The Flexible Round-Optimized Schnorr Threshold (FROST) Protocol for Two‑Round Schnorr SignaturesZcash Foundationdurumcrustulum@gmail.comUniversity of Waterloo, Zcash Foundationckomlo@uwaterloo.caUniversity of Waterlooiang@uwaterloo.caCloudflarecaw@heapingbits.netCrypto ForumThis document specifies the Flexible Round-Optimized Schnorr Threshold
(FROST) signing protocol. FROST signatures can be issued after a threshold
number of entities cooperate to compute a signature, allowing for improved
distribution of trust and redundancy with respect to a secret key. FROST
depends only on a prime-order group and cryptographic hash function. This
document specifies a number of ciphersuites to instantiate FROST using
different prime-order groups and hash functions. This document is a product
of the Crypto Forum Research Group (CFRG) in the IRTF.Status of This Memo
This document is not an Internet Standards Track specification; it is
published for informational purposes.
This document is a product of the Internet Research Task Force
(IRTF). The IRTF publishes the results of Internet-related
research and development activities. These results might not be
suitable for deployment. This RFC represents the consensus of the Crypto Forum
Research Group of the Internet Research Task Force (IRTF).
Documents approved for publication by the IRSG are not
candidates for any level of Internet Standard; see Section 2 of RFC
7841.
Information about the current status of this document, any
errata, and how to provide feedback on it may be obtained at
.
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document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
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Table of Contents
. Introduction
. Conventions and Definitions
. Cryptographic Dependencies
. Prime-Order Group
. Cryptographic Hash Function
. Helper Functions
. Nonce Generation
. Polynomials
. List Operations
. Binding Factors Computation
. Group Commitment Computation
. Signature Challenge Computation
. Two-Round FROST Signing Protocol
. Round One - Commitment
. Round Two - Signature Share Generation
. Signature Share Aggregation
. Identifiable Abort
. Ciphersuites
. FROST(Ed25519, SHA-512)
. FROST(ristretto255, SHA-512)
. FROST(Ed448, SHAKE256)
. FROST(P-256, SHA-256)
. FROST(secp256k1, SHA-256)
. Ciphersuite Requirements
. Security Considerations
. Side-Channel Mitigations
. Optimizations
. Nonce Reuse Attacks
. Protocol Failures
. Removing the Coordinator Role
. Input Message Hashing
. Input Message Validation
. IANA Considerations
. References
. Normative References
. Informative References
. Schnorr Signature Encoding
. Schnorr Signature Generation and Verification for Prime-Order Groups
. Trusted Dealer Key Generation
. Shamir Secret Sharing
. Additional Polynomial Operations
. Verifiable Secret Sharing
. Random Scalar Generation
. Rejection Sampling
. Wide Reduction
. Test Vectors
. FROST(Ed25519, SHA-512)
. FROST(Ed448, SHAKE256)
. FROST(ristretto255, SHA-512)
. FROST(P-256, SHA-256)
. FROST(secp256k1, SHA-256)
Acknowledgments
Authors' Addresses
IntroductionUnlike signatures in a single-party setting, threshold signatures
require cooperation among a threshold number of signing participants, each holding a share
of a common private key. The security of threshold schemes in general assumes
that an adversary can corrupt strictly fewer than a threshold number of signer participants.This document specifies the Flexible Round-Optimized Schnorr Threshold (FROST) signing protocol
based on the original work in . FROST reduces network overhead during
threshold signing operations while employing a novel technique to protect against forgery
attacks applicable to prior Schnorr-based threshold signature constructions. FROST requires
two rounds to compute a signature. Single-round signing variants based on are out of scope.FROST depends only on a prime-order group and cryptographic hash
function. This document specifies a number of ciphersuites to
instantiate FROST using different prime-order groups and hash functions.
Two ciphersuites can be used to produce signatures that are compatible
with Edwards-Curve Digital Signature Algorithm (EdDSA) variants Ed25519
and Ed448 as specified in , i.e., the signatures
can be verified with a verifier that is compliant with .
However, unlike EdDSA, the signatures produced by FROST are
not deterministic, since deriving nonces deterministically allows for a
complete key-recovery attack in multi-party, discrete logarithm-based
signatures.Key generation for FROST signing is out of scope for this document. However, for completeness,
key generation with a trusted dealer is specified in .This document represents the consensus of the Crypto Forum Research
Group (CFRG). It is not an IETF product and is not a standard.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
when, and only when, they appear in all capitals, as shown here.
The following notation is used throughout the document.
byte:
A sequence of eight bits.
random_bytes(n):
Outputs n bytes, sampled uniformly at random
using a cryptographically secure pseudorandom number generator (CSPRNG).
count(i, L):
Outputs the number of times the element i is represented in the list L.
len(l):
Outputs the length of list l, e.g., len([1,2,3]) = 3.
reverse(l):
Outputs the list l in reverse order, e.g., reverse([1,2,3]) = [3,2,1].
range(a, b):
Outputs a list of integers from
a to b-1 in ascending order, e.g., range(1, 4)
= [1,2,3].
pow(a, b):
Outputs the
result, a Scalar, of a to the power of b, e.g.,
pow(2, 3) = 8 modulo the relevant group order
p.
||:
Denotes concatenation of byte strings, i.e., x || y denotes the byte string x, immediately followed by
the byte string y, with no extra separator, yielding xy.
nil:
Denotes an empty byte string.
Unless otherwise stated, we assume that secrets are sampled uniformly at random
using a CSPRNG; see for additional guidance on the generation of random numbers.Cryptographic DependenciesFROST signing depends on the following cryptographic constructs:
Prime-order group ()
Cryptographic hash function ()
The following sections describe these constructs in more detail.Prime-Order GroupFROST depends on an abelian group of prime order p. We represent this
group as the object G that additionally defines helper functions described below. The group operation
for G is addition + with identity element I. For any elements A and B of the group G,
A + B = B + A is also a member of G. Also, for any A in G, there exists an element
-A such that A + (-A) = (-A) + A = I. For convenience, we use - to denote
subtraction, e.g., A - B = A + (-B). Integers, taken modulo the group order p, are called
"Scalars"; arithmetic operations on Scalars are implicitly performed modulo p. Since p is prime,
Scalars form a finite field. Scalar multiplication is equivalent to the repeated
application of the group operation on an element A with itself r-1 times, denoted as
ScalarMult(A, r). We denote the sum, difference, and product of two Scalars using the +, -,
and * operators, respectively. (Note that this means + may refer to group element addition or
Scalar addition, depending on the type of the operands.) For any element A, ScalarMult(A, p) = I.
We denote B as a fixed generator of the group. Scalar base multiplication is equivalent to the repeated application
of the group operation on B with itself r-1 times, denoted as ScalarBaseMult(r). The set of
Scalars corresponds to GF(p), which we refer to as the Scalar field. It is assumed that
group element addition, negation, and equality comparison can be efficiently computed for
arbitrary group elements.This document uses types Element and Scalar to denote elements of the group G and
its set of Scalars, respectively. We denote Scalar(x) as the conversion of integer input x
to the corresponding Scalar value with the same numeric value. For example, Scalar(1) yields
a Scalar representing the value 1. Moreover, we use the type NonZeroScalar to denote a Scalar
value that is not equal to zero, i.e., Scalar(0). We denote equality comparison of these types
as == and assignment of values by =. When comparing Scalar values, e.g., for the purposes
of sorting lists of Scalar values, the least nonnegative representation mod p is used.We now detail a number of member functions that can be invoked on G.
Order():
Outputs the order of G (i.e., p).
Identity():
Outputs the identity Element of the group (i.e., I).
