Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-02
The information below is for an old version of the document.
| Document | Type |
This is an older version of an Internet-Draft whose latest revision state is "Active".
|
|
|---|---|---|---|
| Authors | Deirdre Connolly , Chelsea Komlo , Ian Goldberg , Christopher A. Wood | ||
| Last updated | 2022-02-12 | ||
| RFC stream | Internet Research Task Force (IRTF) | ||
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| IETF conflict review | conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost, conflict-review-irtf-cfrg-frost | ||
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draft-irtf-cfrg-frost-02
CFRG D. Connolly
Internet-Draft Zcash Foundation
Intended status: Informational C. Komlo
Expires: 16 August 2022 University of Waterloo, Zcash Foundation
I. Goldberg
University of Waterloo
C. A. Wood
Cloudflare
12 February 2022
Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-02
Abstract
In this draft, we present a 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.
Discussion Venues
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.
Status of This Memo
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/.
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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 16 August 2022.
Copyright Notice
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
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provided without warranty as described in the Revised BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 4
3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 5
3.1. Prime-Order Group . . . . . . . . . . . . . . . . . . . . 5
3.1.1. Input Validation . . . . . . . . . . . . . . . . . . 6
3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 6
4. Helper functions . . . . . . . . . . . . . . . . . . . . . . 7
4.1. Schnorr Signature Operations . . . . . . . . . . . . . . 7
4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 8
4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 8
4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 9
4.2.3. Deriving the constant term of a polynomial . . . . . 10
4.3. Encoding Operations . . . . . . . . . . . . . . . . . . . 10
5. Two-Round FROST . . . . . . . . . . . . . . . . . . . . . . . 11
5.1. Round One - Commitment . . . . . . . . . . . . . . . . . 13
5.2. Round Two - Signature Share Generation . . . . . . . . . 13
5.3. Signature Share Aggregation . . . . . . . . . . . . . . . 15
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 16
6.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 16
6.2. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 17
6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 17
6.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 18
7. Security Considerations . . . . . . . . . . . . . . . . . . . 19
7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 20
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7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 20
7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 20
8. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 21
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 21
9.1. Normative References . . . . . . . . . . . . . . . . . . 21
9.2. Informative References . . . . . . . . . . . . . . . . . 22
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 22
Appendix B. Trusted Dealer Key Generation . . . . . . . . . . . 23
B.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 23
B.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 25
Appendix C. Wire Format . . . . . . . . . . . . . . . . . . . . 26
C.1. Signing Commitment . . . . . . . . . . . . . . . . . . . 26
C.2. Signing Packages . . . . . . . . . . . . . . . . . . . . 27
C.3. Signature Share . . . . . . . . . . . . . . . . . . . . . 27
Appendix D. Test Vectors . . . . . . . . . . . . . . . . . . . . 27
D.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 28
D.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 30
D.3. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 31
D.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 33
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 34
1. Introduction
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.
In this draft, we present a variant of FROST, a Flexible Round-
Optimized Schnorr Threshold signature scheme. FROST reduces network
overhead during threshold signing operations while employing a novel
technique to protect against forgery attacks applicable to prior
Schnorr-based threshold signature constructions. FROST requires two
rounds to compute a signature.
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, is insecure in a multi-party signature setting.
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Further, this draft implements signing efficiency improvements for
FROST described by Crites, Komlo, and Maller in [Schnorr21].
1.1. Change Log
draft-02
* Fully specify both rounds of FROST, as well as trusted dealer key
generation.
* Add ciphersuites and corresponding test vectors, including suites
for RFC8032 compatibility.
* Refactor document for editorial clarity.
draft-01
* Specify operations, notation and cryptographic dependencies.
draft-00
* Outline CFRG draft based on draft-komlo-frost.
2. Conventions and Definitions
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.
* A participant is an entity that is trusted to hold a secret share.
* 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.
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* I2OSP(x, w): Convert non-negative integer x to a w-length, big-
endian byte string as described in [RFC8017].
* OS2IP(s): Convert byte string s to a non-negative integer as
described in [RFC8017], assuming big-endian byte order.
* || denotes contatenation, i.e., x || y = 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.
3. Cryptographic Dependencies
FROST depends on the following cryptographic constructs:
* Prime-order Group, Section 3.1;
* Cryptographic hash function, Section 3.2;
These are described in the following sections.
