< draft-irtf-cfrg-frost-03.txt   draft-irtf-cfrg-frost-04.txt >
CFRG D. Connolly CFRG D. Connolly
Internet-Draft Zcash Foundation Internet-Draft Zcash Foundation
Intended status: Informational C. Komlo Intended status: Informational C. Komlo
Expires: 8 September 2022 University of Waterloo, Zcash Foundation Expires: 30 September 2022 University of Waterloo, Zcash Foundation
I. Goldberg I. Goldberg
University of Waterloo University of Waterloo
C. A. Wood C. A. Wood
Cloudflare Cloudflare
7 March 2022 29 March 2022
Two-Round Threshold Schnorr Signatures with FROST Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-03 draft-irtf-cfrg-frost-04
Abstract Abstract
In this draft, we present a two-round signing variant of FROST, a In this draft, we present the two-round signing variant of FROST, a
Flexible Round-Optimized Schnorr Threshold signature scheme. FROST Flexible Round-Optimized Schnorr Threshold signature scheme. FROST
signatures can be issued after a threshold number of entities signatures can be issued after a threshold number of entities
cooperate to issue a signature, allowing for improved distribution of cooperate to issue a signature, allowing for improved distribution of
trust and redundancy with respect to a secret key. Further, this trust and redundancy with respect to a secret key. Further, this
draft specifies signatures that are compatible with [RFC8032]. draft specifies signatures that are compatible with [RFC8032].
However, unlike [RFC8032], the protocol for producing signatures in However, unlike [RFC8032], the protocol for producing signatures in
this draft is not deterministic, so as to ensure protection against a this draft is not deterministic, so as to ensure protection against a
key-recovery attack that is possible when even only one participant key-recovery attack that is possible when even only one participant
is malicious. is malicious.
skipping to change at page 2, line 10 skipping to change at page 2, line 10
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This Internet-Draft will expire on 8 September 2022. This Internet-Draft will expire on 30 September 2022.
Copyright Notice Copyright Notice
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Table of Contents Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4 1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 4
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 5 2. Conventions and Definitions . . . . . . . . . . . . . . . . . 5
3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 5 3. Cryptographic Dependencies . . . . . . . . . . . . . . . . . 6
3.1. Prime-Order Group . . . . . . . . . . . . . . . . . . . . 6 3.1. Prime-Order Group . . . . . . . . . . . . . . . . . . . . 6
3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 7 3.2. Cryptographic Hash Function . . . . . . . . . . . . . . . 7
4. Helper functions . . . . . . . . . . . . . . . . . . . . . . 7 4. Helper functions . . . . . . . . . . . . . . . . . . . . . . 7
4.1. Schnorr Signature Operations . . . . . . . . . . . . . . 7 4.1. Schnorr Signature Operations . . . . . . . . . . . . . . 8
4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 9 4.2. Polynomial Operations . . . . . . . . . . . . . . . . . . 9
4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 9 4.2.1. Evaluation of a polynomial . . . . . . . . . . . . . 9
4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 9 4.2.2. Lagrange coefficients . . . . . . . . . . . . . . . . 10
4.2.3. Deriving the constant term of a polynomial . . . . . 10 4.2.3. Deriving the constant term of a polynomial . . . . . 11
4.3. Commitment List Encoding . . . . . . . . . . . . . . . . 11 4.3. Commitment List Encoding . . . . . . . . . . . . . . . . 12
4.4. Binding Factor Computation . . . . . . . . . . . . . . . 11 4.4. Binding Factor Computation . . . . . . . . . . . . . . . 12
4.5. Group Commitment Computation . . . . . . . . . . . . . . 12 4.5. Group Commitment Computation . . . . . . . . . . . . . . 13
4.6. Signature Challenge Computation . . . . . . . . . . . . . 12 4.6. Signature Challenge Computation . . . . . . . . . . . . . 13
5. Two-Round FROST . . . . . . . . . . . . . . . . . . . . . . . 13 5. Two-Round FROST Signing Protocol . . . . . . . . . . . . . . 14
5.1. Round One - Commitment . . . . . . . . . . . . . . . . . 16 5.1. Round One - Commitment . . . . . . . . . . . . . . . . . 17
5.2. Round Two - Signature Share Generation . . . . . . . . . 16 5.2. Round Two - Signature Share Generation . . . . . . . . . 17
5.3. Signature Share Verification and Aggregation . . . . . . 18 5.3. Signature Share Verification and Aggregation . . . . . . 19
6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 20 6. Ciphersuites . . . . . . . . . . . . . . . . . . . . . . . . 21
6.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 21 6.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 22
6.2. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 21 6.2. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 22
6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 22 6.3. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 23
6.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 23 6.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 24
7. Security Considerations . . . . . . . . . . . . . . . . . . . 24 7. Security Considerations . . . . . . . . . . . . . . . . . . . 25
7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 25 7.1. Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . . 26
7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 25 7.2. Protocol Failures . . . . . . . . . . . . . . . . . . . . 26
7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 25 7.3. Removing the Coordinator Role . . . . . . . . . . . . . . 26
7.4. Input Message Validation . . . . . . . . . . . . . . . . 26 7.4. Input Message Validation . . . . . . . . . . . . . . . . 27
8. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 26 8. Contributors . . . . . . . . . . . . . . . . . . . . . . . . 27
9. References . . . . . . . . . . . . . . . . . . . . . . . . . 26 9. References . . . . . . . . . . . . . . . . . . . . . . . . . 27
9.1. Normative References . . . . . . . . . . . . . . . . . . 26 9.1. Normative References . . . . . . . . . . . . . . . . . . 27
9.2. Informative References . . . . . . . . . . . . . . . . . 27 9.2. Informative References . . . . . . . . . . . . . . . . . 28
Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 28 Appendix A. Acknowledgments . . . . . . . . . . . . . . . . . . 29
Appendix B. Trusted Dealer Key Generation . . . . . . . . . . . 28 Appendix B. Trusted Dealer Key Generation . . . . . . . . . . . 29
B.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 29 B.1. Shamir Secret Sharing . . . . . . . . . . . . . . . . . . 30
B.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 31 B.2. Verifiable Secret Sharing . . . . . . . . . . . . . . . . 32
Appendix C. Wire Format . . . . . . . . . . . . . . . . . . . . 32 Appendix C. Wire Format . . . . . . . . . . . . . . . . . . . . 34
C.1. Signing Commitment . . . . . . . . . . . . . . . . . . . 32 C.1. Signing Commitment . . . . . . . . . . . . . . . . . . . 34
C.2. Signing Packages . . . . . . . . . . . . . . . . . . . . 33 C.2. Signing Packages . . . . . . . . . . . . . . . . . . . . 34
C.3. Signature Share . . . . . . . . . . . . . . . . . . . . . 33 C.3. Signature Share . . . . . . . . . . . . . . . . . . . . . 35
Appendix D. Test Vectors . . . . . . . . . . . . . . . . . . . . 33 Appendix D. Test Vectors . . . . . . . . . . . . . . . . . . . . 35
D.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 34 D.1. FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . . 36
D.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 36 D.2. FROST(Ed448, SHAKE256) . . . . . . . . . . . . . . . . . 37
D.3. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 37 D.3. FROST(ristretto255, SHA-512) . . . . . . . . . . . . . . 39
D.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 39 D.4. FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . . 40
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 40 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 41
1. Introduction 1. Introduction
DISCLAIMER: This is a work-in-progress draft of FROST. DISCLAIMER: This is a work-in-progress draft of FROST.
RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this
draft is maintained in GitHub. Suggested changes should be submitted draft is maintained in GitHub. Suggested changes should be submitted
as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost. as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost.
Instructions are on that page as well. Instructions are on that page as well.
Unlike signatures in a single-party setting, threshold signatures Unlike signatures in a single-party setting, threshold signatures
require cooperation among a threshold number of signers each holding require cooperation among a threshold number of signers each holding
a share of a common private key. The security of threshold schemes a share of a common private key. The security of threshold schemes
in general assume that an adversary can corrupt strictly fewer than a in general assume that an adversary can corrupt strictly fewer than a
threshold number of participants. threshold number of participants.
In this draft, we present a variant of FROST, a Flexible Round- This document presents a variant of a Flexible Round-Optimized
Optimized Schnorr Threshold signature scheme. FROST reduces network Schnorr Threshold (FROST) signature scheme originally defined in
overhead during threshold signing operations while employing a novel [FROST20]. FROST reduces network overhead during threshold signing
technique to protect against forgery attacks applicable to prior operations while employing a novel technique to protect against
Schnorr-based threshold signature constructions. FROST requires two forgery attacks applicable to prior Schnorr-based threshold signature
rounds to compute a signature. constructions. The variant of FROST presented in this document
requires two rounds to compute a signature, and implements signing
efficiency improvements described by [Schnorr21]. Single-round
signing with FROST is out of scope.
