| Internet-Draft | FROST | January 2023 |
| Connolly, et al. | Expires 28 July 2023 | [Page] |
This document specifies the Flexible Round-Optimized Schnorr Threshold (FROST) signing protocol. FROST signatures can be issued after a threshold number of entities cooperate to compute a signature, allowing for improved distribution of trust and redundancy with respect to a secret key. FROST depends only on a prime-order group and cryptographic hash function. This document specifies a number of ciphersuites to instantiate FROST using different prime-order groups and hash functions. One such ciphersuite can be used to produce signatures that can be verified with an Edwards-Curve Digital Signature Algorithm (EdDSA, as defined in RFC8032) compliant verifier. However, unlike EdDSA, the signatures produced by FROST are not deterministic. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.¶
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 28 July 2023.¶
Copyright (c) 2023 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.¶
RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this draft is maintained in GitHub. Suggested changes should be submitted as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost. Instructions are on that page as well.¶
Unlike signatures in a single-party setting, threshold signatures require cooperation among a threshold number of signing participants each holding a share of a common private key. The security of threshold schemes in general assumes that an adversary can corrupt strictly fewer than a threshold number of signer participants.¶
This document specifies the Flexible Round-Optimized Schnorr Threshold (FROST) signing protocol based on the original work 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. FROST requires two rounds to compute a signature. Single-round signing variants based on [FROST20] are out of scope.¶
FROST depends only on a prime-order group and cryptographic hash function. This document specifies a number of ciphersuites to instantiate FROST using different prime-order groups and hash functions. Two ciphersuites can be used to produce signatures that are compatible with Edwards-Curve Digital Signature Algorithm (EdDSA) variants Ed25519 and Ed448 as specified in [RFC8032], i.e., the signatures can be verified with an [RFC8032] compliant verifier. However, unlike EdDSA, the 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.¶
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.¶
This document represents the consensus of the Crypto Forum Research Group (CFRG). It is not an IETF product and is not a standard.¶
draft-12¶
draft-11¶
draft-10¶
draft-09¶
draft-08¶
draft-07¶
draft-06¶
draft-05¶
draft-04¶
draft-03¶
draft-02¶
draft-01¶
draft-00¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
The following notation is used throughout the document.¶
random_bytes(n): Outputs n bytes, sampled uniformly at random
using a cryptographically secure pseudorandom number generator (CSPRNG).¶
count(i, L): Outputs the number of times the element i is represented in the list L.¶
len(l): Outputs the length of list l, e.g., len([1,2,3]) = 3.¶
reverse(l): Outputs the list l in reverse order, e.g., reverse([1,2,3]) = [3,2,1].¶
range(a, b): Outputs a list of integers from a to b-1 in ascending order, e.g., range(1, 4) = [1,2,3].¶
pow(a, b): Outputs the result, a Scalar, of a to the power of b, e.g., pow(2, 3) = 8 modulo the relevant group order p.¶
x || y denotes the byte string x, immediately followed by
the byte string y, with no extra separator, yielding xy.¶
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). Integers, taken modulo the group order p, are called
scalars; arithmetic operations on scalars are implicitly performed modulo p. Since p is prime,
scalars form a finite field. Scalar multiplication is equivalent to the repeated
application of the group operation on an element A with itself r-1 times, denoted as
ScalarMult(A, r). We denote the sum, difference, and product of two scalars using the +, -,
and * operators, respectively. (Note that this means + may refer to group element addition or
scalar addition, depending on the type of the operands.) For any element A, ScalarMult(A, p) = I.
We denote B as a fixed generator of the group. Scalar base multiplication is equivalent to the repeated application
of the group operation on 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. It is assumed that
group element addition, negation, and equality comparison can be efficiently computed for
arbitrary group elements.¶
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. Moreover, we use the type NonZeroScalar to denote a Scalar
value that is not equal to zero, i.e., Scalar(0). We denote equality comparison of these types
as == and assignment of values by =. When comparing Scalar values, e.g., for the purposes
of sorting lists of Scalar values, the least nonnegative representation mod p is used.¶
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), i.e., a random scalar in [0, p - 1].¶
A and Scalar k.¶
k and the group generator B.¶
Element A to a canonical byte array buf of fixed length Ne. This
function raises an error if A is the identity element of the group.¶
buf to an Element A,
and fails if the input is not the valid canonical byte representation of an element of
the group. This function raises an error if deserialization fails
or if A is the identity element of the group; see Section 6 for group-specific
input validation steps.¶
s to a canonical byte array buf of fixed length Ns.¶
buf to a Scalar s.
This function raises an error 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 is modeled as a random oracle in security proofs for the protocol (see [FROST20] and [StrongerSec22]). For concrete recommendations on hash functions which SHOULD be used in practice, see Section 6. Using H, we introduce distinct 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:¶
The following sections describe these operations in more detail.¶
To hedge against a bad RNG that outputs predictable values, nonces are
generated with the nonce_generate function by combining fresh randomness
with the secret key as input to a domain-separated hash function built
from the ciphersuite hash function H. This domain-separated hash function
is denoted H3. This function always samples 32 bytes of fresh randomness
to ensure that the probability of nonce reuse is at most 2-128
as long as no more than 264 signatures are computed by a given
signing participant.¶
nonce_generate(secret):
Inputs:
- secret, a Scalar.
Outputs:
- nonce, a Scalar.
def nonce_generate(secret):
random_bytes = random_bytes(32)
secret_enc = G.SerializeScalar(secret)
return H3(random_bytes || secret_enc)
¶
This section defines polynomials over Scalars that are used in the main protocol.
A polynomial of maximum degree t is represented as a list of t+1 coefficients,
where the constant term of the polynomial is in the first position and the
highest-degree coefficient is in the last position. For example, the polynomial
x^2 + 2x + 3 has degree 2 and is represented as a list of 3 coefficients [3, 2, 1].
A point on the polynomial f is a tuple (x, y), where y = f(x).¶
The function derive_interpolating_value derives a value used for polynomial
interpolation. It is provided a list of x-coordinates as input, each of which
cannot equal 0.¶
derive_interpolating_value(x_i, L):
Inputs:
- x_i, an x-coordinate contained in L, a NonZeroScalar.
- L, the set of x-coordinates, each a NonZeroScalar.
Outputs:
- value, a Scalar.
Errors:
- "invalid parameters", if 1) x_i is not in L, or if 2) any
x-coordinate is represented more than once in L.
def derive_interpolating_value(x_i, L):
if x_i not in L:
raise "invalid parameters"
for x_j in L:
if count(x_j, L) > 1:
raise "invalid parameters"
numerator = Scalar(1)
denominator = Scalar(1)
for x_j in L:
if x_j == x_i: continue
numerator *= x_j
denominator *= x_j - x_i
value = numerator / denominator
return value
¶
This section describes helper functions that work on lists of values produced during the FROST protocol. The following function encodes a list of participant commitments into a byte string 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 participant, where each element in the list
indicates a NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by identifier.
Outputs:
- encoded_group_commitment, the serialized representation of commitment_list, a byte string.
def encode_group_commitment_list(commitment_list):
encoded_group_commitment = nil
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
encoded_commitment = G.SerializeScalar(identifier) ||
G.SerializeElement(hiding_nonce_commitment) ||
G.SerializeElement(binding_nonce_commitment)
encoded_group_commitment = encoded_group_commitment || encoded_commitment
return encoded_group_commitment
¶
The following function is used to extract identifiers from a commitment list.¶
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates a NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by identifier.