3.1. Prime-Order Group
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.
* Order(): Outputs the order of G (i.e. p).
* Identity(): Outputs the identity element of the group (i.e. I).
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* RandomScalar(): A member function of G that chooses at random a
non-zero element in GF(p).
* SerializeElement(A): A member function of G that maps a group
element A to a unique byte array buf of fixed length Ne. The
output type of this function is SerializedElement.
* DeserializeElement(buf): A member function of G that maps a byte
array buf to a group element A, or fails if the input is not a
valid byte representation of an element. This function can raise
a DeserializeError if deserialization fails or A is the identity
element of the group; see Section 3.1.1.
3.1.1. Input Validation
The DeserializeElement function recovers a group element from an
arbitrary byte array. This function validates that the element is a
proper member of the group and is not the identity element, and
returns an error if either condition is not met.
For ristretto255, elements are deserialized by invoking the Decode
function from [RISTRETTO], Section 4.3.1, which returns false if the
element is invalid. If this function returns false, deserialization
returns an error.
The DeserializeScalar function recovers a scalar field element from
an arbitrary byte array. Like DeserializeElement, this function
validates that the element is a member of the scalar field and
returns an error if this condition is not met.
For ristretto255, this function ensures that the input, when treated
as a little-endian integer, is a value greater than or equal to 0,
and less than Order().
3.2. Cryptographic Hash Function
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.
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4. Helper functions
Beyond the core dependencies, the protocol in this document depends
on the following helper operations:
* Schnorr signatures, Section 4.1;
* Polynomial operations, Section 4.2;
* Encoding operations, Section 4.3
This sections describes these operations in more detail.
4.1. Schnorr Signature Operations
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
- PK, public key, a group element
Outputs: signature (R, z), a pair of scalar values
def schnorr_signature_generate(msg, SK, PK):
r = G.RandomScalar()
R = G.ScalarBaseMult(r)
msg_hash = H3(msg)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg_hash
c = H2(challenge_input)
z = r + (c * SK)
return (R, z)
The corresponding verification operation is as follows.
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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):
msg_hash = H3(msg)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg_hash
c = H2(challenge_input)
l = ScalarBaseMult(z)
r = R + (c * PK)
if l == r:
return 1
return 0
4.2. Polynomial Operations
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.
4.2.1. Evaluation of a polynomial
This section describes a method for evaluating a polynomial f at a
particular input x, i.e., y = f(x) using Horner's method.
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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
4.2.2. Lagrange coefficients
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
def derive_lagrange_coefficient(x_i, L):
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
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4.2.3. Deriving the constant term of a polynomial
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
4.3. Encoding Operations
This section describes various helper functions used for encoding
data structures into values that can be processed with hash
functions.
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 = I2OSP(index, 2) ||
G.SerializeElement(hiding_nonce_commitment) ||
G.SerializeElement(binding_nonce_commitment)
encoded_group_commitment = encoded_group_commitment || encoded_commitment
return encoded_group_commitment
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5. Two-Round FROST
FROST is a two-round 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:
1. Determining which signers will participate (at least
THRESHOLD_LIMIT in number);
2. Coordinating rounds (receiving and forwarding inputs among
participants); and
3. Aggregating signature shares output by each participant, and
publishing the resulting signature.
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.
In FROST, all signers are assumed to have the group state and their
corresponding signing key shares. In particular, FROST assumes that
each signing participant P_i knows the following:
* Group public key, denoted PK = G.ScalarMultBase(s), corresponding
to the group secret key s.
* Participant is signing key, 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.
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There are two rounds in FROST: commitment and signature share
generation. The first round serves for each participant to issue a
commitment. The second round receives commitments for all signers as
well as the message, and issues a signature share. The Coordinator
performs the coordination of each of these rounds. At the end of the
second round, the Coordinator then performs an aggregation step at
the end 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 |
<-----------+
Figure 1: FROST signature overview
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.
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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.
5.1. Round One - Commitment
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.RandomScalar()
binding_nonce = G.RandomScalar()
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. The public output comm
from Participant P_i is sent to the Coordinator; see Appendix C.1 for
encoding recommendations.
5.2. Round Two - Signature Share Generation
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. Upon receipt, each Signer then runs the
following procedure to produce its own signature share.
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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.
