CFRG D. Connolly
InternetDraft Zcash Foundation
Intended status: Informational C. Komlo
Expires: 30 September 2022 University of Waterloo, Zcash Foundation
I. Goldberg
University of Waterloo
C. A. Wood
Cloudflare
29 March 2022
TwoRound Threshold Schnorr Signatures with FROST
draftirtfcfrgfrost04
Abstract
In this draft, we present the tworound signing variant of FROST, a
Flexible RoundOptimized 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
keyrecovery 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/draftirtfcfrgfrost.
Status of This Memo
This InternetDraft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
InternetDrafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as InternetDrafts. The list of current Internet
Drafts is at https://datatracker.ietf.org/drafts/current/.
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InternetDrafts are draft documents valid for a maximum of six months
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This InternetDraft will expire on 30 September 2022.
Copyright Notice
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document authors. All rights reserved.
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Please review these documents carefully, as they describe your rights
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 5
3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 6
3.1. PrimeOrder Group . . . . . . . . . . . . . . . . . . . . 6
3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 7
4. Helper functions . . . . . . . . . . . . . . . . . . . . . . 7
4.1. Schnorr Signature Operations . . . . . . . . . . . . . . 8
4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 9
4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 9
4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 10
4.2.3. Deriving the constant term of a polynomial . . . . . 11
4.3. Commitment List Encoding . . . . . . . . . . . . . . . . 12
4.4. Binding Factor Computation . . . . . . . . . . . . . . . 12
4.5. Group Commitment Computation . . . . . . . . . . . . . . 13
4.6. Signature Challenge Computation . . . . . . . . . . . . . 13
5. TwoRound FROST Signing Protocol . . . . . . . . . . . . . . 14
5.1. Round One  Commitment . . . . . . . . . . . . . . . . . 17
5.2. Round Two  Signature Share Generation . . . . . . . . . 17
5.3. Signature Share Verification and Aggregation . . . . . . 19
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1. FROST(Ed25519, SHA512) . . . . . . . . . . . . . . . . . 22
6.2. FROST(ristretto255, SHA512) . . . . . . . . . . . . . . 22
6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 23
6.4. FROST(P256, SHA256) . . . . . . . . . . . . . . . . . . 24
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7. Security Considerations . . . . . . . . . . . . . . . . . . . 25
7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 26
7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 26
7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 26
7.4. Input Message Validation . . . . . . . . . . . . . . . . 27
8. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 27
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.1. Normative References . . . . . . . . . . . . . . . . . . 27
9.2. Informative References . . . . . . . . . . . . . . . . . 28
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 29
Appendix B. Trusted Dealer Key Generation . . . . . . . . . . . 29
B.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 30
B.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 32
Appendix C. Wire Format . . . . . . . . . . . . . . . . . . . . 34
C.1. Signing Commitment . . . . . . . . . . . . . . . . . . . 34
C.2. Signing Packages . . . . . . . . . . . . . . . . . . . . 34
C.3. Signature Share . . . . . . . . . . . . . . . . . . . . . 35
Appendix D. Test Vectors . . . . . . . . . . . . . . . . . . . . 35
D.1. FROST(Ed25519, SHA512) . . . . . . . . . . . . . . . . . 36
D.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 37
D.3. FROST(ristretto255, SHA512) . . . . . . . . . . . . . . 39
D.4. FROST(P256, SHA256) . . . . . . . . . . . . . . . . . . 40
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 41
1. Introduction
DISCLAIMER: This is a workinprogress 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/draftirtfcfrgfrost.
Instructions are on that page as well.
Unlike signatures in a singleparty 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 RoundOptimized
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 Schnorrbased 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]. Singleround
signing with FROST is out of scope.
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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 keyrecovery attack in multi
party discrete logarithmbased 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.
1.1. Change Log
draft04
* Added methods to verify VSS commitments and derive group info
(#126, #132).
* Changed check for participants to consider only nonnegative
numbers (#133).
