| Internet-Draft | FROST | September 2022 |
| Connolly, et al. | Expires 31 March 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 signer 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 31 March 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 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 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 signing participants 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 signer participants that started round one of the protocol is the same set of signing participants that produced the signature output by round two.¶
Key generation for FROST signing is out of scope for this document. However, for completeness, key generation with a trusted dealer is specified in Appendix C.¶
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The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
The following 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 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): Outputs the integer result of a to the power of b, e.g., pow(2, 3) = 8.¶
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 types 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 B with itself r-1 times, this is denoted as ScalarBaseMult(r). The set of
scalars corresponds to GF(p), which we refer to as the scalar field. This document uses types
Element and Scalar to denote elements of the group G and its set of scalars, respectively.
We denote Scalar(x) as the conversion of integer input x to the corresponding Scalar value with
the same numeric value. For example, Scalar(1) yields a Scalar representing the value 1.
We denote equality comparison as == and assignment of values by =. Finally, it is assumed that
group element addition, negation, and equality comparisons can be efficiently computed for
arbitrary group elements.¶
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.¶
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 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 canonical byte array buf of fixed length Ns.¶
buf to a Scalar s.
This function can raise a DeserializeError if deserialization fails; see
Section 6 for group-specific input validation steps.¶
FROST requires the use of a cryptographically secure hash function, generically written as H, which functions effectively as a random oracle. For concrete recommendations on hash functions which SHOULD be used in practice, see Section 6. Using H, we introduce separate domain-separated hashes, H1, H2, H3, H4, and H5:¶
The details of H1, H2, H3, H4, and H5 vary based on ciphersuite. See Section 6 for more details about each.¶
Beyond the core dependencies, the protocol in this document depends on the following helper operations:¶
These sections describes these operations in more detail.¶
To hedge against a bad RNG that outputs predictable values, nonces are
generated with the nonce_generate function by combining fresh randomness
and 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 describes operations on and associated with polynomials over Scalars
that are used in the main signing protocol. A polynomial of maximum degree t+1
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 1) any x-coordinate is equal to 0, 2) if x_i
is not in L, or if 3) any x-coordinate is represented more than once 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"
for x_j in L:
if count(x_i, 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
L_i = numerator / denominator
return L_i
¶
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 bytestring for use in the FROST protocol.¶
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...],
a list of commitments issued by each participant, where each element in the list
indicates the participant 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 participant identifier.
Outputs: A byte string containing the serialized representation of commitment_list
def encode_group_commitment_list(commitment_list):
encoded_group_commitment = nil
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
encoded_commitment = 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 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 participant, where each element in the list
indicates the participant 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 participant identifier.
Outputs: A list of 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 participant, where each element in the list
indicates the participant identifier i and their binding factor. This list MUST be sorted
in ascending order by participant identifier.
- identifier, participant identifier, a Scalar.
Outputs: A Scalar value.
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 the participant 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 participant identifier.
- msg, the message to be signed.
Outputs: A list of (identifier, Scalar) tuples representing the binding factors.
def compute_binding_factors(commitment_list, msg):
msg_hash = H4(msg)
encoded_commitment_hash = H5(encode_group_commitment_list(commitment_list))
rho_input_prefix = msg_hash || encoded_commitment_hash
binding_factor_list = []
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
rho_input = rho_input_prefix || 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 the participant 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 participant identifier.
- binding_factor_list = [(i, binding_factor), ...],
a list of (identifier, Scalar) tuples representing the binding factor Scalar
for the given identifier. This list MUST be sorted in ascending order by identifier.
Outputs: An Element in G representing the group commitment
def compute_group_commitment(commitment_list, binding_factor_list):
group_commitment = G.Identity()
for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
binding_factor = binding_factor_for_participant(binding_factors, identifier)
group_commitment = group_commitment +
hiding_nonce_commitment + G.ScalarMult(binding_nonce_commitment, binding_factor)
return group_commitment
¶
This section describes the subroutine for creating the per-message challenge.¶
Inputs:
- group_commitment, an Element in G representing the group commitment
- group_public_key, public key corresponding to the group signing key, an
Element in G.
