| Internet-Draft | FROST | July 2022 |
| Connolly, et al. | Expires 12 January 2023 | [Page] |
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 participant is malicious.¶
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.¶
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/.¶
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 12 January 2023.¶
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 extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
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 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 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 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 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.¶
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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.¶
MAX_SIGNERS denotes the number of participants, and the number of shares that s is split into. This value MUST NOT exceed 2^16-1.¶
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.¶
Additionally, the following notation is used throughout the document.¶
encode_uint16(x): Convert two byte unsigned integer (uint16) x to a 2-byte,
big-endian 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 b-1 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.¶
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.¶
FROST signing depends on the following cryptographic constructs:¶
These are described in the following sections.¶
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 r-1 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 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 equality comparison as == and assignment of values by =.¶
We now detail a number of member functions that can be invoked on G.¶
G (i.e. p).¶
Element of the group (i.e. I).¶
Scalar element in GF(p).¶
Element A to a unique byte array buf of fixed length Ne.¶
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 group-specific
input validation steps.¶
s to a unique byte array buf of fixed length Ns.¶
buf to a Scalar s.
This function can raise a DeserializeError if deserialization fails; see
Section 6 for group-specific input validation steps.¶
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 domain-separated 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.¶
Beyond the core dependencies, the protocol in this document depends on the following helper operations:¶
These sections describes these operations in more detail.¶
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 domain-separated 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)
¶
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 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.¶
This section describes a method for evaluating a polynomial f at a
particular input x, i.e., y = f(x) using Horner's method.¶
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
¶
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.¶
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 any x-coordinate 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
¶
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 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
¶
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.¶
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"
¶
This section describes the subroutine for computing binding factors based on the signer 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 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
¶
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
¶
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
¶
We now present the two-round 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:¶
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.¶
In particular, it is assumed that the Coordinator and each signer participant P_i knows the following
group info:¶
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.¶
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:¶
is signing key share sk_i, which is the i-th 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.¶
(group info) (group info, (group info,
| signing key share) signing key share)
| | |
v v v
Coordinator Signer-1 ... Signer-n
------------------------------------------------------------
message
------------>
|
== Round 1 (Commitment) ==
| signer commitment | |
|<-----------------------+ |
| ... |
| signer commitment |
|<-----------------------------------------+
== Round 2 (Signature Share Generation) ==
|
| signer input | |
+------------------------> |
| signature share | |
|<-----------------------+ |
| ... |
| signer input |
+------------------------------------------>
| signature share |
<------------------------------------------+
|
== Aggregation ==
|
signature |
<-----------+
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.¶
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.¶
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.¶
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.¶
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 application-layer 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.¶
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 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 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.¶
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.¶
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.¶
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
¶
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).¶
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.¶
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 signer.¶
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 empty.¶
Group: edwards25519 [RFC8032]¶
G.Order() - 1.¶
Hash (H): SHA-512, and Nh = 64.¶
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.¶
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-v5".¶
Group: ristretto255 [RISTRETTO]¶
G.Order() - 1.¶
Hash (H): SHA-512, and Nh = 64.¶
Signature verification is as specified in Appendix B.¶
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.¶
G.Order() - 1.¶
Hash (H): SHAKE256, and Nh = 114.¶
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.¶
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-v5".¶
Group: P-256 (secp256r1) [x9.62]¶
G.Order() - 1.¶
Hash (H): SHA-256, and Nh = 32.¶
expand_message_xmd with SHA-256, DST = contextString || "rho", and
prime modulus equal to Order().¶
expand_message_xmd with SHA-256, DST = contextString || "chal", and
prime modulus equal to Order().¶
expand_message_xmd with SHA-256, DST = contextString || "nonce", and
prime modulus equal to Order().¶
Signature verification is as specified in Appendix B.¶
A security analysis of FROST exists in [FROST20] and [Schnorr21]. The protocol as specified in this document assumes the following threat model.¶
(MIN_SIGNERS-1) corrupted signers. So long as an adversary
corrupts fewer than MIN_SIGNERS participants, the scheme remains secure against Existential
Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in [BonehShoup],
Definition 13.2.¶
The protocol as specified in this document does not target the following goals:¶
The rest of this section documents issues particular to implementations or deployments.¶
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 key-recovery 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.¶
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.¶
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.¶
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 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 signers do not operate as signing oracles for arbitrary messages.¶
This document was improved based on input and contributions by the Zcash Foundation engineering team.¶
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 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 + (c * PK)
return l == r
¶
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.¶
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 prime-order 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
¶
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_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
¶
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_SIGNERS-1 is as follows.¶
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.¶
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
¶
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 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 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¶
// 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 // 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 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¶
// 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 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¶
// Configuration information MAX_SIGNERS: 3 MIN_SIGNERS: 2 NUM_SIGNERS: 2 // Group input parameters 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 // Signer round two outputs S1 sig_share: 61e8b9c474df2e66ad19fd80a6e6cec1c6fe43c0a1cffd2d1c28299 e93e1bbdb S3 sig_share: 9651d355ca1dea2557ba1f73e38a9f4ff1f1afc565323ef27f88a9d 14df8370e sig: 02dfba781e17b830229ae4ed22ebe402873683d9dfd945d01762217fb3172c2a 71f83a8d1a3efd188c04d41cf48a716e11b8eff38607023c1f9bb0d36fe1d9f2e9¶