Outputs:
- identifiers, a list of NonZeroScalar values.
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 participant, where each element in the list
indicates a NonZeroScalar identifier i and Scalar binding factor.
- identifier, participant identifier, a NonZeroScalar.
Outputs:
- binding_factor, a Scalar.
Errors:
- "invalid participant", when the designated participant is not known.
def binding_factor_for_participant(binding_factor_list, identifier):
for (i, binding_factor) in binding_factor_list:
if identifier == i:
return binding_factor
raise "invalid participant"
¶
This section describes the subroutine for computing binding factors based on the participant commitment list and message to be signed.¶
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates a NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted
in ascending order by identifier.
- msg, the message to be signed.
Outputs:
- binding_factor_list, a list of (NonZeroScalar, Scalar) tuples representing the binding factors.
def compute_binding_factors(commitment_list, msg):
msg_hash = H4(msg)
encoded_commitment_hash = H5(encode_group_commitment_list(commitment_list))
rho_input_prefix = msg_hash || encoded_commitment_hash
binding_factor_list = []
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
rho_input = rho_input_prefix || G.SerializeScalar(identifier)
binding_factor = H1(rho_input)
binding_factor_list.append((identifier, binding_factor))
return binding_factor_list
¶
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 participant, where each element in the list
indicates a NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be
sorted in ascending order by identifier.
- binding_factor_list = [(i, binding_factor), ...],
a list of (NonZeroScalar, Scalar) tuples representing the binding factor Scalar
for the given identifier.
Outputs:
- group_commitment, an Element.
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_factor_list, 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, the group commitment, an Element.
- group_public_key, the public key corresponding to the group signing key, an
Element.
- msg, the message to be signed, a byte string.
Outputs:
- challenge, a Scalar.
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
¶
This section describes the two-round FROST signing protocol for producing Schnorr signatures.
The protocol is configured to run with a selection of NUM_PARTICIPANTS signer participants and a Coordinator.
NUM_PARTICIPANTS is a positive integer at least MIN_PARTICIPANTS but no larger than
MAX_PARTICIPANTS, where MIN_PARTICIPANTS <= MAX_PARTICIPANTS, MIN_PARTICIPANTS is a positive
non-zero integer and MAX_PARTICIPANTS is a positive integer less than the group order.
A signer participant, or simply participant, is an entity that is trusted to hold and
use a signing key share. The Coordinator is an entity with the following responsibilities:¶
FROST assumes that the Coordinator and the set of signer participants are chosen externally to the protocol. Note that it is possible to deploy the protocol without a distinguished Coordinator; see Section 7.5 for more information.¶
FROST produces signatures that can be verified as if they were produced from a single signer
using a signing key s with corresponding public key PK, where s is a Scalar
value and PK = G.ScalarBaseMult(s). As a threshold signing protocol, the group signing
key s is Shamir secret-shared amongst each of the MAX_PARTICIPANTS participants
and used to produce signatures; see [ShamirSecretSharing] for
more information about Shamir secret sharing. In particular, FROST assumes each participant
is configured with the following information:¶
i in the range [1, MAX_PARTICIPANTS]
and MUST be distinct from the identifier of every other participant.¶
sk_i, which is a Scalar value representing the i-th Shamir secret share
of the group signing key s. In particular, sk_i is the value f(i) on a secret
polynomial f of degree (MIN_PARTICIPANTS - 1), where s is f(0). The public key
corresponding to this signing key share is PK_i = G.ScalarBaseMult(sk_i).¶
The Coordinator and each participant are additionally configured with common group information, denoted "group info," which consists of the following:¶
Element in G denoted PK.¶
PK_i for each participant, which are Element values in G denoted PK_i
for each i in [1, MAX_PARTICIPANTS].¶
This document does not specify how this information, including the signing key shares, are configured and distributed to participants. In general, two possible configuration mechanisms are possible: one that requires a single, trusted dealer, and the other which requires performing a distributed key generation protocol. We highlight key generation mechanism by a trusted dealer in Appendix C for reference.¶
FROST requires two rounds to complete. In the first round, participants generate and publish one-time-use commitments to be used in the second round. In the second round, each participant produces a share of the signature over the Coordinator-chosen message and the other participant commitments. After the second round completes, the Coordinator aggregates the signature shares to produce a final signature. The Coordinator SHOULD abort if the signature is invalid; see Section 5.4 for more information about dealing with invalid signatures and misbehaving participants. This complete interaction, without abort, 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) ==
| participant commitment | |
|<-----------------------+ |
| ... |
| participant commitment (commit state) ==\
|<-----------------------------------------+ |
|
== Round 2 (Signature Share Generation) == |
| |
| participant input | | |
+------------------------> | |
| signature share | | |
|<-----------------------+ | |
| ... | |
| participant 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 participant persists some state between the two rounds, and this state is deleted as described in Section 5.2. The final Aggregation step is described in Section 5.3.¶
FROST assumes that all inputs to each round, especially those of which are received over the network, are validated before use. In particular, this means that any value of type Element or Scalar is deserialized using DeserializeElement and DeserializeScalar, respectively, as these functions perform the necessary input validation steps.¶
FROST assumes reliable message delivery between the Coordinator and participants in order for the protocol to complete. An attacker masquerading as another participant will result only in an invalid signature; see Section 7. However, in order to identify misbehaving participants, we assume that the network channel is additionally authenticated; confidentiality is not required.¶
Round one involves each participant generating nonces and their corresponding public commitments.
A nonce is a pair of Scalar values, and a commitment is a pair of Element values. Each participant's
behavior in this round is described by the commit function below. Note that this function
invokes nonce_generate twice, once for each type of nonce produced. The output of this function is
a pair of secret nonces (hiding_nonce, binding_nonce) and their corresponding public commitments
(hiding_nonce_commitment, binding_nonce_commitment).¶
Inputs:
- sk_i, the secret key share, a Scalar.
Outputs:
- (nonce, comm), a tuple of nonce and nonce commitment pairs,
where each value in the nonce pair is a Scalar and each value in
the nonce commitment pair is an Element.
def commit(sk_i):
hiding_nonce = nonce_generate(sk_i)
binding_nonce = nonce_generate(sk_i)
hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
nonce = (hiding_nonce, binding_nonce)
comm = (hiding_nonce_commitment, binding_nonce_commitment)
return (nonce, comm)
¶
The outputs nonce and comm from participant P_i should both be stored locally and
kept for use in the second round. The nonce value is secret and MUST NOT be shared, whereas
the public output comm is sent to the Coordinator. The nonce values produced by this
function MUST NOT be used in more than one invocation of sign, and the nonces MUST be generated
from a source of secure randomness.¶
In round two, the Coordinator is responsible for sending the message to be signed, and for choosing which participants will participate (of number at least MIN_PARTICIPANTS). Signers additionally require locally held data; specifically, their private key and the nonces corresponding to their commitment issued in round one.¶
The Coordinator begins by sending each participant the message to be signed along with the set of signing commitments for all participants in the participant list. Each participant MUST validate the inputs before processing the Coordinator's request. In particular, the Signer MUST validate commitment_list, deserializing each group Element in the list using DeserializeElement from Section 3.1. If deserialization fails, the Signer MUST abort the protocol. Moreover, each participant MUST ensure that its identifier and commitments (from the first round) appear in commitment_list. Applications which require that participants not process arbitrary input messages are also required to perform relevant application-layer input validation checks; see Section 7.7 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 participant, a NonZeroScalar.