* Changed sampling for secrets and coefficients to allow the zero
element (#130).
* Split test vectors into separate files (#129)
* Update wire structs to remove commitment shares where not
necessary (#128)
* Add failure checks (#127)
* Update group info to include each participant's key and clarify
how public key material is obtained (#120, #121).
* Define cofactor checks for verification (#118)
* Various editorial improvements and add contributors (#124, #123,
#119, #116, #113, #109)
draft03
* Refactor the second round to use state from the first round (#94).
* Ensure that verification of signature shares from the second round
uses commitments from the first round (#94).
* Clarify RFC8032 interoperability based on PureEdDSA (#86).
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* Specify signature serialization based on element and scalar
serialization (#85).
* Fix hash function domain separation formatting (#83).
* Make trusted dealer key generation deterministic (#104).
* Add additional constraints on participant indexes and nonce usage
(#105, #103, #98, #97).
* Apply various editorial improvements.
draft02
* 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.
draft01
* Specify operations, notation and cryptographic dependencies.
draft00
* Outline CFRG draft based on draftkomlofrost.
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 2^161.
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* 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 8byte, bigendian
integer.
* encode_uint16(x): Convert two byte unsigned integer (uint16) x to
a 2byte, bigendian byte string. For example, encode_uint16(310)
= [0x01, 0x36].
*  denotes concatenation, 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 signing depends on the following cryptographic constructs:
* Primeorder Group, Section 3.1;
* Cryptographic hash function, Section 3.2;
These are described in the following sections.
3.1. PrimeOrder 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 r1 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 r1 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 =.
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We now detail a number of member functions that can be invoked on a
primeorder group G.
* Order(): Outputs the order of G (i.e. p).
* Identity(): Outputs the identity element of the group (i.e. I).
* RandomScalar(): A member function of G that chooses at random a
Scalar element in GF(p).
* RandomNonzeroScalar(): A member function of G that chooses at
random a nonzero Scalar element in GF(p).
* SerializeElement(A): A member function of G that maps an Element A
to a unique byte array buf of fixed length Ne.
* DeserializeElement(buf): A member function of 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.
* SerializeScalar(s): A member function of G that maps a Scalar s to
a unique byte array buf of fixed length Ns.
* DeserializeScalar(buf): A member function of 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
groupspecific input validation steps.
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 domainseparated hashes, H1, H2, and H3, where H1 and H2 map
arbitrary inputs to nonzero Scalar elements of the primeorder 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.
4. Helper functions
Beyond the core dependencies, the protocol in this document depends
on the following helper operations:
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* Schnorr signatures, Section 4.1;
* Polynomial operations, Section 4.2;
* Encoding operations, Section 4.3;
* Signature binding Section 4.4, group commitment Section 4.5, and
challenge computation Section 4.6
This sections describes these operations in more detail.
4.1. Schnorr Signature Operations
In the singleparty 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.
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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):
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()
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 xcoordinate and ycoordinate 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.
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derive_lagrange_coefficient(x_i, L):
Inputs:
 x_i, an xcoordinate contained in L, a scalar
 L, the set of xcoordinates, each a scalar
Outputs: L_i, the ith 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
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.
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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. Commitment List Encoding
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
4.4. Binding Factor Computation
This section describes the subroutine for computing the binding
factor based on the signer commitment list and message to be signed.
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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
4.5. Group Commitment Computation
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
4.6. Signature Challenge Computation
This section describes the subroutine for creating the permessage
challenge.
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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
5. TwoRound FROST Signing Protocol
We now present the tworound 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:
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.
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:
* Group public key, denoted 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.
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* Public keys for each signer, denoted 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:
* Participant is signing key share sk_i, which is the ith 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.
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(group info) (group info, (group info,
 signing key share) signing key share)
  
v v v
Coordinator Signer1 ... Signern

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.
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.
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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.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.
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. 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 applicationlayer input validation
checks; see Section 7.4 for more details.
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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 permessage 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.
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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.