RandomScalar():
Outputs a random Scalar element in GF(p), i.e., a random Scalar in [0, p - 1].
ScalarMult(A, k):
Outputs the Scalar multiplication between Element A and Scalar k.
ScalarBaseMult(k):
Outputs the Scalar multiplication between Scalar k and the group generator B.
SerializeElement(A):
Maps an ElementA to a canonical byte array buf of fixed length Ne. This
function raises an error if A is the identity element of the group.
DeserializeElement(buf):
Attempts to map a byte array buf to an ElementA
and fails if the input is not the valid canonical byte representation of an element of
the group. This function raises an error if deserialization fails
or if A is the identity element of the group; see for group-specific
input validation steps.
SerializeScalar(s):
Maps a Scalar s to a canonical byte array buf of fixed length Ns.
DeserializeScalar(buf):
Attempts to map a byte array buf to a Scalars.
This function raises an error if deserialization fails; see
for group-specific input validation steps.
Cryptographic Hash FunctionFROST requires the use of
a cryptographically secure hash function, generically written as H,
which is modeled as a random oracle in security proofs for the
protocol (see and ). For concrete recommendations on hash
functions that SHOULD be used in practice, see . Using H, we introduce distinct
domain-separated hashes H1, H2, H3, H4, and H5:
H1, H2, and H3 map arbitrary byte strings to Scalar elements associated with the prime-order group.
H4 and H5 are aliases for H with distinct domain separators.
The details of H1, H2, H3, H4, and H5 vary based on the ciphersuite used. See
for more details about each.Helper FunctionsBeyond the core dependencies, the protocol in this document depends on the
following helper operations:
Nonce generation ();
Polynomials ();
List operations ();
Binding factors computation ();
Group commitment computation (); and
Signature challenge computation ().
The following sections describe these operations in more detail.Nonce GenerationTo hedge against a bad random number generator (RNG) that outputs predictable values, nonces are
generated with the nonce_generate function by combining fresh randomness
with the secret key as input to a domain-separated hash function built
from the ciphersuite hash function H. This domain-separated hash function
is denoted as H3. This function always samples 32 bytes of fresh randomness
to ensure that the probability of nonce reuse is at most 2^{-128}
as long as no more than 2^{64} signatures are computed by a given
signing participant.
Inputs:
- secret, a Scalar.
Outputs:
- nonce, a Scalar.
def nonce_generate(secret):
random_bytes = random_bytes(32)
secret_enc = G.SerializeScalar(secret)
return H3(random_bytes || secret_enc)
PolynomialsThis section defines polynomials over Scalars that are used in the main protocol.
A polynomial of maximum degree t is represented as a list of t+1 coefficients,
where the constant term of the polynomial is in the first position and the
highest-degree coefficient is in the last position. For example, the polynomial
x^{2} + 2x + 3 has degree 2 and is represented as a list of three coefficients [3, 2, 1].
A point on the polynomial f is a tuple (x, y), where y = f(x).The function derive_interpolating_value derives a value that is used for polynomial
interpolation.
It is provided a list of x-coordinates as input, each of which
cannot equal 0.
Inputs:
- L, the list of x-coordinates, each a NonZeroScalar.
- x_i, an x-coordinate contained in L, a NonZeroScalar.
Outputs:
- value, a Scalar.
Errors:
- "invalid parameters", if 1) x_i is not in L, or if 2) any
x-coordinate is represented more than once in L.
def derive_interpolating_value(L, x_i):
if x_i not in L:
raise "invalid parameters"
for x_j in L:
if count(x_j, L) > 1:
raise "invalid parameters"
numerator = Scalar(1)
denominator = Scalar(1)
for x_j in L:
if x_j == x_i: continue
numerator *= x_j
denominator *= x_j - x_i
value = numerator / denominator
return value
List OperationsThis section describes helper functions that work on lists of values produced
during the FROST protocol. The following function encodes a list of participant
commitments into a byte string for use in the FROST protocol.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
Outputs:
- encoded_group_commitment, the serialized representation of
commitment_list, a byte string.
def encode_group_commitment_list(commitment_list):
encoded_group_commitment = nil
for (identifier, hiding_nonce_commitment,
binding_nonce_commitment) in commitment_list:
encoded_commitment = (
G.SerializeScalar(identifier) ||
G.SerializeElement(hiding_nonce_commitment) ||
G.SerializeElement(binding_nonce_commitment))
encoded_group_commitment = (
encoded_group_commitment ||
encoded_commitment)
return encoded_group_commitment
The following function is used to extract identifiers from a commitment list.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
Outputs:
- identifiers, a list of NonZeroScalar values.
def participants_from_commitment_list(commitment_list):
identifiers = []
for (identifier, _, _) in commitment_list:
identifiers.append(identifier)
return identifiers
The following function is used to extract a binding factor from a list of binding factors.
Inputs:
- binding_factor_list = [(i, binding_factor), ...],
a list of binding factors for each participant, where each element
in the list indicates a NonZeroScalar identifier i and Scalar
binding factor.
- identifier, participant identifier, a NonZeroScalar.
Outputs:
- binding_factor, a Scalar.
Errors:
- "invalid participant", when the designated participant is
not known.
def binding_factor_for_participant(binding_factor_list, identifier):
for (i, binding_factor) in binding_factor_list:
if identifier == i:
return binding_factor
raise "invalid participant"
Binding Factors ComputationThis section describes the subroutine for computing binding factors based
on the participant commitment list, message to be signed, and group public key.
Inputs:
- group_public_key, the public key corresponding to the group signing
key, an Element.
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
- msg, the message to be signed.
Outputs:
- binding_factor_list, a list of (NonZeroScalar, Scalar) tuples
representing the binding factors.
def compute_binding_factors(group_public_key, commitment_list, msg):
group_public_key_enc = G.SerializeElement(group_public_key)
// Hashed to a fixed length.
msg_hash = H4(msg)
// Hashed to a fixed length.
encoded_commitment_hash =
H5(encode_group_commitment_list(commitment_list))
// The encoding of the group public key is a fixed length
// within a ciphersuite.
rho_input_prefix = group_public_key_enc || msg_hash ||
encoded_commitment_hash
binding_factor_list = []
for (identifier, hiding_nonce_commitment,
binding_nonce_commitment) in commitment_list:
rho_input = rho_input_prefix || G.SerializeScalar(identifier)
binding_factor = H1(rho_input)
binding_factor_list.append((identifier, binding_factor))
return binding_factor_list
Group Commitment ComputationThis section describes the subroutine for creating the group commitment
from a commitment list.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
- binding_factor_list = [(i, binding_factor), ...],
a list of (NonZeroScalar, Scalar) tuples representing the binding
factor Scalar for the given identifier.
Outputs:
- group_commitment, an Element.
def compute_group_commitment(commitment_list, binding_factor_list):
group_commitment = G.Identity()
for (identifier, hiding_nonce_commitment,
binding_nonce_commitment) in commitment_list:
binding_factor = binding_factor_for_participant(
binding_factor_list, identifier)
binding_nonce = G.ScalarMult(
binding_nonce_commitment,
binding_factor)
group_commitment = (
group_commitment +
hiding_nonce_commitment +
binding_nonce)
return group_commitment
Note that the performance of this algorithm is defined
naively and scales linearly relative to the number of signers. For improved
performance, the group commitment can be computed using multi-exponentiation
techniques such as Pippinger's algorithm; see for more details.Signature Challenge ComputationThis section describes the subroutine for creating the per-message challenge.
Inputs:
- group_commitment, the group commitment, an Element.
- group_public_key, the public key corresponding to the group signing
key, an Element.
- msg, the message to be signed, a byte string.