For select ciphersuites, the signatures produced by this draft are For select ciphersuites, the signatures produced by this draft are
compatible with [RFC8032]. However, unlike [RFC8032], signatures compatible with [RFC8032]. However, unlike [RFC8032], signatures
produced by FROST are not deterministic, since deriving nonces produced by FROST are not deterministic, since deriving nonces
deterministically, is insecure in a multi-party signature setting. deterministically allows for a complete key-recovery attack in multi-
party discrete logarithm-based signatures, such as FROST.
Further, this draft implements signing efficiency improvements for Key generation for FROST signing is out of scope for this document.
FROST described by Crites, Komlo, and Maller in [Schnorr21]. However, for completeness, key generation with a trusted dealer is
specified in Appendix B.
1.1. Change Log 1.1. Change Log
draft-04
* 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)
draft-03 draft-03
* Refactor the second round to use state from the first round (#94). * Refactor the second round to use state from the first round (#94).
* Ensure that verificaftion of signature shares from the second * Ensure that verification of signature shares from the second round
round uses commitments from the first round (#94). uses commitments from the first round (#94).
* Clarify RFC8032 interoperability based on PureEdDSA (#86). * Clarify RFC8032 interoperability based on PureEdDSA (#86).
* Specify signature serialization based on element and scalar * Specify signature serialization based on element and scalar
serialization (#85). serialization (#85).
* Fix hash function domain separation formatting (#83). * Fix hash function domain separation formatting (#83).
* Make trusted dealer key generation deterministic (#104). * Make trusted dealer key generation deterministic (#104).
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THRESHOLD_LIMIT shares must be combined to issue a valid THRESHOLD_LIMIT shares must be combined to issue a valid
signature. signature.
* len(x) is the length of integer input x as an 8-byte, big-endian * len(x) is the length of integer input x as an 8-byte, big-endian
integer. integer.
* encode_uint16(x): Convert two byte unsigned integer (uint16) x to * encode_uint16(x): Convert two byte unsigned integer (uint16) x to
a 2-byte, big-endian byte string. For example, encode_uint16(310) a 2-byte, big-endian byte string. For example, encode_uint16(310)
= [0x01, 0x36]. = [0x01, 0x36].
* || denotes contatenation, i.e., x || y = xy. * || denotes concatenation, i.e., x || y = xy.
Unless otherwise stated, we assume that secrets are sampled uniformly Unless otherwise stated, we assume that secrets are sampled uniformly
at random using a cryptographically secure pseudorandom number at random using a cryptographically secure pseudorandom number
generator (CSPRNG); see [RFC4086] for additional guidance on the generator (CSPRNG); see [RFC4086] for additional guidance on the
generation of random numbers. generation of random numbers.
3. Cryptographic Dependencies 3. Cryptographic Dependencies
FROST depends on the following cryptographic constructs: FROST signing depends on the following cryptographic constructs:
* Prime-order Group, Section 3.1; * Prime-order Group, Section 3.1;
* Cryptographic hash function, Section 3.2; * Cryptographic hash function, Section 3.2;
These are described in the following sections. These are described in the following sections.
3.1. Prime-Order Group 3.1. Prime-Order Group
FROST depends on an abelian group G of prime order p. The FROST depends on an abelian group G of prime order p. The
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assignment of values by =. assignment of values by =.
We now detail a number of member functions that can be invoked on a We now detail a number of member functions that can be invoked on a
prime-order group G. prime-order group G.
* Order(): Outputs the order of G (i.e. p). * Order(): Outputs the order of G (i.e. p).
* Identity(): Outputs the identity element of the group (i.e. I). * Identity(): Outputs the identity element of the group (i.e. I).
* RandomScalar(): A member function of G that chooses at random a * RandomScalar(): A member function of G that chooses at random a
non-zero Scalar element in GF(p). Scalar element in GF(p).
* RandomNonzeroScalar(): A member function of G that chooses at
random a non-zero Scalar element in GF(p).
* SerializeElement(A): A member function of G that maps an Element A * SerializeElement(A): A member function of G that maps an Element A
to a unique byte array buf of fixed length Ne. to a unique byte array buf of fixed length Ne.
* DeserializeElement(buf): A member function of G that attempts to * 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 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 not a valid byte representation of an element of the group. This
function can raise a DeserializeError if deserialization fails or function can raise a DeserializeError if deserialization fails or
A is the identity element of the group; see Section 6 for group- A is the identity element of the group; see Section 6 for group-
specific input validation steps. specific input validation steps.
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R = G.ScalarBaseMult(k) R = G.ScalarBaseMult(k)
comm_enc = G.SerializeElement(R) comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK) pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input) c = H2(challenge_input)
z = k + (c * SK) z = k + (c * SK)
return (R, z) return (R, z)
The corresponding verification operation is as follows. The corresponding verification operation is as follows. Here, h is
the cofactor for the group being operated over, e.g. h=8 for the case
of Curve25519, h=4 for Ed448, and h=1 for groups such as ristretto255
and secp256k1, etc. This final scalar multiplication MUST be
performed when h>1.
schnorr_signature_verify(msg, sig, PK): schnorr_signature_verify(msg, sig, PK):
Inputs: Inputs:
- msg, signed message, an octet string - msg, signed message, an octet string
- sig, a tuple (R, z) output from schnorr_signature_generate or FROST - sig, a tuple (R, z) output from schnorr_signature_generate or FROST
- PK, public key, a group element - PK, public key, a group element
Outputs: 1 if signature is valid, and 0 otherwise Outputs: 1 if signature is valid, and 0 otherwise
def schnorr_signature_verify(msg, sig = (R, z), PK): def schnorr_signature_verify(msg, sig = (R, z), PK):
comm_enc = G.SerializeElement(R) comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK) pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input) c = H2(challenge_input)
l = G.ScalarBaseMult(z) l = G.ScalarBaseMult(z)
r = R + (c * PK) r = R + (c * PK)
if l == r: check = (l - r) * h
return 1 return check == G.Identity()
return 0
4.2. Polynomial Operations 4.2. Polynomial Operations
This section describes operations on and associated with polynomials This section describes operations on and associated with polynomials
that are used in the main signing protocol. A polynomial of degree t 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 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 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 convenience, we refer to the x-coordinate and y-coordinate of a point
p as p.x and p.y, respectively. p as p.x and p.y, respectively.
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Inputs: Inputs:
- x_i, an x-coordinate contained in L, a scalar - x_i, an x-coordinate contained in L, a scalar
- L, the set of x-coordinates, each a scalar - L, the set of x-coordinates, each a scalar
Outputs: L_i, the i-th Lagrange coefficient Outputs: L_i, the i-th Lagrange coefficient
Errors: Errors:
- "invalid parameters", if any coordinate is less than or equal to 0 - "invalid parameters", if any coordinate is less than or equal to 0
def derive_lagrange_coefficient(x_i, L): def derive_lagrange_coefficient(x_i, L):
if x_i <= 0: if x_i = 0:
raise "invalid parameters" raise "invalid parameters"
for x_j in L: for x_j in L:
if x_j <= 0: if x_j = 0:
raise "invalid parameters" raise "invalid parameters"
numerator = 1 numerator = 1
denominator = 1 denominator = 1
for x_j in L: for x_j in L:
if x_j == x_i: continue if x_j == x_i: continue
numerator *= x_j numerator *= x_j
denominator *= x_j - x_i denominator *= x_j - x_i
L_i = numerator / denominator L_i = numerator / denominator
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Outputs: a challenge Scalar value Outputs: a challenge Scalar value
def compute_challenge(group_commitment, group_public_key, msg): def compute_challenge(group_commitment, group_public_key, msg):
group_comm_enc = G.SerializeElement(group_commitment) group_comm_enc = G.SerializeElement(group_commitment)
group_public_key_enc = G.SerializeElement(group_public_key) group_public_key_enc = G.SerializeElement(group_public_key)
challenge_input = group_comm_enc || group_public_key_enc || msg challenge_input = group_comm_enc || group_public_key_enc || msg
challenge = H2(challenge_input) challenge = H2(challenge_input)
return challenge return challenge
5. Two-Round FROST 5. Two-Round FROST Signing Protocol
FROST is a two-round threshold signature protocol for producing We now present the two-round variant of the FROST threshold signature
Schnorr signatures. It involves signer participants and a protocol for producing Schnorr signatures. It involves signer
coordinator. Signing participants are entities with signing key participants and a coordinator. Signing participants are entities
shares that participate in the threshold signing protocol. The with signing key shares that participate in the threshold signing
coordinator is a distinguished signer with the following protocol. The coordinator is a distinguished signer with the
responsibilities: following responsibilities:
1. Determining which signers will participate (at least 1. Determining which signers will participate (at least
THRESHOLD_LIMIT in number); THRESHOLD_LIMIT in number);
2. Coordinating rounds (receiving and forwarding inputs among 2. Coordinating rounds (receiving and forwarding inputs among
participants); and participants); and
3. Aggregating signature shares output by each participant, and 3. Aggregating signature shares output by each participant, and
publishing the resulting signature. publishing the resulting signature.
