| Internet-Draft | FROST | March 2022 |
| Connolly, et al. | Expires 30 September 2022 | [Page] |
In this draft, we present the two-round signing variant of FROST, a Flexible Round-Optimized Schnorr Threshold signature scheme. FROST signatures can be issued after a threshold number of entities cooperate to issue a signature, allowing for improved distribution of trust and redundancy with respect to a secret key. Further, this draft specifies signatures that are compatible with [RFC8032]. However, unlike [RFC8032], the protocol for producing signatures in this draft is not deterministic, so as to ensure protection against a key-recovery attack that is possible when even only one participant is malicious.¶
This note is to be removed before publishing as an RFC.¶
Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.¶
Source for this draft and an issue tracker can be found at https://github.com/cfrg/draft-irtf-cfrg-frost.¶
This Internet-Draft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
Internet-Drafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as Internet-Drafts. The list of current Internet-Drafts is at https://datatracker.ietf.org/drafts/current/.¶
Internet-Drafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use Internet-Drafts as reference material or to cite them other than as "work in progress."¶
This Internet-Draft will expire on 30 September 2022.¶
Copyright (c) 2022 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/license-info) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
DISCLAIMER: This is a work-in-progress draft of FROST.¶
RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this draft is maintained in GitHub. Suggested changes should be submitted as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost. Instructions are on that page as well.¶
Unlike signatures in a single-party setting, threshold signatures require cooperation among a threshold number of signers each holding a share of a common private key. The security of threshold schemes in general assume that an adversary can corrupt strictly fewer than a threshold number of participants.¶
This document presents a variant of a Flexible Round-Optimized Schnorr Threshold (FROST) signature scheme originally defined in [FROST20]. FROST reduces network overhead during threshold signing operations while employing a novel technique to protect against forgery attacks applicable to prior Schnorr-based threshold signature constructions. The variant of FROST presented in this document requires two rounds to compute a signature, and implements signing efficiency improvements described by [Schnorr21]. 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.¶
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 B.¶
draft-04¶
draft-03¶
draft-02¶
draft-01¶
draft-00¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
The following notation and terminology are used throughout this document.¶
NUM_SIGNERS denotes the number of participants, and the number of shares that s is split into.
This value MUST NOT exceed 2^16-1.¶
THRESHOLD_LIMIT denotes the threshold number of participants required to issue a signature. More specifically,
at least THRESHOLD_LIMIT shares must be combined to issue a valid signature.¶
len(x) is the length of integer input x as an 8-byte, big-endian integer.¶
encode_uint16(x): Convert two byte unsigned integer (uint16) x to a 2-byte,
big-endian byte string. For example, encode_uint16(310) = [0x01, 0x36].¶
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 G of prime order p. The fundamental group operation
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 GG, there exists an element
-A such that A + (-A) = (-A) + A = I. Scalar multiplication is equivalent to the repeated
application of the group operation on an element A with itself r-1 times, this is denoted
as r*A = A + ... + A. For any element A, p * A = I. We denote B as the 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 refer to as the scalar field. This document uses types
Element and Scalar to denote elements of the group G and its set of scalars, respectively.
We denote equality comparison as == and assignment of values by =.¶
We now detail a number of member functions that can be invoked on a prime-order group G.¶
G (i.e. p).¶
I).¶
G that chooses at random a
Scalar element in GF(p).¶
G that chooses at random a
non-zero Scalar element in GF(p).¶
G that maps an Element A
to a unique byte array buf of fixed length Ne.¶
G that attempts to map a
byte array buf to an Element A, and fails if the input is not a
valid byte representation of an element of the group. This function can
raise a DeserializeError if deserialization fails or A is the identity
element of the group; see Section 6 for group-specific input validation
steps.¶
G that maps a Scalar s
to a unique byte array buf of fixed length Ns.¶
G that attempts to map a
byte array 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 three separate domain-separated hashes, H1, H2, and H3, where H1 and H2 map arbitrary inputs to non-zero Scalar elements of the prime-order group scalar field, and H3 is an alias for H with domain separation applied. The details of H1, H2, and H3 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:¶
This sections describes these operations in more detail.¶
In the single-party setting, a Schnorr signature is generated with the following operation.¶
schnorr_signature_generate(msg, SK):
Inputs:
- msg, message to be signed, an octet string
- SK, private key, a scalar
Outputs: signature (R, z), a pair of scalar values
def schnorr_signature_generate(msg, SK):
PK = G.ScalarBaseMult(SK)
k = G.RandomScalar()
R = G.ScalarBaseMult(k)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
z = k + (c * SK)
return (R, z)
¶
The corresponding verification operation is as follows. Here, h is
the cofactor for the group being operated over, e.g. h=8 for the
case of Curve25519, h=4 for Ed448, and h=1 for groups such as