Outputs:
- challenge, a Scalar.
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
Two-Round FROST Signing ProtocolThis section describes the two-round FROST signing protocol for producing Schnorr signatures.
The protocol is configured to run with a selection of NUM_PARTICIPANTS signer participants and a Coordinator.
NUM_PARTICIPANTS is a positive and
non-zero integer that MUST be at least
MIN_PARTICIPANTS, but MUST NOT be larger than
MAX_PARTICIPANTS, where MIN_PARTICIPANTS <=
MAX_PARTICIPANTS and MIN_PARTICIPANTS is a positive and non-zero
integer. Additionally, MAX_PARTICIPANTSMUST be a positive
integer less than the group order.
A signer participant, or simply
"participant", is an entity that is trusted to hold and use a signing key
share. The Coordinator is an entity with the following responsibilities:
Determining the participants that will participate (at least MIN_PARTICIPANTS in number);
Coordinating rounds (receiving and forwarding inputs among participants);
Aggregating signature shares output by each participant; and
Publishing the resulting signature.
FROST assumes that the Coordinator and the set of signer participants are chosen
externally to the protocol. Note that it is possible to deploy the protocol without
designating a single Coordinator; see for more information.FROST produces signatures that can be verified as if they were produced from a single signer
using a signing key s with corresponding public key PK, where s is a Scalar
value and PK = G.ScalarBaseMult(s). As a threshold signing protocol, the group signing
key s is Shamir secret-shared amongst each of the MAX_PARTICIPANTS participants
and is used to produce signatures; see for more information about Shamir secret sharing.
In particular, FROST assumes each participant is configured with the following information:
An identifier, which is a NonZeroScalar value denoted as i in the range [1, MAX_PARTICIPANTS]
and MUST be distinct from the identifier of every other participant.
A signing key sk_i, which is a Scalar value representing the i-th Shamir secret share
of the group signing key s. In particular, sk_i is the value f(i) on a secret
polynomial f of degree (MIN_PARTICIPANTS - 1), where s is f(0). The public key
corresponding to this signing key share is PK_i = G.ScalarBaseMult(sk_i).
Additionally, the Coordinator and each participant are configured with common group
information, denoted as "group info," which consists of the following:
Group public key, which is an Element in G denoted as PK.
Public keys PK_i for each participant, which are Element values in G denoted as PK_i
for each i in [1, MAX_PARTICIPANTS].
This document does not specify how this information, including the signing key shares,
are configured and distributed to participants.
In general, two configuration
mechanisms are possible: one that requires a single trusted dealer and one
that requires performing a distributed key generation protocol. We highlight
the key generation mechanism by a trusted dealer in for reference.FROST requires two rounds to complete. In the first round,
participants generate and publish one-time-use commitments to be used in
the second round. In the second round, each participant produces a share
of the signature over the Coordinator-chosen message and the other
participant commitments. After the second round is completed, the
Coordinator aggregates the signature shares to produce a final
signature. The Coordinator SHOULD abort the protocol if
the signature is invalid; see for more
information about dealing with invalid signatures and misbehaving
participants. This complete interaction (without being aborted) is shown
in .Details for round one are described in and details for round two
are described in .
Note that each participant persists some state between
the two rounds; this state is deleted as described in . The final
Aggregation step is described in .FROST assumes that all inputs to each round, especially those that are received
over the network, are validated before use.
In particular, this means that any value of type Element or Scalar received
over the network MUST be deserialized using DeserializeElement
and DeserializeScalar, respectively, as these functions perform the necessary
input validation steps. Additionally, all messages sent over the wire
MUST be encoded using their respective functions, e.g., Scalars
and Elements are encoded using SerializeScalar and SerializeElement.FROST assumes reliable message delivery between the Coordinator and participants in
order for the protocol to complete. An attacker masquerading as another participant
will result only in an invalid signature; see . However, in order
to identify misbehaving participants,
we assume that the network channel is additionally authenticated; confidentiality is
not required.Round One - CommitmentRound one involves each participant generating nonces and their corresponding public commitments.
A nonce is a pair of Scalar values, and a commitment is a pair of Element values. Each participant's behavior in this round is described by the commit function below. Note that this function invokes nonce_generate twice, once for each type of nonce produced. The output of this function is a pair of secret nonces (hiding_nonce, binding_nonce) and their corresponding public commitments
(hiding_nonce_commitment, binding_nonce_commitment).
Inputs:
- sk_i, the secret key share, a Scalar.
Outputs:
- (nonce, comm), a tuple of nonce and nonce commitment pairs,
where each value in the nonce pair is a Scalar and each value in
the nonce commitment pair is an Element.
def commit(sk_i):
hiding_nonce = nonce_generate(sk_i)
binding_nonce = nonce_generate(sk_i)
hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
nonces = (hiding_nonce, binding_nonce)
comms = (hiding_nonce_commitment, binding_nonce_commitment)
return (nonces, comms)
The outputs nonce and comm from participant P_i are both stored locally and
kept for use in the second round. The nonce value is secret and MUST NOT be shared, whereas
the public output comm is sent to the Coordinator. The nonce values produced by this function MUST NOT be used in more than one invocation of sign, and the nonces MUST be generated
from a source of secure randomness.Round Two - Signature Share GenerationIn round two, the Coordinator is responsible for sending the message to be signed and choosing the participants that will participate (a number of at least MIN_PARTICIPANTS). Signers
additionally require locally held data, specifically their private key and the
nonces corresponding to their commitment issued in round one.The Coordinator begins by sending each participant the message to be signed along with the set of signing commitments for all participants in the participant list. Each participant
MUST validate the inputs before processing the Coordinator's request. In particular,
the signer MUST validate commitment_list, deserializing each group Element in the
list using DeserializeElement from . If deserialization fails, the signer
MUST abort the protocol. Moreover, each participant MUST ensure that
its identifier and commitments (from the first round) appear in commitment_list.
Applications that restrict participants from processing arbitrary input
messages are also required to perform relevant application-layer input
validation checks; see for more
details.Upon receipt and successful input validation, each signer then runs the following
procedure to produce its own signature share.
Inputs:
- identifier, identifier i of the participant, a NonZeroScalar.
- sk_i, signer secret key share, a Scalar.
- group_public_key, public key corresponding to the group signing
key, an Element.
- nonce_i, pair of Scalar values (hiding_nonce, binding_nonce)
generated in round one.
- msg, the message to be signed, a byte string.
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
Outputs:
- sig_share, a signature share, a Scalar.
def sign(identifier, sk_i, group_public_key,
nonce_i, msg, commitment_list):
# Compute the binding factor(s)
binding_factor_list = compute_binding_factors(group_public_key,
commitment_list, msg)
binding_factor = binding_factor_for_participant(
binding_factor_list, identifier)
# Compute the group commitment
group_commitment = compute_group_commitment(
commitment_list, binding_factor_list)
# Compute the interpolating value
participant_list = participants_from_commitment_list(
commitment_list)
lambda_i = derive_interpolating_value(participant_list, identifier)
# Compute the per-message challenge
challenge = compute_challenge(
group_commitment, group_public_key, msg)
# Compute the signature share
(hiding_nonce, binding_nonce) = nonce_i
sig_share = hiding_nonce + (binding_nonce * binding_factor) +
(lambda_i * sk_i * challenge)
return sig_share
The output of this procedure is a signature share. Each participant sends
these shares back to the Coordinator. Each participant MUST delete the nonce and
corresponding commitment after completing sign and MUST NOT use the nonce
as input more than once to sign.Note that the lambda_i value derived during this procedure does not change
across FROST signing operations for the same signing group. As such, participants
can compute it once and store it for reuse across signing sessions.Signature Share AggregationAfter participants perform round two and send their signature shares to the Coordinator,
the Coordinator aggregates each share to produce a final signature. Before aggregating,
the Coordinator MUST validate each signature share using DeserializeScalar. If validation
fails, the Coordinator MUST abort the protocol, as the resulting signature will be invalid.