- sk_i, Signer secret key share, a Scalar.
- group_public_key, public key corresponding to the group signing key,
an Element.
- nonce_i, pair of Scalar values (hiding_nonce, binding_nonce) generated in
round one.
- msg, the message to be signed, a byte string.
- commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each participant and sent by the Coordinator.
Each element in the list indicates a NonZeroScalar identifier j and two commitment
Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by identifier.
Outputs:
- sig_share, a signature share, a Scalar.
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 the interpolating value
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_interpolating_value(identifier, participant_list)
# Compute the per-message challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge)
return sig_share
¶
The output of this procedure is a signature share. Each participant then sends
these shares back to the Coordinator. Each participant MUST delete the nonce and
corresponding commitment after completing sign, and MUST NOT use the nonce
as input more than once to sign.¶
Note that the lambda_i value derived during this procedure does not change
across FROST signing operations for the same signing group. As such, participants
can compute it once and store it for reuse across signing sessions.¶
After participants perform round two and send their signature shares to the Coordinator, the Coordinator aggregates each share to produce a final signature. Before aggregating, the Coordinator MUST validate each signature share using DeserializeScalar. If validation fails, the Coordinator MUST abort the protocol as the resulting signature will be invalid. If all signature shares are valid, the Coordinator then aggregates them to produce the final signature using the following procedure.¶
Inputs:
- commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each participant, where each element
in the list indicates a NonZeroScalar identifier j and two commitment
Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by identifier.
- msg, the message to be signed, a byte string.
- sig_shares, a set of signature shares z_i, Scalar values, for each participant,
of length NUM_PARTICIPANTS, where MIN_PARTICIPANTS <= NUM_PARTICIPANTS <= MAX_PARTICIPANTS.
Outputs:
- (R, z), a Schnorr signature consisting of an Element R and Scalar z.
def aggregate(commitment_list, msg, sig_shares):
# Compute the binding factors
binding_factor_list = compute_binding_factors(commitment_list, msg)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor_list)
# Compute aggregated signature
z = Scalar(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). This signature encoding is the same for all FROST ciphersuites
specified in Section 6.¶
The Coordinator SHOULD verify this signature using the group public key before publishing or releasing the signature. Signature verification is as specified for the corresponding ciphersuite; see Section 6 for details. The aggregate signature will verify successfully if all signature shares are valid. Moreover, subsets of valid signature shares will themselves not yield a valid aggregate signature.¶
If the aggregate signature verification fails, the Coordinator can verify each signature share individually to identify and act on misbehaving participants. The mechanism for acting on a misbehaving participant is out of scope for this specification; see Section 5.4 for more information about dealing with invalid signatures and misbehaving participants.¶
The function for verifying a signature share, denoted verify_signature_share, is described below.
Recall that the Coordinator is configured with "group info" which contains
the group public key PK and public keys PK_i for each participant, so the group_public_key and
PK_i function arguments should come from that previously stored group info.¶
Inputs:
- identifier, identifier i of the participant, a NonZeroScalar.
- PK_i, the public key for the i-th participant, where PK_i = G.ScalarBaseMult(sk_i),
an Element.
- comm_i, pair of Element values in G (hiding_nonce_commitment, binding_nonce_commitment)
generated in round one from the i-th participant.
- sig_share_i, a Scalar value indicating the signature share as produced in
round two from the i-th participant.
- commitment_list =
[(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each participant, where each element
in the list indicates a NonZeroScalar identifier j and two commitment
Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by identifier.
- group_public_key, public key corresponding to the group signing key,
an Element.
- msg, the message to be signed, a byte string.
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 the interpolating value
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_interpolating_value(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
¶
The Coordinator can verify each signature share before first aggregating and verifying the signature under the group public key. However, since the aggregate signature is valid if all signature shares are valid, this order of operations is more expensive if the signature is valid.¶
FROST does not provide robustness; i.e, all participants are required to complete the protocol honestly in order to generate a valid signature. When the signing protocol does not produce a valid signature, the Coordinator SHOULD abort; see Section 7 for more information about FROST's security properties and the threat model.¶
As a result of this property, a misbehaving participant can cause a denial-of-service on the signing protocol by contributing malformed signature shares or refusing to participate. FROST assumes the network channel is authenticated to identify which signer misbehaved. FROST allows for identifying misbehaving participants that produce invalid signature shares as described in Section 5.3. FROST does not provide accommodations for identifying participants that refuse to participate, though applications are assumed to detect when participants fail to engage in the signing protocol.¶
In both cases, preventing this type of attack requires the Coordinator to identify misbehaving participants such that applications can take corrective action. The mechanism for acting on misbehaving participants is out of scope for this specification. However, one reasonable approach would be to remove the misbehaving participant from the set of allowed participants in future runs of FROST.¶
A FROST ciphersuite must specify the underlying prime-order group details
and cryptographic hash function. Each ciphersuite is denoted as (Group, Hash),
e.g., (ristretto255, SHA-512). This section contains some ciphersuites.
Each ciphersuite also includes a context string, denoted contextString,
which is an ASCII string literal (with no NULL terminating character).¶
The RECOMMENDED ciphersuite is (ristretto255, SHA-512) as described in Section 6.2. The (Ed25519, SHA-512) and (Ed448, SHAKE256) ciphersuites are included for compatibility with Ed25519 and Ed448 as defined in [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 of the ciphersuites below. Future ciphersuites MUST describe how input validation is done for DeserializeElement and DeserializeScalar.¶
Each ciphersuite includes explicit instructions for verifying signatures produced by FROST. Note that these instructions are equivalent to those produced by a single participant.¶
Each ciphersuite adheres to the requirements in Section 6.6. Future ciphersuites MUST also adhere to these requirements.¶
This ciphersuite uses edwards25519 for the Group and SHA-512 for the Hash function H
meant to produce Ed25519-compliant signatures as specified in Section 5.1 of [RFC8032].
The value of the contextString parameter is "FROST-ED25519-SHA512-v11".¶
Group: edwards25519 [RFC8032]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1]. Note that this means the
top three bits of the input MUST be zero.¶
Hash (H): SHA-512, which has 64 bytes of output¶
Normally H2 would also include a domain separator, but for 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][z]B = [8]R + [8][c]PK
(changed to use the notation in this document).¶
This ciphersuite uses ristretto255 for the Group and SHA-512 for the Hash function H.
The value of the contextString parameter is "FROST-RISTRETTO255-SHA512-v11".¶
Group: ristretto255 [RISTRETTO]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1]. Note that this means the
top three bits of the input MUST be zero.¶
Hash (H): SHA-512, which has 64 bytes of output¶
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 Ed448-compliant signatures as specified in Section 5.2 of [RFC8032]. Note that this
ciphersuite does not allow applications to specify a context string as is allowed for Ed448
in [RFC8032], and always sets the [RFC8032] context string to the empty string.
The value of the (internal to FROST) contextString parameter is "FROST-ED448-SHAKE256-v11".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHAKE256 with 114 bytes of output¶
Normally H2 would also include a domain separator, but for 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][z]B = [4]R + [4][c]PK
(changed to use the notation in this document).¶
This ciphersuite uses P-256 for the Group and SHA-256 for the Hash function H.
