| Internet-Draft | FROST | August 2022 |
| Connolly, et al. | Expires 20 February 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 20 February 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, denoted as
ScalarBaseMult(A, r). We denote the sum, difference, and product of two scalars using the +, -,
and * operators, respectively. (Note that this means + may refer to group element addition or
scalar addition, depending on types of the operands.) For any element A, ScalarBaseMult(A, p) = I.
We denote B as a fixed generator of the group. Scalar base multiplication is equivalent to the repeated application
of the group operation B with itself r-1 times, this is denoted as ScalarBaseMult(r). The set of
scalars corresponds to GF(p), which we refer to as the scalar field. This document uses types
Element and Scalar to denote elements of the group G and its set of scalars, respectively.
We denote Scalar(x) as the conversion of integer input x to the corresponding Scalar value with
the same numeric value. For example, Scalar(1) yields a Scalar representing the value 1.
We denote equality comparison as == and assignment of values by =.¶
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).¶
A and Scalar k.¶
A and the group generator B.¶
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 separate domain-separated hashes, H1, H2, H3, H4, and H5:¶
The details of H1, H2, H3, H4, and H5 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, H3:¶
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 H3(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 = H4(msg)
encoded_commitment_hash = H5(encode_group_commitment_list(commitment_list))
rho_input_prefix = msg_hash || encoded_commitment_hash
binding_factor_list = []
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
rho_input = rho_input_prefix || 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 + G.ScalarMult(binding_nonce_commitment, binding_factor)
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 (commit state) ==\
|<-----------------------------------------+ |
|
== 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. Note that each signer persists some state between both rounds, and this state is deleted as described in Section 5.2. The final Aggregation step is described in Section 5.3.¶
FROST assumes that all inputs to each round, especially those of which are received over the network, are validated before use. In particular, this means that any value of type Element or Scalar is deserialized using DeserializeElement and DeserializeScalar, respectively, as these functions perform the necessary input validation steps.¶
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(Scalar(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 MUST 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 + G.ScalarMult(binding_nonce_commitment, binding_factor)
# Compute the challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute Lagrange coefficient
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_lagrange_coefficient(Scalar(identifier), participant_list)
# Compute relation values
l = G.ScalarBaseMult(sig_share_i)
r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i)
return l == r
¶
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 shares from signers that participated in Rounds 1 and 2 are valid, the Coordinator
performs the aggregate operation and publishes the resulting signature.¶
Inputs:
- group_commitment, the group commitment returned by compute_group_commitment,
an Element in G.
- sig_shares, a set of signature shares z_i, Scalar values, for each 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 "FROST-ED25519-SHA512-v8".¶
Group: edwards25519 [RFC8032]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
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 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-v8".¶
Group: ristretto255 [RISTRETTO]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
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 "FROST-ED448-SHAKE256-v8".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
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 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-v8".¶
Group: P-256 (secp256r1) [x9.62]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
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.¶
This ciphersuite uses secp256k1 for the Group and SHA-256 for the Hash function H.
The value of the contextString parameter is "FROST-secp256k1-SHA256-v8".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
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.¶
Section 4.1 describes the procedure that signers use to produce nonces during
the first round of singing. The randomness produced in this procedure MUST be sampled
uniformly at random. The resulting nonces produced via nonce_generate are indistinguishable
from values sampled uniformly at random. This requirement is necessary to avoid
replay attacks initiated by other 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 + G.ScalarMult(PK, c)
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 tuple
consisting of the participant identifier and the key share (a Scalar).
- 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, each a tuple
consisting of the participant identifier and the key share (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(Scalar(x_i), coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_share.append(secret_key_share_i)
return secret_key_shares, coefficients
¶
Let points be the output of this function. The i-th element in points is
the share for the i-th participant, which is the randomly generated polynomial
evaluated at coordinate i. We denote a secret share as the tuple (i, points[i]),
and the list of these shares as shares.