ristretto255 and secp256k1, etc. This final scalar multiplication
MUST be performed when h>1.¶
schnorr_signature_verify(msg, sig, PK):
Inputs:
- msg, signed message, an octet string
- sig, a tuple (R, z) output from schnorr_signature_generate or FROST
- PK, public key, a group element
Outputs: 1 if signature is valid, and 0 otherwise
def schnorr_signature_verify(msg, sig = (R, z), PK):
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
l = G.ScalarBaseMult(z)
r = R + (c * PK)
check = (l - r) * h
return check == G.Identity()
¶
This section describes operations on and associated with polynomials
that are used in the main signing protocol. A polynomial of degree t
is represented as a sorted list of t coefficients. 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 (counter, coeff) in coeffs.reverse():
if counter == coeffs.len() - 1:
value += coeff // add the constant term
else:
value += coeff
value *= x
return value
¶
Lagrange coefficients are used in FROST to evaluate a polynomial
f at f(0), given a set of t other points, where f is
represented as a set of coefficients.¶
derive_lagrange_coefficient(x_i, L):
Inputs:
- x_i, an x-coordinate contained in L, a scalar
- L, the set of x-coordinates, each a scalar
Outputs: L_i, the i-th Lagrange coefficient
Errors:
- "invalid parameters", if any coordinate is less than or equal to 0
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"
numerator = 1
denominator = 1
for x_j in L:
if x_j == x_i: continue
numerator *= x_j
denominator *= x_j - x_i
L_i = numerator / denominator
return L_i
¶
Secret sharing requires "splitting" a secret, which is represented as
a constant term of some polynomial f of degree t. Recovering the
constant term occurs with a set of t points using polynomial
interpolation, defined as follows.¶
Inputs:
- points, a set of `t` points on a polynomial f, each a tuple of two
scalar values representing the x and y coordinates
Outputs: The constant term of f, i.e., f(0)
def polynomial_interpolation(points):
L = []
for point in points:
L.append(point.x)
f_zero = F(0)
for point in points:
delta = point.y * derive_lagrange_coefficient(point.x, L)
f_zero = f_zero + delta
return f_zero
¶
This section describes the subroutine used for encoding a list of signer commitments into a bytestring that is used in the FROST protocol.¶
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...], a list of commitments issued by each signer,
where each element in the list indicates the signer index i and their
two commitment Element values (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted in ascending order
by signer index.
Outputs: A byte string containing the serialized representation of commitment_list.
def encode_group_commitment_list(commitment_list):
encoded_group_commitment = nil
for (index, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
encoded_commitment = encode_uint16(index) ||
G.SerializeElement(hiding_nonce_commitment) ||
G.SerializeElement(binding_nonce_commitment)
encoded_group_commitment = encoded_group_commitment || encoded_commitment
return encoded_group_commitment
¶
This section describes the subroutine for computing the binding factor based on the signer commitment list and message to be signed.¶
Inputs:
- encoded_commitment_list, an encoded commitment list (as computed
by encode_group_commitment_list)
- msg, the message to be signed (sent by the Coordinator).
Outputs: binding_factor, a Scalar representing the binding factor
def compute_binding_factor(encoded_commitment_list, msg):
msg_hash = H3(msg)
rho_input = encoded_commitment_list || msg_hash
binding_factor = H1(rho_input)
return binding_factor
¶
This section describes the subroutine for creating the group commitment from a commitment list.¶
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...], a list of
commitments issued by each signer, where each element in the list indicates the signer index i and their
two commitment Element values (hiding_nonce_commitment_i, binding_nonce_commitment_i).
This list MUST be sorted in ascending order by signer index.
- binding_factor, a Scalar
Outputs: An Element representing the group commitment
def compute_group_commitment(commitment_list, binding_factor):
group_commitment = G.Identity()
for (_, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
group_commitment = group_commitment + (hiding_nonce_commitment + (binding_nonce_commitment * binding_factor))
return group_commitment
¶
This section describes the subroutine for creating the per-message challenge.¶
Inputs:
- group_commitment, an Element representing the group commitment
- group_public_key, public key corresponding to the signer secret key share.
- msg, the message to be signed (sent by the Coordinator).
Outputs: a challenge Scalar value
def compute_challenge(group_commitment, group_public_key, msg):
group_comm_enc = G.SerializeElement(group_commitment)
group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg
challenge = H2(challenge_input)
return challenge
¶
We now present the two-round variant of the FROST threshold signature protocol for producing Schnorr signatures. It involves signer participants and a coordinator. Signing participants are entities with signing key shares that participate in the threshold signing protocol. The coordinator is a distinguished signer with the following responsibilities:¶
FROST assumes the selection of all participants, including the dealer, signer, and Coordinator are all chosen external to the protocol. Note that it is possible to deploy the protocol without a distinguished Coordinator; see Section 7.3 for more information.¶
Because key generation is not specified, all signers are assumed to have the (public) group state that we refer to as "group info" below, and their corresponding signing key shares.¶
In particular, it is assumed that the coordinator and each signing participant P_i knows the following
group info:¶
PK = G.ScalarMultBase(s), corresponding to the group secret key s.