The value of the contextString parameter is "FROST-P256-SHA256-v11".¶
Group: P-256 (secp256r1) [x9.62]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHA-256, which has 32 bytes of output¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "rho",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "chal",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "nonce",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
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-v11".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHA-256, which has 32 bytes of output¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "rho",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "chal",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
expand_message_xmd with SHA-256 with parameters DST = contextString || "nonce",
F set to the scalar field, p set to G.Order(), m = 1, and L = 48.¶
Signature verification is as specified in Appendix B.¶
Future documents that introduce new ciphersuites MUST adhere to the following requirements.¶
A security analysis of FROST exists in [FROST20] and [StrongerSec22]. At a high level, FROST provides security against Existential Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in [StrongerSec22]. Satisfying this requirement requires the ciphersuite to adhere to the requirements in Section 6.6, as well as the following assumptions to hold.¶
(MIN_PARTICIPANTS-1) participants may be corrupted.¶
Note that the Coordinator is not trusted with any private information and communication at the time of signing can be performed over a public channel, as long as it is authenticated and reliable.¶
FROST provides security against denial of service attacks under the following assumptions:¶
FROST does not aim to achieve the following goals:¶
The rest of this section documents issues particular to implementations or deployments.¶
Several routines process secret values (nonces, signing keys / shares), and depending
on the implementation and deployment environment, mitigating side-channels may be
pertinent. Mitigating these side-channels requires implementing G.ScalarMult(), G.ScalarBaseMult(),
G.SerializeScalar(), and G.DeserializeScalar() in constant (value-independent) time.
The various ciphersuites lend themselves differently to specific implementation techniques
and ease of achieving side-channel resistance, though ultimately avoiding value-dependent
computation or branching is the goal.¶
[StrongerSec22] presented an optimization to FROST that reduces the total number of scalar multiplications from linear in the number of signing participants to a constant. However, as described in [StrongerSec22], this optimization removes the guarantee that the set of signer participants that started round one of the protocol is the same set of signing participants that produced the signature output by round two. As such, the optimization is NOT RECOMENDED, and it is not covered in this document.¶
Section 4.1 describes the procedure that participants use to produce nonces during
the first round of signing. The randomness produced in this procedure MUST be sampled
uniformly at random. The resulting nonces produced via nonce_generate are indistinguishable
from values sampled uniformly at random. This requirement is necessary to avoid
replay attacks initiated by other participants, which allow for a complete key-recovery attack.
The Coordinator MAY further hedge against nonce reuse attacks by tracking participant nonce
commitments used for a given group key, at the cost of additional state.¶
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 participants without this coordinator role. We assume that every participant receives as input from an external source the message to be signed prior to performing the protocol.¶
Every participant begins by performing commit() as is done in the setting
where a Coordinator is used. However, instead of sending the commitment
to the Coordinator, every participant instead will publish
this commitment to every other participant. Then, in the second round, participants will already have
sufficient information to perform signing. They will directly perform sign().
All participants will then publish their signature shares to one another. After having
received all signature shares from all other participants, each participant will then perform
verify_signature_share and then aggregate directly.¶
The requirements for the underlying network channel remain the same in the setting where all participants play the role of the Coordinator, in that all messages that are exchanged are public and so the channel simply must be reliable. However, in the setting that a player attempts to split the view of all other players by sending disjoint values to a subset of players, the signing operation will output an invalid signature. To avoid this denial of service, implementations may wish to define a mechanism where messages are authenticated, so that cheating players can be identified and excluded.¶
FROST signatures do not pre-hash message inputs. This means that the entire message
must be known in advance of invoking the signing protocol. Applications can apply
pre-hashing in settings where storing the full message is prohibitively expensive.
In such cases, pre-hashing MUST use a collision-resistant hash function with a security
level commensurate with the security inherent to the ciphersuite chosen. It is
RECOMMENDED that applications which choose to apply pre-hashing use the hash function
(H) associated with the chosen ciphersuite in a manner similar to how H4 is defined.
In particular, a different prefix SHOULD be used to differentiate this pre-hash from
H4. For example, if a fictional protocol Quux decided to pre-hash its input messages,
one possible way to do so is via H(contextString || "Quux-pre-hash" || m).¶
Message validation varies by application. For example, some applications may require that participants only process messages of a certain structure. In digital currency applications, wherein multiple participants may collectively sign a transaction, it is reasonable to require that each participant check the input message to be a syntactically valid transaction.¶
As another example, some applications may require that participants only process messages with permitted content according to some policy. In digital currency applications, this might mean that a transaction being signed is allowed and intended by the relevant stakeholders. Another instance of this type of message validation is in the context of [TLS], wherein implementations may use threshold signing protocols to produce signatures of transcript hashes. In this setting, signing participants might require the raw TLS handshake messages to validate before computing the transcript hash that is signed.¶
In general, input message validation is an application-specific consideration that varies based on the use case and threat model. However, it is RECOMMENDED that applications take additional precautions and validate inputs so that participants do not operate as signing oracles for arbitrary messages.¶
This document was improved based on input and contributions by the Zcash Foundation engineering team. In addition, the authors of this document would like to thank Isis Lovecruft, Alden Torres, T. Wilson-Brown, and Conrado Gouvea for their inputs and contributions.¶
This section contains descriptions of functions for generating and verifying Schnorr signatures.
It is included to complement the routines present in [RFC8032] for prime-order groups, including
ristretto255, P-256, and secp256k1. The functions for generating and verifying signatures are
prime_order_sign and prime_order_verify, respectively.¶
The function prime_order_sign produces a Schnorr signature over a message given a full secret signing
key as input (as opposed to a key share.)¶
prime_order_sign(msg, sk):
Inputs:
- msg, message to sign, a byte string.
- sk, secret key, a Scalar.
Outputs:
- (R, z), a Schnorr signature consisting of an Element R and Scalar z.
def prime_order_sign(msg, sk):
r = G.RandomScalar()
R = G.ScalarBaseMult(r)
PK = G.ScalarBaseMult(sk)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
z = r + (c * sk) // Scalar addition and multiplication
return (R, z)
¶
The function prime_order_verify verifies Schnorr signatures with validated inputs.
Specifically, it assumes that signature R component and public key belong to the prime-order group.¶
prime_order_verify(msg, sig, PK):
Inputs:
- msg, signed message, a byte string.
- sig, a tuple (R, z) output from signature generation.
- PK, public key, an Element.
Outputs:
- True if signature is valid, and False 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, denoted s_i for i = 1, ..., MAX_PARTICIPANTS, to be sent to all MAX_PARTICIPANTS participants.
This operation is specified in the trusted_dealer_keygen algorithm. The mathematical relation
between the secret key s and the MAX_PARTICIPANTS secret shares is formalized in the secret_share_combine(shares)
algorithm, defined in Appendix C.1.¶
The dealer that performs trusted_dealer_keygen is trusted to 1) generate good randomness, and 2) delete secret values after distributing shares to each participant, and 3) keep secret values confidential.¶
Inputs:
- secret_key, a group secret, a Scalar, that MUST be derived from at least Ns bytes of entropy.
- MAX_PARTICIPANTS, the number of shares to generate, an integer.
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer.
Outputs:
- participant_private_keys, MAX_PARTICIPANTS shares of the secret key s, each a tuple
consisting of the participant identifier (a NonZeroScalar) and the key share (a Scalar).
- group_public_key, public key corresponding to the group signing key, an
Element.