FROST assumes the selection of all participants, including the FROST assumes the selection of all participants, including the
dealer, signer, and Coordinator are all chosen external to the dealer, signer, and Coordinator are all chosen external to the
protocol. Note that it is possible to deploy the protocol without a protocol. Note that it is possible to deploy the protocol without a
distinguished Coordinator; see Section 7.3 for more information. distinguished Coordinator; see Section 7.3 for more information.
In FROST, all signers are assumed to have the group state and their Because key generation is not specified, all signers are assumed to
corresponding signing key shares. In particular, FROST assumes that have the (public) group state that we refer to as "group info" below,
each signing participant P_i knows the following: 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 * Group public key, denoted PK = G.ScalarMultBase(s), corresponding
to the group secret key s. to the group secret key s. PK is an output from the group's key
generation protocol, such as trusted_dealer_keygenor a DKG.
* Participant is signing key, which is the i-th secret share of s. * 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 i-th secret
share of s.
The exact key generation mechanism is out of scope for this The exact key generation mechanism is out of scope for this
specification. In general, key generation is a protocol that outputs specification. In general, key generation is a protocol that outputs
(1) a shared, group public key PK owned by each Signer, and (2) (1) a shared, group public key PK owned by each Signer, and (2)
individual shares of the signing key owned by each Signer. In individual shares of the signing key owned by each Signer. In
general, two possible key generation mechanisms are possible, one general, two possible key generation mechanisms are possible, one
that requires a single, trusted dealer, and the other which requires that requires a single, trusted dealer, and the other which requires
performing a distributed key generation protocol. We highlight key performing a distributed key generation protocol. We highlight key
generation mechanism by a trusted dealer in Appendix B, for generation mechanism by a trusted dealer in Appendix B, for
reference. reference.
There are two rounds in FROST: commitment and signature share This signing variant of FROST requires signers to perform two network
generation. The first round serves for each participant to issue a rounds: 1) generating and publishing commitments, and 2) signature
commitment. The second round receives commitments for all signers as share generation and publication. The first round serves for each
well as the message, and issues a signature share. The Coordinator participant to issue a commitment to a nonce. The second round
performs the coordination of each of these rounds. At the end of the receives commitments for all signers as well as the message, and
second round, the Coordinator then performs an aggregation step at issues a signature share with respect to that message. The
the end and outputs the final signature. This complete interaction Coordinator performs the coordination of each of these rounds. At
is shown in Figure 1. the end of the second round, the Coordinator then performs an
aggregation step and outputs the final signature. This complete
interaction is shown in Figure 1.
(group info) (group info, (group info, (group info) (group info, (group info,
| signing key share) signing key share) | signing key share) signing key share)
| | | | | |
v v v v v v
Coordinator Signer-1 ... Signer-n Coordinator Signer-1 ... Signer-n
------------------------------------------------------------ ------------------------------------------------------------
message message
------------> ------------>
| |
== Round 1 (Commitment) == == Round 1 (Commitment) ==
| signer commitment | | | signer commitment | |
skipping to change at page 16, line 20 skipping to change at page 17, line 20
Each signer in round one generates a nonce nonce = (hiding_nonce, Each signer in round one generates a nonce nonce = (hiding_nonce,
binding_nonce) and commitment comm = (hiding_nonce_commitment, binding_nonce) and commitment comm = (hiding_nonce_commitment,
binding_nonce_commitment). binding_nonce_commitment).
Inputs: None Inputs: None
Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs. Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs.
def commit(): def commit():
hiding_nonce = G.RandomScalar() hiding_nonce = G.RandomNonzeroScalar()
binding_nonce = G.RandomScalar() binding_nonce = G.RandomNonzeroScalar()
hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce) hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
binding_nonce_commitment = G.ScalarBaseMult(binding_nonce) binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
nonce = (hiding_nonce, binding_nonce) nonce = (hiding_nonce, binding_nonce)
comm = (hiding_nonce_commitment, binding_nonce_commitment) comm = (hiding_nonce_commitment, binding_nonce_commitment)
return (nonce, comm) return (nonce, comm)
The private output nonce from Participant P_i is stored locally and 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 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 reused in more than one invocation of FROST, and it MUST be generated
from a source of secure randomness. The public output comm from from a source of secure randomness. The public output comm from
skipping to change at page 17, line 45 skipping to change at page 18, line 45
# Compute the per-message challenge # Compute the per-message challenge
challenge = compute_challenge(group_commitment, group_public_key, msg) challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute the signature share # Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i (hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge) sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge)
return sig_share return sig_share
The output of this procedure is a signature share and group The output of this procedure is a signature share. Each signer then
commitment share. Each signer then sends these shares back to the sends these shares back to the collector; see Appendix C.3 for
collector; see Appendix C.3 for encoding recommendations. Each encoding recommendations. Each signer MUST delete the nonce and
signer MUST delete the nonce and corresponding commitment after this corresponding commitment after this round completes.
round completes.
Upon receipt from each Signer, the Coordinator MUST validate the Upon receipt from each Signer, the Coordinator MUST validate the
input signature and commitment shares using DeserializeElement for input signature using DeserializeElement. If validation fails, the
each. If validation fails, the Coordinator MUST abort the protocol. Coordinator MUST abort the protocol. If validation succeeds, the
If validation succeeds, the Coordinator then verifies the set of Coordinator then verifies the set of signature shares using the
signature shares using the following procedure. following procedure.
5.3. Signature Share Verification and Aggregation 5.3. Signature Share Verification and Aggregation
After signers perform round two and send their signature shares to After signers perform round two and send their signature shares to
the Coordinator, the Coordinator verifies each signature share for the Coordinator, the Coordinator verifies each signature share for
correctness. In particular, for each signer, the Coordinator uses correctness. In particular, for each signer, the Coordinator uses
commitment pairs generated during round one and the signature share commitment pairs generated during round one and the signature share
generated during round two, along with other group parameters, to generated during round two, along with other group parameters, to
check that the signature share is valid using the following check that the signature share is valid using the following
procedure. procedure.
Inputs: Inputs:
- index, Index `i` of the signer. Note index will never equal `0`. - index, Index `i` of the signer. Note index will never equal `0`.
- public_key_share_i, the public key for the ith signer, where public_key_share_i = G.ScalarBaseMult(s[i]) - 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 - comm_i, pair of Element values (hiding_nonce_commitment, binding_nonce_commitment) generated
in round one from the ith signer. 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. - 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 - 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 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). two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
This list MUST be sorted in ascending order by signer index. This list MUST be sorted in ascending order by signer index.
- participant_list, a set containing identifiers for each signer, similarly of length - participant_list, a set containing identifiers for each signer, similarly of length
NUM_SIGNERS (sent by the Coordinator). NUM_SIGNERS (sent by the Coordinator).
- group_public_key, the public key for the group - group_public_key, the public key for the group
- msg, the message to be signed - msg, the message to be signed
Outputs: True if the signature share is valid, and False otherwise. Outputs: True if the signature share is valid, and False otherwise.
def verify_signature_share(index, public_key_share_i, comm_i, sig_share_i, commitment_list, def verify_signature_share(index, PK_i, comm_i, sig_share_i, commitment_list,
participant_list, group_public_key, msg): participant_list, group_public_key, msg):
# Encode the commitment list # Encode the commitment list
encoded_commitments = encode_group_commitment_list(commitment_list) encoded_commitments = encode_group_commitment_list(commitment_list)
# Compute the binding factor # Compute the binding factor
binding_factor = compute_binding_factor(encoded_commitments, msg) binding_factor = compute_binding_factor(encoded_commitments, msg)
# Compute the group commitment # Compute the group commitment
group_commitment = compute_group_commitment(commitment_list, binding_factor) group_commitment = compute_group_commitment(commitment_list, binding_factor)
skipping to change at page 19, line 45 skipping to change at page 20, line 45
comm_share = hiding_nonce_commitment + (binding_nonce_commitment * binding_factor) comm_share = hiding_nonce_commitment + (binding_nonce_commitment * binding_factor)
# Compute the challenge # Compute the challenge
challenge = compute_challenge(group_commitment, group_public_key, msg) challenge = compute_challenge(group_commitment, group_public_key, msg)
# Compute Lagrange coefficient # Compute Lagrange coefficient
lambda_i = derive_lagrange_coefficient(index, participant_list) lambda_i = derive_lagrange_coefficient(index, participant_list)
# Compute relation values # Compute relation values
l = G.ScalarBaseMult(sig_share_i) l = G.ScalarBaseMult(sig_share_i)
r = comm_share + (public_key_share_i * challenge * lambda_i) r = comm_share + (PK_i * challenge * lambda_i)
return l == r return l == r
If any signature share fails to verify, i.e., if If any signature share fails to verify, i.e., if
verify_signature_share returns False for any signer share, the verify_signature_share returns False for any signer share, the
Coordinator MUST abort the protocol. Otherwise, if all signer shares Coordinator MUST abort the protocol. Otherwise, if all signer shares
are valid, the Coordinator performs the aggregate operation and are valid, the Coordinator performs the aggregate operation and
publishes the resulting signature. publishes the resulting signature.