If all signature shares are valid, the Coordinator aggregates them to produce the final
signature using the following procedure.
Inputs:
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
- msg, the message to be signed, a byte string.
- group_public_key, public key corresponding to the group signing
key, an Element.
- sig_shares, a set of signature shares z_i, Scalar values, for each
participant, of length NUM_PARTICIPANTS, where
MIN_PARTICIPANTS <= NUM_PARTICIPANTS <= MAX_PARTICIPANTS.
Outputs:
- (R, z), a Schnorr signature consisting of an Element R and
Scalar z.
def aggregate(commitment_list, msg, group_public_key, sig_shares):
# Compute the binding factors
binding_factor_list = compute_binding_factors(group_public_key,
commitment_list, msg)
# Compute the group commitment
group_commitment = compute_group_commitment(
commitment_list, binding_factor_list)
# Compute aggregated signature
z = Scalar(0)
for z_i in sig_shares:
z = z + z_i
return (group_commitment, z)
The output from the aggregation step is the output signature (R, z). The canonical encoding
of this signature is specified in .The Coordinator SHOULD verify this signature using the group public key before publishing or
releasing the signature. Signature verification is as specified for the corresponding
ciphersuite; see for details. The aggregate signature will verify successfully
if all signature shares are valid. Moreover, subsets of valid signature shares will not yield
a valid aggregate signature themselves.If the aggregate signature verification fails, the Coordinator MAY verify each signature
share individually to identify and act on misbehaving participants. The mechanism for acting on
a misbehaving participant is out of scope for this specification; see for more information
about dealing with invalid signatures and misbehaving participants.The function for verifying a signature share, denoted as verify_signature_share, is described below.
Recall that the Coordinator is configured with "group info" that contains
the group public key PK and public keys PK_i for each participant. The group_public_key and
PK_i function arguments MUST come from that previously stored group info.
Inputs:
- identifier, identifier i of the participant, a NonZeroScalar.
- PK_i, the public key for the i-th participant, where
PK_i = G.ScalarBaseMult(sk_i), an Element.
- comm_i, pair of Element values in G
(hiding_nonce_commitment, binding_nonce_commitment) generated in
round one from the i-th participant.
- sig_share_i, a Scalar value indicating the signature share as
produced in round two from the i-th participant.
- commitment_list = [(i, hiding_nonce_commitment_i,
binding_nonce_commitment_i), ...], a list of commitments issued by
each participant, where each element in the list indicates a
NonZeroScalar identifier i and two commitment Element values
(hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
MUST be sorted in ascending order by identifier.
- group_public_key, public key corresponding to the group signing
key, an Element.
- msg, the message to be signed, a byte string.
Outputs:
- True if the signature share is valid, and False otherwise.
def verify_signature_share(
identifier, PK_i, comm_i, sig_share_i, commitment_list,
group_public_key, msg):
# Compute the binding factors
binding_factor_list = compute_binding_factors(group_public_key,
commitment_list, msg)
binding_factor = binding_factor_for_participant(
binding_factor_list, identifier)
# Compute the group commitment
group_commitment = compute_group_commitment(
commitment_list, binding_factor_list)
# Compute the commitment share
(hiding_nonce_commitment, binding_nonce_commitment) = comm_i
comm_share = hiding_nonce_commitment + G.ScalarMult(
binding_nonce_commitment, binding_factor)
# Compute the challenge
challenge = compute_challenge(
group_commitment, group_public_key, msg)
# Compute the interpolating value
participant_list = participants_from_commitment_list(
commitment_list)
lambda_i = derive_interpolating_value(participant_list, identifier)
# Compute relation values
l = G.ScalarBaseMult(sig_share_i)
r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i)
return l == r
The Coordinator can verify each signature share before aggregating and
verifying the signature under the group public key. However, since the
aggregate signature is valid if all signature shares are valid, this order of
operations is more expensive if the signature is valid.Identifiable AbortFROST does not provide robustness; i.e, all participants are required to complete the
protocol honestly in order to generate a valid signature. When the signing protocol
does not produce a valid signature, the Coordinator SHOULD abort; see
for more information about FROST's security properties and the threat model.As a result of this property, a misbehaving participant can cause a denial of service (DoS) on
the signing protocol by contributing malformed signature shares or refusing to participate.
Identifying misbehaving participants that produce invalid shares can be done by checking
signature shares from each participant using verify_signature_share as described in .
FROST assumes the network channel is authenticated to identify the signer that misbehaved.
FROST allows for identifying misbehaving participants that produce invalid signature shares
as described in . FROST does not provide accommodations for identifying
participants that refuse to participate, though applications are assumed to detect when participants
fail to engage in the signing protocol.In both cases, preventing this type of attack requires the Coordinator to identify
misbehaving participants such that applications can take corrective action. The mechanism
for acting on misbehaving participants is out of scope for this specification. However,
one reasonable approach would be to remove the misbehaving participant from the set of allowed
participants in future runs of FROST.CiphersuitesA FROST ciphersuite must specify the underlying prime-order group details
and cryptographic hash function. Each ciphersuite is denoted as (Group, Hash),
e.g., (ristretto255, SHA-512). This section contains some ciphersuites.
Each ciphersuite also includes a context string, denoted as contextString,
which is an ASCII string literal (with no terminating NUL character).The RECOMMENDED ciphersuite is (ristretto255, SHA-512) as described in .
The (Ed25519, SHA-512) and (Ed448, SHAKE256) ciphersuites are included
for compatibility with Ed25519 and Ed448 as defined in .The DeserializeElement and DeserializeScalar functions instantiated for a
particular prime-order group corresponding to a ciphersuite MUST adhere
to the description in . Validation steps for these functions
are described for each of the ciphersuites below. Future ciphersuites MUST
describe how input validation is done for DeserializeElement and DeserializeScalar.Each ciphersuite includes explicit instructions for verifying signatures produced
by FROST. Note that these instructions are equivalent to those produced by a single
participant.Each ciphersuite adheres to the requirements in . Future
ciphersuites MUST also adhere to these requirements.FROST(Ed25519, SHA-512)This ciphersuite uses edwards25519 for the Group and SHA-512 for the hash function H
meant to produce Ed25519-compliant signatures as specified in .
The value of the contextString parameter is "FROST-ED25519-SHA512-v1".
Group:
edwards25519 , where Ne = 32 and Ns = 32.
Order():
Return 2^{252} + 27742317777372353535851937790883648493 (see ).
Identity():
As defined in .
RandomScalar():
Implemented by returning a uniformly random Scalar in the range
[0, G.Order() - 1]. Refer to for implementation guidance.
SerializeElement(A):
Implemented as specified in .
Additionally, this function validates that the input element is not the group
identity element.
DeserializeElement(buf):
Implemented as specified in .
Additionally, this function validates that the resulting element is not the group
identity element and is in the prime-order subgroup. If any of these checks fail,
deserialization returns an error. The latter check can
be implemented by multiplying the resulting point by the order of the group and
checking that the result is the identity element. Note that optimizations for
this check exist; see .
SerializeScalar(s):
Implemented by outputting the little-endian 32-byte encoding of
the Scalar value with the top three bits set to zero.
DeserializeScalar(buf):
Implemented by attempting to deserialize a Scalar from a
little-endian 32-byte string. This function can fail if the input does not
represent a Scalar in the range [0, G.Order() - 1]. Note that this means the
top three bits of the input MUST be zero.