PK is an output from the group's key generation protocol, such as trusted_dealer_keygenor a DKG.¶
PK_i = G.ScalarMultBase(sk_i), which are similarly
outputs from the group's key generation protocol.¶
And that each participant with identifier i additionally knows the following:¶
is signing key share sk_i, which is the i-th secret share of s.¶
The exact key generation mechanism is out of scope for this specification. In general, key generation is a protocol that outputs (1) a shared, group public key PK owned by each Signer, and (2) individual shares of the signing key owned by each Signer. In general, two possible key generation mechanisms are possible, one that requires a single, trusted dealer, and the other which requires performing a distributed key generation protocol. We highlight key generation mechanism by a trusted dealer in Appendix B, for reference.¶
This signing variant of FROST requires signers to perform two network rounds: 1) generating and publishing commitments, and 2) signature share generation and publication. The first round serves for each participant to issue a commitment to a nonce. The second round receives commitments for all signers as well as the message, and issues a signature share with respect to that message. The Coordinator performs the coordination of each of these rounds. At the end of the second round, the Coordinator then performs an aggregation step and outputs the final signature. This complete interaction is shown in Figure 1.¶
(group info) (group info, (group info,
| signing key share) signing key share)
| | |
v v v
Coordinator Signer-1 ... Signer-n
------------------------------------------------------------
message
------------>
|
== Round 1 (Commitment) ==
| signer commitment | |
|<-----------------------+ |
| ... |
| signer commitment |
|<-----------------------------------------+
== Round 2 (Signature Share Generation) ==
|
| signer input | |
+------------------------> |
| signature share | |
|<-----------------------+ |
| ... |
| signer input |
+------------------------------------------>
| signature share |
<------------------------------------------+
|
== Aggregation ==
|
signature |
<-----------+
Details for round one are described in Section 5.1, and details for round two are described in Section 5.2. The final Aggregation step is described in Section 5.3.¶
FROST assumes reliable message delivery between Coordinator and signing participants in order for the protocol to complete. Messages exchanged during signing operations are all within the public domain. An attacker masquerading as another participant will result only in an invalid signature; see Section 7.¶
Round one involves each signer generating a pair of nonces and their corresponding public commitments. A nonce is a pair of Scalar values, and a commitment is a pair of Element values.¶
Each signer in round one generates a nonce nonce = (hiding_nonce, binding_nonce) and commitment
comm = (hiding_nonce_commitment, binding_nonce_commitment).¶
Inputs: None
Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs.
def commit():
hiding_nonce = G.RandomNonzeroScalar()
binding_nonce = G.RandomNonzeroScalar()
hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
nonce = (hiding_nonce, binding_nonce)
comm = (hiding_nonce_commitment, binding_nonce_commitment)
return (nonce, comm)
¶
The private output nonce from Participant P_i is stored locally and kept private
for use in the second round. This nonce MUST NOT be reused in more than one invocation
of FROST, and it MUST be generated from a source of secure randomness. The public output
comm from Participant P_i is sent to the Coordinator; see Appendix C.1
for encoding recommendations.¶
In round two, the Coordinator is responsible for sending the message to be signed, and for choosing which signers will participate (of number at least THRESHOLD_LIMIT). Signers additionally require locally held data; specifically, their private key and the nonces corresponding to their commitment issued in round one.¶
The Coordinator begins by sending each signer the message to be signed along with the set of signing commitments for other signers in the participant list. Each signer MUST validate the inputs before processing the Coordinator's request. In particular, the Signer MUST validate commitment_list, deserializing each group Element in the list using DeserializeElement from Section 3.1. If deserialization fails, the Signer MUST abort the protocol. Applications which require that signers not process arbitrary input messages are also required to also perform relevant application-layer input validation checks; see Section 7.4 for more details.¶
Upon receipt and successful input validation, each Signer then runs the following procedure to produce its own signature share.¶
Inputs:
- index, Index `i` of the signer. Note index will never equal `0`.
- sk_i, Signer secret key share.
- group_public_key, public key corresponding to the signer secret key share.
- nonce_i, pair of Scalar values (hiding_nonce, binding_nonce) generated in round one.
- msg, the message to be signed (sent by the Coordinator).
- commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
list of commitments issued in Round 1 by each signer, where each element in the list indicates the signer index j and their
two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by signer index.
- participant_list, a set containing identifiers for each signer, similarly of length
NUM_SIGNERS (sent by the Coordinator).
Outputs: a Scalar value representing the signature share
def sign(index, sk_i, group_public_key, nonce_i, msg, commitment_list, participant_list):
# Encode the commitment list
encoded_commitments = encode_group_commitment_list(commitment_list)
# Compute the binding factor
binding_factor = compute_binding_factor(encoded_commitments, msg)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor)
# Compute Lagrange coefficient
lambda_i = derive_lagrange_coefficient(index, participant_list)
# Compute the per-message challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge)
return sig_share
¶
The output of this procedure is a signature share. Each signer then sends these shares back to the collector; see Appendix C.3 for encoding recommendations. Each signer MUST delete the nonce and corresponding commitment after this round completes.¶
Upon receipt from each Signer, the Coordinator MUST validate the input signature using DeserializeElement. If validation fails, the Coordinator MUST abort the protocol. If validation succeeds, the Coordinator then verifies the set of signature shares using the following procedure.¶
After signers perform round two and send their signature shares to the Coordinator, the Coordinator verifies each signature share for correctness. In particular, for each signer, the Coordinator uses commitment pairs generated during round one and the signature share generated during round two, along with other group parameters, to check that the signature share is valid using the following procedure.¶
Inputs:
- index, Index `i` of the signer. Note index will never equal `0`.