Inputs: Inputs:
skipping to change at page 21, line 14 skipping to change at page 22, line 14
6.1. FROST(Ed25519, SHA-512) 6.1. FROST(Ed25519, SHA-512)
This ciphersuite uses edwards25519 for the Group and SHA-512 for the This ciphersuite uses edwards25519 for the Group and SHA-512 for the
Hash function H meant to produce signatures indistinguishable from Hash function H meant to produce signatures indistinguishable from
Ed25519 as specified in [RFC8032]. The value of the contextString Ed25519 as specified in [RFC8032]. The value of the contextString
parameter is empty. parameter is empty.
* Group: edwards25519 [RFC8032] * Group: edwards25519 [RFC8032]
- Cofactor (h): 8
- SerializeElement: Implemented as specified in [RFC8032], - SerializeElement: Implemented as specified in [RFC8032],
Section 5.1.2. Section 5.1.2.
- DeserializeElement: Implemented as specified in [RFC8032], - DeserializeElement: Implemented as specified in [RFC8032],
Section 5.1.3. Additionally, this function validates that the Section 5.1.3. Additionally, this function validates that the
resulting element is not the group identity element. resulting element is not the group identity element.
- SerializeScalar: Implemented by outputting the little-endian - SerializeScalar: Implemented by outputting the little-endian
32-byte encoding of the Scalar value. 32-byte encoding of the Scalar value.
skipping to change at page 22, line 4 skipping to change at page 23, line 6
Normally H2 would also include a domain separator, but for backwards Normally H2 would also include a domain separator, but for backwards
compatibility with [RFC8032], it is omitted. compatibility with [RFC8032], it is omitted.
6.2. FROST(ristretto255, SHA-512) 6.2. FROST(ristretto255, SHA-512)
This ciphersuite uses ristretto255 for the Group and SHA-512 for the This ciphersuite uses ristretto255 for the Group and SHA-512 for the
Hash function H. The value of the contextString parameter is "FROST- Hash function H. The value of the contextString parameter is "FROST-
RISTRETTO255-SHA512". RISTRETTO255-SHA512".
* Group: ristretto255 [RISTRETTO] * Group: ristretto255 [RISTRETTO]
- Cofactor (h): 1
- SerializeElement: Implemented using the 'Encode' function from - SerializeElement: Implemented using the 'Encode' function from
[RISTRETTO]. [RISTRETTO].
- DeserializElement: Implemented using the 'Decode' function from - DeserializeElement: Implemented using the 'Decode' function
[RISTRETTO]. from [RISTRETTO].
- SerializeScalar: Implemented by outputting the little-endian - SerializeScalar: Implemented by outputting the little-endian
32-byte encoding of the Scalar value. 32-byte encoding of the Scalar value.
- DeserializeScalar: Implemented by attempting to deserialize a - DeserializeScalar: Implemented by attempting to deserialize a
Scalar from a 32-byte string. This function can fail if the Scalar from a 32-byte string. This function can fail if the
input does not represent a Scalar between the value 0 and input does not represent a Scalar between the value 0 and
G.Order() - 1. G.Order() - 1.
* Hash (H): SHA-512, and Nh = 64. * Hash (H): SHA-512, and Nh = 64.
- H1(m): Implemented by computing H(contextString || "rho" || m) - H1(m): Implemented by computing H(contextString || "rho" || m)
and mapping the the output to a Scalar as described in and mapping the output to a Scalar as described in [RISTRETTO],
[RISTRETTO], Section 4.4. Section 4.4.
- H2(m): Implemented by computing H(contextString || "chal" || m) - H2(m): Implemented by computing H(contextString || "chal" || m)
and mapping the the output to a Scalar as described in and mapping the output to a Scalar as described in [RISTRETTO],
[RISTRETTO], Section 4.4. Section 4.4.
- H3(m): Implemented by computing H(contextString || "digest" || - H3(m): Implemented by computing H(contextString || "digest" ||
m). m).
6.3. FROST(Ed448, SHAKE256) 6.3. FROST(Ed448, SHAKE256)
This ciphersuite uses edwards448 for the Group and SHA256 for the This ciphersuite uses edwards448 for the Group and SHA256 for the
Hash function H meant to produce signatures indistinguishable from Hash function H meant to produce signatures indistinguishable from
Ed448 as specified in [RFC8032]. The value of the contextString Ed448 as specified in [RFC8032]. The value of the contextString
parameter is empty. parameter is empty.
* Group: edwards448 [RFC8032] * Group: edwards448 [RFC8032]
- Cofactor (h): 4
- SerializeElement: Implemented as specified in [RFC8032], - SerializeElement: Implemented as specified in [RFC8032],
Section 5.2.2. Section 5.2.2.
- DeserializeElement: Implemented as specified in [RFC8032], - DeserializeElement: Implemented as specified in [RFC8032],
Section 5.2.3. Additionally, this function validates that the Section 5.2.3. Additionally, this function validates that the
resulting element is not the group identity element. resulting element is not the group identity element.
- SerializeScalar: Implemented by outputting the little-endian - SerializeScalar: Implemented by outputting the little-endian
48-byte encoding of the Scalar value. 48-byte encoding of the Scalar value.
skipping to change at page 23, line 35 skipping to change at page 24, line 38
compatibility with [RFC8032], it is omitted. compatibility with [RFC8032], it is omitted.
6.4. FROST(P-256, SHA-256) 6.4. FROST(P-256, SHA-256)
This ciphersuite uses P-256 for the Group and SHA-256 for the Hash This ciphersuite uses P-256 for the Group and SHA-256 for the Hash
function H. The value of the contextString parameter is "FROST- function H. The value of the contextString parameter is "FROST-
P256-SHA256". P256-SHA256".
* Group: P-256 (secp256r1) [x9.62] * Group: P-256 (secp256r1) [x9.62]
- Cofactor (h): 1
- SerializeElement: Implemented using the compressed Elliptic- - SerializeElement: Implemented using the compressed Elliptic-
Curve-Point-to-Octet-String method according to [SECG]. Curve-Point-to-Octet-String method according to [SECG].
- DeserializeElement: Implemented by attempting to deserialize a - DeserializeElement: Implemented by attempting to deserialize a
public key using the compressed Octet-String-to-Elliptic-Curve- public key using the compressed Octet-String-to-Elliptic-Curve-
Point method according to [SECG], and then performs partial Point method according to [SECG], and then performs partial
public-key validation as defined in section 5.6.2.3.4 of public-key validation as defined in section 5.6.2.3.4 of
[KEYAGREEMENT]. This includes checking that the coordinates of [KEYAGREEMENT]. This includes checking that the coordinates of
the resulting point are in the correct range, that the point is 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. on the curve, and that the point is not the point at infinity.
skipping to change at page 24, line 25 skipping to change at page 25, line 28
- H2(m): Implemented using hash_to_field from [HASH-TO-CURVE], - H2(m): Implemented using hash_to_field from [HASH-TO-CURVE],
Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST
= contextString || "chal", and prime modulus equal to Order(). = contextString || "chal", and prime modulus equal to Order().
- H3(m): Implemented by computing H(contextString || "digest" || - H3(m): Implemented by computing H(contextString || "digest" ||
m). m).
7. Security Considerations 7. Security Considerations
A security analysis of FROST exists in [FROST20]. The protocol as A security analysis of FROST exists in [FROST20] and [Schnorr21].
specified in this document assumes the following threat model. The protocol as specified in this document assumes the following
threat model.
