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 2^{64} 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 participants, 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)
.¶
Each participant, including the Coordinator, 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 signer, 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.¶
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-v8".¶
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-v8".¶
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-v8".¶
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-v8".¶
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-v8".¶
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, 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 this 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: 5c8f4d74076647069566b460154525c0d80bf278a 49ef0da7e5b56ab1cceac08 P1 binding_nonce_randomness: 90cda7316703b5f5e0c890ac456a1f64c0db1e9b 025493b50f795a38ec373b69 P1 hiding_nonce: 8fa57cf7b4f6476d1f0ee1ee972ea9acc79633fc6a46a1ab7223 4e8f4cc65b00 P1 binding_nonce: 305458d8109e95c71d5e61fb985a8e00126276dcd02ba9814fa 2a5674f36f70d P1 hiding_nonce_commitment: 9edff055e77f1b7addf4bf48a58beea392a0032ac f2744196c61f3b3c3b1cb47 P1 binding_nonce_commitment: 4a03fb80acea2712f2ff1548e990457cdd58fb17 1a69f70fcdf53d1b0ea83008 P1 binding_factor_input: 25120720c3a416e292edfb0510780bc84eb734347a5f d84dd46d0dcbf3a21d21a23a776628859678a968acc8c8564c4641a1fd4b29a12d536 7ca12ab10b6b497c2385b35930a34d98238021b25ab0641e39f8cf4277f90e3c03161 b1755e7b08b7c5e5334a57e89c5e841186e9cca95214851ed99dcbd07f104fc399194 fd8dc0100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: febb7ae5b20ffe3c1d2bcf08f7e6104a78b3136ad3b7df35d5 f84878ce7bb50d P3 hiding_nonce_randomness: a85d5ce57b709e59b07b1fefc80a50a928b409881 7e482fa38f831ab453da14d P3 binding_nonce_randomness: b74a04b3a7e7311c974204c1e142952d1e93cb1e e0791c90126d2c4eb375e0e1 P3 hiding_nonce: 59b0da2f402a69956b4c904737afefe29ef0aaa4401330f56dec ebe450204a08 P3 binding_nonce: 1a54642c6b975f1b39a1b391bebc4f72b6dc6ca551cbebedee6 ce5a60069fd0d P3 hiding_nonce_commitment: 5e11b65802153b03332533a3018d22fe760098508 a250832a457e6004848411b P3 binding_nonce_commitment: 9ef535b69175da3833d330c1a43823b4b08dc1fb 38fdc3af99f6804d8b250a09 P3 binding_factor_input: 25120720c3a416e292edfb0510780bc84eb734347a5f d84dd46d0dcbf3a21d21a23a776628859678a968acc8c8564c4641a1fd4b29a12d536 7ca12ab10b6b497c2385b35930a34d98238021b25ab0641e39f8cf4277f90e3c03161 b1755e7b08b7c5e5334a57e89c5e841186e9cca95214851ed99dcbd07f104fc399194 fd8dc0300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: 6662bfe16e045aba72954ef54b6dabfc61db322698b57d5a40 94b39d4ce01f05 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: d21fd9d6f793c8efef622ae139ec94071dc1b1f47624f638f4f8615 e1934bc09 P3 sig_share: 7b599a881fc39829b2f69b194fd64bd6ee0df8b34e3d067be5612fa c4da8ac0b sig: 7c64be7528effeab65503e813ee28f94c53a4d8e45a0a71e936c840c78e42233 60a57d02fdf34ec1cbbcce57aac801c90bcfa9a8c561fcb3d95a910a67dc6805¶
