282308b1a22eb8efa13d4655f795f1cbf6525d88
63ac0d60c4e164b7436d41778
S3 binding_nonce_commitment: 036549bda4158ec5f76611275360a57e6ad5007d
6c072462feb42c8f2a25ec94ea
S3 binding_factor_input: 7a753fed12531fbcd151e1d84702927c39063e780e91
c01f02bd11b60d7632bf44df5a9e0d49f359549018a13a586b5ede02cadef80472f75
d195b82160f43ea0003
S3 binding_factor: 57a1061da0837cc0cd7e901a1d33f46efa18af9c3e6468cca8
8edd2d4a16e78d
// Round two parameters
participants: 1,3
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// Signer round two outputs
S1 sig_share: 61e8b9c474df2e66ad19fd80a6e6cec1c6fe43c0a1cffd2d1c28299
e93e1bbdb
S3 sig_share: 9651d355ca1dea2557ba1f73e38a9f4ff1f1afc565323ef27f88a9d
14df8370e
sig: 02dfba781e17b830229ae4ed22ebe402873683d9dfd945d01762217fb3172c2a
71f83a8d1a3efd188c04d41cf48a716e11b8eff38607023c1f9bb0d36fe1d9f2e9
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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