i MUST never equal 0; recall that f(0) = s, where f is the polynomial defined in a Shamir secret sharing operation.¶
The procedure for combining a shares list of length MIN_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))
where i is an identifier and f(i) is a Scalar
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"
scalar_shares = [(Scalar(x), y) for x, y in shares]
s = polynomial_interpolation(scalar_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' += G.ScalarMult(vss_commitment[j], pow(i, 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 += G.ScalarMult(vss_commitment[j], pow(i, j))
signer_public_keys.append(PK_i)
return PK, signer_public_keys
¶
Two popular algorithms for generating a random integer uniformly distributed in the range [0, G.Order() -1] are as follows:¶
Generate a random byte array with Ns bytes, and attempt to map to a Scalar
by calling DeserializeScalar. If it succeeds, return the result. If it fails,
try again with another random byte array, until the procedure succeeds.¶
Note the that the Scalar size might be some bits smaller than the array size,
which can result in the loop iterating more times than required. In that case
it's acceptable to set the high-order bits to 0 before calling DeserializeScalar,
but care must be taken to not set to zero more bits than required. For example,
in the FROST(Ed25519, SHA-512) ciphersuite, the order has 253 bits while
the array has 256; thus the top 3 bits of the last byte can be set to zero.¶
Generate a random byte array with L = ceil(((3 * ceil(log2(G.Order()))) / 2) / 8)
bytes, and interpret it as an integer; reduce the integer modulo G.Order() and return the
result. See Section 5 of [HASH-TO-CURVE] for the underlying derivation of L.¶
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: 1c406170127e33142b8611bc02bf14d5909e49d5cb87150eff3e c9804212920c S1 binding_nonce: 5be4edde8b7acd79528721191626810c94fbc2bcc814b7a67d3 01fbd7fc16e07 S1 hiding_nonce_commitment: eab073cf90278e1796c2db197566c8d1f62f9992d 399a5329239481f9cbb5811 S1 binding_nonce_commitment: 13172c94dec7b22eb0a910e93fa1af8a79e27f61 b69981e1ebb227438ca3be84 S1 binding_factor_input: 25120720c3a416e292edfb0510780bc84eb734347a5f d84dd46d0dcbf3a21d21a23a776628859678a968acc8c8564c4641a1fd4b29a12d536 7ca12ab10b6b497980a7514dabb157e25a78ab2f02572efad70a677a398de943abb9a f16d2decc1197f36000ae9db37db3b39f3fbf9d854bbed9cb84d41973ac81af76ad63 7166a0001 S1 binding_factor: c538ab7707e484ba5d29bb80d9ac795e0542e8089debbaca4d f090e92a6d5504 S3 hiding_nonce: 795f87122f05b7efc4b1a52f435c3d28597411b1a6fec198ce9c 818c5451c401 S3 binding_nonce: c9193aaef37bc074ea3286d0361c815f7201bf764cd9e7d8bb4 eb5ecca840a09 S3 hiding_nonce_commitment: 049e0a8d62db8fd2f8401fb027e0a51374f5c4c79 6a1765ecf14467df8c4829a S3 binding_nonce_commitment: eeb691d3dc19e0dbc33471c7a7681a51801c481d a34a8f55efe3070a75e9991d S3 binding_factor_input: 25120720c3a416e292edfb0510780bc84eb734347a5f d84dd46d0dcbf3a21d21a23a776628859678a968acc8c8564c4641a1fd4b29a12d536 