- comm_i, pair of Element values (hiding_nonce_commitment, binding_nonce_commitment) 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 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 signature share sig_share and commitment share comm_share, which
are Scalar and Element values respectively.
def sign(index, sk_i, group_public_key, nonce_i, comm_i, msg, commitment_list, participant_list):
# Compute the binding factor
encoded_commitments = encode_group_commitment_list(commitment_list)
msg_hash = H3(msg)
binding_factor = H1(encoded_commitments || msg_hash)
# Compute the group commitment
R = G.Identity()
for (_, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
R = R + (hiding_nonce_commitment + (binding_nonce_commitment * binding_factor))
lambda_i = derive_lagrange_coefficient(index, participant_list)
# Compute the per-message challenge
group_comm_enc = G.SerializeElement(R)
group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg_hash
c = H2(challenge_input)
# Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * c)
# Compute the commitment share
(hiding_nonce_commitment, binding_nonce_commitment) = comm_i
comm_share = hiding_nonce_commitment + (binding_nonce_commitment * binding_factor)
return sig_share, comm_share
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The output of this procedure is a signature share and group
commitment share. Each signer then sends these shares back to the
collector; see Appendix C.3 for encoding recommendations.
The Coordinator MUST verify the set of signature shares using the
following procedure.
Inputs:
- index, Index `i` of the signer. Note index will never equal `0`.
- PK, the public key for the group
- PK_i, the public key for the ith signer, where PK_i = ScalarBaseMult(s[i])
- sig_share, the signature share for the ith signer, computed from the signer
- comm_share, the commitment for the ith signer, computed from the signer
- R, the group commitment
- msg, the message to be signed
- participant_list, a set containing identifiers for each signer, similarly of length
NUM_SIGNERS (sent by the Coordinator).
Outputs: 1 if the signature share is valid, and 0 otherwise.
def verify_signature_share(index, PK, PK_i, sig_share, comm_share, R, msg, participant_list):
msg_hash = H3(msg)
group_comm_enc = G.SerializeElement(R)
group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg_hash
c = H2(challenge_input)
l = G.ScalarbaseMult(sig_share)
lambda_i = derive_lagrange_coefficient(index, participant_list)
r = comm_share + (sig_share * c * lambda_i)
return l == r
5.3. Signature Share Aggregation
After signers perform round two and send their signature shares to
the Coordinator, the Coordinator performs the aggregate operation and
publishes the resulting signature. Note that here we do not specify
the Coordinator as validating each signature schare, as if any
signature share is invalid, the resulting joint signature will
similarly be invalid. Deployments that wish to validate signature
shares can do so using the verify_signature_share function in
Section 5.2
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Inputs:
- R: the 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(R, sig_shares):
z = 0
for z_i in sig_shares:
z = z + z_i
return (R, z)
6. Ciphersuites
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].
6.1. FROST(Ed25519, SHA-512)
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]
- SerializeElement: Implemented as specified in [RFC8032],
Section 5.1.2.
- DeserializeElement: Implemented as specified in [RFC8032],
Section 5.1.3.
* Hash (H): SHA-512, and Nh = 64.
- H1(m): Implemented by computing H("rho" || m), interpreting the
lower 32 bytes as a little-endian integer, and reducing the
resulting integer modulo L =
2^252+27742317777372353535851937790883648493.
- H2(m): Implemented by computing H(m), interpreting the lower 32
bytes as a little-endian integer, and reducing the resulting
integer modulo L =
2^252+27742317777372353535851937790883648493.
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- H3(m): Implemented as an alias for H, i.e., H(m).
Normally H2 would also include a domain separator, but for backwards
compatibility with [RFC8032], it is omitted.
6.2. FROST(ristretto255, SHA-512)
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]
- SerializeElement: Implemented using the 'Encode' function from
[RISTRETTO].
- DeserializElement: Implemented using the 'Decode' function from
[RISTRETTO].
* Hash (H): SHA-512, and Nh = 64.
- H1(m): Implemented by computing H(contextString || "rho" || m)
and mapping the the output to a Scalar as described in
[RISTRETTO], Section 4.4.
- H2(m): Implemented by computing H(contextString || "chal" || m)
and mapping the the output to a Scalar as described in
[RISTRETTO], Section 4.4.
- H3(m): Implemented by computing H(contextString || "digest" ||
m).
6.3. FROST(Ed448, SHAKE256)
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.