- vss_commitment, a vector commitment of Elements in G, to each of the coefficients
in the polynomial defined by secret_key_shares and whose first element is
G.ScalarBaseMult(s).
def trusted_dealer_keygen(secret_key, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
# Generate random coefficients for the polynomial
coefficients = []
for i in range(0, MIN_PARTICIPANTS - 1):
coefficients.append(G.RandomScalar())
participant_private_keys, coefficients = secret_share_shard(secret_key, coefficients, MAX_PARTICIPANTS)
vss_commitment = vss_commit(coefficients):
return participant_private_keys, vss_commitment[0], vss_commitment
¶
It is assumed the dealer then sends one secret key share to each of the NUM_PARTICIPANTS participants, along with vss_commitment.
After receiving their secret key share and vss_commitment, participants MUST abort if they do not have the same view of vss_commitment.
The dealer can use a secure broadcast channel to ensure each participant has a consistent view of this 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 at least MIN_PARTICIPANTS participants can recover the
secret. There are two basic steps in this scheme: (1) splitting a secret into
multiple shares, and (2) combining shares to reveal the resulting secret.¶
This secret sharing scheme works over any field F. In this specification, F is
the scalar field of the prime-order group G.¶
The procedure for splitting a secret into shares is as follows.
The algorithm polynomial_evaluate is defined in Appendix C.1.1.¶
secret_share_shard(s, coefficients, MAX_PARTICIPANTS):
Inputs:
- s, secret value to be shared, a Scalar.
- coefficients, an array of size MIN_PARTICIPANTS - 1 with randomly generated
Scalars, not including the 0th coefficient of the polynomial.
- MAX_PARTICIPANTS, the number of shares to generate, an integer less than 2^16.
Outputs:
- secret_key_shares, A list of MAX_PARTICIPANTS number of secret shares, each a tuple
consisting of the participant identifier (a NonZeroScalar) and the key share (a Scalar).
- coefficients, a vector of MIN_PARTICIPANTS coefficients which uniquely determine a polynomial f.
def secret_share_shard(s, coefficients, MAX_PARTICIPANTS):
# Prepend the secret to the coefficients
coefficients = [s] + coefficients
# Evaluate the polynomial for each point x=1,...,n
secret_key_shares = []
for x_i in range(1, MAX_PARTICIPANTS + 1):
y_i = polynomial_evaluate(Scalar(x_i), coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_shares.append(secret_key_share_i)
return secret_key_shares, coefficients
¶
Let points be the output of this function. The i-th element in points is
the share for the i-th participant, which is the randomly generated polynomial
evaluated at coordinate i. We denote a secret share as the tuple (i, points[i]),
and the list of these shares as shares. i MUST never equal 0; recall that
f(0) = s, where f is the polynomial defined in a Shamir secret sharing operation.¶
The procedure for combining a shares list of length MIN_PARTICIPANTS to recover the
secret s is as follows; the algorithm polynomial_interpolate_constant is defined in Appendix C.1.1.¶
secret_share_combine(shares):
Inputs:
- shares, a list of at minimum MIN_PARTICIPANTS secret shares, each a tuple (i, f(i))
where i and f(i) are Scalars.
Outputs:
- s, the resulting secret that was previously split into shares, a Scalar.
Errors:
- "invalid parameters", if fewer than MIN_PARTICIPANTS input shares are provided.
def secret_share_combine(shares):
if len(shares) < MIN_PARTICIPANTS:
raise "invalid parameters"
s = polynomial_interpolate_constant(shares)
return s
¶
This section describes two functions. One function, denoted polynomial_evaluate,
is for evaluating a polynomial f(x) at a particular point x using Horner's method,
i.e., computing y = f(x). The other function, polynomial_interpolate_constant, is for
recovering the constant term of an interpolating polynomial defined by a set of points.¶
The function polynomial_evaluate is defined as follows.¶
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 = Scalar(0)
for coeff in reverse(coeffs):
value *= x
value += coeff
return value
¶
The function polynomial_interpolate_constant is defined as follows.¶
Inputs:
- points, a set of t points with distinct x coordinates on a polynomial f,
each a tuple of two Scalar values representing the x and y coordinates.
Outputs:
- f_zero, the constant term of f, i.e., f(0), a Scalar.
def polynomial_interpolate_constant(points):
x_coords = []
for (x, y) in points:
x_coords.append(x)
f_zero = Scalar(0)
for (x, y) in points:
delta = y * derive_interpolating_value(x, x_coords)
f_zero += delta
return f_zero
¶
Feldman's Verifiable Secret Sharing (VSS) [FeldmanSecretSharing]
builds upon Shamir secret sharing, adding a verification step to demonstrate the consistency of a participant's
share with a public commitment to the polynomial f for which the secret s
is the constant term. This check ensures that all participants have a point
(their share) on the same polynomial, ensuring that they can later reconstruct
the correct secret.¶
The procedure for committing to a polynomial f of degree at most MIN_PARTICIPANTS-1 is as follows.¶
vss_commit(coeffs):
Inputs:
- coeffs, a vector of the MIN_PARTICIPANTS coefficients which uniquely determine
a polynomial f.
Outputs:
- vss_commitment, a vector commitment to each of the coefficients in coeffs, where
each item of the vector commitment is an Element.
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 (a NonZeroScalar), 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:
- True if sk_i is valid, and False otherwise.
vss_verify(share_i, vss_commitment)
(i, sk_i) = share_i
S_i = G.ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
S_i' += G.ScalarMult(vss_commitment[j], pow(i, j))
return S_i == S_i'
¶
We now define how the Coordinator and participants can derive group info, which is an input into the FROST signing protocol.¶
derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS, vss_commitment):
Inputs:
- MAX_PARTICIPANTS, the number of shares to generate, an integer.
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer.
- vss_commitment, a VSS commitment to a secret polynomial f, a vector commitment to each of the
coefficients in coeffs, where each element of the vector commitment is an Element.
Outputs:
- PK, the public key representing the group, an Element.
- participant_public_keys, a list of MAX_PARTICIPANTS public keys PK_i for i=1,...,MAX_PARTICIPANTS,
where each PK_i is the public key, an Element, for participant i.
derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS, vss_commitment)
PK = vss_commitment[0]
participant_public_keys = []
for i in range(1, MAX_PARTICIPANTS+1):
PK_i = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
PK_i += G.ScalarMult(vss_commitment[j], pow(i, j))
participant_public_keys.append(PK_i)
return PK, participant_public_keys
¶
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 in constant time. If it succeeds, return the
result. If it fails, try again with another random byte array, until the
procedure succeeds. Failure to implement DeserializeScalar in constant time
can leak information about the underlying corresponding Scalar.¶
As an optimization, if the group order is very close to a power of
2, it is acceptable to omit the rejection test completely. In
particular, if the group order is p, and there is an integer b
such that |p - 2b| is less than 2(b/2), then
RandomScalar can simply return a uniformly random integer of at
most b bits.¶
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.¶
share_polynomial_coefficients[1] is the coefficient
of the first term in the polynomial. Note that the 0-th coefficient is omitted as this
is equal to the group secret key. All values are encoded as hexadecimal strings.¶
nonce_generate; the resulting group
binding factor input computed in part from the group commitment list encoded as
described in Section 4.3; and group binding factor as computed in Section 5.2).¶
// Configuration information MAX_PARTICIPANTS: 3 MIN_PARTICIPANTS: 2 NUM_PARTICIPANTS: 2 // Group input parameters group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7 90c6e13a98304 group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040 d380fb9738673 message: 74657374 share_polynomial_coefficients[1]: 178199860edd8c62f5212ee91eff1295d0d 670ab4ed4506866bae57e7030b204 // Signer input parameters P1 participant_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f 4a033f2ec83d93509 P2 participant_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685 c07eed76bf409e80d P3 participant_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c 46da8bdea643a9a02 // Round one parameters participant_list: 1,3 // Signer round one outputs P1 hiding_nonce_randomness: 9d06a6381c7a4493929761a73692776772b274236 fb5cfcc7d1b48ac3a9c249f P1 binding_nonce_randomness: db184d7bc01a3417fe1f2eb3cf5479bb027145e6 369a5f879f32d334ab256b23 P1 hiding_nonce: 70652da3e8d7533a0e4b9e9104f01b48c396b5b553717784ed8d 05c6a36b9609 P1 binding_nonce: 4f9e1ad260b5c0e4fe0e0719c6324f89fecd053758f77c957f5 6967e634a710e P1 hiding_nonce_commitment: 44105304351ceddc58e15ddea35b2cb48e60ced54 ceb22c3b0e5d42d098aa1d8 P1 binding_nonce_commitment: b8274b18a12f2cef74ae42f876cec1e31daab5cb 162f95a56cd2487409c9d1dd P1 binding_factor_input: c5b95020cba31a9035835f074f718d0c3af02a318d6b 4723bbd1c088f4889dd7b9ff8e79f9a67a9d27605144259a7af18b7cca2539ffa5c4f 1366a98645da8f4e077d604fff64f20e2377a37e5a10ce152194d62fe856ef4cd935d 4f1cb0088c2083a2722ad3f5a84d778e257da0df2a7cadb004b1f5528352af778b94e e1c2a0100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: 2d5630c36d33258b1208c4205fa759b762d09bfa06b29cf792 cf98758c0b3305 P3 hiding_nonce_randomness: 31ca9b07936d6b342a43d97f23b7bec5a5f5a0957 5a075393868dd8df5c05a54 P3 binding_nonce_randomness: c1db96a85d8b593e14fdb869c0955625478afa6a 987ad217e7f2261dcab26819 P3 hiding_nonce: 233adcb0ec0eddba5f1cc5268f3f4e6fc1dd97fb1e4a1754e6dd c92ed834ca0b P3 binding_nonce: b59fc8a32fe02ec0a44c4671f3d1f82ea3924b7c7c0179398fc 9137e82757803 P3 hiding_nonce_commitment: d31bd81ce216b1c83912803a574a0285796275cb8 b14f6dc92c8b09a6951f0a2 P3 binding_nonce_commitment: e1c863cfd08df775b6747ef2456e9bf9a03cc281 a479a95261dc39137fcf0967 P3 binding_factor_input: c5b95020cba31a9035835f074f718d0c3af02a318d6b 4723bbd1c088f4889dd7b9ff8e79f9a67a9d27605144259a7af18b7cca2539ffa5c4f 1366a98645da8f4e077d604fff64f20e2377a37e5a10ce152194d62fe856ef4cd935d 4f1cb0088c2083a2722ad3f5a84d778e257da0df2a7cadb004b1f5528352af778b94e e1c2a0300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: 1137be5cdf3d18e44367acee8485e9a66c3164077af80619b6 291e3943bbef04 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: c4b26af1e91fbc8440a0dad253e72620da624553c5b625fd51e6ea1 79fc09f05 P3 sig_share: 9369640967d0cb98f4dedfde58a845e0e18e0a7164396358439060e d282b4e08 sig: ae11c539fdc709b78fef5ee1f5a2250297e3e1b62a86a86c26d93c389934ba0e 571ccffa50f0871d357fbab1ac8f6c00bcf14fc429f0885595764b05c8ebed0d¶
// Configuration information MAX_PARTICIPANTS: 3 MIN_PARTICIPANTS: 2 NUM_PARTICIPANTS: 2 // Group input parameters group_secret_key: 6298e1eef3c379392caaed061ed8a31033c9e9e3420726f23b4 04158a401cd9df24632adfe6b418dc942d8a091817dd8bd70e1c72ba52f3c00 group_public_key: 3832f82fda00ff5365b0376df705675b63d2a93c24c6e81d408 01ba265632be10f443f95968fadb70d10786827f30dc001c8d0f9b7c1d1b000 message: 74657374 share_polynomial_coefficients[1]: dbd7a514f7a731976620f0436bd135fe8dd dc3fadd6e0d13dbd58a1981e587d377d48e0b7ce4e0092967c5e85884d0275a7a740b 6abdcd0500 // Signer input parameters P1 participant_share: 4a2b2f5858a932ad3d3b18bd16e76ced3070d72fd79ae44 02df201f525e754716a1bc1b87a502297f2a99d89ea054e0018eb55d39562fd0100 P2 participant_share: 2503d56c4f516444a45b080182b8a2ebbe4d9b2ab509f25 308c88c0ea7ccdc44e2ef4fc4f63403a11b116372438a1e287265cadeff1fcb0700 P3 participant_share: 00db7a8146f995db0a7cf844ed89d8e94c2b5f259378ff6 6e39d172828b264185ac4decf7219e4aa4478285b9c0eef4fccdf3eea69dd980d00 // Round one parameters participant_list: 1,3 // Signer round one outputs P1 hiding_nonce_randomness: 89bf16040081ff2990336b200613787937ebe1f02 4b8cdff90eb6f1c741d91c1 P1 binding_nonce_randomness: cd646348bb98fd2a4b2f27fb7d6da18201c16184 7352576b4bf125190e965483 P1 hiding_nonce: 67a6f023e77361707c6e894c625e809e80f33fdb310810053ae2 9e28e7011f3193b9020e73c183a98cc3a519160ed759376dd92c9483162200 