- PK_i, the public key for the ith signer, where `PK_i = G.ScalarBaseMult(sk_i)`
- comm_i, pair of Element values (hiding_nonce_commitment, binding_nonce_commitment) generated
in round one from the ith signer.
- sig_share_i, a Scalar value indicating the signature share as produced in round two from the ith signer.
- commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a list of commitments
issued in Round 1 by each signer, where each element in the list indicates the signer index j and their
two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by signer index.
- participant_list, a set containing identifiers for each signer, similarly of length
NUM_SIGNERS (sent by the Coordinator).
- group_public_key, the public key for the group
- msg, the message to be signed
Outputs: True if the signature share is valid, and False otherwise.
def verify_signature_share(index, PK_i, comm_i, sig_share_i, commitment_list,
participant_list, group_public_key, msg):
# Encode the commitment list
encoded_commitments = encode_group_commitment_list(commitment_list)
# Compute the binding factor
binding_factor = compute_binding_factor(encoded_commitments, msg)
# Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor)
# Compute the commitment share
(hiding_nonce_commitment, binding_nonce_commitment) = comm_i
comm_share = hiding_nonce_commitment + (binding_nonce_commitment * binding_factor)
# Compute the challenge
challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute Lagrange coefficient
lambda_i = derive_lagrange_coefficient(index, participant_list)
# Compute relation values
l = G.ScalarBaseMult(sig_share_i)
r = comm_share + (PK_i * challenge * lambda_i)
return l == r
¶
If any signature share fails to verify, i.e., if verify_signature_share returns False for
any signer share, the Coordinator MUST abort the protocol. Otherwise, if all signer shares
are valid, the Coordinator performs the aggregate operation and publishes the resulting
signature.¶
Inputs:
- group_commitment, the group commitment returned by compute_group_commitment
- sig_shares, a set of signature shares z_i for each signer, of length NUM_SIGNERS,
where THRESHOLD_LIMIT <= NUM_SIGNERS <= MAX_SIGNERS.
Outputs: (R, z), a Schnorr signature consisting of an Element and Scalar value.
def frost_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:¶
struct {
opaque R_encoded[Ne];
opaque z_encoded[Ns];
} Signature;
¶
Where Signature.R_encoded is G.SerializeElement(R) and Signature.z_encoded is
G.SerializeScalar(z).¶
A FROST ciphersuite must specify the underlying prime-order group details and cryptographic hash function. Each ciphersuite is denoted as (Group, Hash), e.g., (ristretto255, SHA-512). This section contains some ciphersuites.¶
The RECOMMENDED ciphersuite is (ristretto255, SHA-512) Section 6.2. The (Ed25519, SHA-512) ciphersuite is included for backwards compatibility with [RFC8032].¶
The DeserializeElement and DeserializeScalar functions instantiated for a particular prime-order group corresponding to a ciphersuite MUST adhere to the description in Section 3.1. Validation steps for these functions are described for each the ciphersuites below. Future ciphersuites MUST describe how input validation is done for DeserializeElement and DeserializeScalar.¶
This ciphersuite uses edwards25519 for the Group and SHA-512 for the Hash function H
meant to produce signatures indistinguishable from Ed25519 as specified in [RFC8032].
The value of the contextString parameter is empty.¶
Group: edwards25519 [RFC8032]¶
h): 8¶
G.Order() - 1.¶
Hash (H): SHA-512, and Nh = 64.¶
Normally H2 would also include a domain separator, but for backwards compatibility with [RFC8032], it is omitted.¶
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".¶
Group: ristretto255 [RISTRETTO]¶
h): 1¶
G.Order() - 1.¶
Hash (H): SHA-512, and Nh = 64.¶
This ciphersuite uses edwards448 for the Group and SHA256 for the Hash function H
meant to produce signatures indistinguishable from Ed448 as specified in [RFC8032].
The value of the contextString parameter is empty.¶
h): 4¶
G.Order() - 1.¶
Hash (H): SHAKE256, and Nh = 117.¶
Normally H2 would also include a domain separator, but for backwards compatibility with [RFC8032], it is omitted.¶
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".¶
Group: P-256 (secp256r1) [x9.62]¶
h): 1¶
G.Order() - 1.¶
Hash (H): SHA-256, and Nh = 32.¶
expand_message_xmd with SHA-256, DST = contextString || "rho", and
prime modulus equal to Order().¶
expand_message_xmd with SHA-256, DST = contextString || "chal", and
prime modulus equal to Order().¶
A security analysis of FROST exists in [FROST20] and [Schnorr21]. The protocol as specified in this document assumes the following threat model.¶
(t-1) corrupted signers. So long as an adverary
corrupts fewer than (t-1) participants, the scheme remains secure against EUF-CMA attacks.¶
The protocol as specified in this document does not target the following goals:¶
The rest of this section documents issues particular to implementations or deployments.¶
Nonces generated by each participant in the first round of signing must be sampled uniformly at random and cannot be derived from some deterministic function. This is to avoid replay attacks initiated by other signers, which allows for a complete key-recovery attack. Coordinates MAY further hedge against nonce reuse attacks by tracking signer nonce commitments used for a given group key, at the cost of additional state.¶
We do not specify what implementations should do when the protocol fails, other than requiring that the protocol abort. Examples of viable failure include when a verification check returns invalid or if the underlying transport failed to deliver the required messages.¶
In some settings, it may be desirable to omit the role of the coordinator entirely. Doing so does not change the security implications of FROST, but instead simply requires each participant to communicate with all other participants. We loosely describe how to perform FROST signing among signers without this coordinator role. We assume that every participant receives as input from an external source the message to be signed prior to performing the protocol.¶
Every participant begins by performing frost_commit() as is done in the setting
where a coordinator is used. However, instead of sending the commitment
SigningCommitment to the coordinator, every participant instead will publish
this commitment to every other participant. Then, in the second round, instead of
receiving a SigningPackage from the coordinator, signers will already have
sufficient information to perform signing. They will directly perform frost_sign.
