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.¶
draft-06¶
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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 a helper function for extracting signer participant identifiers from a commitment list. It also describes a helper function for extracting a given participant's binding factor from a list.¶
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) binding_factor_list = [] 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) rho_input = encoded_commitment + msg_hash 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:¶
i
s 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.¶
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: 424fb678c3cd26ae86813ed1e22f9074b2dec6b0fb39d748101a 9434d6c4c60a S1 binding_nonce: 49c957d98fec37b81a491fcbc3b16af925d393d734acc765446 574bc7b453109 S1 hiding_nonce_commitment: 6ca4bc3e03fa91bd15ec29993bc0582779c1d0c60 2c3141b22ea76abf9384290 S1 binding_nonce_commitment: 03dc2bc626fc5fa41168dee456ec17bac8673691 e27547c534a559c3d4d7fc6b S1 binding_factor_input: 00016ca4bc3e03fa91bd15ec29993bc0582779c1d0c6 02c3141b22ea76abf938429003dc2bc626fc5fa41168dee456ec17bac8673691e2754 7c534a559c3d4d7fc6bee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f88 19a5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f 50028a8ff S1 binding_factor: 1af00d46730628637df21dd56023489d6691468b2515c28c22 052abc63e4d405 S3 hiding_nonce: bcca3084a26538abd5326825f7a1203907c09ae7d12160810c4c 8254aac02e03 S3 binding_nonce: 9063464ee91ee0981f0073783c7ca1590ed6b1cb7382bad842f 6a5a4dc7ba607 S3 hiding_nonce_commitment: 7786dc3db9823b1b2c81d71957d2464302ffc5b31 e586f9e754385e7d216a2d1 S3 binding_nonce_commitment: 21b0c507a779f3288c74506838e8f084cd378eab 4b66ce185fe842a2262ac899 S3 binding_factor_input: 00037786dc3db9823b1b2c81d71957d2464302ffc5b3 1e586f9e754385e7d216a2d121b0c507a779f3288c74506838e8f084cd378eab4b66c e185fe842a2262ac899ee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f88 19a5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f 50028a8ff S3 binding_factor: 7702669b34a9534bce0e73ed97c3d002433d969f4ee70c4a30 c6f56e661efb0e // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 95bac1faca14c3f85d9f3b5d515009a2db7715cdafaa414bb5d271a 2f1300c0c S3 sig_share: 636280b4886b8120e2aeffd6637f82e64a4616c1b65682bb4a68863 5c4bd3107 sig: ad05ed58d78f26dba585d4e452b099126f733bc35c698210ce8590691582e387 0b494c52391d32c169b14391d6d5ac7326be2b8e6601c406003bf8d7b5ee3d03¶
// 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: d1c4212b1b0987135b686fc1e0c26d464609e3b2cb0363f4ce81 676d9798c10161775894edae39f91731083a8751b754ce7faa9b51bd3c1800 S1 binding_nonce: c1f429e6b853dfefac16acebba0691916b5e41ea62122a7d565 b16c98069202f6e18cba21790f26a351253ce03ff41556435ac7a1ce6960d00 S1 hiding_nonce_commitment: cc1cb80545250515c06b5b5c2ae124973a232d7bf 74fd2b3d4f9ba0505b67ee3eee3f10b1c8a739b5eedd49158e0b4a1789ec3b4aef2f6 4280 S1 binding_nonce_commitment: 5d0766016f294cc27af4a94a1f6f0c1e4074377d 