5.3. Signature Share Verification and Aggregation
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.
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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
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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).
6. Ciphersuites
A FROST ciphersuite must specify the underlying primeorder group
details and cryptographic hash function. Each ciphersuite is denoted
as (Group, Hash), e.g., (ristretto255, SHA512). This section
contains some ciphersuites.
The RECOMMENDED ciphersuite is (ristretto255, SHA512) Section 6.2.
The (Ed25519, SHA512) ciphersuite is included for backwards
compatibility with [RFC8032].
The DeserializeElement and DeserializeScalar functions instantiated
for a particular primeorder 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.
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6.1. FROST(Ed25519, SHA512)
This ciphersuite uses edwards25519 for the Group and SHA512 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]
 Cofactor (h): 8
 SerializeElement: Implemented as specified in [RFC8032],
Section 5.1.2.
 DeserializeElement: Implemented as specified in [RFC8032],
Section 5.1.3. Additionally, this function validates that the
resulting element is not the group identity element.
 SerializeScalar: Implemented by outputting the littleendian
32byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHA512, and Nh = 64.
 H1(m): Implemented by computing H("rho"  m), interpreting the
lower 32 bytes as a littleendian 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 littleendian integer, and reducing the resulting
integer modulo L =
2^252+27742317777372353535851937790883648493.
 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, SHA512)
This ciphersuite uses ristretto255 for the Group and SHA512 for the
Hash function H. The value of the contextString parameter is "FROST
RISTRETTO255SHA512".
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* Group: ristretto255 [RISTRETTO]
 Cofactor (h): 1
 SerializeElement: Implemented using the 'Encode' function from
[RISTRETTO].
 DeserializeElement: Implemented using the 'Decode' function
from [RISTRETTO].
 SerializeScalar: Implemented by outputting the littleendian
32byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHA512, and Nh = 64.
 H1(m): Implemented by computing H(contextString  "rho"  m)
and mapping the output to a Scalar as described in [RISTRETTO],
Section 4.4.
 H2(m): Implemented by computing H(contextString  "chal"  m)
and mapping 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]
 Cofactor (h): 4
 SerializeElement: Implemented as specified in [RFC8032],
Section 5.2.2.
 DeserializeElement: Implemented as specified in [RFC8032],
Section 5.2.3. Additionally, this function validates that the
resulting element is not the group identity element.
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 SerializeScalar: Implemented by outputting the littleendian
48byte encoding of the Scalar value.
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 48byte string. This function can fail if the
input does not represent a Scalar between the value 0 and
G.Order()  1.
* Hash (H): SHAKE256, and Nh = 117.
 H1(m): Implemented by computing H("rho"  m), interpreting the
lower 57 bytes as a littleendian 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 littleendian 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(P256, SHA256)
This ciphersuite uses P256 for the Group and SHA256 for the Hash
function H. The value of the contextString parameter is "FROST
P256SHA256".
* Group: P256 (secp256r1) [x9.62]
 Cofactor (h): 1
 SerializeElement: Implemented using the compressed Elliptic
CurvePointtoOctetString method according to [SECG].
 DeserializeElement: Implemented by attempting to deserialize a
public key using the compressed OctetStringtoEllipticCurve
Point method according to [SECG], and then performs partial
publickey validation as defined in section 5.6.2.3.4 of
[KEYAGREEMENT]. This includes checking that the coordinates of
the resulting point are in the correct range, that the point is
on the curve, and that the point is not the point at infinity.
Additionally, this function validates that the resulting
element is not the group identity element. If these checks
fail, deserialization returns an error.
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 SerializeScalar: Implemented using the FieldElementtoOctet
String conversion according to [SECG].
 DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32byte string using OctetStringtoField
Element from [SECG]. This function can fail if the input does
not represent a Scalar between the value 0 and G.Order()  1.
* Hash (H): SHA256, and Nh = 32.
 H1(m): Implemented using hash_to_field from [HASHTOCURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA256, DST
= contextString  "rho", and prime modulus equal to Order().