P1 binding_nonce: 4812e8d7c8b7a50ced80b507902d074ef8647bc1146979683da 8d0fecd93fa3c8230cade2fb4344600aa04bd4b7a21d046c5b63ee865b12a00 P1 hiding_nonce_commitment: 649c6a53b109897d962d033f23d01fd4e1053dddf 3746d2ddce9bd66aea38ccfc3df061df03ca399eb806312ab3037c0c31523142956ad a780 P1 binding_nonce_commitment: 0064cc729a8e2fcf417e43788ecec37b10e9e1dc b3ae90854efbfaad00a0ef3cdd52e18d56f073c8ff0947cb71ff0bb17c3d45d096409 ddb00 P1 binding_factor_input: 106dadce87ca867018702d69a02effd165e1ac1a511c 957cff1897ceff2e34ca212fe798d84f0bde6054bf0fa77fd4cd4bc4853d6dc8dbd19 d340923f0ebbbb35172df4ab865a45d55af31fa0e6606ea97cf8513022b2b133d0f9f 6b8d3be184221fc4592bf12bd7fb4127bb67e51a6dc9e5f1ed5243362fb46a6da5524 18ca967d43d9bc811a21917a3018de58f11c25f6b9ad8bec3699e06b87dd3ab67a732 6c30878c7c55ec1a45802af65da193ce99634158539e38c232a627895c5f14e2e20d4 87382ccc9c99cd0a0df266a292f283bb9b6854e344ecc32d5e1852fdde5fde7779801 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P1 binding_factor: 3412ac894a91a6bc0e3e7c790f3e8ef5d1288e54de780aba38 4cbb3081b602dd188010e5b0c9ac2b5dca0aae54cfd0f5c391cece8092131d00 P3 hiding_nonce_randomness: 3718dabb4fd3d7dd9adad4878c6de8b33c8841cfe 7cc95a85592952a2c9c554d P3 binding_nonce_randomness: 3becbc90798211a0f52543dd1f24869a143fdf74 3409581af4db30f045773d64 P3 hiding_nonce: 4f2666770317d14ec9f7fd6690c075c34b4cde7f6d9bceda9e94 33ec8c0f2dc983ff1622c3a54916ce7c161381d263fad62539cddab2101600 P3 binding_nonce: 88f66df8bb66389932721a40de4aa5754f632cac114abc10526 88104d19f3b1a010880ebcd0c4c0f8cf567d887e5b0c3c0dc78821166550f00 P3 hiding_nonce_commitment: 8dcf049167e28d5f53fa7ebbbd136abcaf2be9f2c 02448c8979002f92577b22027640def7ddd5b98f9540c2280f36a92d4747bbade0b0c 4280 P3 binding_nonce_commitment: 12e837b89a2c085481fcf0ca640a17a24b6fc96b 032d40e4301c78e7232a9f49ffdcad2c21acbc992e79dfc3c6c07cb94e4680b3dcc99 35580 P3 binding_factor_input: 106dadce87ca867018702d69a02effd165e1ac1a511c 957cff1897ceff2e34ca212fe798d84f0bde6054bf0fa77fd4cd4bc4853d6dc8dbd19 d340923f0ebbbb35172df4ab865a45d55af31fa0e6606ea97cf8513022b2b133d0f9f 6b8d3be184221fc4592bf12bd7fb4127bb67e51a6dc9e5f1ed5243362fb46a6da5524 18ca967d43d9bc811a21917a3018de58f11c25f6b9ad8bec3699e06b87dd3ab67a732 6c30878c7c55ec1a45802af65da193ce99634158539e38c232a627895c5f14e2e20d4 87382ccc9c99cd0a0df266a292f283bb9b6854e344ecc32d5e1852fdde5fde7779803 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P3 binding_factor: 6aa48a3635d7b962489283ee1ccda8ea66e5677b1e17f2f475 eb565e3ae8ea73360f24c04e3775dadd1f2923adcda3d105536ad28c3c561100 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: c5057c80d13e565545dac6f3aa333065c809a14a94fea3c8e4e87e3 86a9cb89602de7355c5d19ebb09d553b100ef1858104fc7c43992d83400 P3 sig_share: 2b490ea08411f78c620c668fff8ba70b25b7c89436f20cc45419213 de70f93fb6c9094c79293697d72e741b68d2e493446005145d0b7fc3500 sig: 83ac141d289a5171bc894b058aee2890316280719a870fc5c1608b7740302315 5d7a9dc15a2b7920bb5826dd540bf76336be99536cebe36280fd093275c38dd4be525 767f537fd6a4f5d8a9330811562c84fded5f851ac4b926f6e081d586508397cbc9567 8e1d628c564f180a0a4ad52a00¶
// Configuration information MAX_PARTICIPANTS: 3 MIN_PARTICIPANTS: 2 NUM_PARTICIPANTS: 2 // Group input parameters group_secret_key: 1b25a55e463cfd15cf14a5d3acc3d15053f08da49c8afcf3ab2 65f2ebc4f970b group_public_key: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3dd9c3c0d 05083c7254f57 message: 74657374 share_polynomial_coefficients[1]: 410f8b744b19325891d73736923525a4f59 6c805d060dfb9c98009d34e3fec02 // Signer input parameters P1 participant_share: 5c3430d391552f6e60ecdc093ff9f6f4488756aa6cebdba d75a768010b8f830e P2 participant_share: b06fc5eac20b4f6e1b271d9df2343d843e1e1fb03c4cbb6 73f2872d459ce6f01 P3 participant_share: f17e505f0e2581c6acfe54d3846a622834b5e7b50cad9a2 109a97ba7a80d5c04 // Round one parameters participant_list: 1,3 // Signer round one outputs P1 hiding_nonce_randomness: 81800157bb554f299fe0b6bd658e4c4591d74168b 5177bf55e8dceed59dc80c7 P1 binding_nonce_randomness: e9b37de02fde28f601f09051ed9a277b02ac81c8 03a5c72492d58635001fe355 P1 hiding_nonce: 40f58e8df202b21c94f826e76e4647efdb0ea3ca7ae7e3689bc0 cbe2e2f6660c P1 binding_nonce: 373dd42b5fe80e88edddf82e03744b6a12d59256f546de612d4 bbd91a6b1df06 P1 hiding_nonce_commitment: b8c7319a56b296537436e5a6f509a871a3c74eff1 534ec1e2f539ccd8b322411 P1 binding_nonce_commitment: 7af5d4bece8763ce3630370adbd978699402f624 fd3a7d2c71ea5839efc3cf54 P1 binding_factor_input: 9c245d5fc2e451c5c5a617cc6f2a20629fb317d9b1c1 915ab4bfa319d4ebf922c54dd1a5b3b754550c72734ac9255db8107a2b01f361754d9 f13f428c2f6de9e4f609ae0dbe8bd1f95bee9f9ea219154d567ef174390bac737bb67 ee1787c8a34279728d4aa99a6de2d5ce6deb86afe6bc68178f01223bb5eb934c8a23b 6354e0100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: 607df5e2e3a8b5e2704716693e18f548100a32b86a5685d393 2a774c3f107e06 P3 hiding_nonce_randomness: daeb223c4a913943cff2fb0b0e638dfcc281e1e89 36ee6c3fef4d49ad9cbfaa0 P3 binding_nonce_randomness: c425768d952ab8f18b9720c54b93e612ba2cca17 0bb7518cac080896efa7429b P3 hiding_nonce: 491477c9dbe8717c77c6c1e2c5f4cec636c7c154313a44c91fea 63e309f3e100 P3 binding_nonce: 3ae1bba7d6f2076f81596912dd916efae5b3c2ef89695632119 4fdd2e52ebc0f P3 hiding_nonce_commitment: e4466b7670ac4f9d9b7b67655860dd1ab341be18a 654bb1966df53c76c85d511 P3 binding_nonce_commitment: ce47cd595d25d7effc3c095efa2a687a1728a5ec ab402b39e0c0ad9a525ea54f P3 binding_factor_input: 9c245d5fc2e451c5c5a617cc6f2a20629fb317d9b1c1 915ab4bfa319d4ebf922c54dd1a5b3b754550c72734ac9255db8107a2b01f361754d9 f13f428c2f6de9e4f609ae0dbe8bd1f95bee9f9ea219154d567ef174390bac737bb67 ee1787c8a34279728d4aa99a6de2d5ce6deb86afe6bc68178f01223bb5eb934c8a23b 6354e0300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: 2bd27271c28746eb93e2114d6778c12b44c9287d84b85dc780 eb08da6f689900 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: c38f438c325ce6bfa4272b37e7707caaeb57fa8c7ddcc05e0725acb 8a7d9cd0c P3 sig_share: 4cb9917be3bd53f1d60f1c3d1a3ff563565fa15a391133e7f980e55 d3aeb7904 sig: 204d5d93aa486192ecf2f64ce7dbc1db76948fb1077d1a719ae1ecca6143501e 2275dfaafbb62759a59a4fd122b692f941b79be7b6edf34501a69116e2c44701¶