- msg, the message to be signed.
Outputs: A Scalar representing the challenge
def compute_challenge(group_commitment, group_public_key, msg):
group_comm_enc = G.SerializeElement(group_commitment)
group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg
challenge = H2(challenge_input)
return challenge
¶
This section describes the two-round variant of the FROST threshold signature
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
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 all parties and their roles, including the Coordinator and the set of participants, are chosen externally to the protocol. Note that it is possible to deploy the protocol without a distinguished Coordinator; see Section 7.3 for more information.¶
FROST produces signatures that are indistinguishable from those produced with a single
participant using a signing key s with corresponding public key PK, where s is a Scalar
value and PK = G.ScalarMultBase(s). As a threshold signing protocol, the group signing
key s is secret-shared amongst each participant and used to produce signatures. 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 secret share
of the group signing key s. The public key corresponding to this signing key share
is PK_i = G.ScalarMultBase(sk_i).¶
The Coordinator and each participant is additionally configured with common group information, denoted "group info," which consists of the following information:¶
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.¶
The signing variant of FROST in this document requires participants 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 participants as well as the message, and issues a signature share with respect to that message. The Coordinator performs the coordination of each of these rounds. At the end of the second round, the Coordinator then performs an aggregation step and outputs the final signature. This complete interaction is shown in Figure 1.¶
(group info) (group info, (group info,
| signing key share) signing key share)
| | |
v v v
Coordinator Signer-1 ... Signer-n
------------------------------------------------------------
message
------------>
|
== Round 1 (Commitment) ==
| 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 both rounds, and this state is deleted as described in Section 5.2. The final Aggregation step is described in Section 5.3.¶
FROST assumes that all inputs to each round, especially those of which are received over the network, are validated before use. In particular, this means that any value of type Element or Scalar is deserialized using DeserializeElement and DeserializeScalar, respectively, as these functions perform the necessary input validation steps.¶
FROST assumes reliable message delivery between the Coordinator and 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 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 reused in more than one invocation of FROST, and it 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 their identifier as well as their commitment as from the first round appears in commitment_list. Applications which require that participants not process arbitrary input messages are also required to also perform relevant application-layer input validation checks; see Section 7.5 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. 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 participant and sent by the Coordinator.
Each element in the list indicates the participant 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 participant 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 participant then sends these shares back to the Coordinator. Each participant 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, participants
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 participants perform round two and send their signature shares to the Coordinator, the Coordinator verifies each signature share for correctness. In particular, for each participant, 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 participant. Note: identifier MUST never equal 0.
- PK_i, the public key for the ith participant, 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 participant.
- sig_share_i, a Scalar value indicating the signature share as produced in
round two from the ith 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 the participant 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 participant identifier.
- group_public_key, public key corresponding to the group signing key,
an Element in G.
- msg, the message to be signed.
Outputs: True if the signature share is valid, and False otherwise.
def verify_signature_share(identifier, PK_i, comm_i, sig_share_i, commitment_list,
group_public_key, msg):
# Compute the binding factors
binding_factor_list = compute_binding_factors(commitment_list, msg)
binding_factor = binding_factor_for_participant(binding_factor_list, identifier)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor_list)
# Compute the commitment share
(hiding_nonce_commitment, binding_nonce_commitment) = comm_i
comm_share = hiding_nonce_commitment + G.ScalarMult(binding_nonce_commitment, binding_factor)
# Compute the challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute Lagrange coefficient
participant_list = participants_from_commitment_list(commitment_list)
lambda_i = derive_lagrange_coefficient(identifier, participant_list)
# Compute relation values
l = G.ScalarBaseMult(sig_share_i)
r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i)
return l == r
¶
If any signature share fails to verify, i.e., if verify_signature_share returns False for
any participant share, the Coordinator MUST abort the protocol for correctness reasons
(this is true regardless of the size or makeup of the signing set selected by
the Coordinator).