Hash (H):
SHA-512, which has an output of 64 bytes.
H1(m):
Implemented by computing H(contextString || "rho" || m), interpreting the 64-byte digest
as a little-endian integer, and reducing the resulting integer modulo
2^{252} + 27742317777372353535851937790883648493.
H2(m):
Implemented by computing H(m), interpreting the 64-byte digest
as a little-endian integer, and reducing the resulting integer modulo
2^{252} + 27742317777372353535851937790883648493.
H3(m):
Implemented by computing H(contextString || "nonce" || m), interpreting the 64-byte digest
as a little-endian integer, and reducing the resulting integer modulo
2^{252} + 27742317777372353535851937790883648493.
H4(m):
Implemented by computing H(contextString || "msg" || m).
H5(m):
Implemented by computing H(contextString || "com" || m).
Normally, H2 would also include a domain separator; however, for compatibility with , it is omitted.Signature verification is as specified in with the
constraint that implementations MUST check the group equation [8][z]B = [8]R + [8][c]PK
(changed to use the notation in this document).Canonical signature encoding is as specified in .FROST(ristretto255, SHA-512)This ciphersuite uses ristretto255 for the Group and SHA-512 for the hash function H.
The value of the contextString parameter is "FROST-RISTRETTO255-SHA512-v1".
Group:
ristretto255 , where Ne = 32 and Ns = 32.
Order():
Return 2^{252} + 27742317777372353535851937790883648493 (see ).
Identity():
As defined in .
RandomScalar():
Implemented by returning a uniformly random Scalar in the range
[0, G.Order() - 1]. Refer to for implementation guidance.
SerializeElement(A):
Implemented using the "Encode" function from .
Additionally, this function validates that the input element is not the group
identity element.
DeserializeElement(buf):
Implemented using the "Decode" function from .
Additionally, this function validates that the resulting element is not the group
identity element. If either the "Decode" function or the check fails, deserialization returns an error.
SerializeScalar(s):
Implemented by outputting the little-endian 32-byte encoding of
the Scalar value with the top three bits set to zero.
DeserializeScalar(buf):
Implemented by attempting to deserialize a Scalar from a
little-endian 32-byte string. This function can fail if the input does not
represent a Scalar in the range [0, G.Order() - 1]. Note that this means the
top three bits of the input MUST be zero.
Hash (H):
SHA-512, which has 64 bytes of output.
H1(m):
Implemented by computing H(contextString || "rho" || m) and mapping the
output to a Scalar as described in .
H2(m):
Implemented by computing H(contextString || "chal" || m) and mapping the
output to a Scalar as described in .
H3(m):
Implemented by computing H(contextString || "nonce" || m) and mapping the
output to a Scalar as described in .
H4(m):
Implemented by computing H(contextString || "msg" || m).
H5(m):
Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in .Canonical signature encoding is as specified in .FROST(Ed448, SHAKE256)This ciphersuite uses edwards448 for the Group and SHAKE256 for the hash function H
meant to produce Ed448-compliant signatures as specified in .
Unlike Ed448 in , this ciphersuite does not allow applications to
specify a context string and always sets the context of to the empty string.
Note that this ciphersuite does not allow applications to specify a context
string as is allowed for Ed448 in , and always sets
the context string to the empty string. The value of
the (internal to FROST) contextString parameter is
"FROST-ED448-SHAKE256-v1".
Implemented by returning a uniformly random Scalar in the range
[0, G.Order() - 1]. Refer to for implementation guidance.
SerializeElement(A):
Implemented as specified in .
Additionally, this function validates that the input element is not the group
identity element.
DeserializeElement(buf):
Implemented as specified in .
Additionally, this function validates that the resulting element is not the group
identity element and is in the prime-order subgroup. If any of these checks fail,
deserialization returns an error. The latter check can
be implemented by multiplying the resulting point by the order of the group and
checking that the result is the identity element. Note that optimizations for
this check exist; see .
SerializeScalar(s):
Implemented by outputting the little-endian 57-byte encoding of
the Scalar value.
DeserializeScalar(buf):
Implemented by attempting to deserialize a Scalar from a
little-endian 57-byte string. This function can fail if the input does not
represent a Scalar in the range [0, G.Order() - 1].
Hash (H):
SHAKE256 with 114 bytes of output.
H1(m):
Implemented by computing H(contextString || "rho" || m), interpreting the
114-byte digest as a little-endian integer, and reducing the resulting integer modulo
2^{446} - 13818066809895115352007386748515426880336692474882178609894547503885.
H2(m):
Implemented by computing H("SigEd448" || 0 || 0 || m), interpreting
the 114-byte digest as a little-endian integer, and reducing the resulting integer
modulo 2^{446} - 13818066809895115352007386748515426880336692474882178609894547503885.
H3(m):
Implemented by computing H(contextString || "nonce" || m), interpreting the
114-byte digest as a little-endian integer, and reducing the resulting integer modulo
2^{446} - 13818066809895115352007386748515426880336692474882178609894547503885.
H4(m):
Implemented by computing H(contextString || "msg" || m).
H5(m):
Implemented by computing H(contextString || "com" || m).
Normally, H2 would also include a domain separator. However, it is omitted for compatibility with .Signature verification is as specified in with the
constraint that implementations MUST check the group equation [4][z]B = [4]R + [4][c]PK
(changed to use the notation in this document).Canonical signature encoding is as specified in .FROST(P-256, SHA-256)This ciphersuite uses P-256 for the Group and SHA-256 for the hash function H.
The value of the contextString parameter is "FROST-P256-SHA256-v1".
Implemented by returning a uniformly random Scalar in the range
[0, G.Order() - 1]. Refer to for implementation guidance.
SerializeElement(A):
Implemented using the compressed Elliptic-Curve-Point-to-Octet-String
method according to , yielding a 33-byte output. Additionally, this function validates
that the input element is not the group identity element.
DeserializeElement(buf):
Implemented by attempting to deserialize a 33-byte input string to
a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to and then performing public key validation as defined in Section 3.2.2.1 of .
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. (As noted in the specification, validation of the point
order is not required since the cofactor is 1.) If any of these checks fail,
deserialization returns an error.
SerializeScalar(s):
Implemented using the Field-Element-to-Octet-String conversion
according to .
DeserializeScalar(buf):
Implemented by attempting to deserialize a Scalar from a 32-byte
string using Octet-String-to-Field-Element from . This function can fail if the
input does not represent a Scalar in the range [0, G.Order() - 1].
Hash (H):
SHA-256, which has 32 bytes of output.
H1(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "rho",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H2(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "chal",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H3(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "nonce",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H4(m):
Implemented by computing H(contextString || "msg" || m).
H5(m):
Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in .Canonical signature encoding is as specified in .FROST(secp256k1, SHA-256)This ciphersuite uses secp256k1 for the Group and SHA-256 for the hash function H.
The value of the contextString parameter is "FROST-secp256k1-SHA256-v1".
Implemented by returning a uniformly random Scalar in the range
[0, G.Order() - 1]. Refer to for implementation guidance.
SerializeElement(A):
Implemented using the compressed Elliptic-Curve-Point-to-Octet-String
method according to , yielding a 33-byte output. Additionally, this function validates
that the input element is not the group identity element.
DeserializeElement(buf):
Implemented by attempting to deserialize a 33-byte input string to
a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to and then performing public key validation as defined in Section 3.2.2.1 of .
This includes checking that the coordinates of the resulting point are
in the correct range, the point is on the curve, and the point is not
the point at infinity. (As noted in the specification, validation of the point
order is not required since the cofactor is 1.) If any of these checks fail,
deserialization returns an error.