7850133fa6e56ee6073e47fe0e3e0b9af5a8a4c24d179f2711702d9232557e72e7f04 9ec00 S1 binding_factor_input: 0001cc1cb80545250515c06b5b5c2ae124973a232d7b f74fd2b3d4f9ba0505b67ee3eee3f10b1c8a739b5eedd49158e0b4a1789ec3b4aef2f 642805d0766016f294cc27af4a94a1f6f0c1e4074377d7850133fa6e56ee6073e47fe 0e3e0b9af5a8a4c24d179f2711702d9232557e72e7f049ec00b54ff7255705a71ee29 25e4a3e30e41aed489a579d5595e0df13e32e1e4dd202a7c7f68b31d6418d9845eb4d 757adda6ab189e1bb340db818e5b3bc725d992faf63e9b0500db10517fe09d3f566fb a3a80e46a403e0c7d41548fbf75cf2662b00225b502961f98d8c9ff937de0b24c2318 45 S1 binding_factor: 2648c00456a49c45068225a53daf4141c49d94f24c91baa0c4 709b929652b678533533da023bc86546d2bf27103401069e82e7d76e45d80700 S3 hiding_nonce: 27c798a453fdc42aee7302b02140622fe17a99aa57f61dc5f9ce 0e55cf64034cfe41954d8e4d4bc46bad25b67058e9d8de87cee9dcea340f00 S3 binding_nonce: 2fdf07df954ce0378e47551841db9826e8ceec79c083f4e1e70 5d1f14824784dc82dade69b01065e9483d8c6cc577afa15c60484fe670e0500 S3 hiding_nonce_commitment: 7f4e108990f82c411317c5e1f945b46e3c115a52c 60e57a360fbc69c3101ca4b3151b855f7c28574ce2e6a4150fcb351d50bb81fec3d03 f500 S3 binding_nonce_commitment: ac79cf898abd0896a1468944f40b38cfb82cb191 c737681d0b8b2d1a6bebb559ae0221cda231238caf603e2ee85658bc28782083ecd3a ab900 S3 binding_factor_input: 00037f4e108990f82c411317c5e1f945b46e3c115a52 c60e57a360fbc69c3101ca4b3151b855f7c28574ce2e6a4150fcb351d50bb81fec3d0 3f500ac79cf898abd0896a1468944f40b38cfb82cb191c737681d0b8b2d1a6bebb559 ae0221cda231238caf603e2ee85658bc28782083ecd3aab900b54ff7255705a71ee29 25e4a3e30e41aed489a579d5595e0df13e32e1e4dd202a7c7f68b31d6418d9845eb4d 757adda6ab189e1bb340db818e5b3bc725d992faf63e9b0500db10517fe09d3f566fb a3a80e46a403e0c7d41548fbf75cf2662b00225b502961f98d8c9ff937de0b24c2318 45 S3 binding_factor: b165cef8a42b605d192930206ffcdf48cf242e968340c9e13f 58962be27bccb2af4d674e67c3e97ff719ba87216565544edb1f862c18433e00 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: c0c5055e6429cf24c60fd19cb7e887caed243db821f81fc899a30f2 dae833bf6c373acb0a0dbd9d7d3bd3de36e1b33b21ea9c19dce74d23b00 S3 sig_share: 495e6971b5ab6bf800860296eeb8e0caa4a8b2137f390b53366bb97 432dbf0fc07b315fa9df67eda1417720fc98f9325340d7fb8fe708a2f00 sig: 6d244abe9f23024c58b1962baec45a0cce035d7a73034ca39642ed58df8dd444 f8f2a3e08b187645fecdd93fbd151af076e608b976206bbe8016df16248712c2f9710 60ea533dffb730297191d5756dc56e6eafe24e15e2cf3cb26c2aa3ed258b2e8d4aff2 37abc6d752b64056cde55c2b00¶
// 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: 82fd8f26f0b5f86d93d3da8162c23641aa1b16ab803be1a4d850 0f143b592605 S1 binding_nonce: c9ee2f764434e263808722a3cb9886b7ce25ade80ce00bac4ac 4d9548ac2450a S1 hiding_nonce_commitment: c4f0ea06e07ee8e41bd3de3a63dc77f40d65a7ede 329263db410eb0e455c9a7a S1 binding_nonce_commitment: c2ab142e4c2572aea1dcd0044e864544c49874b2 e39ed5f969b6133372a23643 S1 binding_factor_input: 0001c4f0ea06e07ee8e41bd3de3a63dc77f40d65a7ed e329263db410eb0e455c9a7ac2ab142e4c2572aea1dcd0044e864544c49874b2e39ed 5f969b6133372a23643678630bf982c566949d7f22d2aefb94f252c664216d332f34e 