Excluding one participant means that their nonce will not be included in the joint response z
and consequently the output signature will not verify. This is because the
group commitment will be with respect to a different signing set than the
the aggregated response.¶
Otherwise, if all shares from participants that participated in Rounds 1 and 2 are valid, the Coordinator
performs the aggregate operation and publishes the resulting signature.¶
Inputs:
- group_commitment, the group commitment returned by compute_group_commitment,
an Element in G.
- sig_shares, a set of signature shares z_i, Scalar values, for each 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(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.
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) 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 participant.¶
This ciphersuite uses edwards25519 for the Group and SHA-512 for the Hash function H
meant to produce signatures indistinguishable from Ed25519 as specified in [RFC8032].
The value of the contextString parameter is "FROST-ED25519-SHA512-v10".¶
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¶
Normally H2 would also include a domain separator, but for backwards compatibility with [RFC8032], it is omitted.¶
Signature verification is as specified in Section 5.1.7 of [RFC8032] with the constraint that implementations MUST check the group equation [8][S]B = [8]R + [8][k]A'. The alternative check [S]B = R + [k]A' is not safe or interoperable in practice. Note that optimizations for this check exist; see [Pornin22].¶
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-v10".¶
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¶
Signature verification is as specified in Appendix B.¶
This ciphersuite uses edwards448 for the Group and SHAKE256 for the Hash function H
meant to produce signatures indistinguishable from Ed448 as specified in [RFC8032].
The value of the contextString parameter is "FROST-ED448-SHAKE256-v10".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHAKE256¶
Normally H2 would also include a domain separator, but for backwards compatibility with [RFC8032], it is omitted.¶
Signature verification is as specified in Section 5.2.7 of [RFC8032] with the constraint that implementations MUST check the group equation [4][S]B = [4]R + [4][k]A'. The alternative check [S]B = R + [k]A' is not safe or interoperable in practice. Note that optimizations for this check exist; see [Pornin22].¶
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-v10".¶
Group: P-256 (secp256r1) [x9.62]¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHA-256¶
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-v10".¶
G.Order() - 1]. Refer to Appendix D for implementation guidance.¶
G.Order() - 1].¶
Hash (H): SHA-256¶
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.¶
A security analysis of FROST exists in [FROST20] and [Schnorr21]. The protocol as specified in this document assumes the following threat model.¶
(MIN_PARTICIPANTS-1) corrupted participants. So long as an adversary
corrupts fewer than MIN_PARTICIPANTS participants, the scheme remains secure against Existential
Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in [BonehShoup],
Definition 13.2.¶
The protocol as specified in this document does not target the following goals:¶
The rest of this section documents issues particular to implementations or deployments.¶
Section 4.1 describes the procedure that participants use to produce nonces during
the first round of singing. The randomness produced in this procedure MUST be sampled
uniformly at random. The resulting nonces produced via nonce_generate are indistinguishable
from values sampled uniformly at random. This requirement is necessary to avoid
replay attacks initiated by other 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 in 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. One possible example is to construct this pre-hash over message m as
H(contextString \|\| "pre-hash" \|\| m).¶
Some applications may require that participants only process messages of a certain structure. For example, 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, use of threshold signatures in [TLS] to produce signatures of transcript hashes might require the participants receive the source handshake messages themselves, and recompute the transcript hash which is used as input message to the signature generation process, so that they can verify that they are signing a proper TLS transcript hash and not some other data.¶
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: 1 if signature is valid, and 0 otherwise
def prime_order_verify(msg, sig = (R, z), PK):
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
l = G.ScalarBaseMult(z)
r = R + G.ScalarMult(PK, c)
return l == r
¶
One possible key generation mechanism is to depend on a trusted dealer, wherein the
dealer generates a group secret s uniformly at random and uses Shamir and Verifiable
Secret Sharing as described in Appendix C.1 and Appendix C.2 to create secret
shares of s, denoted s_i for i = 0, ..., 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_SIGNER 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 and the key share (a Scalar).
- group_public_key, public key corresponding to the group signing key, an
Element in G.
- 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, MIN_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.