SerializeScalar(s):
Implemented using the Field-Element-to-Octet-String conversion
according to .
DeserializeScalar(buf):
Implemented by attempting to deserialize a Scalar from a 32-byte
string using Octet-String-to-Field-Element from . This function can fail if the
input does not represent a Scalar in the range [0, G.Order() - 1].
Hash (H):
SHA-256, which has 32 bytes of output.
H1(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "rho",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H2(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "chal",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H3(m):
Implemented as hash_to_field(m, 1) (see )
using expand_message_xmd with SHA-256 with parameters DST = contextString || "nonce",
F set to the Scalar field, p set to G.Order(), m = 1, and L = 48.
H4(m):
Implemented by computing H(contextString || "msg" || m).
H5(m):
Implemented by computing H(contextString || "com" || m).
Signature verification is as specified in .Canonical signature encoding is as specified in .Ciphersuite RequirementsFuture documents that introduce new ciphersuites MUST adhere to
the following requirements.
H1, H2, and H3 all have output distributions that are close to
(indistinguishable from) the uniform distribution.
All hash functions MUST be domain-separated with a per-suite context
string. Note that the FROST(Ed25519, SHA-512) ciphersuite does not
adhere to this requirement for H2 alone in order to maintain compatibility
with .
The group MUST be of prime order and all deserialization functions MUST
output elements that belong to their respective sets of Elements or Scalars, or else fail.
The canonical signature encoding details are clearly specified.
Security ConsiderationsA security analysis of FROST is documented in and . At a high level, FROST provides security against Existential Unforgeability Under Chosen Message
Attacks (EUF-CMA) as defined in .
To satisfy this requirement, the ciphersuite needs to adhere to the
requirements in and the following
assumptions must hold.
The signer key shares are generated and distributed securely, e.g., via a trusted dealer
that performs key generation (see ) or through a distributed key generation protocol.
The Coordinator and at most (MIN_PARTICIPANTS-1) participants may be corrupted.
Note that the Coordinator is not trusted with any private information, and communication at the time of signing can be performed over a public channel as long as it is authenticated and reliable.FROST provides security against DoS attacks under the following assumptions:
The Coordinator does not perform a DoS attack.
The Coordinator identifies misbehaving participants such that they can be removed from
future invocations of FROST. The Coordinator may also abort upon detecting a misbehaving
participant to ensure that invalid signatures are not produced.
FROST does not aim to achieve the following goals:
Post-quantum security. FROST, like plain Schnorr signatures, requires the hardness of the Discrete Logarithm Problem.
Robustness. Preventing DoS attacks against misbehaving participants requires the Coordinator
to identify and act on misbehaving participants; see for more information.
While FROST
does not provide robustness, is a wrapper protocol around FROST that does.
Downgrade prevention. All participants in the protocol are assumed to agree on which algorithms to use.
Metadata protection. If protection for metadata is desired, a higher-level communication
channel can be used to facilitate key generation and signing.
The rest of this section documents issues particular to implementations or deployments.Side-Channel MitigationsSeveral routines process secret values (nonces, signing keys / shares), and depending
on the implementation and deployment environment, mitigating side-channels may be
pertinent. Mitigating these side-channels requires implementing G.ScalarMult(), G.ScalarBaseMult(),
G.SerializeScalar(), and G.DeserializeScalar() in constant (value-independent) time.
The various ciphersuites lend themselves differently to specific implementation techniques
and ease of achieving side-channel resistance, though ultimately avoiding value-dependent
computation or branching is the goal.Optimizations presented an optimization to FROST that reduces the total number of Scalar multiplications
from linear in the number of signing participants to a constant. However, as described in ,
this optimization removes the guarantee that the set of signer participants that started round one of
the protocol is the same set of signing participants that produced the signature output by round two.
As such, the optimization is NOT RECOMMENDED and is not covered in this document.Nonce Reuse Attacks describes the procedure that participants use to produce nonces during
the first round of signing. The randomness produced in this procedure MUST be sampled
uniformly at random. The resulting nonces produced via nonce_generate are indistinguishable
from values sampled uniformly at random. This requirement is necessary to avoid
replay attacks initiated by other participants that allow for a complete key-recovery attack.
The Coordinator MAY further hedge against nonce reuse attacks by tracking participant nonce
commitments used for a given group key at the cost of additional state.Protocol FailuresWe do not specify what implementations should do when the protocol fails other than requiring
the protocol to abort. Examples of viable failures include when a verification check returns invalid or the underlying transport failed to deliver the required messages.Removing the Coordinator RoleIn some settings, it may be desirable to omit the role of the Coordinator entirely.
Doing so does not change the security implications of FROST; instead, it simply
requires each participant to communicate with all other participants. We loosely
describe how to perform FROST signing among participants without this coordinator role.
We assume that every participant receives a message to be signed from an external source as input prior to performing the protocol.Every participant begins by performing commit() as is done in the setting
where a Coordinator is used. However, instead of sending the commitment
to the Coordinator, every participant will publish this commitment to every other participant. In the second round, participants will already have sufficient information to perform signing, and they will directly perform sign(). All participants will then publish their signature shares to one another. After having received all signature shares from all other participants, each participant will then perform verify_signature_share and then aggregate directly.The requirements for the underlying network channel remain the same in the
setting where all participants play the role of the Coordinator, in that all
exchanged messages are public and the channel must be reliable. However, in the setting
where 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 DoS, implementations may wish to define a
mechanism where messages are authenticated so that cheating players can be
identified and excluded.Input Message HashingFROST signatures do not pre-hash message inputs. This means that
the entire message must be known in advance of invoking the signing
protocol. Applications can apply pre-hashing in settings where storing
the full message is prohibitively expensive. In such cases,
pre-hashing MUST use a collision-resistant hash
function with a security level commensurate with the security inherent
to the ciphersuite chosen. For applications that choose to apply
pre-hashing, it is RECOMMENDED that they use the hash
function (H) associated with the chosen ciphersuite in a
manner similar to how H4 is defined. In particular, a
different prefix SHOULD be used to differentiate this
pre-hash from H4. For example, if a fictional protocol Quux
decided to pre-hash its input messages, one possible way to do so is
via H(contextString || "Quux-pre-hash" || m).Input Message ValidationMessage validation varies by application. For example, some applications may
require that participants only process messages of a certain structure. In digital
currency applications, wherein multiple participants may collectively sign a transaction,
it is reasonable to require each participant to check that the input message is a
syntactically valid transaction.As another example, some applications may require that participants only process
messages with permitted content according to some policy. In digital currency
applications, this might mean that a transaction being signed is allowed and
intended by the relevant stakeholders. Another instance of this type of message
validation is in the context of , wherein implementations may
use threshold signing protocols to produce signatures of transcript hashes. In
this setting, signing participants might require the raw TLS handshake messages
to validate before computing the transcript hash that is signed.In general, input message validation is an application-specific consideration
that varies based on the use case and threat model. However, it is RECOMMENDED
that applications take additional precautions and validate inputs so that