2c8fdcd7045f207f854504d0daa534a5b31dbdf4183be30eb4fdba4f962d8a6b69cf2 0c2734043 S1 binding_factor: 1bed7e23dbe3066085a132da86ccf1e58291cec6571249cd4f 16a053c8409b04 S3 hiding_nonce: d2478a9a47594005cb6de147f4c98b92b5e27e157fddb37ef122 a75dfd7b8e06 S3 binding_nonce: 5f2193dc5f3e5537fba742786fec22e18e40b059e50d6ef2e31 60a3faf11dd0e S3 hiding_nonce_commitment: 9a543be2b522d20a9d7e920388bb32f954bcfc695 d145736de09f09646c6ae4c S3 binding_nonce_commitment: debe73bc386b4dba161d72fa46ecc86765d6ff36 27cfe07d56891d21399f1e4c S3 binding_factor_input: 00039a543be2b522d20a9d7e920388bb32f954bcfc69 5d145736de09f09646c6ae4cdebe73bc386b4dba161d72fa46ecc86765d6ff3627cfe 07d56891d21399f1e4c678630bf982c566949d7f22d2aefb94f252c664216d332f34e 2c8fdcd7045f207f854504d0daa534a5b31dbdf4183be30eb4fdba4f962d8a6b69cf2 0c2734043 S3 binding_factor: f611130beed6aa69de48fc1c05e7a005c7d5232b2852fe721a 1d711c35a28600 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 2aeac152cff93a82233c1ac6bece654af750f75c47cd5c87fcf6109 a562c8903 S3 sig_share: a9487edfb9349f792693278fa9f8abd57c2caaac4f901fb03da8534 c56574303 sig: a6fa7f7553307cfa36e55a705a2e59f808a1bcda888efcf4f26ba0c4080cc400 d3324032892edafb49cf415568c71120747da109975d7c373a9f64e6ac83cc06¶
// 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: e1bec1511c502e6b6866ef35469dac6df913efa8042a70b39aa6 494307d6ee41 S1 binding_nonce: 55682738caade073b207abdec93014bc111bc5d6a8977f3a24d f58bebf650894 S1 hiding_nonce_commitment: 0257ea635b11b63eb125ed1c983082a406e0185e5 df6bd4c2a708f7d4b1ffe72d2 S1 binding_nonce_commitment: 03aec5dfcd35c5ad7a7242552efe9c45878a0b2f 3f576b8f60fa172d380881f040 S1 binding_factor_input: 00010257ea635b11b63eb125ed1c983082a406e0185e 5df6bd4c2a708f7d4b1ffe72d203aec5dfcd35c5ad7a7242552efe9c45878a0b2f3f5 76b8f60fa172d380881f0407a753fed12531fbcd151e1d84702927c39063e780e91c0 1f02bd11b60d7632bf S1 binding_factor: 846723bf0898010ebb3d0a0c193c682a22946bb99c9551f5e3 a0c29008aa8891 S3 hiding_nonce: a284c689cd1d8ec33dcfe840d18ba0ddcbb69571744116912d7a 5c20631ea003 S3 binding_nonce: 9f3ca4297e01ce82a5c7301a8fc39e6ec0588087483c912e269 92f0d00009f0b S3 hiding_nonce_commitment: 02fc8fe2f5d1bfa6b027ff1184c57cc7b880e050f bf370a4e23d17eb3b249f45d9 S3 binding_nonce_commitment: 02288d41169a97462b323dd61658eb134f8dd802 af7d1932f602f2532afd9d1903 S3 binding_factor_input: 000302fc8fe2f5d1bfa6b027ff1184c57cc7b880e050 fbf370a4e23d17eb3b249f45d902288d41169a97462b323dd61658eb134f8dd802af7 d1932f602f2532afd9d19037a753fed12531fbcd151e1d84702927c39063e780e91c0 1f02bd11b60d7632bf S3 binding_factor: 739b96f69c0ec5fbe05c2f5191ec349f835d9a38f8cf1a5632 87b3547540c4b0 // Round two parameters participants: 1,3 // Signer round two outputs S1 sig_share: 38f57b8c861a9eb03eeeb381bae58f0f7fce6f0d0da431a0ae58a00 a760aadf4 S3 sig_share: e4f1da04a879bd5b23c51662c22cbd71a80129db5e387895c769727 a15176556 sig: 02b72e3a186d719efb18f4341d7b5d983f782d31c57f884a5fe79040d30ddd39 381de755922e945c0a62b3c9e47d124c816ae89e3ac4c50bb1820847c18ebeedf9¶