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_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.¶
secret_share_shard(s, coefficients, MAX_PARTICIPANTS, MIN_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
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme, an integer greater than 0
Outputs:
- secret_key_shares, A list of MAX_PARTICIPANTS number of secret shares, each a tuple
consisting of the participant identifier and the key share (a Scalar)
- coefficients, a vector of MIN_PARTICIPANTS coefficients which uniquely determine a polynomial f.
Errors:
- "invalid parameters", if MIN_PARTICIPANTS > MAX_PARTICIPANTS or if MIN_PARTICIPANTS is less than 2
def secret_share_shard(s, coefficients, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
if MIN_PARTICIPANTS > MAX_PARTICIPANTS:
raise "invalid parameters"
if MIN_PARTICIPANTS < 2:
raise "invalid parameters"
# 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_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_PARTICIPANTS to recover the
secret s is as follows; the algorithm polynomial_interpolation is defined in {{dep-polynomial-interpolate}}.¶
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: The resulting secret s, a Scalar, that was previously split into shares
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_interpolation(shares)
return s
¶
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 distinct 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):
x_coords = []
for point in points:
x_coords.append(point.x)
f_zero = Scalar(0)
for point in points:
delta = point.y * derive_lagrange_coefficient(point.x, x_coords)
f_zero = f_zero + delta
return f_zero
¶
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 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: 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, vss_commitment)
(i, sk_i) = share_i
S_i = ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
S_i' += G.ScalarMult(vss_commitment[j], pow(i, j))
if S_i == S_i':
return 1
return 0
¶
We now define how the Coordinator and 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 in G.
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 rejecting sampling in constant time can
leak information about the underlying corresponding Scalar.¶
Note the that the Scalar size might be some bits smaller than the array size,
which can result in the loop iterating more times than required. In that case
it's acceptable to set the high-order bits to 0 before calling DeserializeScalar,
but care must be taken to not set to zero more bits than required. For example,
in the FROST(Ed25519, SHA-512) ciphersuite, the order has 253 bits while
the array has 256; thus the top 3 bits of the last byte can be set to zero.¶
Generate a random byte array with l = ceil(((3 * ceil(log2(G.Order()))) / 2) / 8)
bytes, and interpret it as an integer; reduce the integer modulo G.Order() and return the
result. See Section 5 of [HASH-TO-CURVE] for the underlying derivation of l.¶
This section contains test vectors for all ciphersuites listed in Section 6.