All participants will then publish a SignatureShare to one another. After having
received all signature shares from all other signers, each signer will then perform
frost_verify and then frost_aggregate directly.¶
The requirements for the underlying network channel remain the same in the setting where all participants play the role of the coordinator, in that all messages that are exchanged are public and so the channel simply must be reliable. However, in the setting that a player attempts to split the view of all other players by sending disjoint values to a subset of players, the signing operation will output an invalid signature. To avoid this denial of service, implementations may wish to define a mechanism where messages are authenticated, so that cheating players can be identified and excluded.¶
Some applications may require that signers only process messages of a certain structure. For example, in digital currency applications wherein multiple signers may collectively sign a transaction, it is reasonable to require that each signer check the input message to be a syntactically valid transaction. As another example, use of threshold signatures in TLS [TLS] to produce signatures of transcript hashes might require that signers check that the input message is a valid TLS transcript from which the corresponding transcript hash can be derived.¶
In general, input message validation is an application-specific consideration that varies based on the use case and threat model. However, it is RECOMMENDED that applications take additional precautions and validate inputs so that signers do not operate as signing oracles for arbitrary messages.¶
The Zcash Foundation engineering team designed a serialization format for FROST messages which we employ a slightly adapted version here.¶
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 Sections Appendix B.1 and Appendix B.2 to create secret
shares of s to be sent to all other participants. We highlight at a high level how this
operation can be performed.¶
Inputs:
- s, a group secret that MUST be derived from at least `Ns` bytes of entropy
- n, the number of shares to generate, an integer
- t, the threshold of the secret sharing scheme, an integer
Outputs:
- signer_private_keys, `n` shares of the secret key `s`, each a Scalar value.
- vss_commitment, a vector commitment to each of the coefficients in the polynomial defined by secret_key_shares and whose constant term is s.
def trusted_dealer_keygen(s, n, t):
signer_private_keys, coefficients = secret_share_shard(secret_key, n, t)
vss_commitment = vss_commit(coefficients):
PK = G.ScalarBaseMult(secret_key)
return signer_private_keys, vss_commitment
¶
It is assumed the dealer then sends one secret key share to each of the NUM_SIGNERS participants, along with C.
After receiving their secret key share and C each participant MUST perform vss_verify(secret_key_share_i, C).
It is assumed that all participant have the same view of C.
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 s to n participants
in such a way that any cooperating subset of t 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, n, t):
Inputs:
- s, secret to be shared, an element of F
- n, the number of shares to generate, an integer
- t, the threshold of the secret sharing scheme, an integer
Outputs:
- secret_key_shares, A list of n secret shares, which is a tuple
consisting of the participant identifier and the key share, each of
which is an element of F
- coefficients, a vector of the t coefficients which uniquely determine
a polynomial f.
Errors:
- "invalid parameters", if t > n or if t is less than 2
def secret_share_shard(s, n, t):
if t > n:
raise "invalid parameters"
if t < 2:
raise "invalid parameters"
# Generate random coefficients for the polynomial, yielding
# a polynomial of degree (t - 1)
coefficients = [s]
for i in range(t - 1):
coefficients.append(G.RandomScalar())
# Evaluate the polynomial for each point x=1,...,n
secret_key_shares = []
for x_i in range(1, n + 1):
y_i = polynomial_evaluate(x_i, coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_share.append(secret_key_share_i)
return secret_key_shares, coefficients
¶
Let points be the output of this function. The i-th element in points is
the share for the i-th participant, which is the randomly generated polynomial
evaluated at coordinate i. We denote a secret share as the tuple (i, points[i]),
and the list of these shares as shares.
i MUST never equal 0; recall that f(0) = s, where f is the polynomial defined in a Shamir secret sharing operation.¶
The procedure for combining a shares list of length t to recover the
secret s is as follows.¶
secret_share_combine(shares):
Inputs:
- shares, a list of t secret shares, each a tuple (i, f(i))
Outputs: The resulting secret s, that was previously split into shares
Errors:
- "invalid parameters", if less than t input shares are provided
def secret_share_combine(shares):
if len(shares) < t:
raise "invalid parameters"
s = polynomial_interpolation(shares)
return s
¶
Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret sharing,
adding a verification step to demonstrate the consistency of a participant's
share with a public commitment to the polynomial f for which the secret s
is the constant term. This check ensure that all participants have a point
(their share) on the same polynomial, ensuring that they can later reconstruct
the correct secret.¶
The procedure for committing to a polynomial f of degree t-1 is as follows.¶
vss_commit(coeffs):
Inputs:
- coeffs, a vector of the t coefficients which uniquely determine
a polynomial f.