// 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: b5eb1b4e369491e7d09427b01285803215bbc4c44 0edebf8a337bfa3bc75a931 P1 binding_nonce_randomness: c73b102813418576607ed61e726233f5df303f96 38af479b05fbc8dda6739eeb P1 hiding_nonce: aa4606f9b2e5cf331e473833087769579685c4582a80ae45b01e 3ffe83c0455b6f0eb05b320db0de0223509e8f8d30f356173b1ff2383a0100 P1 binding_nonce: 90c118161d17804126d474af8cc1e2358319b704b8adb4088a8 a36a3f48e8ec1c6b5106d394a36bd9e62d7dd7ee2a9978c866931d8da792300 P1 hiding_nonce_commitment: 55d2fe1b9b91d8bd8511417ff9b315c752944acdf a483f134d2004c441bf6da055219dd0632ccaf8f35970fac9bea116f7269855833453 9800 P1 binding_nonce_commitment: 8ac0e4cf4d9fb584d5372d963341d6144d5d0310 b6029c34c7c7a2bba4b259a3bb803da16e024764de4612aafc8d63d48f4e2a97df3f4 0e800 P1 binding_factor_input: 766a004ac6e87a2fa70f2095b19596ac33b94e2f6803 e1a5b8fa8ea5adaf3e7989b2c167a38a42a1693ad69cfd674e089a498672753563d53 54654ba106d5fdffb134a8917fae91d412164436f734b95572af6208605744400c6ff 9a60fa2ce8fb7f3213414c32e347ee2e29e3d17654ef0277d99dbfcd70bf9162c46ca 7f45754d1eabb8ab239f4fb88cc8af91e5f1af5d81f1f2f8dfb564c6ddb0a75bdf681 b2f403d7d903000c647602978b2e714360a3847ca469b932d4dcdff6ea421c27a382c c230625e7c9b45e662056ef73687412f705f77fc4bc20e3d902888c02c2fd2ff07801 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P1 binding_factor: 4d58a6a888309da8b8d2d2068a44ad3c597591b1b806cc8ac1 2d2c1202c66ae9c4e68b6bfd2ef3b3a5968a288fa5a2aed86161ebeb2aa60e00 P3 hiding_nonce_randomness: 516a0e41f6b2f1f9a6661b5549db7486c6073249b 6dcc5bbdca69e7fd6bcbd12 P3 binding_nonce_randomness: 28f89d1c037cf61ccd08f7e06fa2289bb5a31a60 5455bad6bec61923d60fe811 P3 hiding_nonce: 4fd3566260ecc3f57f1cfb7c84724d2fc80f90dab5ee3613eb9b 278e71fd9d88c8dae06ec1bbf2fd535ec3fca393dd011d63ee5c3db7383000 P3 binding_nonce: c0a583112b13e238f249bac41a01ccb1ad679c4feaab14fc7ab 0df3511066cd117fadfc1ec3a0fd31820e8857943675262ebc5d16194cd2300 P3 hiding_nonce_commitment: 148325fe0941216938cd6bcbe45a68c3184b1f087 1f2b17f3bbdc4e2aef9c03d27845bd6f45c05a810b0dd2582af3e11518015405a4ffb d480 P3 binding_nonce_commitment: 2a34a68cfb05f20f0e2437daf4480cbafdab3243 9a6798c1f7b36e0e47714c570242ef8972914ec65b8ed49784dc62fbdc8f7838faa75 63700 P3 binding_factor_input: 766a004ac6e87a2fa70f2095b19596ac33b94e2f6803 e1a5b8fa8ea5adaf3e7989b2c167a38a42a1693ad69cfd674e089a498672753563d53 54654ba106d5fdffb134a8917fae91d412164436f734b95572af6208605744400c6ff 9a60fa2ce8fb7f3213414c32e347ee2e29e3d17654ef0277d99dbfcd70bf9162c46ca 7f45754d1eabb8ab239f4fb88cc8af91e5f1af5d81f1f2f8dfb564c6ddb0a75bdf681 b2f403d7d903000c647602978b2e714360a3847ca469b932d4dcdff6ea421c27a382c c230625e7c9b45e662056ef73687412f705f77fc4bc20e3d902888c02c2fd2ff07803 000000000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000 P3 binding_factor: 7bfb1317cd530ba478544ee911687d0d0fa5e4c49e40a79347 2601f9c587440478eebb90bbec84c98055783a32e442a3b40503b2a9389d3d00 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: d982cdbb5d14fb53c8cb28cfb690ec38fc4f8f44c666de910703ee7 ba2183989286c614c9bb5319a3a70bf22ee68ca54b3a6b90f33a33a1a00 P3 sig_share: adc2d8301b5f5597af72c51cce53ab6d265e8e788ce4738b1e2fbdc e24eaa283b0bfbd7b587a407844c20e958c21a5ba9895b51fdfbb3e3700 sig: 22943fef7107155064b0baa1d6ad78ee6c5d909ca88ac8bd1002bde99e124a34 514f826d24d83fbdecf7752be1fd166fd31752fd4eb3a3280093004e41e6b0d7c722a f285e12222b859277470e097003593c0ee1cdc702dc0cd92b1fc8f32f72127f32ceb7 7a8a6f0f4c3c6f2f125f791100¶