All Element and Scalar values are represented in serialized form and encoded in
hexadecimal strings. Signatures are represented as the concatenation of their
constituent parts. The input message to be signed is also encoded as a hexadecimal
string.¶
Each test vector consists of the following information.¶
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: 6c69be7a46a0e3be20833f864c77788bb05d6c239 8ceadb84dbafa9522520049 P1 binding_nonce_randomness: f6e164329c66be6fa75954dde46967171be80dda fd1d1defe51f03f6e3e6876e P1 hiding_nonce: 4e64f59e90a3b9cdce346fae68eb0e459532c8ca1ad59a566c3e e2c67bf0100b P1 binding_nonce: 470c660895c6db164ee6564120eec71023fa5297f09c663bb81 71646c5632d00 P1 hiding_nonce_commitment: f0ff219693d61f164ca785e03d6209ef94d57047c fc5c14d2cab4eb70c5fb0f4 P1 binding_nonce_commitment: 71269d5f41d79e3d13059170875bf437c28369c6 7aa6ae7478faf1334550c8d0 P1 binding_factor_input: 736c1aac57730ec200b2e1518606c711d9ea10906582 8e5ff26e81eb0247093047543440063365d5442a86234a8c9808720b77e094ace1f0c 924183e30605b86dae87916da6a9544612f403ea4789735698142db05b44519ce717d a7cee249174689aa30845e44d8f93d5993259112e5f17e293ece62de4a844f79ef1f4 61db00100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: 02e82a18a7de829a93e63114d56841e060affd3595c8079fe7 dea00eec053806 P3 hiding_nonce_randomness: a2de7b16419a23bf116e6b89f75264802063944d1 131ec7d9e789e01ab002aa1 P3 binding_nonce_randomness: 177cec6de656b2c53f53198e74fefcb387753b45 470ab373dc93a1adfbaafabf P3 hiding_nonce: 6fc516495dbb364b807cdd0c2e5e3f58aa4914a53fed33cc3400 33979bb07304 P3 binding_nonce: 0837e770a88147d41ff39138ca23b35d6cf303a4f148294755e de4b7e760d701 P3 hiding_nonce_commitment: a98ebfeab6684035cc51983f72e682837c70dd8a8 f6dfd52f116680ba35e9d54 P3 binding_nonce_commitment: bb6f0926361c9ef40a171bfcee67c3a693fc25a6 0e8a8d40d35f31829fda7799 P3 binding_factor_input: 736c1aac57730ec200b2e1518606c711d9ea10906582 8e5ff26e81eb0247093047543440063365d5442a86234a8c9808720b77e094ace1f0c 924183e30605b86dae87916da6a9544612f403ea4789735698142db05b44519ce717d a7cee249174689aa30845e44d8f93d5993259112e5f17e293ece62de4a844f79ef1f4 61db00300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: f573b30ea1920f52dab3ec741430ce7b8838cd16ed0ff9b9d1 32e13e62c01601 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 3f2eb12735e5b39da97e884a6caadf6bb83f1efcec709d6f66333d0 d67ebe707 P3 sig_share: 79e572b8632fbb928519dd2eff793de8784a56d582ae48c807d39b0 dc5b93509 sig: e31e69a4e10d5ca2307c4a0d12cd86e3fceee550e55cb5b3f47c7ad6dbb38884 cb3f2e837eb15cd858fb6dd68c2a3e3f318a74d16f1fe6376e06d91a2ca51d01¶
// 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: 924ca0a20602aecf5b46a44191beae0f82662a934 5d8bba273e752d49cf09d38 P1 binding_nonce_randomness: 53cd669e849b4d2a357f122f5ba3b99242bde269 791ed1194a855e10befdaad3 P1 hiding_nonce: 06f2e15b05d29a50f0686a890259f4dcf66147a80809ed9e5092 6f5f173fe23a0627561efa003724dc270effc47a30bc4d80aba30725401d00 P1 binding_nonce: e0482e611c34f191d1c13a09bc8bbf4bda68db4de32aa790884 9b02ba912cfba46c805e2d8560ab9437e343e1dde6b481a2bae527e111b2c00 P1 hiding_nonce_commitment: 