Outputs: a commitment vss_commitment, which is a vector commitment to each of the
coefficients in coeffs.
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, where sk_i is a secret share of
the constant term of f.
- vss_commitment: A VSS commitment to a secret polynomial f.
Outputs: 1 if sk_i is valid, and 0 otherwise
vss_verify(share_i, commitment)
(i, sk_i) = share_i
S_i = ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, THRESHOLD_LIMIT-1):
S_i' += vss_commitment_j * i^j
if S_i == S_i':
return 1
return 0
¶
We now define how the coordinator and signing participants can derive group info, which is an input into the FROST signing protocol.¶
derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment):
Inputs:
- MAX_SIGNERS, the number of shares to generate, an integer
- THRESHOLD_LIMIT, the threshold of the secret sharing scheme, an integer
- vss_commitment: A VSS commitment to a secret polynomial f.
Outputs:
- PK, the public key representing the group
- signer_public_keys, a list of MAX_SIGNERS public keys PK_i for i=1,...,MAX_SIGNERS, where PK_i is the public key for participant i.
derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment)
PK = vss_commitment[0]
signer_public_keys = []
for i in range(1, MAX_SIGNERS):
PK_i = G.Identity()
for j in range(0, THRESHOLD_LIMIT-1):
PK_i += vss_commitment_j * i^j
signer_public_keys.append(PK_i)
return PK, signer_public_keys
¶
Applications are responsible for encoding protocol messages between peers. This section contains RECOMMENDED encodings for different protocol messages as described in Section 5.¶
A commitment from a signer is a pair of Element values. It can be encoded in the following manner.¶
SignerID uint64;
struct {
SignerID id;
opaque D[Ne];
opaque E[Ne];
} SigningCommitment;
¶
The Coordinator sends "signing packages" to each Signer in Round two. Each package contains the list of signing commitments generated during round one along with the message to sign. This package can be encoded in the following manner.¶
struct {
SigningCommitment signing_commitments<1..2^16-1>;
opaque msg<0..2^16-1>;
} SigningPackage;
¶
An list of SIGNING_COUNT SigningCommitment values, where THRESHOLD_LIMIT <= SIGNING_COUNT <= NUM_SIGNERS, ordered in ascending order by SigningCommitment.id. This list MUST NOT contain more than one SigningCommitment value corresponding to each signer. Signers MUST ignore SigningPackage values with duplicate SignerIDs.¶
The message to be signed.¶
This section contains test vectors for all ciphersuites listed in Section 6. All Element and Scalar values are represented in serialized form and encoded in hexadecimal strings. Signatures are represented as the concatenation of their constituent parts. The input message to be signed is also encoded as a hexadecimal string.¶
Each test vector consists of the following information.¶
// Configuration information MAX_SIGNERS: 3 THRESHOLD_LIMIT: 2 NUM_SIGNERS: 2 // Group input parameters group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7 90c6e13a98304 group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040 d380fb9738673 message: 74657374 // Signer input parameters S1 signer_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033 f2ec83d93509 S2 signer_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee d76bf409e80d S3 signer_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8 bdea643a9a02 // Round one parameters participants: 1,2 group_binding_factor_input: 000178e175d15cb5cec1257e0d84d797ba8c3dd9b 4c7bc50f3fa527c200bcc6c4a954cdad16ae67ac5919159d655b681bd038574383bab 423614f8967396ee12ca62000288a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea98 981bfb25375996c5767c932bbf10c41feb17d41cc6433e69f16cceccc42a00aedf72f eb5f44929fdf2e2fee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f8819a 5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f500 28a8ff group_binding_factor: c4d7668d793ff4c6ec424fb493cdab3ef5b625eefffe775 71ff28a345e5f700a // Signer round one outputs S1 hiding_nonce: 570f27bfd808ade115a701eeee997a488662bca8c2a073143e66 2318f1ed8308 S1 binding_nonce: 6720f0436bd135fe8dddc3fadd6e0d13dbd58a1981e587d377d 48e0b8f1c3c01 S1 hiding_nonce_commitment: 78e175d15cb5cec1257e0d84d797ba8c3dd9b4c7b c50f3fa527c200bcc6c4a95 S1 binding_nonce_commitment: 