* Trusted dealer. The dealer that performs key generation is * Trusted dealer. The dealer that performs key generation is
trusted to follow the protocol, although participants still are trusted to follow the protocol, although participants still are
able to verify the consistency of their shares via a VSS able to verify the consistency of their shares via a VSS
(verifiable secret sharing) step; see Appendix B.2. (verifiable secret sharing) step; see Appendix B.2.
* Unforgeability assuming less than (t-1) corrupted signers. So * Unforgeability assuming less than (t-1) corrupted signers. So
long as an adverary corrupts fewer than (t-1) participants, the long as an adverary corrupts fewer than (t-1) participants, the
scheme remains secure against EUF-CMA attacks. scheme remains secure against EUF-CMA attacks.
skipping to change at page 24, line 50 skipping to change at page 26, line 5
completing or causing the resulting signature to be invalid. Such completing or causing the resulting signature to be invalid. Such
actions for the latter include sending inconsistent values to actions for the latter include sending inconsistent values to
signing participants, such as messages or the set of individual signing participants, such as messages or the set of individual
commitments. Note that the Coordinator is _not_ trusted with any commitments. Note that the Coordinator is _not_ trusted with any
private information and communication at the time of signing can private information and communication at the time of signing can
be performed over a public but reliable channel. be performed over a public but reliable channel.
The protocol as specified in this document does not target the The protocol as specified in this document does not target the
following goals: following goals:
* Post quantum security. FROST, like generic Schnorr signatures, * Post quantum security. FROST, like plain Schnorr signatures,
requires the hardness of the Discrete Logarithm Problem. requires the hardness of the Discrete Logarithm Problem.
* Robustness. In the case of failure, FROST requires aborting the * Robustness. In the case of failure, FROST requires aborting the
protocol. protocol.
* Downgrade prevention. The sender and receiver are assumed to * Downgrade prevention. The sender and receiver are assumed to
agree on what algorithms to use. agree on what algorithms to use.
* Metadata protection. If protection for metadata is desired, a * Metadata protection. If protection for metadata is desired, a
higher-level communication channel can be used to facilitate key higher-level communication channel can be used to facilitate key
generation and signing. generation and signing.
The rest of this section documents issues particular to The rest of this section documents issues particular to
implementations or deployments. implementations or deployments.
7.1. Nonce Reuse Attacks 7.1. Nonce Reuse Attacks
Nonces generated by each participant in the first round of signing Nonces generated by each participant in the first round of signing
must be sampled uniformly at random and cannot be derived from some must be sampled uniformly at random and cannot be derived from some
determinstic function. This is to avoid replay attacks initiated by deterministic function. This is to avoid replay attacks initiated by
other signers, which allows for a complete key-recovery attack. other signers, which allows for a complete key-recovery attack.
Coordinates MAY further hedge against nonce reuse attacks by tracking Coordinates MAY further hedge against nonce reuse attacks by tracking
signer nonce commitments used for a given group key, at the cost of signer nonce commitments used for a given group key, at the cost of
additional state. additional state.
7.2. Protocol Failures 7.2. Protocol Failures
We do not specify what implementations should do when the protocol We do not specify what implementations should do when the protocol
fails, other than requiring that the protocol abort. Examples of fails, other than requiring that the protocol abort. Examples of
viable failure include when a verification check returns invalid or viable failure include when a verification check returns invalid or
skipping to change at page 26, line 43 skipping to change at page 27, line 46
However, it is RECOMMENDED that applications take additional However, it is RECOMMENDED that applications take additional
precautions and validate inputs so that signers do not operate as precautions and validate inputs so that signers do not operate as
signing oracles for arbitrary messages. signing oracles for arbitrary messages.
8. Contributors 8. Contributors
* Isis Lovecruft * Isis Lovecruft
* T. Wilson-Brown * T. Wilson-Brown
* Alden Torres
9. References 9. References
9.1. Normative References 9.1. Normative References
[HASH-TO-CURVE] [HASH-TO-CURVE]
Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S., Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
and C. A. Wood, "Hashing to Elliptic Curves", Work in and C. A. Wood, "Hashing to Elliptic Curves", Work in
Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve- Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
14, 18 February 2022, 14, 18 February 2022,
<https://datatracker.ietf.org/doc/html/draft-irtf-cfrg- <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
skipping to change at page 28, line 42 skipping to change at page 29, line 42
random and uses Shamir and Verifiable Secret Sharing as described in 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 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 to be sent to all other participants. We highlight at a high level
how this operation can be performed. how this operation can be performed.
Inputs: Inputs:
- s, a group secret that MUST be derived from at least `Ns` bytes of entropy - s, a group secret that MUST be derived from at least `Ns` bytes of entropy
- n, the number of shares to generate, an integer - n, the number of shares to generate, an integer
- t, the threshold of the secret sharing scheme, an integer - t, the threshold of the secret sharing scheme, an integer
Outputs: a public key Element, along with `n`shares of the secret key, each a Scalar value. 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): def trusted_dealer_keygen(s, n, t):
secret_key_shares = secret_share_shard(secret_key, n, t) signer_private_keys, coefficients = secret_share_shard(secret_key, n, t)
public_key = G.ScalarBaseMult(secret_key) vss_commitment = vss_commit(coefficients):
return public_key, secret_key_shares PK = G.ScalarBaseMult(secret_key)
return signer_private_keys, vss_commitment
It is assumed the dealer then sends one secret key share to each of It is assumed the dealer then sends one secret key share to each of
the NUM_SIGNERS participants. The trusted dealer MUST delete the the NUM_SIGNERS participants, along with C. After receiving their
secret_key and secret_key_shares upon completion. 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 Use of this method for key generation requires a mutually
authenticated secure channel between the dealer and participants to authenticated secure channel between the dealer and participants to
send secret key shares, wherein the channel provides confidentiality send secret key shares, wherein the channel provides confidentiality
and integrity. Mutually authenticated TLS is one possible deployment and integrity. Mutually authenticated TLS is one possible deployment
option. option.
B.1. Shamir Secret Sharing B.1. Shamir Secret Sharing
In Shamir secret sharing, a dealer distributes a secret s to n In Shamir secret sharing, a dealer distributes a secret s to n
participants in such a way that any cooperating subset of t participants in such a way that any cooperating subset of t
participants can recover the secret. There are two basic steps in participants can recover the secret. There are two basic steps in
this scheme: (1) splitting a secret into multiple shares, and (2) this scheme: (1) splitting a secret into multiple shares, and (2)
combining shares to reveal the resulting secret. combining shares to reveal the resulting secret.
This secret sharing scheme works over any field F. In this This secret sharing scheme works over any field F. In this
specification, F is the scalar field of the prime-order group G. specification, F is the scalar field of the prime-order group G.
The procedure for splitting a secret into shares is as follows. The procedure for splitting a secret into shares is as follows.
secret_share_shard(s, n, t): secret_share_shard(s, n, t):
Inputs: Inputs:
- s, secret to be shared, an element of F - s, secret to be shared, an element of F
- n, the number of shares to generate, an integer - n, the number of shares to generate, an integer
- t, the threshold of the secret sharing scheme, 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 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: Errors:
- "invalid parameters", if t > n or if t is less than 2 - "invalid parameters", if t > n or if t is less than 2
def secret_share_shard(s, n, t): def secret_share_shard(s, n, t):
if t > n: if t > n:
raise "invalid parameters" raise "invalid parameters"
if t < 2: if t < 2:
raise "invalid parameters" raise "invalid parameters"
# Generate random coefficients for the polynomial, yielding # Generate random coefficients for the polynomial, yielding
# a polynomial of degree (t - 1) # a polynomial of degree (t - 1)
coefficients = [s] coefficients = [s]
for i in range(t - 1): for i in range(t - 1):
coefficients.append(G.RandomScalar()) coefficients.append(G.RandomScalar())
# Evaluate the polynomial for each point x=1,...,n # Evaluate the polynomial for each point x=1,...,n
points = [] secret_key_shares = []
for x_i in range(1, n + 1): for x_i in range(1, n + 1):
y_i = polynomial_evaluate(x_i, coefficients) y_i = polynomial_evaluate(x_i, coefficients)
point_i = (x_i, y_i) secret_key_share_i = (x_i, y_i)
points.append(point_i) secret_key_share.append(secret_key_share_i)
return points return secret_key_shares, coefficients
Let points be the output of this function. The i-th element in 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 points is the share for the i-th participant, which is the randomly
generated polynomial evaluated at coordinate i. We denote a secret generated polynomial evaluated at coordinate i. We denote a secret
share as the tuple (i, points[i]), and the list of these shares as 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 shares. i MUST never equal 0; recall that f(0) = s, where f is the
polynomial defined in a Shamir secret sharing operation. polynomial defined in a Shamir secret sharing operation.