7034e3ad2c5641259d481fcc32b56b4552d3d0369 11dc456f948858bfca51c2c231d407222b9d6d997ff3c093309895c0ec510272b7cf0 bd00 P1 binding_nonce_commitment: 90d557d616fd4d495044fef2cd9da9ef7e962950 0888bc715d68702510f4757eb2eebe85493d2cc73c1d785b57984ee220da72fdedc35 7fc00 P1 binding_factor_input: 80c3dfd62df16cc677ad6772d130badcd6dbd9754fd1 939a3db6252646088570c91470beec2cf5e32dac40f5ab582cbd21f538f48a69ad053 ffac2ddbb2c73e2b503edb94c3e86c1b71d114dda9980d87e7ce96172719c6de68869 f9552edcf4a909bb2010822c236cff0a52afb5e85f897137f4d456aef79b8212fb9a6 c22e8ab92b5455a01423a2f158324749f1828384d9dc96169753c7f19b206f7fd0a55 fd2a85311170c5e4953de3d48894d4aa978e29df2332293ca97390068753aee1e4008 a99d699d3a56451caeaed1b54be56a47a84c277b228f000232e75b32610db0c366001 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P1 binding_factor: d4f54f478b17d57cb62beced74b1e1d04124ecd955445645e9 37bda9d501b06f2c029495267759b7ddb5a127fff61e913467dc045bd2721200 P3 hiding_nonce_randomness: 59eb2ee60798a0df135361845963f0e4f16529158 e89c2e5144c6fe9bf407087 P3 binding_nonce_randomness: 70814edfac5db2ab76c7e216859bbd87e5e2c092 e903e82fb8b4dc920378056c P3 hiding_nonce: 295c56447c070157e6bc3c83ed2afca194569e07d0ad27d28a40 dec2c4107c07d507db20da1be62ea6976b8e53ab5d26e225c663f2e7151100 P3 binding_nonce: b97303a6c5ab12b6ad310834361033a19d99dfdf93109da721d a35c3abbc5f29df33b3402692bef9f005bb8ea00af5ba20cc688360fd883100 P3 hiding_nonce_commitment: ee8acb4b1c466d539ffac6d7a66b934c2ba929fd0 4f45398d18e7086953a7e064a1bd61cfcf04d7cb3f2efc71fac2769a445ba187fe0e9 5280 P3 binding_nonce_commitment: e81d75c6d11d14cf22387dc1e3f1734dc6c8cb77 44a0473c6af46b133a5f94a7f9fba511c5f95b1ac7ab6542ff98603c5b2e0fc0554bb bd800 P3 binding_factor_input: 80c3dfd62df16cc677ad6772d130badcd6dbd9754fd1 939a3db6252646088570c91470beec2cf5e32dac40f5ab582cbd21f538f48a69ad053 ffac2ddbb2c73e2b503edb94c3e86c1b71d114dda9980d87e7ce96172719c6de68869 f9552edcf4a909bb2010822c236cff0a52afb5e85f897137f4d456aef79b8212fb9a6 c22e8ab92b5455a01423a2f158324749f1828384d9dc96169753c7f19b206f7fd0a55 fd2a85311170c5e4953de3d48894d4aa978e29df2332293ca97390068753aee1e4008 a99d699d3a56451caeaed1b54be56a47a84c277b228f000232e75b32610db0c366003 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P3 binding_factor: 4525e6d697a2b8e5a373d051061f3ecb3f1cf2fb462fe6ee95 75c3e71e15412244667b5406ed76c39c99f4b24d5e88d564082e926f5db12400 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 5b65641e27007ec71509c6af5cf8527eb01fee5b2b07d8beecf6646 eb7e7e27d85119b74f895b56ba7561834a1b0c42639b122160a0b620800 P3 sig_share: 821b7ac04d7c01d970b0b3ba4ae8f737a5bac934aed1600b1cad760 11c240629bce6a4671a1b6f572cec708ec161a72a5ca04e50eabdfc2500 sig: c7ad7ad9fcfeef9d1492361ba641400bd3a3c8335a83cdffbdd8867d2849bb44 19dcc3e594baa731081a1a00cd3dea9219a81ecba4646e9500dd80dede747c7fa086b 9796aa7e04ab655dab790d9d838ca08a4db6fd30be9a641f83fdc12b124c3d34289c2 62126c5195517166f4c85e2e00¶