* Group: edwards448 [RFC8032]
- SerializeElement: Implemented as specified in [RFC8032],
Section 5.2.2.
- DeserializeElement: Implemented as specified in [RFC8032],
Section 5.2.3.
* Hash (H): SHAKE256, and Nh = 117.
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- H1(m): Implemented by computing H("rho" || m), interpreting the
lower 57 bytes as a little-endian integer, and reducing the
resulting integer modulo L = 2^446 - 13818066809895115352007386
748515426880336692474882178609894547503885.
- H2(m): Implemented by computing H(m), interpreting the lower 57
bytes as a little-endian integer, and reducing the resulting
integer modulo L = 2^446 - 138180668098951153520073867485154268
80336692474882178609894547503885.
- H3(m): Implemented as an alias for H, i.e., H(m).
Normally H2 would also include a domain separator, but for backwards
compatibility with [RFC8032], it is omitted.
6.4. FROST(P-256, SHA-256)
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]
- SerializeElement: Implemented using the compressed Elliptic-
Curve-Point-to-Octet-String method according to [SECG].
- DeserializeElement: Implemented by attempting to deserialize a
public key using the compressed Octet-String-to-Elliptic-Curve-
Point method according to [SECG], and then performs partial
public-key validation as defined in section 5.6.2.3.4 of
[KEYAGREEMENT].
- Serialization: Elements are serialized as Ne = 33 byte string
* Hash (H): SHA-256, and Nh = 32.
- H1(m): Implemented using hash_to_field from [HASH-TO-CURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST
= contextString || "rho", and prime modulus equal to Order().
- H2(m): Implemented using hash_to_field from [HASH-TO-CURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST
= contextString || "chal", and prime modulus equal to Order().
- H3(m): Implemented by computing H(contextString || "digest" ||
m).
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7. Security Considerations
A security analysis of FROST exists in [FROST20]. The protocol as
specified in this document assumes the following threat model.
* Trusted dealer. The dealer that performs key generation is
trusted to follow the protocol, although participants still are
able to verify the consistency of their shares via a VSS
(verifiable secret sharing) step; see Appendix B.2.
* Unforgeability assuming less than (t-1) corrupted signers. So
long as an adverary corrupts fewer than (t-1) participants, the
scheme remains secure against EUF-CMA attacks.
* Coordinator. We assume the Coordinator at the time of signing
does not perform a denial of service attack. A denial of service
would include any action which either prevents the protocol from
completing or causing the resulting signature to be invalid. Such
actions for the latter include sending inconsistent values to
signing participants, such as messages or the set of individual
commitments. Note that the Coordinator is _not_ trusted with any
private information and communication at the time of signing can
be performed over a public but reliable channel.
The protocol as specified in this document does not target the
following goals:
* Post quantum security. FROST, like generic Schnorr signatures,
requires the hardness of the Discrete Logarithm Problem.
* Robustness. In the case of failure, FROST requires aborting the
protocol.
* Downgrade prevention. The sender and receiver are assumed to
agree on what algorithms to use.
* Metadata protection. If protection for metadata is desired, a
higher-level communication channel can be used to facilitate key
generation and signing.
The rest of this section documents issues particular to
implementations or deployments.
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7.1. Nonce Reuse Attacks
Nonces generated by each participant in the first round of signing
must be sampled uniformly at random and cannot be derived from some
determinstic 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.
7.2. Protocol Failures
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.
7.3. Removing the Coordinator Role
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.
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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.
8. Contributors
* Isis Lovecruft
* T. Wilson-Brown
9. References
9.1. Normative References
[HASH-TO-CURVE]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
13, 10 November 2021,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
hash-to-curve-13>.
[KEYAGREEMENT]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
Davis, "Recommendation for pair-wise key-establishment
schemes using discrete logarithm cryptography", National
Institute of Standards and Technology report,
DOI 10.6028/nist.sp.800-56ar3, April 2018,
<https://doi.org/10.6028/nist.sp.800-56ar3>.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/rfc/rfc2119>.
[RFC8017] Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
"PKCS #1: RSA Cryptography Specifications Version 2.2",
RFC 8017, DOI 10.17487/RFC8017, November 2016,
<https://www.rfc-editor.org/rfc/rfc8017>.