The procedure for combining a shares list of length t to recover the The procedure for combining a shares list of length t to recover the
secret s is as follows. secret s is as follows.
skipping to change at page 31, line 28 skipping to change at page 32, line 28
s = polynomial_interpolation(shares) s = polynomial_interpolation(shares)
return s return s
B.2. Verifiable Secret Sharing B.2. Verifiable Secret Sharing
Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret
sharing, adding a verification step to demonstrate the consistency of sharing, adding a verification step to demonstrate the consistency of
a participant's share with a public commitment to the polynomial f 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 for which the secret s is the constant term. This check ensure that
all participants have a point (their share) on the same polynomial, all participants have a point (their share) on the same polynomial,
ensuring that they can later reconstruct the correct secret. If the ensuring that they can later reconstruct the correct secret.
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 The procedure for committing to a polynomial f of degree t-1 is as
follows. follows.
vss_commit(coeffs): vss_commit(coeffs):
Inputs: Inputs:
- coeffs, a vector of the t coefficients which uniquely determine - coeffs, a vector of the t coefficients which uniquely determine
a polynomial f. a polynomial f.
Outputs: a commitment C, which is a vector commitment to each of the Outputs: a commitment vss_commitment, which is a vector commitment to each of the
coefficients in coeffs. coefficients in coeffs.
def vss_commit(coeffs): def vss_commit(coeffs):
C = [] vss_commitment = []
for coeff in coeffs: for coeff in coeffs:
A_i = ScalarBaseMult(coeff) A_i = G.ScalarBaseMult(coeff)
C.append(A_i) vss_commitment.append(A_i)
return C return vss_commitment
The procedure for verification of a participant's share is as The procedure for verification of a participant's share is as
follows. follows. If vss_verify fails, the participant MUST abort the
protocol, and failure should be investigated out of band.
vss_verify(sk_i, C): vss_verify(share_i, vss_commitment):
Inputs: Inputs:
- sk_i: A participant's secret key, the tuple sk_i = (i, s[i]), - share_i: A tuple of the form (i, sk_i), where i indicates the participant
where s[i] is a secret share of the constant term of f. identifier, and sk_i the participant's secret key, where sk_i is a secret share of
- C: A VSS commitment to a secret polynomial f. the constant term of f.
- vss_commitment: A VSS commitment to a secret polynomial f.
Outputs: 1 if s[i] is valid, and 0 otherwise Outputs: 1 if sk_i is valid, and 0 otherwise
vss_verify(sk_i, commitment) vss_verify(share_i, commitment)
S_i = ScalarBaseMult(s[i]) (i, sk_i) = share_i
S_i' = SUM(commitment[0], commitment[t-1]){A_j}: A_j*(i^j) S_i = ScalarBaseMult(sk_i)
if S_i == S_i': S_i' = G.Identity()
return 1 for j in range(0, THRESHOLD_LIMIT-1):
return 0 S_i' += vss_commitment_j * i^j
if S_i == S_i':
return 1
return 0
We now define how the coordinator and signing participants can derive
group info, which is an input into the FROST signing protocol.
derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment):
Inputs:
- MAX_SIGNERS, the number of shares to generate, an integer
- THRESHOLD_LIMIT, the threshold of the secret sharing scheme, an integer
- vss_commitment: A VSS commitment to a secret polynomial f.
Outputs:
- PK, the public key representing the group
- signer_public_keys, a list of MAX_SIGNERS public keys PK_i for i=1,...,MAX_SIGNERS, where PK_i is the public key for participant i.
derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment)
PK = vss_commitment[0]
signer_public_keys = []
for i in range(1, MAX_SIGNERS):
PK_i = G.Identity()
for j in range(0, THRESHOLD_LIMIT-1):
PK_i += vss_commitment_j * i^j
signer_public_keys.append(PK_i)
return PK, signer_public_keys
Appendix C. Wire Format Appendix C. Wire Format
Applications are responsible for encoding protocol messages between Applications are responsible for encoding protocol messages between
peers. This section contains RECOMMENDED encodings for different peers. This section contains RECOMMENDED encodings for different
protocol messages as described in Section 5. protocol messages as described in Section 5.
C.1. Signing Commitment C.1. Signing Commitment
A commitment from a signer is a pair of Element values. It can be A commitment from a signer is a pair of Element values. It can be
skipping to change at page 33, line 18 skipping to change at page 34, line 45
Each package contains the list of signing commitments generated Each package contains the list of signing commitments generated
during round one along with the message to sign. This package can be during round one along with the message to sign. This package can be
encoded in the following manner. encoded in the following manner.
struct { struct {
SigningCommitment signing_commitments<1..2^16-1>; SigningCommitment signing_commitments<1..2^16-1>;
opaque msg<0..2^16-1>; opaque msg<0..2^16-1>;
} SigningPackage; } SigningPackage;
signing_commitments An list of SIGNING_COUNT SigningCommitment signing_commitments An list of SIGNING_COUNT SigningCommitment
values, where THRESHOLD_LIIMT <= SIGNING_COUNT <= NUM_SIGNERS, values, where THRESHOLD_LIMIT <= SIGNING_COUNT <= NUM_SIGNERS,
ordered in ascending order by SigningCommitment.id. This list ordered in ascending order by SigningCommitment.id. This list
MUST NOT contain more than one SigningCommitment value MUST NOT contain more than one SigningCommitment value
corresponding to each signer. Signers MUST ignore SigningPackage corresponding to each signer. Signers MUST ignore SigningPackage
values with duplicate SignerIDs. values with duplicate SignerIDs.
msg The message to be signed. msg The message to be signed.
C.3. Signature Share C.3. Signature Share
The output of each signer is a signature share which is sent to the The output of each signer is a signature share which is sent to the
Coordinator. This can be constructed as follows. Coordinator. This can be constructed as follows.
struct { struct {
SignerID id; SignerID id;
opaque signature_share[Ns]; opaque signature_share[Ns];
opaque commitment_share[Ne];
} SignatureShare; } SignatureShare;
id The SignerID. id The SignerID.
signature_share The signature share from this signer encoded as a signature_share The signature share from this signer encoded as a
serialized scalar. serialized scalar.
commitment_share The signature commitment share from this signer
encoded as a serialized element.
Appendix D. Test Vectors Appendix D. Test Vectors
This section contains test vectors for all ciphersuites listed in This section contains test vectors for all ciphersuites listed in
Section 6. All Element and Scalar values are represented in Section 6. All Element and Scalar values are represented in
serialized form and encoded in hexadecimal strings. Signatures are serialized form and encoded in hexadecimal strings. Signatures are
represented as the concatenation of their constituent parts. The represented as the concatenation of their constituent parts. The
input message to be signed is also encoded as a hexadecimal string. input message to be signed is also encoded as a hexadecimal string.
Each test vector consists of the following information. Each test vector consists of the following information.
skipping to change at page 34, line 29 skipping to change at page 36, line 7
* Round one parameters and outputs: This lists the NUM_SIGNERS * Round one parameters and outputs: This lists the NUM_SIGNERS
participants engaged in the protocol, identified by their integer participants engaged in the protocol, identified by their integer
index, the hiding and binding commitment values produced in index, the hiding and binding commitment values produced in
Section 5.1, as well as the resulting group binding factor input, Section 5.1, as well as the resulting group binding factor input,
computed in part from the group commitment list encoded as computed in part from the group commitment list encoded as
described in Section 4.3, and group binding factor as computed in described in Section 4.3, and group binding factor as computed in
Section 5.2). Section 5.2).
* Round two parameters and outputs: This lists the NUM_SIGNERS * Round two parameters and outputs: This lists the NUM_SIGNERS
participants engaged in the protocol, identified by their integer participants engaged in the protocol, identified by their integer
index, along with their corresponding output signature share and index, along with their corresponding output signature share as
group commitment share as produced in Section 5.2. produced in Section 5.2.
* Final output: This lists the aggregate signature as produced in * Final output: This lists the aggregate signature as produced in
Section 5.3. Section 5.3.