participants do not operate as signing oracles for arbitrary messages.IANA ConsiderationsThis document has no IANA actions.ReferencesNormative ReferencesHashing to Elliptic CurvesThis document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.Key words for use in RFCs to Indicate Requirement LevelsIn many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements.Edwards-Curve Digital Signature Algorithm (EdDSA)This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm is instantiated with recommended parameters for the edwards25519 and edwards448 curves. An example implementation and test vectors are provided.Ambiguity of Uppercase vs Lowercase in RFC 2119 Key WordsRFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings.The ristretto255 and decaf448 GroupsThis memo specifies two prime-order groups, ristretto255 and decaf448, suitable for safely implementing higher-level and complex cryptographic protocols. The ristretto255 group can be implemented using Curve25519, allowing existing Curve25519 implementations to be reused and extended to provide a prime-order group. Likewise, the decaf448 group can be implemented using edwards448.This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF.SEC 1: Elliptic Curve CryptographyStandards for Efficient CryptographyVersion 2.0SEC 2: Recommended Elliptic Curve Domain ParametersStandards for Efficient CryptographyVersion 2.0Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)American National Standards InstituteInformative ReferencesA practical scheme for non-interactive verifiable secret sharingIEEE28th Annual Symposium on Foundations of Computer Science (sfcs 1987)FROST: Flexible Round-Optimized Schnorr Threshold SignaturesSpeeding up FROST with multi-scalar multiplicationPoint-Halving and Subgroup Membership in Twisted Edwards CurvesRandomness Requirements for SecuritySecurity systems are built on strong cryptographic algorithms that foil pattern analysis attempts. However, the security of these systems is dependent on generating secret quantities for passwords, cryptographic keys, and similar quantities. The use of pseudo-random processes to generate secret quantities can result in pseudo-security. A sophisticated attacker may find it easier to reproduce the environment that produced the secret quantities and to search the resulting small set of possibilities than to locate the quantities in the whole of the potential number space.Choosing random quantities to foil a resourceful and motivated adversary is surprisingly difficult. This document points out many pitfalls in using poor entropy sources or traditional pseudo-random number generation techniques for generating such quantities. It recommends the use of truly random hardware techniques and shows that the existing hardware on many systems can be used for this purpose. It provides suggestions to ameliorate the problem when a hardware solution is not available, and it gives examples of how large such quantities need to be for some applications. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements.Elliptic Curves for SecurityThis memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128-bit and ~224-bit security level, respectively, and are generated deterministically based on a list of required properties.ROAST: Robust Asynchronous Schnorr Threshold SignaturesPaper 2022/550How to share a secretMassachusetts Institute of TechnologyAssociation for Computing Machinery (ACM)Communications of the ACM, Vol. 22, Issue 11, pp. 612-613Better than Advertised Security for Non-interactive Threshold SignaturesThe Transport Layer Security (TLS) Protocol Version 1.3This document specifies version 1.3 of the Transport Layer Security (TLS) protocol. TLS allows client/server applications to communicate over the Internet in a way that is designed to prevent eavesdropping, tampering, and message forgery.This document updates RFCs 5705 and 6066, and obsoletes RFCs 5077, 5246, and 6961. This document also specifies new requirements for TLS 1.2 implementations.Schnorr Signature EncodingThis section describes one possible canonical encoding of FROST signatures. Using notation
from , the encoding of a FROST signature (R, z) is as follows:
struct {
opaque R_encoded[Ne];
opaque z_encoded[Ns];
} Signature;
Where Signature.R_encoded is G.SerializeElement(R), Signature.z_encoded is
G.SerializeScalar(z), and G is determined by ciphersuite.Schnorr Signature Generation and Verification for Prime-Order GroupsThis section contains descriptions of functions for generating and verifying Schnorr signatures.
It is included to complement the routines present in for prime-order groups, including
ristretto255, P-256, and secp256k1. The functions for generating and verifying signatures are
prime_order_sign and prime_order_verify, respectively.The function prime_order_sign produces a Schnorr signature over a message given a full secret signing
key as input (as opposed to a key share).
Inputs:
- msg, message to sign, a byte string.
- sk, secret key, a Scalar.
Outputs:
- (R, z), a Schnorr signature consisting of an Element R and
Scalar z.
def prime_order_sign(msg, sk):
r = G.RandomScalar()
R = G.ScalarBaseMult(r)
PK = G.ScalarBaseMult(sk)
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
z = r + (c * sk) // Scalar addition and multiplication
return (R, z)
The function prime_order_verify verifies Schnorr signatures with validated inputs.
Specifically, it assumes that the signature R component and public key belong to the prime-order group.
Inputs:
- msg, signed message, a byte string.
- sig, a tuple (R, z) output from signature generation.
- PK, public key, an Element.
Outputs:
- True if signature is valid, and False otherwise.
def prime_order_verify(msg, sig = (R, z), PK):
comm_enc = G.SerializeElement(R)
pk_enc = G.SerializeElement(PK)
challenge_input = comm_enc || pk_enc || msg
c = H2(challenge_input)
l = G.ScalarBaseMult(z)
r = R + G.ScalarMult(PK, c)
return l == r
Trusted Dealer Key GenerationOne 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 Appendices and to create secret
shares of s, denoted as s_i for i = 1, ..., MAX_PARTICIPANTS, to be sent to all MAX_PARTICIPANTS participants.
This operation is specified in the trusted_dealer_keygen algorithm. The mathematical relation
between the secret key s and the MAX_PARTICIPANTS secret shares is formalized in the secret_share_combine(shares)
algorithm, defined in .The dealer that performs trusted_dealer_keygen is trusted to 1) generate good randomness, 2) delete secret values after distributing shares to each participant, and 3) keep secret values confidential.
Inputs:
- secret_key, a group secret, a Scalar, that MUST be derived from at
least Ns bytes of entropy.
- MAX_PARTICIPANTS, the number of shares to generate, an integer.
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme,
an integer.
Outputs:
- participant_private_keys, MAX_PARTICIPANTS shares of the secret
key s, each a tuple consisting of the participant identifier
(a NonZeroScalar) and the key share (a Scalar).
- group_public_key, public key corresponding to the group signing
key, an Element.
- vss_commitment, a vector commitment of Elements in G, to each of
the coefficients in the polynomial defined by secret_key_shares and
whose first element is G.ScalarBaseMult(s).
def trusted_dealer_keygen(
secret_key, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
# Generate random coefficients for the polynomial
coefficients = []
for i in range(0, MIN_PARTICIPANTS - 1):
coefficients.append(G.RandomScalar())
participant_private_keys, coefficients = secret_share_shard(
secret_key, coefficients, MAX_PARTICIPANTS)
vss_commitment = vss_commit(coefficients):
return participant_private_keys, vss_commitment[0], vss_commitment
It is assumed that the dealer then sends one secret key share to each of the NUM_PARTICIPANTS participants, along with vss_commitment.
After receiving their secret key share and vss_commitment, participants MUST abort if they do not have the same view of vss_commitment.
The dealer can use a secure broadcast channel to ensure each participant has a consistent view of this commitment.
Furthermore, each participant MUST perform vss_verify(secret_key_share_i, vss_commitment) and abort if the check fails.
The trusted dealer MUST delete the secret_key and secret_key_shares upon completion.Use of this method for key generation requires a mutually authenticated secure channel
between the dealer and participants to send secret key shares, wherein the channel provides confidentiality and integrity. Mutually authenticated TLS is one possible deployment option.Shamir Secret SharingIn Shamir secret sharing, a dealer distributes a secret Scalars to n participants
in such a way that any cooperating subset of at least MIN_PARTICIPANTS participants can recover the
secret. There are two basic steps in this scheme: 1) splitting a secret into
multiple shares and 2) combining shares to reveal the resulting secret.This secret sharing scheme works over any field F. In this specification, F is
the Scalar field of the prime-order group G.The procedure for splitting a secret into shares is as follows.
The algorithm polynomial_evaluate is defined in .
Inputs:
- s, secret value to be shared, a Scalar.
- coefficients, an array of size MIN_PARTICIPANTS - 1 with randomly
generated Scalars, not including the 0th coefficient of the
polynomial.