// 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: b8f4ed5c201cb19151d091945eb0011db6bf4b816 debfc21fc6833a99dca87b8 P1 binding_nonce_randomness: 2588621bdf623b8a687f4def4f4f720b8da24f6b 15c64a2a11dc74ffa04c9c02 P1 hiding_nonce: 05a9f136274e02107502f5073e91b72bd42165f277d524b2eb3f 9c6fa7a38a08 P1 binding_nonce: 8fe91bee76721535b6ac5b84ea5474892cb476404217a9b6129 d1974cd211703 P1 hiding_nonce_commitment: 4ce0c9d3de33b7876122a3490cef6a5b8cc22acc2 9e96195629b5b8cfb1b784f P1 binding_nonce_commitment: 26d80e82877fd0fd78a803996bf4847ee9d1ffd8 0072c9f16d34c5dde4138115 P1 binding_factor_input: fe9082dcc1ae1ae11380ac4cf0b6e2770af565ff5af9 016254dc7c9d4869cbae0f6e4d94b23e5781b91bc74a25e0c773446b2640290d07c83 f0b067ff870a801f846633bc5505172ee8dc7bb109916b61ea54ddc715684bb50f612 8cec164ab16da6bcebddb83b78493ef85fca52ca460199609523f3eb877d46c399d28 a00900100000000000000000000000000000000000000000000000000000000000000 P1 binding_factor: 38915e5ab3b90122785515aa89d20c8e592de9f04cb0f42c6a e7cf99d3990d0b P3 hiding_nonce_randomness: 7f417eadf8651b6043eece2965411bfb237b0a468 9c88a8469a2a66b79a6d7f9 P3 binding_nonce_randomness: 53dd06d89cdeb574fc32e83fa138f73f668a99aa ccfa03f82e68ff6b845b7a8a P3 hiding_nonce: 2efc0a895c5788cb3b65f1032f8d64cf59ed71637353a7103fce ea669a169f04 P3 binding_nonce: 6e60f3eac7359a4c19a22b618aa4531bd49a1ec83d2c3f503d7 5474ddb15b406 P3 hiding_nonce_commitment: 064f733cb26f728ef185d40e75c50b717cd543abe 8057e38a96831a7de47b17f P3 binding_nonce_commitment: 6a47acc247a4cbf5b753a1d1f2dcb43241f2df71 3468bf9a7a00e6257442bc46 P3 binding_factor_input: fe9082dcc1ae1ae11380ac4cf0b6e2770af565ff5af9 016254dc7c9d4869cbae0f6e4d94b23e5781b91bc74a25e0c773446b2640290d07c83 f0b067ff870a801f846633bc5505172ee8dc7bb109916b61ea54ddc715684bb50f612 8cec164ab16da6bcebddb83b78493ef85fca52ca460199609523f3eb877d46c399d28 a00900300000000000000000000000000000000000000000000000000000000000000 P3 binding_factor: 3f83b7811980bb032a3f85d857af755a00f4b16a10cdf88f32 d13ea908d3b107 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 86879530bc8d700ec20e6211d0c8d27d1b518f5aed00b9eef1b04ed 625c1ab05 P3 sig_share: 66f01933240c000ab4abb203f49bf8bb7138ffe17d5cd6c943ea30c f9f32410c sig: 8adca3c885d6ffa07429de8df9ba9bc72a3b0dfa0f6139035deadc72e13c645e ffa3b906c6365ec09f1d1d72e56aec248d898e3c6b5d8fb8359b7fa5c5f3ec01¶
// 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: 3c3cce19a46e2539a2a3f0fc1302c04d643c80b61 beff7545f09bf3379ce763f P1 binding_nonce_randomness: 7550f8224653dec3179586846a156e08911cad54 e785e7a8db13539ad1fd7ac6 P1 hiding_nonce: f07d7e697f7db6b0eaa29e8303e718440197ff47e1918cd4e630 c202ab323223 P1 binding_nonce: 248eb6983093337c1dae193643e8e4618cdafd189fb8289239b 1b54a1778e0cf P1 hiding_nonce_commitment: 02cc0ef1b65d7284c0be9c555e9877fac2dad008d f7df6133928f900ca34d82bbe P1 binding_nonce_commitment: 0281413660fa8a78ae1ca4a65fdad3259246e19e 74f81687b9812246c8b174b1f2 P1 binding_factor_input: 3617acb73b44df565fbcbbbd1824142c473ad1d6c800 7c4b72a298d1eaae57664a317d123a18b12922413ebd80944a0816821b2694b4aea22 16ebbe38eb2daf3000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: 0f2c1ea6da0fceb9cbb63bfe81d80bd22c434a5d9810b1cc97 7f7eab7f0e9da4 P3 hiding_nonce_randomness: 