D.1. FROST(Ed25519, SHA-512) D.1. FROST(Ed25519, SHA-512)
// Configuration information // Configuration information
MAX_SIGNERS: 3 MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2 THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2 NUM_SIGNERS: 2
// Group input parameters // Group input parameters
group_secret_key: 7c1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7 group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
90c6e13a98304 90c6e13a98304
group_public_key: 377a6acb3b9b5f642c5ce355d23cac0568aad0da63c633d59d4 group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040
168bdcbce35af d380fb9738673
message: 74657374 message: 74657374
// Signer input parameters // Signer input parameters
S1 signer_share: 949dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033 S1 signer_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033
f2ec83d93509 f2ec83d93509
S2 signer_share: ac1e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee S2 signer_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee
d76bf409e80d d76bf409e80d
S3 signer_share: d7cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8 S3 signer_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8
bdea643a9a02 bdea643a9a02
// Round one parameters // Round one parameters
participants: 1,2 participants: 1,2
group_binding_factor_input: 000178e175d15cb5cec1257e0d84d797ba8c3dd9b group_binding_factor_input: 000178e175d15cb5cec1257e0d84d797ba8c3dd9b
4c7bc50f3fa527c200bcc6c4a954cdad16ae67ac5919159d655b681bd038574383bab 4c7bc50f3fa527c200bcc6c4a954cdad16ae67ac5919159d655b681bd038574383bab
423614f8967396ee12ca62000288a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea98 423614f8967396ee12ca62000288a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea98
981bfb25375996c5767c932bbf10c41feb17d41cc6433e69f16cceccc42a00aedf72f 981bfb25375996c5767c932bbf10c41feb17d41cc6433e69f16cceccc42a00aedf72f
eb5f44929fdf2e2fee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f8819a eb5f44929fdf2e2fee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f8819a
5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f500 5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f500
skipping to change at page 35, line 42 skipping to change at page 37, line 20
5d2d5dfc5620c 5d2d5dfc5620c
S2 hiding_nonce_commitment: 88a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea S2 hiding_nonce_commitment: 88a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea
98981bfb25375996c5767c9 98981bfb25375996c5767c9
S2 binding_nonce_commitment: 32bbf10c41feb17d41cc6433e69f16cceccc42a0 S2 binding_nonce_commitment: 32bbf10c41feb17d41cc6433e69f16cceccc42a0
0aedf72feb5f44929fdf2e2f 0aedf72feb5f44929fdf2e2f
// Round two parameters // Round two parameters
participants: 1,2 participants: 1,2
// Signer round two outputs // Signer round two outputs
S1 sig_share: f8bbaf924e1c90e11dec1eb679194aade084f92fbf52fdd436ba0a0 S1 sig_share: b7e8f03a1a1149adacb96f952dbc39b6034facceafe4a70d6963592
7f71ab708 fce75570c
S1 group_commitment_share: 11bb9777aa393b92e814e415039adf62687a0be543 S2 sig_share: cd388f9aff4376397c5ad231713fe6b167bed9cc88a1cc97b0b6bbe
c2322d817e4934bc5a7cf2 0316a7909
S2 sig_share: 80f589405714ca0e6adc87c2c0186a0ae4d6e352f7b248b23149a5d
cd3fe4704
S2 group_commitment_share: 4af85179d17ed031b767ab579e59c7018dac09ae40
0b1700623d0af1129a9c55
sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e
78b139d3a5305af087c8a6783a32b4b7c45bdd82b60546876803b0e3ca19ff0c 974d8a78fff1ac8e52774a24c00141536b0d869b388674a5191a151000e0d005
D.2. FROST(Ed448, SHAKE256) D.2. FROST(Ed448, SHAKE256)
// Configuration information // Configuration information
MAX_SIGNERS: 3 MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2 THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2 NUM_SIGNERS: 2
// Group input parameters // Group input parameters
group_secret_key: cef4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35 group_secret_key: cdf4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35
4a9a3af62b8c7c89753055949cab8fd044c17c94211f167672b053659420b00 4a9a3af62b8c7c89753055949cab8fd044c17c94211f167672b053659420b00
group_public_key: 005f23508a78131aee4d6cb027f967d89557ec5f24dc3ebeede group_public_key: 800e9b495543b04aaebdba2813de65d1aefe78e8b219d38966b
b550466fcc1411283ff5d9c605d9a8b36e6eea36b67ceba047d57968896db80 c0afa1d5d9d685c740c8ab720bff3c84cd9f4a701c1588e40d981f4abb19600
message: 74657374 message: 74657374
// Signer input parameters // Signer input parameters
S1 signer_share: d408a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08 S1 signer_share: d208a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08
175d4298dd6bbe2e1c0b29aaa972c78555ea3b3d7308b248994780219e0800 175d4298dd6bbe2e1c0b29aaa972c78555ea3b3d7308b248994780219e0800
S2 signer_share: da1c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc S2 signer_share: d71c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc
8416d5cd0210b4c5e410f90a892c91065fbdae37d51ffc29078acae9f90500 8416d5cd0210b4c5e410f90a892c91065fbdae37d51ffc29078acae9f90500
S3 signer_share: e03094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f S3 signer_share: dc3094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f
f2cf670328b4a95cad16c96b68e65a87689021323737460b75cc14b2550300 f2cf670328b4a95cad16c96b68e65a87689021323737460b75cc14b2550300
// Round one parameters // Round one parameters
participants: 1,2 participants: 1,2
group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b
6841aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c8 6841aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c8
9a870400a0c29f750605b10c52e347fc538af0d4ebddd23a1e0300482a7d98a39d408 9a870400a0c29f750605b10c52e347fc538af0d4ebddd23a1e0300482a7d98a39d408
356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c1753a80000242c2fdc11e5f 356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c1753a80000242c2fdc11e5f
726d4c897ed118f668a27bfb0d5946b5f513e975638b7c4b0a46cf5184d4a9c1f6310 726d4c897ed118f668a27bfb0d5946b5f513e975638b7c4b0a46cf5184d4a9c1f6310
fd3c10f84d9de704a33aab2af976d60804fa4ecba88458bcf7677a3952f540e20556d fd3c10f84d9de704a33aab2af976d60804fa4ecba88458bcf7677a3952f540e20556d
skipping to change at page 37, line 22 skipping to change at page 38, line 44
6b5f513e975638b7c4b0a46cf5184d4a9c1f6310fd3c10f84d9de704a33aab2af976d 6b5f513e975638b7c4b0a46cf5184d4a9c1f6310fd3c10f84d9de704a33aab2af976d
6080 6080
S2 binding_nonce_commitment: 4fa4ecba88458bcf7677a3952f540e20556d5e90 S2 binding_nonce_commitment: 4fa4ecba88458bcf7677a3952f540e20556d5e90
d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e1b0b d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e1b0b
28a80 28a80
// Round two parameters // Round two parameters
participants: 1,2 participants: 1,2
// Signer round two outputs // Signer round two outputs
S1 sig_share: ad41cd3320c82edd20c344769bd7b250105d9d0516109b7f774c297 S1 sig_share: c5ab0a80c561d1a616ac70f4f13d993156f65f2b44a4a90f37f0640
faaf8b3b6065b19bbae2afb6c34cce460b40e15655fb8ad0bcc26e21e00 7a1b62e3940bf14199301d128358b812bef32cb4bffaf03030238772000
S1 group_commitment_share: 086d4d2ff2555fab65afc8eb473cc708f37cdb9c5d S2 sig_share: 15211cb96d6aa73de803d46caf2043859fd796a6282f9adb00033f1
e74d8e12a1a9d1a086a8914175e4db77e5d281f10441913aa680fedb207c954afdd88 4f4827f23f8cc792c2e322a1f30631ec7690ac587e5eb9c2afd323e3300
380
S2 sig_share: 5dcc0aec7d0a71eddd5ba2dd0f185ba7990bcd39b6fc0e4b0470c35
6ed0deb736d7f2652e87e932a0c176cc4bc5ba0ef756cc62081e4f51900
S2 group_commitment_share: 7e91f66097b6450c52c89c14400a506ee1d37f5e52
a8d4c3fc9733c23d0b27cd6cfce55a8aee692262e5815be341e8d0b9d240a9630c9f0
600
sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353 sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353