- MAX_PARTICIPANTS, the number of shares to generate, an integer less
than the group order.
Outputs:
- secret_key_shares, A list of MAX_PARTICIPANTS number of secret
shares, each a tuple consisting of the participant identifier
(a NonZeroScalar) and the key share (a Scalar).
- coefficients, a vector of MIN_PARTICIPANTS coefficients which
uniquely determine a polynomial f.
def secret_share_shard(s, coefficients, MAX_PARTICIPANTS):
# Prepend the secret to the coefficients
coefficients = [s] + coefficients
# Evaluate the polynomial for each point x=1,...,n
secret_key_shares = []
for x_i in range(1, MAX_PARTICIPANTS + 1):
y_i = polynomial_evaluate(Scalar(x_i), coefficients)
secret_key_share_i = (x_i, y_i)
secret_key_shares.append(secret_key_share_i)
return secret_key_shares, coefficients
Let points be the output of this function. The i-th element in points is
the share for the i-th participant, which is the randomly generated polynomial
evaluated at coordinate i. We denote a secret share as the tuple (i, points[i])
and the list of these shares as shares. iMUST never equal 0; recall that
f(0) = s, where f is the polynomial defined in a Shamir secret sharing operation.The procedure for combining a shares list of length MIN_PARTICIPANTS to recover the
secret s is as follows; the algorithm polynomial_interpolate_constant is defined in .
Inputs:
- shares, a list of at minimum MIN_PARTICIPANTS secret shares, each a
tuple (i, f(i)) where i and f(i) are Scalars.
Outputs:
- s, the resulting secret that was previously split into shares,
a Scalar.
Errors:
- "invalid parameters", if fewer than MIN_PARTICIPANTS input shares
are provided.
def secret_share_combine(shares):
if len(shares) < MIN_PARTICIPANTS:
raise "invalid parameters"
s = polynomial_interpolate_constant(shares)
return s
Additional Polynomial OperationsThis section describes two functions. One function, denoted as polynomial_evaluate,
is for evaluating a polynomial f(x) at a particular point x using Horner's method,
i.e., computing y = f(x). The other function, polynomial_interpolate_constant, is for
recovering the constant term of an interpolating polynomial defined by a set of points.The function polynomial_evaluate is defined as follows.
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 = Scalar(0)
for coeff in reverse(coeffs):
value *= x
value += coeff
return value
The function polynomial_interpolate_constant is defined as follows.
Inputs:
- points, a set of t points with distinct x coordinates on
a polynomial f, each a tuple of two Scalar values representing the
x and y coordinates.
Outputs:
- f_zero, the constant term of f, i.e., f(0), a Scalar.
def polynomial_interpolate_constant(points):
x_coords = []
for (x, y) in points:
x_coords.append(x)
f_zero = Scalar(0)
for (x, y) in points:
delta = y * derive_interpolating_value(x_coords, x)
f_zero += delta
return f_zero
Verifiable Secret SharingFeldman's Verifiable Secret Sharing (VSS)
builds upon Shamir secret sharing, adding a verification step to demonstrate the consistency of a participant's
share with a public commitment to the polynomial f for which the secret s
is the constant term. This check ensures that all participants have a point
(their share) on the same polynomial, ensuring that they can reconstruct
the correct secret later.The procedure for committing to a polynomial f of degree at most MIN_PARTICIPANTS-1 is as follows.
Inputs:
- coeffs, a vector of the MIN_PARTICIPANTS coefficients that
uniquely determine a polynomial f.
Outputs:
- vss_commitment, a vector commitment to each of the coefficients in
coeffs, where each item of the vector commitment is an Element.
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 the failure should be investigated out of band.
Inputs:
- share_i: A tuple of the form (i, sk_i), where i indicates the
participant identifier (a NonZeroScalar), and sk_i the
participant's secret key, a secret share of the constant term of f,
where sk_i is a Scalar.
- vss_commitment, a VSS commitment to a secret polynomial f, a vector
commitment to each of the coefficients in coeffs, where each
element of the vector commitment is an Element.
Outputs:
- True if sk_i is valid, and False otherwise.
def vss_verify(share_i, vss_commitment)
(i, sk_i) = share_i
S_i = G.ScalarBaseMult(sk_i)
S_i' = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
S_i' += G.ScalarMult(vss_commitment[j], pow(i, j))
return S_i == S_i'
We now define how the Coordinator and participants can derive group info,
which is an input into the FROST signing protocol.
Inputs:
- MAX_PARTICIPANTS, the number of shares to generate, an integer.
- MIN_PARTICIPANTS, the threshold of the secret sharing scheme,
an integer.
- vss_commitment, a VSS commitment to a secret polynomial f, a vector
commitment to each of the coefficients in coeffs, where each
element of the vector commitment is an Element.
Outputs:
- PK, the public key representing the group, an Element.
- participant_public_keys, a list of MAX_PARTICIPANTS public keys
PK_i for i=1,...,MAX_PARTICIPANTS, where each PK_i is the public
key, an Element, for participant i.
def derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS,
vss_commitment):
PK = vss_commitment[0]
participant_public_keys = []
for i in range(1, MAX_PARTICIPANTS+1):
PK_i = G.Identity()
for j in range(0, MIN_PARTICIPANTS):
PK_i += G.ScalarMult(vss_commitment[j], pow(i, j))
participant_public_keys.append(PK_i)
return PK, participant_public_keys
Random Scalar GenerationTwo popular algorithms for
generating a random integer uniformly distributed in the range [0,
G.Order() -1] are described in the sections that follow.Rejection SamplingGenerate a random byte array with Ns bytes and attempt to map to a Scalar
by calling DeserializeScalar in constant time. If it succeeds, return the
result. If it fails, try again with another random byte array, until the
procedure succeeds. Failure to implement DeserializeScalar in constant time
can leak information about the underlying corresponding Scalar.As an optimization, if the group order is very close to a power of
2, it is acceptable to omit the rejection test completely. In particular, if the group order is p and there is an integer b such that |p - 2^{b}| is less than 2^{(b/2)}, then
RandomScalar can simply return a uniformly random integer of at
most b bits.Wide ReductionGenerate a random byte array with l = ceil(((3 * ceil(log2(G.Order()))) / 2) / 8)
bytes and interpret it as an integer; reduce the integer modulo G.Order() and return the
result. See for the underlying derivation of l.Test VectorsThis section contains test vectors for all ciphersuites listed in .
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_PARTICIPANTS, MIN_PARTICIPANTS, and NUM_PARTICIPANTS.
Group input parameters. This lists the group secret key and shared public key,
generated by a trusted dealer as described in , as well as the
input message to be signed. The randomly generated coefficients produced by the
trusted dealer to share the group signing secret are also listed. Each coefficient
is identified by its index, e.g., share_polynomial_coefficients[1] is the coefficient
of the first term in the polynomial. Note that the 0-th coefficient is omitted, as this
is equal to the group secret key. All values are encoded as hexadecimal strings.
Signer input parameters. This lists the signing key share for each of the
NUM_PARTICIPANTS participants.
Round one parameters and outputs. This lists the NUM_PARTICIPANTS participants engaged
in the protocol, identified by their NonZeroScalar identifier, and the following for each participant: the hiding and binding commitment values produced in ; the randomness
values used to derive the commitment nonces in nonce_generate; the resulting group
binding factor input computed in part from the group commitment list encoded as
described in ; and the group binding factor as computed in .
Round two parameters and outputs. This lists the NUM_PARTICIPANTS participants engaged
in the protocol, identified by their NonZeroScalar identifier, along with their corresponding
output signature share as produced in .
Final output. This lists the aggregate signature as produced in .