 H2(m): Implemented using hash_to_field from [HASHTOCURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA256, DST
= contextString  "chal", and prime modulus equal to Order().
 H3(m): Implemented by computing H(contextString  "digest" 
m).
7. Security Considerations
A security analysis of FROST exists in [FROST20] and [Schnorr21].
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 (t1) corrupted signers. So
long as an adverary corrupts fewer than (t1) participants, the
scheme remains secure against EUFCMA 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:
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* Post quantum security. FROST, like plain 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
higherlevel communication channel can be used to facilitate key
generation and signing.
The rest of this section documents issues particular to
implementations or deployments.
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
deterministic function. This is to avoid replay attacks initiated by
other signers, which allows for a complete keyrecovery 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
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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.
7.4. Input Message Validation
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 applicationspecific
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.
8. Contributors
* Isis Lovecruft
* T. WilsonBrown
* Alden Torres
9. References
9.1. Normative References
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[HASHTOCURVE]
FazHernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, InternetDraft, draftirtfcfrghashtocurve
14, 18 February 2022,
.
[KEYAGREEMENT]
Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
Davis, "Recommendation for pairwise keyestablishment
schemes using discrete logarithm cryptography", National
Institute of Standards and Technology report,
DOI 10.6028/nist.sp.80056ar3, April 2018,
.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
.
[RFC8032] Josefsson, S. and I. Liusvaara, "EdwardsCurve Digital
Signature Algorithm (EdDSA)", RFC 8032,
DOI 10.17487/RFC8032, January 2017,
.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, .
[RISTRETTO]
Valence, H. D., Grigg, J., Hamburg, M., Lovecruft, I.,
Tankersley, G., and F. Valsorda, "The ristretto255 and
decaf448 Groups", Work in Progress, InternetDraft, draft
irtfcfrgristretto255decaf44803, 25 February 2022,
.
[SECG] "Elliptic Curve Cryptography, Standards for Efficient
Cryptography Group, ver. 2", 2009,
.
[x9.62] ANSI, "Public Key Cryptography for the Financial Services
Industry: the Elliptic Curve Digital Signature Algorithm
(ECDSA)", ANSI X9.621998, September 1998.
9.2. Informative References
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[FROST20] Komlo, C. and I. Goldberg, "TwoRound Threshold Signatures
with FROST", 22 December 2020,
.
[RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker,
"Randomness Requirements for Security", BCP 106, RFC 4086,
DOI 10.17487/RFC4086, June 2005,
.
[Schnorr21]
Crites, E., Komlo, C., and M. Maller, "How to Prove
Schnorr Assuming Schnorr", 11 October 2021,
.
[TLS] Rescorla, E., "The Transport Layer Security (TLS) Protocol
Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
.
Appendix A. Acknowledgments
The Zcash Foundation engineering team designed a serialization format
for FROST messages which we employ a slightly adapted version here.
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:
 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
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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.
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 primeorder group G.
The procedure for splitting a secret into shares is as follows.
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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 ith element in
points is the share for the ith 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.
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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.
The procedure for committing to a polynomial f of degree t1 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.
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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_LIMIT1):
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_LIMIT1):
PK_i += vss_commitment_j * i^j
signer_public_keys.append(PK_i)
return PK, signer_public_keys
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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.
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..2^161>;
opaque msg<0..2^161>;
} SigningPackage;
signing_commitments 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.
msg The message to be signed.
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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];
} 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.
* 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).
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* 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 as
produced in Section 5.2.
* Final output: This lists the aggregate signature as produced in
Section 5.3.
D.1. FROST(Ed25519, SHA512)
// 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
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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
D.2. FROST(Ed448, SHAKE256)
// 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
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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
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
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D.3. FROST(ristretto255, SHA512)
// 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
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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
D.4. FROST(P256, SHA256)
// 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
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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
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
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Christopher A. Wood
Cloudflare
Email: caw@heapingbits.net
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