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[RFC8032] Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
<https://www.rfc-editor.org/rfc/rfc8032>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[RISTRETTO]
Valence, H. D., Grigg, J., Tankersley, G., Valsorda, F.,
Lovecruft, I., and M. Hamburg, "The ristretto255 and
decaf448 Groups", Work in Progress, Internet-Draft, draft-
irtf-cfrg-ristretto255-decaf448-01, 4 August 2021,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
ristretto255-decaf448-01>.
[SECG] "Elliptic Curve Cryptography, Standards for Efficient
Cryptography Group, ver. 2", 2009,
<https://secg.org/sec1-v2.pdf>.
[x9.62] ANSI, "Public Key Cryptography for the Financial Services
Industry: the Elliptic Curve Digital Signature Algorithm
(ECDSA)", ANSI X9.62-1998, September 1998.
9.2. Informative References
[FROST20] Komlo, C. and I. Goldberg, "Two-Round Threshold Signatures
with FROST", 22 December 2020,
<https://eprint.iacr.org/2020/852.pdf>.
[RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106, RFC 4086,
DOI 10.17487/RFC4086, June 2005,
<https://www.rfc-editor.org/rfc/rfc4086>.
[Schnorr21]
Crites, E., Komlo, C., and M. Maller, "How to Prove
Schnorr Assuming Schnorr", 11 October 2021,
<https://eprint.iacr.org/2021/1375>.
Appendix A. Acknowledgments
The Zcash Foundation engineering team designed a serialization format
for FROST messages which we employ a slightly adapted version here.
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Appendix B. Trusted Dealer Key Generation
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:
- n, the number of shares to generate, an integer
- t, the threshold of the secret sharing scheme, an integer
Outputs: a secret key Scalar, public key Element, along with `n`
shares of the secret key, each a Scalar value.
def trusted_dealer_keygen(n, t):
secret_key = G.RandomScalar()
secret_key_shares = secret_share_split(secret_key, n, t)
public_key = G.ScalarBaseMult(s)
return secret_key, public_key, secret_key_shares
It is assumed the dealer then sends one secret key to each of the
NUM_SIGNERS participants, and afterwards deletes the secrets from
their local device.
Use of this method for key generation requires a mutually
authenticated secure channel between Coordinator and participants,
wherein the channel provides confidentiality and integrity. Mutually
authenticated TLS is one possible deployment option.
B.1. Shamir Secret Sharing
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.
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secret_share_split(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: A list of n secret shares, each of which is an element of F
Errors:
- "invalid parameters", if t > n
def secret_share(s, n, t):
if t > n:
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(RandomScalar())
# Evaluate the polynomial for each point x=1,...,n
points = []
for x_i in range(1, n+1):
y_i = polynomial_evaluate(x_i, coefficients)
point_i = (x_i, y_i)
points.append(point_i)
return points
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.
The procedure for combining a shares list of length t to recover the
secret s is as follows.
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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
B.2. Verifiable Secret Sharing
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. If the
validation fails, the participant can issue a complaint against the
dealer, and take actions such as broadcasting this complaint to all
other participants. We do not specify the complaint procedure in
this draft, as it will be implementation-specific.
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 C, which is a vector commitment to each of the
coefficients in coeffs.
def vss_commit(coeffs):
C = []
for coeff in coeffs:
A_i = ScalarBaseMult(coeff)
C.append(A_i)
return C
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The procedure for verification of a participant's share is as
follows.
vss_verify(sk_i, C):
Inputs:
- sk_i: A participant's secret key, the tuple sk_i = (i, s[i]),
where s[i] is a secret share of the constant term of f.
- C: A VSS commitment to a secret polynomial f.
Outputs: 1 if s[i] is valid, and 0 otherwise
vss_verify(sk_i, commitment)
S_i = ScalarBaseMult(s[i])
S_i' = SUM(commitment[0], commitment[t-1]){A_j}: A_j*(i^j)
if S_i == S_i':
return 1
return 0
Appendix C. Wire Format
Applications are responsible for encoding protocol messages between
peers. This section contains RECOMMENDED encodings for different
protocol messages as described in Section 5.
C.1. Signing Commitment
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;
id The SignerID.
D The commitment hiding factor encoded as a serialized group
element.
E The commitment binding factor encoded as a serialized group
element.
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C.2. Signing Packages
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;
signing_commitments An list of SIGNING_COUNT SigningCommitment
values, where THRESHOLD_LIIMT <= 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.
msg The message to be signed.