4cdad16ae67ac5919159d655b681bd038574383b ab423614f8967396ee12ca62 S2 hiding_nonce: 2a67c5e85884d0275a7a740ba8f53617527148418797345071dd cf1a1bd37206 S2 binding_nonce: a0609158eeb448abe5b0df27f5ece96196df5722c01a999e8a4 5d2d5dfc5620c S2 hiding_nonce_commitment: 88a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea 98981bfb25375996c5767c9 S2 binding_nonce_commitment: 32bbf10c41feb17d41cc6433e69f16cceccc42a0 0aedf72feb5f44929fdf2e2f // Round two parameters participants: 1,2 // Signer round two outputs S1 sig_share: b7e8f03a1a1149adacb96f952dbc39b6034facceafe4a70d6963592 fce75570c S2 sig_share: cd388f9aff4376397c5ad231713fe6b167bed9cc88a1cc97b0b6bbe 0316a7909 sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e 974d8a78fff1ac8e52774a24c00141536b0d869b388674a5191a151000e0d005¶
// Configuration information MAX_SIGNERS: 3 THRESHOLD_LIMIT: 2 NUM_SIGNERS: 2 // Group input parameters group_secret_key: cdf4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35 4a9a3af62b8c7c89753055949cab8fd044c17c94211f167672b053659420b00 group_public_key: 800e9b495543b04aaebdba2813de65d1aefe78e8b219d38966b c0afa1d5d9d685c740c8ab720bff3c84cd9f4a701c1588e40d981f4abb19600 message: 74657374 // Signer input parameters S1 signer_share: d208a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08 175d4298dd6bbe2e1c0b29aaa972c78555ea3b3d7308b248994780219e0800 S2 signer_share: d71c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc 8416d5cd0210b4c5e410f90a892c91065fbdae37d51ffc29078acae9f90500 S3 signer_share: dc3094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f f2cf670328b4a95cad16c96b68e65a87689021323737460b75cc14b2550300 // Round one parameters participants: 1,2 group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b 6841aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c8 9a870400a0c29f750605b10c52e347fc538af0d4ebddd23a1e0300482a7d98a39d408 356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c1753a80000242c2fdc11e5f 726d4c897ed118f668a27bfb0d5946b5f513e975638b7c4b0a46cf5184d4a9c1f6310 fd3c10f84d9de704a33aab2af976d60804fa4ecba88458bcf7677a3952f540e20556d 5e90d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e 1b0b28a80b54ff7255705a71ee2925e4a3e30e41aed489a579d5595e0df13e32e1e4d d202a7c7f68b31d6418d9845eb4d757adda6ab189e1bb340db818e5b3bc725d992faf 63e9b0500db10517fe09d3f566fba3a80e46a403e0c7d41548fbf75cf2662b00225b5 02961f98d8c9ff937de0b24c231845 group_binding_factor: 2716e157c3da80b65149b1c2cb546723516272ccf75e111 334533e2840a9bf85f3c71478ade11be26d26d8e4b9a1667af88f7df61670f60a00 // Signer round one outputs S1 hiding_nonce: 04eccfe12348a5a2e4b30e95efcf4e494ce64b89f6504de46b3d 67a5341baaa931e455c57c6c5c81f4895e333da9d71f7d119fcfbd0d7d2000 S1 binding_nonce: 80bcd1b09e82d7d2ff6dd433b0f81e012cadd4661011c44d929 1269cf24820f5c5086d4363dc67450f24ebe560eb4c2059883545d54aa43a00 S1 hiding_nonce_commitment: 6d8ef55145bab18c129311f1d07bef2110d0b6841 aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c89a87 0400 S1 binding_nonce_commitment: a0c29f750605b10c52e347fc538af0d4ebddd23a 1e0300482a7d98a39d408356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c17 53a80 S2 hiding_nonce: 3b3bbe82babf2a67ded81b308ba45f73b88f6cf3f6aaa4442256 b7a0354d1567478cfde0a2bba98ba4c3e65645e1b77386eb4063f925e00700 S2 binding_nonce: bcbd112a88bebf463e3509076c5ef280304cb4f1b3a7499cca1 d5e282cc2010a92ff56a3bdcf5ba352e0f4241ba2e54c1431a895c19fff0600 S2 hiding_nonce_commitment: 42c2fdc11e5f726d4c897ed118f668a27bfb0d594 6b5f513e975638b7c4b0a46cf5184d4a9c1f6310fd3c10f84d9de704a33aab2af976d 6080 S2 binding_nonce_commitment: 4fa4ecba88458bcf7677a3952f540e20556d5e90 d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e1b0b 28a80 // Round two parameters participants: 1,2 // Signer round two outputs S1 sig_share: c5ab0a80c561d1a616ac70f4f13d993156f65f2b44a4a90f37f0640 7a1b62e3940bf14199301d128358b812bef32cb4bffaf03030238772000 S2 sig_share: 15211cb96d6aa73de803d46caf2043859fd796a6282f9adb00033f1 4f4827f23f8cc792c2e322a1f30631ec7690ac587e5eb9c2afd323e3300 sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353 37e4d1aae5fd1c73a55701133238602f423886fc134a3c6580e787ce8da00900c1a92 07fd32e9c6f956597202323f8f4264ecfd99e9539ae5c388c8e45c133fb4765ee9ff2 583d90d3e49ba02dff6ab51300¶