7ca12ab10b6b497980a7514dabb157e25a78ab2f02572efad70a677a398de943abb9a f16d2decc1197f36000ae9db37db3b39f3fbf9d854bbed9cb84d41973ac81af76ad63 7166a0003 S3 binding_factor: 9e4474b925576c54c8e50ec27e09a04537c837f38b0f71312a 58a8c12861b408 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 1f16a3989b4aa2cc3782a503331b9a21d7ba56c9c5455d06981b542 5306c9d01 S3 sig_share: 4c8f33c301c05871b434a686847d5818417a01e50a59e9e7fddaefd e7d244207 sig: 1aff2259ecb59cfcbb36ae77e02a9b134422abeae47cf7ff56c85fdf90932b18 6ba5d65b9d0afb3decb64b8ab798f239183558aed09e46ee95f64304ae90df08¶
// 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: bc449f3c5a05ab89eca0578a83ec16541c53867e56ff6bed852d fda64785047245a0aa539a0ca7f4ba04d788e20f81210c9ee4688d25a82800 S1 binding_nonce: 7b34c981d62440777a1481dc62ec3629742a72b59cb1987209a 2d9be358ed0980f5695666c1a43837f471e0f5c311a6a223502bbea3fe21900 S1 hiding_nonce_commitment: 6bc549d7741bc4816d2baa27b8092b4e3b723d185 8cb706f618ec5ba0d1e9f22a1737a9a0659478dcb1d55e5270e7b2d5af6a0ba8d9358 7f00 S1 binding_nonce_commitment: 5fba493104d0ce08fb78bf1cfa600f9858183aa0 b561744449e2f069a6711f9736c65bd07145a0b82d9afb58b117a6e61b0bb7c9bf508 66200 S1 binding_factor_input: 766a004ac6e87a2fa70f2095b19596ac33b94e2f6803 e1a5b8fa8ea5adaf3e7989b2c167a38a42a1693ad69cfd674e089a498672753563d53 54654ba106d5fdffb134a8917fae91d412164436f734b95572af6208605744400c6ff 9a60fa2ce8fb7f3213414c32e347ee2e29e3d17654ef02da59085fa87ee0f01ff5b94 2dd66846e07ff61868d5f8ab2e11e6e59682cb8d9fec6bb1d3559b84facb63cac87e0 d9ef46411497f1866830953485da9c5ed2492c0cd23d775bcb160bc43427cf1d2a2e6 fd32488f37fc41c27e0efdf4be9f7d843dbb6673faef7f981e5b40795608d80d74700 01 S1 binding_factor: c1b2ad1032bf0fec6504d2af7d09b1b39691f33b7f5a64f6da 61322f7a8d88d8af1f2c2a474909812a956fbfa26441e08f86eab74d1b342900 S3 hiding_nonce: a1546a8083c9ccc1c5553e4376a4be3f90f5ab6d44be636032b0 645afabedd70ba4d84c5845b3b7b7c07c2d027263c4ed40c501e19076c2900 S3 binding_nonce: b8ae00851628a94ee932227f8b7b61c1a7ff878d034bc682087 dd7460b49c26956c68a9ad6303a003ae11b87bb76dd6a4e4af8073e0daf2300 S3 hiding_nonce_commitment: b7fbafb73633ca009e40671a926996bda341ce50b 6473013e865a445c64b8097c314603f2996b94a564f71322e51f4c224710252bf708f 3800 S3 binding_nonce_commitment: a9e8016ebac078f133ed18ccbaff75cb8c213cbe 03809dd2e6480e8c0b9c743af242eec49a4e1d38808956c7d6f1fb96795ed949f02cd e5880 S3 binding_factor_input: 766a004ac6e87a2fa70f2095b19596ac33b94e2f6803 e1a5b8fa8ea5adaf3e7989b2c167a38a42a1693ad69cfd674e089a498672753563d53 54654ba106d5fdffb134a8917fae91d412164436f734b95572af6208605744400c6ff 9a60fa2ce8fb7f3213414c32e347ee2e29e3d17654ef02da59085fa87ee0f01ff5b94 2dd66846e07ff61868d5f8ab2e11e6e59682cb8d9fec6bb1d3559b84facb63cac87e0 d9ef46411497f1866830953485da9c5ed2492c0cd23d775bcb160bc43427cf1d2a2e6 fd32488f37fc41c27e0efdf4be9f7d843dbb6673faef7f981e5b40795608d80d74700 03 S3 binding_factor: 470a1fbfd4ab8f72a976b3f3a583fb9f7b7b45691a519efec8 54fa47db6335e9ab41cc95d26066a6809722523b1583f02469950bca37680c00 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 