C.3. Signature Share
The output of each signer is a signature share which is sent to the
Coordinator. This can be constructed as follows.
struct {
SignerID id;
opaque signature_share[Ns];
opaque commitment_share[Ne];
} SignatureShare;
id The SignerID.
signature_share The signature share from this signer encoded as a
serialized scalar.
Appendix D. Test Vectors
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.
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* Configuration: This lists the fixed parameters for the particular
instantiation of FROST, including MAX_SIGNERS, THRESHOLD_LIMIT,
and NUM_SIGNERS.
* Group input parameters: This lists the group secret key and shared
public key, generated by a trusted dealer as described in
Appendix B, as well as the input message to be signed. All values
are encoded as hexadecimal strings.
* Signer input parameters: This lists the signing key share for each
of the NUM_SIGNERS signers.
* Round one parameters and outputs: This lists the NUM_SIGNERS
participants engaged in the protocol, identified by their integer
index, the hiding and binding commitment values produced in
Section 5.1, as well as 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).
* Round two parameters and outputs: This lists the NUM_SIGNERS
participants engaged in the protocol, identified by their integer
index, along with their corresponding output signature share and
group commitment share as produced in Section 5.2.
* Final output: This lists the aggregate signature as produced in
Section 5.3.
D.1. FROST(Ed25519, SHA-512)
// Configuration information
MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: 7c1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
90c6e13a98304
group_public_key: 377a6acb3b9b5f642c5ce355d23cac0568aad0da63c633d59d4
168bdcbce35af
message: 74657374
// Signer input parameters
S1 signer_share: 949dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033
f2ec83d93509
S2 signer_share: ac1e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee
d76bf409e80d
S3 signer_share: d7cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8
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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: f8bbaf924e1c90e11dec1eb679194aade084f92fbf52fdd436ba0a0
7f71ab708
S1 group_commitment_share: 11bb9777aa393b92e814e415039adf62687a0be543
c2322d817e4934bc5a7cf2
S2 sig_share: 80f589405714ca0e6adc87c2c0186a0ae4d6e352f7b248b23149a5d
cd3fe4704
S2 group_commitment_share: 4af85179d17ed031b767ab579e59c7018dac09ae40
0b1700623d0af1129a9c55
sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e
78b139d3a5305af087c8a6783a32b4b7c45bdd82b60546876803b0e3ca19ff0c
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D.2. FROST(Ed448, SHAKE256)
// Configuration information
MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: cef4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35
4a9a3af62b8c7c89753055949cab8fd044c17c94211f167672b053659420b00
group_public_key: 005f23508a78131aee4d6cb027f967d89557ec5f24dc3ebeede
b550466fcc1411283ff5d9c605d9a8b36e6eea36b67ceba047d57968896db80
message: 74657374
// Signer input parameters
S1 signer_share: d408a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08
175d4298dd6bbe2e1c0b29aaa972c78555ea3b3d7308b248994780219e0800
S2 signer_share: da1c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc
8416d5cd0210b4c5e410f90a892c91065fbdae37d51ffc29078acae9f90500
S3 signer_share: e03094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f
f2cf670328b4a95cad16c96b68e65a87689021323737460b75cc14b2550300
// Round one parameters
participants: 1,2
group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b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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
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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: ad41cd3320c82edd20c344769bd7b250105d9d0516109b7f774c297
faaf8b3b6065b19bbae2afb6c34cce460b40e15655fb8ad0bcc26e21e00
S1 group_commitment_share: 086d4d2ff2555fab65afc8eb473cc708f37cdb9c5d
e74d8e12a1a9d1a086a8914175e4db77e5d281f10441913aa680fedb207c954afdd88
380
S2 sig_share: 5dcc0aec7d0a71eddd5ba2dd0f185ba7990bcd39b6fc0e4b0470c35