89c096201f0aeb36b0b8a33763ac92df1169ead49 fcf2dec8209f858852a345f P3 binding_nonce_randomness: 8785c0b2a5817cf40c978e28f7804eff570c3c1c f9df11b37496c318fa38d107 P3 hiding_nonce: dc34074d1f8f427ee8eabc546cb9931cc44371be94ad66f7a27f dc4a665db373 P3 binding_nonce: 0a3e836f36b4011be4f6ef111596838fdfccf89c431d66da2ca 75ea55dcf3b44 P3 hiding_nonce_commitment: 02261bf8649d9943d5f664694219742e9abc501c6 7be6b112e833b144462a662e3 P3 binding_nonce_commitment: 03603f38cefae47af6be8dbeeef53e7b27b5af8d 4159b968d489a1f7295581b0e5 P3 binding_factor_input: 3617acb73b44df565fbcbbbd1824142c473ad1d6c800 7c4b72a298d1eaae57664a317d123a18b12922413ebd80944a0816821b2694b4aea22 16ebbe38eb2daf3000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: 7598b94e95da5f81f9b3ad0c40499526429ec80de9cc0a8240 0bbbd2725ea947 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 35e4cfe05008a064ad66f2ab62005632b5c44d4daea5cf1ffd193c5 56ec4ddad P3 sig_share: 79aa1cc540bacfc5a96d4eb1940aa85ec56ec880885748c3de11ff3 55d4e77cf sig: 03fe31f09eb37b31956749b77eb3d5cf88ac6c2192481e2eaa6e729d4ed8ba0f 28af8eeca590c3702a56d4415cf60afe917b3315ce36fd17e3db2b3b8acc13557c¶
// 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: cfeb82944dbbddde7863519b623acdaf0962217ae fa5a4ad08b07f176600c070 P1 binding_nonce_randomness: 11b45cbcc8eec8877741fc2622ff0c7fbf68a6bd 9b3f8ee313150d6e0bcb5d83 P1 hiding_nonce: b97f0b3fa40a05cbbda2d0acce6170f046520bbf884748e1949b 007c6f11b410 P1 binding_nonce: 6f0c660fb917a39a8b61893ebf8d5e6000da5e766efe717bf40 20c444327a4a0 P1 hiding_nonce_commitment: 03e030abe3c24a6258fe5b15abcdfbb539f146998 d86396e2d3f3cf6d5198f60de P1 binding_nonce_commitment: 0264617bf0828fa5eadc831b2fa4b4038d7e39c7 01708282d707d5f58e6e41d28e P1 binding_factor_input: d759fa818c284537bbb2efa2d7247eac9232b7b992cd 49237106acab251dd954b5398f26122974fbb2a00ebfa06caafe44bc289dc964b83dd 4f46411ebd2bce9000000000000000000000000000000000000000000000000000000 0000000001 P1 binding_factor: d80f1df14cfd6803b8501866ca1667297f7867c21886bc04a4 0b56964efa6af6 P3 hiding_nonce_randomness: c8e243917b9d06df6a92c6beca63d6be08325e7b9 29114ec4670f54c07529b47 P3 binding_nonce_randomness: 920d7850ac0d04049eae8e107ee0d86d416d9887 23c505fde8e68ef3e6c36153 P3 hiding_nonce: 9e73856e36f0ac77165af15ced03c71c4b0b3df9e63e1fc550d1 38effb4cf827 P3 binding_nonce: d1620add724f71f28248822ce5fad517428c6b7247dee8d6a46 56b795d058173 P3 hiding_nonce_commitment: 02ba01e4f0a2ead5da318536893cd20f658a51e78 0dfc96b7604fe8ab73b34a00f P3 binding_nonce_commitment: 0317ec2be255f4647a98b3573582c48ac5c5f087 fdaa057a92ed363f1a1a642056 P3 binding_factor_input: d759fa818c284537bbb2efa2d7247eac9232b7b992cd 49237106acab251dd954b5398f26122974fbb2a00ebfa06caafe44bc289dc964b83dd 4f46411ebd2bce9000000000000000000000000000000000000000000000000000000 0000000003 P3 binding_factor: edd9bb6d85b3610a73e9eaeb2941924b25497d97cfdec17e47 9ef5e9b2a7dcd7 // Round two parameters participant_list: 1,3 // Signer round two outputs P1 sig_share: 8a89a17b72e7235d6fbbd0c25db85a1ede389f84d290258c1a42397 dff111626 P3 sig_share: 100275a38484954d345e238f92d4161bf8064a2c4767e916f406ada b0a734d26 sig: 02652105e8b9d3b30e6868c8cb4cb86d858201fc0f7cef211b6c8f0426d8ac63 f69a8c171ef76bb8aaa419f451f08c703ad63ee9b119f80ea30e48e7290984634c¶