7aee2e4005d5ad2ea252cd954d42376a5f962973211dfca267eeea5 f8dadc2b729ea136817e8f7652ac7beb49983ffdcef1d75ffa69c821a00 S3 sig_share: dc8075aee647361bfa613637fcc5a276705b1e79acff7dadcfecd4f f9f73b32b78cbed3c633ee22eae9d023f17488624088cfaf54599dd2d00 sig: e8832eb0488e7a2b4a063eff722da98549c2ca3ffc4dec1ae2e9e5704c0880f4 f7d545bb9f47d97c62d2eec9be19027879367f6b276046db80632a4c43595a6b26472 53e3fd7456dbf3fbb713d84412b8c4db7f5e22d2176e3a1b501a57a26da94d864c1f3 b0cb8501f8a96ff5ec35600800¶
// 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: 1eaee906e0554a5e533415e971eefa909f3c614c7c75e27f381b 0270a9afe308 S1 binding_nonce: 16175fc2e7545baf7180e8f5b6e1e73c4f2769323cc76754bdd 79fe93ab0bd0b S1 hiding_nonce_commitment: 80d35700fda011d9e2b2fad4f237bf88f2978d954 382dfd36a517ab0497a474f S1 binding_nonce_commitment: 40f0fecaf94e656b3f802ba9827fca9fa994c13c 98a5ff257973f8bdbc733324 S1 binding_factor_input: fe9082dcc1ae1ae11380ac4cf0b6e2770af565ff5af9 016254dc7c9d4869cbae0f6e4d94b23e5781b91bc74a25e0c773446b2640290d07c83 f0b067ff870a80179b2d1262816a7a4fad96b747bd6b35ccf4a912a793c5701d54852 db80904a767cbbd6e37377eec77f407b22890c01190995066cce59d88a14ac56ac40b 3bdc90001 S1 binding_factor: c0f5ee2613c448137bae256a4e95d56deb8c59f934332c0c00 41720b8819680f S3 hiding_nonce: 48d78b8c2de1a515513f9d3fc464a19a72304fac522f17cc6477 06cb22c21403 S3 binding_nonce: 5c0f10966b3f1386660a87de0fafd69decbe9ffae1a152a88b7 d83bb4fb1c908 S3 hiding_nonce_commitment: 20dec6ad0795f82009a1a94b6ad79f01a1e95ae8e 308d8d8fae8285982308113 S3 binding_nonce_commitment: 98437dafb20fdb18255464072bee514889aeeec3 24f149d49747143c3613056d S3 binding_factor_input: fe9082dcc1ae1ae11380ac4cf0b6e2770af565ff5af9 016254dc7c9d4869cbae0f6e4d94b23e5781b91bc74a25e0c773446b2640290d07c83 f0b067ff870a80179b2d1262816a7a4fad96b747bd6b35ccf4a912a793c5701d54852 db80904a767cbbd6e37377eec77f407b22890c01190995066cce59d88a14ac56ac40b 3bdc90003 S3 binding_factor: 8ea449e545706bb3b42c66423005451457e4bb4dea2c2d0b1d 157e6bb652ec09 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 5ae13621ebeef844e39454eb3478a50c4531d25939e1065f44f5b04 a8535090e S3 sig_share: aa432dcf274a9441c205e76fe43497be99efe374f9853477bd5add2 075f6970c sig: 9c407badb8cacf10f306d94e31fb2a71d6a8398039802b4d80a1278472397206 17516e93f8d57a2ecffd43b83ab35db6de20b6ce32673bd601508e6bfa2ba10a¶
// 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: e9165dad654fc20a9e31ca6f32ac032ec327b551a50e8ac5cf25 f5c4c9e20757 S1 binding_nonce: e9059a232598a0fba0e495a687580e624ab425337c3221246fb 2c716905bc9e7 S1 hiding_nonce_commitment: 0228df2e7f6c254b40a9f8853cf6c4f21eacbb6f0 663027384966816b57e513304 S1 binding_nonce_commitment: 02f5b7f48786f8b83ebefed6249825650c4fa657 da66ae0da1b2613dedbe122ec8 S1 binding_factor_input: 3617acb73b44df565fbcbbbd1824142c473ad1d6c800 7c4b72a298d1eaae5766b730d2e6594ea697a5971f15e989ac47ecc015692ad88b615 a41e652a306c7e50001 S1 binding_factor: 95f987c0ab590507a8c4deaf506ffc182d3626e30386306f7a b3aaf0b0013cd3 S3 hiding_nonce: b9d136e29eb758bd77cb83c317ac4e336cf8cda830c089deddf6 d5ec81da9884 S3 