// 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: 0a016efd0abf8e556fd67288950bb7fc0843be63e 306c7264bc9d24d1d65e0ee P1 binding_nonce_randomness: 35b6bab19e3e931e36c612ccc6b3c9d3a3479d27 04aac3324b79c7bb6665acfb P1 hiding_nonce: de3e8f526dcb51a1b9b48cc284aeca27c385aa3ba1a92a0c8440 d51e1a1d2f00 P1 binding_nonce: fa8dca5ec7a05d5a7b782be847ba3dde1509de1dbcf0569fc98 0cff795db5404 P1 hiding_nonce_commitment: 3677297a5df660bf63bb8fcae79b7f98cf4f2e99f 61bc762de9795cacd1cba62 P1 binding_nonce_commitment: 142aece8aa8b16766664d8aaa5a5e709404bb844 3309ef1ea9ad9254794a1f09 P1 binding_factor_input: c70ac0b3effa113b8f4d8a6b1393ef7f0910862d143f de83e410db94f3818295ff49ed5aed0e57b2712f2ce0f9166f1ffdce282786c7ee8c2 db2df295c61dc5fd0f93a769d09d44352c4e709c2e239fc34a1b89db44cb241060228 5ffd70f3fa0a62dd70cfdb369ac0a7efc587f6f671a88412b2570280da24bd36f8ffd a6d280100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: dbaa0ae3c5663816cdc646281be46b0b09eca6a1ecf7781f29 475be27d30fd08 P3 hiding_nonce_randomness: ac4e65529397de3a868a902e9040e38b26547c18b 7267fa1d1bbfe4ed14d6b5f P3 binding_nonce_randomness: 74213c820b7266c4990a0758f4c520685375cb98 822499406654bdb1a426582e P3 hiding_nonce: e07061a9ab6735de9a75b0c64f086c5b999894611d0cdc03f85c 4e87c8aae602 P3 binding_nonce: 38b17578e8e6ad4077071ce6b0bf9cb85ac35fee7868dcb6d9b fa97f0e153e0e P3 hiding_nonce_commitment: f8d758ad9373754c1d2bca9c38478e4eb857aa032 836ade6eb0726f5e1d08037 P3 binding_nonce_commitment: 529823e80220849c195072a26acca88f65639d41 81927bb7fcd96e43d9a34649 P3 binding_factor_input: c70ac0b3effa113b8f4d8a6b1393ef7f0910862d143f de83e410db94f3818295ff49ed5aed0e57b2712f2ce0f9166f1ffdce282786c7ee8c2 db2df295c61dc5fd0f93a769d09d44352c4e709c2e239fc34a1b89db44cb241060228 5ffd70f3fa0a62dd70cfdb369ac0a7efc587f6f671a88412b2570280da24bd36f8ffd a6d280300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: aa076fec41410f6c0667e47443fcd1ed828854d84b19d1d086 24d084720c7d05 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: a5f046916a6a111672111e47f9825586e1188da8a0f3b7c61f2b6b4 32c636e07 P3 sig_share: 4c175c7e43bd197980c2021774036eb288f54179f079fbf21b7d2f9 f52846401 sig: 94b11def3f919503c3544452ad2a59f198f64cc323bd758bb1c65b42032a7473 f107a30fae272b8ff2d3205e6d86c3386a0ecf21916db3b93ba89ae27ee7d208¶
// 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: 3029ae05a266703f618e60c26653f6b8f35a759ec 2adecf8b7d9e1719375494e P1 binding_nonce_randomness: 86755fd9be109ff0549833931080ac344b0d775a 029fca0329f8ce732060f81e P1 hiding_nonce: 9aa66350b0f72b27ce4668323b4280cd49709177ed8373977c22 a75546c9995d P1 binding_nonce: bd8b05d7fd0ff5a5ed65b1f105478f7718a981741fa8fa9b55a c6d3c8fc59a05 P1 hiding_nonce_commitment: 03071549b356988df0f7187585e2d82d6f916700c fdd49634d0c27965139fd53ec P1 binding_nonce_commitment: 02151f45451b719bf68f6c609967ebea3c78c9ec e4c04a564a0c50d22f0f534112 P1 binding_factor_input: 47d0b1c45754dd58dc369bc4c1a9b24ffbb67ceb6d6e 25c302e9875202f7d2b4755d9beaba0a02b01315bd42fa11590d5a4d531d1f7f81c5f c70a82ecada72e9000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: 0e9709d66649a0a245f28666bd01c863a6a647f213fd49eeaa cfeca15402ddf4 P3 hiding_nonce_randomness: 2741900f778d51f4431644a62a69f1623d7569ecf 2d628d60cb28e27db949161 P3 binding_nonce_randomness: a62404370cb2a2e0aebef27ec72c1433a627dfcc 5f0cdf5ba4799fc326a66a3f P3 hiding_nonce: 4c1aec8e84c496b80af98415fada2e6a4b1f902d4bc6c9682699 b8aeffd97419 P3 binding_nonce: eeaf5ef7af01e55050fb8acafc9c9306ef1cc13214677ba33e7 bc51e8677e892 P3 hiding_nonce_commitment: 0351cd636672cac59d384498dd9db2b72ea8e701a 702867c17e3ecf675d9a9fc91 P3 binding_nonce_commitment: 032bddd1ab4bfda79c707742f0e314ff2be95940 58ba590613ba9840886bab1a59 P3 binding_factor_input: 47d0b1c45754dd58dc369bc4c1a9b24ffbb67ceb6d6e 25c302e9875202f7d2b4755d9beaba0a02b01315bd42fa11590d5a4d531d1f7f81c5f c70a82ecada72e9000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: 0b5c759331915b25c5eb5307617e01aa99bc5c89a403d9c6b5 9949045a4c0a77 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: ec5b8ab47d55903698492a07bb322ab6e7d3cf32581dcedf43c4fa1 8b46f3e10 P3 sig_share: c97da3580560e88725a8e393d46fee18ecd2e00148e5e303d4a510f ae9c11da5 sig: 036b3eba585ff5d40df29893fb6f60572803aef97800cfaaaa5cf0f0f19d8237 f7b5d92e0d82b678bcbdf20d9b8fa218d017bfb485f9ec135e24b04050a1cd3664¶
// 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: 753a4f3d424de828c2604f54a807677e4383a184b f373da9855af385e9014693 P1 binding_nonce_randomness: 4f4b423493e7e676d30630c0b165cf6ea428bd43 1af382377981a7f3971ba3f0 P1 hiding_nonce: 36d5c4185c40b02b5e4673e2531a10e6ff9883840a68ec08dbeb 896467e21355 P1 binding_nonce: 7b3f573ca0a28f9f94522be4748df0ed04de8a83085aff4be7b 01aa53fb6ac1b P1 hiding_nonce_commitment: 029ed3b6b8c3f7c1bca427160579ea1cdf47671c7 76ed1933fb617c7d8ebd018cd P1 binding_nonce_commitment: 0217c99463f8ced8a2f2927a85d9a5fcc893bcf9 a6dc0507418e6f511feea0964d P1 binding_factor_input: 8263cf41514645b3d57a9738502a0522c8d7b410ec42 b6ba8cca0830faacfa01cabeb24393904d4051842b28fd09f3f53dd6812c1331a225d c9c9c6f872fe734000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: f995eeab0b6c02673f1e5e338652eac32729d9e827518f8350 934261f7f8a118 P3 hiding_nonce_randomness: 94b70f287aa5c9961a012e00c46f384cf7cc4b385 f7d6f093e53c60117aae9a2 P3 binding_nonce_randomness: a6b5ebff4fe4539535c33bcc267bb5f25c40e300 8bf118ba8a97a12e131d0e8a P3 hiding_nonce: ba4f8b8e587b2c9fc61a6156885f0bc67654b5e068c9e7749f75 c09a98f17c13 P3 binding_nonce: 316de06639051ac7869e5ac4458eda1fef90ce93fa3c490556c 4192e4fa550d0 P3 hiding_nonce_commitment: 037d38246f1ac0d713a79956723e7766861ce624e 7e3bfc98c786d5e20c6ae232a P3 binding_nonce_commitment: 03342d96ef52ad1f95423f32ef17805595393268 f606c3a6d179e4f99910a41dfe P3 binding_factor_input: 8263cf41514645b3d57a9738502a0522c8d7b410ec42 b6ba8cca0830faacfa01cabeb24393904d4051842b28fd09f3f53dd6812c1331a225d c9c9c6f872fe734000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: 73dce3f4ade1d8a357bb4b219840da9ef6f99551cafcfe174e 5d93f1994f5891 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: f9ee00d5ac0c746b751dde99f71d86f8f0300a81bd0336ca6649ef5 97239e13f P3 sig_share: 61048ca334ac6a6cb59d6b3ea2b25b7098e204adc09e2f88b024531 b081d1d6f sig: 023cf76388f92d403aa937af2e3cb3e7a2350e40400c16a282e330af2c60eeb8 5a5af28d78e0b8ded82abb49d899cfe26ace633248ce58c617569be3e7aa20bd6d¶