// Configuration information MAX_PARTICIPANTS: 3 MIN_PARTICIPANTS: 2 NUM_PARTICIPANTS: 2 // Group input parameters group_secret_key: 8ba9bba2e0fd8c4767154d35a0b7562244a4aaf6f36c8fb8735 fa48b301bd8de group_public_key: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99cfbd02b4 dc00585ec45ab70 message: 74657374 share_polynomial_coefficients[1]: 80f25e6c0709353e46bfbe882a11bdbb1f8 097e46340eb8673b7e14556e6c3a4 // Signer input parameters P1 participant_share: 0c9c1a0fe806c184add50bbdcac913dda73e482daf95dcb 9f35dbb0d8a9f7731 P2 participant_share: 8d8e787bef0ff6c2f494ca45f4dad198c6bee01212d6c84 067159c52e1863ad5 P3 participant_share: 0e80d6e8f6192c003b5488ce1eec8f5429587d48cf00154 1e713b2d53c09d928 // Round one parameters participant_list: 1,3 // Signer round one outputs P1 hiding_nonce_randomness: f4e8cf80aec3f888d997900ac7e3e349944b5a6b4 7649fc32186d2f1238103c6 P1 binding_nonce_randomness: a7f220770b6f10ff54ec6afa55f99bd08cc92fa1 a488c86e9bf493e9cb894cdf P1 hiding_nonce: f871dfcf6bcd199342651adc361b92c941cb6a0d8c8c1a3b91d7 9e2c1bf3722d P1 binding_nonce: bd3ece3634a1b303dea0586ed67a91fe68510f11ebe66e88683 09b1551ef2388 P1 hiding_nonce_commitment: 03987febbc67a8ed735affdff4d3a5adf22c05c80 f97f311ab7437a3027372deb3 P1 binding_nonce_commitment: 02a1960477d139035b986d6adcb06491378beb92 ccd097ad94e76291c52343849d P1 binding_factor_input: 350c8b523feea9bb35720e9fbe0405ed48d78caa4fb6 0869f34367e144c68bb0fc77bf512409ad8b91e2ace4909229891a446c45683f5eb2f 843dbec224527dc000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: cb415dd1d866493ee7d2db7cb33929d7e430e84d80c58070e2 bbb1fdbf76a9c8 P3 hiding_nonce_randomness: 1b6149d252a0a0a6618b8d22a1c49897f9b0d23a4 8f19598e191e05dc7b7ae33 P3 binding_nonce_randomness: e13994bb75aafe337c32afdbfd08ae60dd108fc7 68845edaa871992044cabf1b P3 hiding_nonce: 802e9321f9f63688c6c1a9681a4a4661f71770e0cef92b8a5997 155d18fb82ef P3 binding_nonce: 8b6b692ae634a24536f45dda95b2398af71cd605fb7a0bbdd94 08d211ab99eba P3 hiding_nonce_commitment: 0212cac45ebd4100c97506939391f9be4ffc3ca29 60e2ef95aeaa38abdede204ca P3 binding_nonce_commitment: 03017ce754d310eabda0f5681e61ce3d713cdd33 7070faa6a68471af49694a4e7e P3 binding_factor_input: 350c8b523feea9bb35720e9fbe0405ed48d78caa4fb6 0869f34367e144c68bb0fc77bf512409ad8b91e2ace4909229891a446c45683f5eb2f 843dbec224527dc000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: dfd82467569334e952edecb10d92adf85b8e299db0b40be313 1a12efdfa3e796 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: c5acd980310aaf87cb7a9a90428698ef3e6b1e5860f7fb06329bc0e fe3f14ca5 P3 sig_share: 1e064fbd35467377eb3fe161ff975e9ec3ed8e2e0d4c73f3a6b0a02 3777e1264 sig: 029e07d4171dbf9a730ed95e9d95bda06fa4db76c88c519f7f3ca5483019f46c b0e3b3293d665122ffb6ba7bf2421df78e0258ac866e446ef9d94c61135b6f5f09¶
// Configuration information MAX_PARTICIPANTS: 3 MIN_PARTICIPANTS: 2 NUM_PARTICIPANTS: 2 // Group input parameters group_secret_key: 0d004150d27c3bf2a42f312683d35fac7394b1e9e318249c1bf e7f0795a83114 group_public_key: 02f37c34b66ced1fb51c34a90bdae006901f10625cc06c4f646 63b0eae87d87b4f message: 74657374 share_polynomial_coefficients[1]: fbf85eadae3058ea14f19148bb72b45e439 9c0b16028acaf0395c9b03c823579 // Signer input parameters P1 participant_share: 08f89ffe80ac94dcb920c26f3f46140bfc7f95b493f8310 f5fc1ea2b01f4254c P2 participant_share: 04f0feac2edcedc6ce1253b7fab8c86b856a797f44d83d8 2a385554e6e401984 P3 participant_share: 00e95d59dd0d46b0e303e500b62b7ccb0e555d49f5b849f 5e748c071da8c0dbc // Round one parameters participant_list: 1,3 // Signer round one outputs P1 hiding_nonce_randomness: 80cbea5e405d169999d8c4b30b755fedb26ab07ec 8198cda4873ed8ce5e16773 P1 binding_nonce_randomness: f6d5b38197843046b68903048c1feba433e35001 45281fa8bb1e26fdfeef5e7f P1 hiding_nonce: acc83278035223c1ba464e2d11bfacfc872b2b23e1041cf5f613 0da21e4d8068 P1 binding_nonce: c3ef169995bc3d2c2d48f30b83d0c63751e67ceb057695bcb2a 6aa40ed5d926b P1 hiding_nonce_commitment: 036673d68a928793c33ae07776908eae8ea15dd94 7ed81284e939aaba118573a5e P1 binding_nonce_commitment: 03d2a96dd4ec1ee29dc22067109d1290dabd8016 cb41856ee8ff9281c3fa1baffd P1 binding_factor_input: a645d8249457bbcac34fa7b740f66bcce08fc39506b8 bbf1a1c81092f6272eda82ae39234d714f87a7b91dd67d124a06561a36817c1ecaa25 5c3527d694fc4f1000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: d7bcbd29408dedc9e138262d99b09d8b5705d76eb5de2369d9 103e4423f8ac79 P3 hiding_nonce_randomness: b9794047604beda0c5c0529ac9dfd83c0a80399a7 bdf4c3e23cef2faf69cdcc3 P3 binding_nonce_randomness: c28ce6252631620b84c2702b34774fab365e286e bc77030a112ebccccbffa78b P3 hiding_nonce: cb3387defef07fc9010c0564ba6495ed41876626ed86b886ca26 cbbd3566ffbc P3 binding_nonce: 4559459735eb68e8c16319a9fd9a14016053957cb8cea273a24 b7c7bc1ee26f6 P3 hiding_nonce_commitment: 030278e6e6055fb963b40e0c3c37099f803f3f389 30fc89092517f8ce1b47e8d6b P3 binding_nonce_commitment: 028eb6d238c6c0fc6216906706ad0ff9943c6c1d 6079cdf74f674481ebb2485db3 P3 binding_factor_input: a645d8249457bbcac34fa7b740f66bcce08fc39506b8 bbf1a1c81092f6272eda82ae39234d714f87a7b91dd67d124a06561a36817c1ecaa25 5c3527d694fc4f1000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: ecc057259f3c8b195308c9b73aaaf840660a37eb264ebce342 412c58102ee437 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 1750b2a314a81b66fd81366583617aaafcffa68f14495204795aa04 34b907aa3 P3 sig_share: e4dbbbbbcb035eb3512918b0368c4ab2c836a92dccff3251efa7a4a acc7d3790 sig: 0259696aac722558e8638485d252bb2556f6241a7adfdf284c8c87a3428d4644 8dfc2c6e5edfab7a1a4eaa4f15b9edc55dc5364fbce1488456690244ee180db233¶