binding_nonce: 5261e2d00ce227e67bb9b38990294e2c82970f335b2e6d9f1d0 7a72ba43d01f0 S3 hiding_nonce_commitment: 02f87bd95ab5e08ea292a96e21caf9bdc5002ebf6 e3ce14f922817d26a4d08144d S3 binding_nonce_commitment: 0263cb513e347fcf8492c7f97843ed4c3797f2f3 fe925b1e68f65fb90826fe9597 S3 binding_factor_input: 3617acb73b44df565fbcbbbd1824142c473ad1d6c800 7c4b72a298d1eaae5766b730d2e6594ea697a5971f15e989ac47ecc015692ad88b615 a41e652a306c7e50003 S3 binding_factor: 2f21db4f811b13f938a13b8f2633467d250703fe5bd63cd24f 08bef6fd2f3c29 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: bdaa275f10ca57e3a3a9a7a0d95aeabb517897d8482873a8f9713d4 58f94756f S3 sig_share: 0e8fd85386939e8974a8748e66641df0fe043323c52487a2b10b8a3 97897de21 sig: 03c41521412528dce484c35b6b9b7cc8150102ab3e4bdf858d702270c05098e6 c6cc39ffb2975df66d18521c2f3fbf08ac4f7ccafc0d4cfb4baa7cc77f082c5390¶
// Configuration information MAX_SIGNERS: 3 MIN_SIGNERS: 2 NUM_SIGNERS: 2 // Group input parameters group_secret_key: 0d004150d27c3bf2a42f312683d35fac7394b1e9e318249c1bf e7f0795a83114 group_public_key: 02f37c34b66ced1fb51c34a90bdae006901f10625cc06c4f646 63b0eae87d87b4f message: 74657374 // Signer input parameters S1 signer_share: 08f89ffe80ac94dcb920c26f3f46140bfc7f95b493f8310f5fc1 ea2b01f4254c S2 signer_share: 04f0feac2edcedc6ce1253b7fab8c86b856a797f44d83d82a385 554e6e401984 S3 signer_share: 00e95d59dd0d46b0e303e500b62b7ccb0e555d49f5b849f5e748 c071da8c0dbc // Round one parameters participants: 1,3 // Signer round one outputs S1 hiding_nonce: 95f352cf568508bce96ef3cb816bf9229eb521ca9c2aff6a4fe8 b86bf49ae16f S1 binding_nonce: c675aea50ff2510ae6b0fcb55432b97ad0b55a28b959bacb0e8 b466dbf43dd26 S1 hiding_nonce_commitment: 028acf8c9e345673e2544248006f4ba7ead5e94e1 70062b86eb532a74c26f79f98 S1 binding_nonce_commitment: 0314c33f75948224dd39cdffc68fa0faeeb42f7e f94f1552c920196d53fbda04ce S1 binding_factor_input: d759fa818c284537bbb2efa2d7247eac9232b7b992cd 49237106acab251dd9543f613ca4fea19d29cc54b4aa618e93289eff0da1a87fcebd1 d690283016126580001 S1 binding_factor: 6c7933abb7bc86bcc5c549ba984b9526dca099f9d9b787cedd e20c70d36f5fc1 S3 hiding_nonce: b5089ebf363630d3477711005173c1419f4f40514f7287b4ca6f f110967a2d70 S3 binding_nonce: 5e50ce9975cfc6164e85752f52094b11091fdbca846a9c245fd bfa4bab1ae28c S3 hiding_nonce_commitment: 039121f05be205b6a52ffdfdcd5f9cdc3b074a7f0 f031dac294e747b7ca83567d5 S3 binding_nonce_commitment: 0265c40f57bdcdcd0dfa43a8d353301e1474517b 70da29ddb1cb4461cd09eee1ce S3 binding_factor_input: d759fa818c284537bbb2efa2d7247eac9232b7b992cd 49237106acab251dd9543f613ca4fea19d29cc54b4aa618e93289eff0da1a87fcebd1 d690283016126580003 S3 binding_factor: 1b18e710a470fe513e4387c613321aa41151990f65a8577343 b45d6883ab877d // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 280c44c6c37cd64c7f5a552ae8416a57d21c115cab524dbff5fbceb bf5c0019d S3 sig_share: e372bca35133a80ca140dcac2125c966b763a934678f40e09fb8b0a e9d4aee1b sig: 0364b02292a4b0e61f849f4d6fac0e67c2f698a21e1cba9e4a5b8fa535f2f931 0d0b7f016a14b07e59209b31d7096733bfced0ddaa6398ee64d5e220ddc2d4ae77¶