6ed0deb736d7f2652e87e932a0c176cc4bc5ba0ef756cc62081e4f51900
S2 group_commitment_share: 7e91f66097b6450c52c89c14400a506ee1d37f5e52
a8d4c3fc9733c23d0b27cd6cfce55a8aee692262e5815be341e8d0b9d240a9630c9f0
600
sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353
37e4d1aae5fd1c73a55701133238602f423886fc134a3c65800a0ed81f9ed29fcafe1
ee753abef0df8a9686a3fcc0caaca7bbcecd597069f2a74da3f0d97a98e9740e35025
716ab554d524742c4d0bd83800
D.3. FROST(ristretto255, SHA-512)
// Configuration information
MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: b120be204b5e758960458ca9c4675b56b12a8faff2be9c94891
d5e1cd75c880e
group_public_key: 563b80013f337deaa2a282af7b281bd70d2f501928a89c1aa48
b379a5ac4202b
message: 74657374
// Signer input parameters
S1 signer_share: 94ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f
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4693ddcb6705
S2 signer_share: 641003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521
2e0ae43a470c
S3 signer_share: 479eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22
1681eaa92603
// Round one parameters
participants: 1,2
group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca
094f65b814a8bca6013d9cc312774c7e1271d2939a84a9a867e3a06579b4d25659b42
7439ccf0d745b43f75b76600028013834ff4d48e7d6b76c2e732bc611f54720ef8933
c4ca4de7eaaa77ff5cd125e056ecc4f7c4657d3a742354430d768f945db229c335d25
8e9622ad99f3e758f226c1530c93fbfe1a29f34aa2e13da14ace01b6e6412e36d5e01
baba2c78e4921dc1c0b7143210bb0fc42553c3a9490ba011e30250727c0189372a386
32591f
group_binding_factor: f49fbf1a092173b9338394b5818966480c0413c5f90e2a7
65aabc1a10cfb3908
// 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: 503c943f6af38f3ad2271de3d792c4756c5d9c483989e2005add0e8
9b504dc01
S1 group_commitment_share: 40587ff15e9cba4fe601c4378b40ae4babe2ea6f0b
549a6712d84837166b954d
S2 sig_share: eee969c17354ec84933723b45ac9965d2874e21c3a50cd31648f708
bfb9f160b
S2 group_commitment_share: 70661957faf398410a9da3e20cfd6f1233b368548e
44ffccee3143a42b691844
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sig: 90ac875023a311624948b2660bc5524f690ae9e14fcd6541959bee2e868d8c32
3e26fe00de477cbf655f4097325c5bd394d17e6573d9af32be6c7f14b1a4f20c
D.4. FROST(P-256, SHA-256)
// Configuration information
MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2
// Group input parameters
group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0
3a6a51a5a5929
group_public_key: 03db0945167b62e6472ad46373b6cbbca88e2a9a4883071f0b3
fde4b2b6d7b6ba6
message: 74657374
// Signer input parameters
S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678
6f1e410dcecf
S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed
379767c14475
S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62
00108e74ba1b
// Round one parameters
participants: 1,2
group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7
f702906930766b86c462bb8ff7f3402b724640ea9e262469f401c9006991ba3247c2c
91b97cdb1f0eeab1a777e24e1e0002037f8a998dfc2e60a7ad63bc987cb27b8abf78a
68bd924ec6adb9f251850cbe711024a4e90422a19dd8463214e997042206c39d3df56
168b458592462090c89dbcf8bce8e9dd076882537d47858b7ed704e0029ea004fbeb2
8a46d1ba698cc0099a3
group_binding_factor: 9c649ba6084d89db49dafb28f4b50fda14fd8263b3fe907
97ca4258714d581eb
// 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
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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: 0400c1ba4351343192777323e20847b4fd9067af35e0261f6af0413
c7e4bbd26
S1 group_commitment_share: 03208d89db28e4946b02c9942a5245140014c930d8
7ec9b89b1a783abb2422df9b
S2 sig_share: 54d1dbea644a643ca91948398c40f20f12a00f15075b9614095ecfb
f685e421d
S2 group_commitment_share: 026842e4516527c3f47b9c5300d231ae0b61a13ede
858174596855753575567a08
sig: 03f8bda19543758dab107009bd44f2bfb0f6192c8fd1d0eded6bdbee3169a78e
0658d29da4a79b986e3b90bb5d6e4939c4103076c43d3bbc33744f10fbe6a9ff43
Authors' Addresses
Deirdre Connolly
Zcash Foundation
Email: durumcrustulum@gmail.com
Chelsea Komlo
University of Waterloo, Zcash Foundation
Email: ckomlo@uwaterloo.ca
Ian Goldberg
University of Waterloo
Email: iang@uwaterloo.ca
Christopher A. Wood
Cloudflare
Email: caw@heapingbits.net
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