37e4d1aae5fd1c73a55701133238602f423886fc134a3c65800a0ed81f9ed29fcafe1 37e4d1aae5fd1c73a55701133238602f423886fc134a3c6580e787ce8da00900c1a92
ee753abef0df8a9686a3fcc0caaca7bbcecd597069f2a74da3f0d97a98e9740e35025 07fd32e9c6f956597202323f8f4264ecfd99e9539ae5c388c8e45c133fb4765ee9ff2
716ab554d524742c4d0bd83800 583d90d3e49ba02dff6ab51300
D.3. FROST(ristretto255, SHA-512) D.3. FROST(ristretto255, SHA-512)
// Configuration information // Configuration information
MAX_SIGNERS: 3 MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2 THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2 NUM_SIGNERS: 2
// Group input parameters // Group input parameters
group_secret_key: b120be204b5e758960458ca9c4675b56b12a8faff2be9c94891 group_secret_key: b020be204b5e758960458ca9c4675b56b12a8faff2be9c94891
d5e1cd75c880e d5e1cd75c880e
group_public_key: 563b80013f337deaa2a282af7b281bd70d2f501928a89c1aa48 group_public_key: e22ac4850672021eac8e0a36dfc4811466fb01108c3427d2347
b379a5ac4202b 827467ba02a34
message: 74657374 message: 74657374
// Signer input parameters // Signer input parameters
S1 signer_share: 94ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f S1 signer_share: 92ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f
4693ddcb6705 4693ddcb6705
S2 signer_share: 641003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521 S2 signer_share: 611003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521
2e0ae43a470c 2e0ae43a470c
S3 signer_share: 479eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22 S3 signer_share: 439eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22
1681eaa92603 1681eaa92603
// Round one parameters // Round one parameters
participants: 1,2 participants: 1,2
group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca
094f65b814a8bca6013d9cc312774c7e1271d2939a84a9a867e3a06579b4d25659b42 094f65b814a8bca6013d9cc312774c7e1271d2939a84a9a867e3a06579b4d25659b42
7439ccf0d745b43f75b76600028013834ff4d48e7d6b76c2e732bc611f54720ef8933 7439ccf0d745b43f75b76600028013834ff4d48e7d6b76c2e732bc611f54720ef8933
c4ca4de7eaaa77ff5cd125e056ecc4f7c4657d3a742354430d768f945db229c335d25 c4ca4de7eaaa77ff5cd125e056ecc4f7c4657d3a742354430d768f945db229c335d25
8e9622ad99f3e7582d07b35bd9849ce4af6ad403090d69a7d0eb88bba669a9f985175 8e9622ad99f3e7582d07b35bd9849ce4af6ad403090d69a7d0eb88bba669a9f985175
d70cd15ad5f1ef5b734c98a32b4aab7b43a57e93fc09281f2e7a207076b31e416ba63 d70cd15ad5f1ef5b734c98a32b4aab7b43a57e93fc09281f2e7a207076b31e416ba63
skipping to change at page 38, line 46 skipping to change at page 40, line 14
19c43f48ce306 19c43f48ce306
S2 hiding_nonce_commitment: 8013834ff4d48e7d6b76c2e732bc611f54720ef89 S2 hiding_nonce_commitment: 8013834ff4d48e7d6b76c2e732bc611f54720ef89
33c4ca4de7eaaa77ff5cd12 33c4ca4de7eaaa77ff5cd12
S2 binding_nonce_commitment: 5e056ecc4f7c4657d3a742354430d768f945db22 S2 binding_nonce_commitment: 5e056ecc4f7c4657d3a742354430d768f945db22
9c335d258e9622ad99f3e758 9c335d258e9622ad99f3e758
// Round two parameters // Round two parameters
participants: 1,2 participants: 1,2
// Signer round two outputs // Signer round two outputs
S1 sig_share: ec6b075f17c5670e80b1fda8f6de1cfe3c79db06a852f8d5650fb71 S1 sig_share: 6a539c3a4ee281879a6fb350d20d53e17473f28cd3409ffc238dafe
eaad69501 8d9330605
S1 group_commitment_share: bc7e792fce347a15d547935652377c406cc721965c S2 sig_share: 1d4e59636ee089bfaf548834b07658216649a37f87f0818d5190aa9
58d3003dbd947a6dfddc0c b90957505
S2 sig_share: 87ceccc477069aa9b751b307f25955daaf943a3abc51f214a114781
de0f58e03
S2 group_commitment_share: 92c4f352ec392ba779271dc2ed09cda37f38d8c283
747d4a85b4c9ce7289cb07
sig: 7e92309bf40993141acd5f2c7680a302cc5aa5dd291a833906da8e35bc39b03e sig: 7e92309bf40993141acd5f2c7680a302cc5aa5dd291a833906da8e35bc39b03e
733ad4238fcb01b83703b1b0e83872d8ec0d164164a4eaea06242f3c8acc2405 87a1f59dbcc20b474ac43b858284ab02dbbc950c5b31218a751d5a846ac97b0a
D.4. FROST(P-256, SHA-256) D.4. FROST(P-256, SHA-256)
// Configuration information // Configuration information
MAX_SIGNERS: 3 MAX_SIGNERS: 3
THRESHOLD_LIMIT: 2 THRESHOLD_LIMIT: 2
NUM_SIGNERS: 2 NUM_SIGNERS: 2
// Group input parameters // Group input parameters
group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0 group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0
3a6a51a5a5929 3a6a51a5a5928
group_public_key: 03db0945167b62e6472ad46373b6cbbca88e2a9a4883071f0b3 group_public_key: 033a2a83f9c9fdfdab7d620f48238a5e6157a8eb1d6c382c7b0
fde4b2b6d7b6ba6 ba95b7c9f69679c
message: 74657374 message: 74657374
// Signer input parameters // Signer input parameters
S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678 S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678
6f1e410dcecf 6f1e410dcecd
S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed
379767c14475 379767c14472
S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62 S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62
00108e74ba1b 00108e74ba17
// Round one parameters // Round one parameters
participants: 1,2 participants: 1,2
group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7 group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7
f702906930766b86c462bb8ff7f3402b724640ea9e262469f401c9006991ba3247c2c f702906930766b86c462bb8ff7f3402b724640ea9e262469f401c9006991ba3247c2c
91b97cdb1f0eeab1a777e24e1e0002037f8a998dfc2e60a7ad63bc987cb27b8abf78a 91b97cdb1f0eeab1a777e24e1e0002037f8a998dfc2e60a7ad63bc987cb27b8abf78a
68bd924ec6adb9f251850cbe711024a4e90422a19dd8463214e997042206c39d3df56 68bd924ec6adb9f251850cbe711024a4e90422a19dd8463214e997042206c39d3df56
168b458592462090c89dbcf84efca0c54f70a585d6aae28679482b4aed03ae5d38297 168b458592462090c89dbcf84efca0c54f70a585d6aae28679482b4aed03ae5d38297
b9092ab3376d46fdf55 b9092ab3376d46fdf55
group_binding_factor: 9df349a9f34bf01627f6b4f8b376e8c8261d55508d1cac2 group_binding_factor: 9df349a9f34bf01627f6b4f8b376e8c8261d55508d1cac2
skipping to change at page 40, line 17 skipping to change at page 41, line 29
5a7c02f11e3e0 5a7c02f11e3e0
S2 hiding_nonce_commitment: 037f8a998dfc2e60a7ad63bc987cb27b8abf78a68 S2 hiding_nonce_commitment: 037f8a998dfc2e60a7ad63bc987cb27b8abf78a68
bd924ec6adb9f251850cbe711 bd924ec6adb9f251850cbe711
S2 binding_nonce_commitment: 024a4e90422a19dd8463214e997042206c39d3df S2 binding_nonce_commitment: 024a4e90422a19dd8463214e997042206c39d3df
56168b458592462090c89dbcf8 56168b458592462090c89dbcf8
// Round two parameters // Round two parameters
participants: 1,2 participants: 1,2
// Signer round two outputs // Signer round two outputs
S1 sig_share: 120a8ef8a5936444d8087cb10df5648629895e94582720760a10c8c S1 sig_share: 0a658fe198caddf5ddc407ad58c4615458f02a58d0c1f7a38e25692
217e3417b 98dc41df0
S1 group_commitment_share: 0314cc3c03885953b3a15482d0ef4716eba9aca439 S2 sig_share: e84d948cfec74b5e7540ad09fd69dcd1570f708f2d8573dbbf08cb0
fa541a37489a146dd383d07e 2bc872c75
S2 sig_share: 2a7ff42d849f1bcc0e5f75d5810900a3e8f68ab717ff10d7a6da89f
8bb0c16aa
S2 group_commitment_share: 034f4516676fe414883397fc5e150375c9937b3b77
3c9ac57a11b2b6574e6bf452
sig: 035cfbd148da711bbc823455b682ed01a1be3c5415cf692f4a91b7fe22d1dec3 sig: 035cfbd148da711bbc823455b682ed01a1be3c5415cf692f4a91b7fe22d1dec3
453c8a83262a328010e667f2868efe652a127fe94b7026314db0eb52bad2ef5825 45f2b3246e979229545304b4b7562e3e25afff9ae7fe476b7f4d2e342c4a4b4a65
Authors' Addresses Authors' Addresses
Deirdre Connolly Deirdre Connolly
Zcash Foundation Zcash Foundation
Email: durumcrustulum@gmail.com Email: durumcrustulum@gmail.com
Chelsea Komlo Chelsea Komlo
University of Waterloo, Zcash Foundation University of Waterloo, Zcash Foundation
Email: ckomlo@uwaterloo.ca Email: ckomlo@uwaterloo.ca
 End of changes. 96 change blocks. 
235 lines changed or deleted 312 lines changed or added

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