// Configuration information MAX_SIGNERS: 3 THRESHOLD_LIMIT: 2 NUM_SIGNERS: 2 // Group input parameters group_secret_key: b020be204b5e758960458ca9c4675b56b12a8faff2be9c94891 d5e1cd75c880e group_public_key: e22ac4850672021eac8e0a36dfc4811466fb01108c3427d2347 827467ba02a34 message: 74657374 // Signer input parameters S1 signer_share: 92ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f 4693ddcb6705 S2 signer_share: 611003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521 2e0ae43a470c S3 signer_share: 439eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22 1681eaa92603 // Round one parameters participants: 1,2 group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca 094f65b814a8bca6013d9cc312774c7e1271d2939a84a9a867e3a06579b4d25659b42 7439ccf0d745b43f75b76600028013834ff4d48e7d6b76c2e732bc611f54720ef8933 c4ca4de7eaaa77ff5cd125e056ecc4f7c4657d3a742354430d768f945db229c335d25 8e9622ad99f3e7582d07b35bd9849ce4af6ad403090d69a7d0eb88bba669a9f985175 d70cd15ad5f1ef5b734c98a32b4aab7b43a57e93fc09281f2e7a207076b31e416ba63 f53d9d group_binding_factor: f00ae6007f2d74a1507c962cf30006be77596106db28f2d 5443fd66d755e780c // Signer round one outputs S1 hiding_nonce: 349b3bb8464a1d87f7d6b56f4559a3f9a6335261a3266089a9b1 2d9d6f6ce209 S1 binding_nonce: ce7406016a854be4291f03e7d24fe30e77994c3465de031515a 4c116f22ca901 S1 hiding_nonce_commitment: 824e9eddddf02b2a9caf5859825e999d791ca094f 65b814a8bca6013d9cc3127 S1 binding_nonce_commitment: 74c7e1271d2939a84a9a867e3a06579b4d25659b 427439ccf0d745b43f75b766 S2 hiding_nonce: 4d66d319f20a728ec3d491cbf260cc6be687bd87cc2b5fdb4d5f 528f65fd650d S2 binding_nonce: 278b9b1e04632e6af3f1a3c144d07922ffcf5efd3a341b47abc 19c43f48ce306 S2 hiding_nonce_commitment: 8013834ff4d48e7d6b76c2e732bc611f54720ef89 33c4ca4de7eaaa77ff5cd12 S2 binding_nonce_commitment: 5e056ecc4f7c4657d3a742354430d768f945db22 9c335d258e9622ad99f3e758 // Round two parameters participants: 1,2 // Signer round two outputs S1 sig_share: 6a539c3a4ee281879a6fb350d20d53e17473f28cd3409ffc238dafe 8d9330605 S2 sig_share: 1d4e59636ee089bfaf548834b07658216649a37f87f0818d5190aa9 b90957505 sig: 7e92309bf40993141acd5f2c7680a302cc5aa5dd291a833906da8e35bc39b03e 87a1f59dbcc20b474ac43b858284ab02dbbc950c5b31218a751d5a846ac97b0a¶
// Configuration information MAX_SIGNERS: 3 THRESHOLD_LIMIT: 2 NUM_SIGNERS: 2 // Group input parameters group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0 3a6a51a5a5928 group_public_key: 033a2a83f9c9fdfdab7d620f48238a5e6157a8eb1d6c382c7b0 ba95b7c9f69679c message: 74657374 // Signer input parameters S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678 6f1e410dcecd S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed 379767c14472 S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62 00108e74ba17 // Round one parameters participants: 1,2 group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7 f702906930766b86c462bb8ff7f3402b724640ea9e262469f401c9006991ba3247c2c 91b97cdb1f0eeab1a777e24e1e0002037f8a998dfc2e60a7ad63bc987cb27b8abf78a 68bd924ec6adb9f251850cbe711024a4e90422a19dd8463214e997042206c39d3df56 168b458592462090c89dbcf84efca0c54f70a585d6aae28679482b4aed03ae5d38297 b9092ab3376d46fdf55 group_binding_factor: 9df349a9f34bf01627f6b4f8b376e8c8261d55508d1cac2 919cdaf7f9cb20e70 // Signer round one outputs S1 hiding_nonce: 3da92a503cf7e3f72f62dabedbb3ffcc9f555f1c1e78527940fe 3fed6d45e56f S1 binding_nonce: ec97c41fc77ae7e795067976b2edd8b679f792abb062e4d0c33 f0f37d2e363eb S1 hiding_nonce_commitment: 02f34caab210d59324e12ba41f0802d9545f7f702 906930766b86c462bb8ff7f34 S1 binding_nonce_commitment: 02b724640ea9e262469f401c9006991ba3247c2c 91b97cdb1f0eeab1a777e24e1e S2 hiding_nonce: 06cb4425031e695d1f8ac61320717d63918d3edc7a02fcd3f23a de47532b1fd9 S2 binding_nonce: 2d965a4ea73115b8065c98c1d95c7085db247168012a834d828 5a7c02f11e3e0 S2 hiding_nonce_commitment: 037f8a998dfc2e60a7ad63bc987cb27b8abf78a68 bd924ec6adb9f251850cbe711 S2 binding_nonce_commitment: 024a4e90422a19dd8463214e997042206c39d3df 56168b458592462090c89dbcf8 // Round two parameters participants: 1,2 // Signer round two outputs S1 sig_share: 0a658fe198caddf5ddc407ad58c4615458f02a58d0c1f7a38e25692 98dc41df0 S2 sig_share: e84d948cfec74b5e7540ad09fd69dcd1570f708f2d8573dbbf08cb0 2bc872c75 sig: 035cfbd148da711bbc823455b682ed01a1be3c5415cf692f4a91b7fe22d1dec3 45f2b3246e979229545304b4b7562e3e25afff9ae7fe476b7f4d2e342c4a4b4a65¶