]>
Two-Round Threshold Schnorr Signatures with FROSTZcash Foundationdurumcrustulum@gmail.comUniversity of Waterloo, Zcash Foundationckomlo@uwaterloo.caUniversity of Waterlooiang@uwaterloo.caCloudflarecaw@heapingbits.net
General
CFRGInternet-DraftIn this draft, we present the two-round signing variant of FROST, a Flexible Round-Optimized
Schnorr Threshold signature scheme. FROST signatures can be issued after a threshold number
of entities cooperate to issue a signature, allowing for improved distribution of trust and
redundancy with respect to a secret key. Further, this draft specifies signatures that are
compatible with . However, unlike , the protocol for producing
signatures in this draft is not deterministic, so as to ensure protection against a
key-recovery attack that is possible when even only one signer participant is malicious.Discussion VenuesDiscussion of this document takes place on the
Crypto Forum Research Group mailing list (cfrg@ietf.org),
which is archived at .Source for this draft and an issue tracker can be found at
.IntroductionDISCLAIMER: This is a work-in-progress draft of FROST.RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this draft is
maintained in GitHub. Suggested changes should be submitted as pull requests
at https://github.com/cfrg/draft-irtf-cfrg-frost. Instructions are on that page as
well.Unlike signatures in a single-party setting, threshold signatures
require cooperation among a threshold number of 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 presents a variant of a Flexible Round-Optimized Schnorr Threshold (FROST)
signature scheme originally defined 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. The variant of
FROST presented in this document requires two rounds to compute a signature. Single-round
signing with FROST is out of scope.For select ciphersuites, the signatures produced by this draft are compatible with
. However, unlike , signatures produced by FROST are not
deterministic, since deriving nonces deterministically allows for a complete key-recovery
attack in multi-party discrete logarithm-based signatures, such as FROST.While an optimization to FROST was shown in that reduces scalar multiplications
from linear in the number of signing participants to constant, this draft does not specify that optimization
due to the malleability that this optimization introduces, as shown in .
Specifically, 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.Key generation for FROST signing is out of scope for this document. However, for completeness,
key generation with a trusted dealer is specified in .Change Logdraft-09
Add single-signer signature generation to complement RFC8032 functions (#293)
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
Refactor the second round to use state from the first round (#94).
Ensure that verification of signature shares from the second round uses commitments from the first round (#94).
Clarify RFC8032 interoperability based on PureEdDSA (#86).
Specify signature serialization based on element and scalar serialization (#85).
Fix hash function domain separation formatting (#83).
Make trusted dealer key generation deterministic (#104).
Add additional constraints on participant indexes and nonce usage (#105, #103, #98, #97).
Apply various editorial improvements.
draft-02
Fully specify both rounds of FROST, as well as trusted dealer key generation.
Add ciphersuites and corresponding test vectors, including suites for RFC8032 compatibility.
Refactor document for editorial clarity.
draft-01
Specify operations, notation and cryptographic dependencies.
draft-00
Outline CFRG draft based on draft-komlo-frost.
Conventions and DefinitionsThe 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.
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 input 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 integer result of a to the power of b, e.g., pow(2, 3) = 8.
|| 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 cryptographically secure pseudorandom number generator (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, ;
These are described in the following sections.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 types 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 B with itself r-1 times, this is denoted as ScalarBaseMult(r). The set of
scalars corresponds to GF(p), which we refer to as the scalar field. This document uses types
Element and Scalar to denote elements of the group G and its set of scalars, respectively.
We denote 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.
We denote equality comparison as == and assignment of values by =. Finally, it is assumed that
group element addition, negation, and equality comparisons can be efficiently computed for
arbitrary group elements.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): Output the scalar multiplication between Element A and Scalar k.
ScalarBaseMult(k): Output 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.
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 can raise a DeserializeError if deserialization fails
or 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 can raise a DeserializeError 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 functions effectively as a random oracle. For concrete
recommendations on hash functions which SHOULD be used in practice, see
. Using H, we introduce separate domain-separated hashes,
H1, H2, H3, H4, and H5:
H1, H2, and H3 map arbitrary byte strings to Scalar elements of the prime-order group scalar field.
H4 and H5 are aliases for H with distinct domain separators.
The details of H1, H2, H3, H4, and H5 vary based on ciphersuite. See
for more details about each.Helper FunctionsBeyond the core dependencies, the protocol in this document depends on the
following helper operations:
Nonce generation, ;
Polynomial operations, ;
Encoding operations, ;
Signature binding , group commitment , and challenge computation .
These sections describes these operations in more detail.Nonce generationTo hedge against a bad RNG that outputs predictable values, nonces are
generated with the nonce_generate function by combining fresh randomness
and 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 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.Polynomial OperationsThis section describes operations on and associated with polynomials over Scalars
that are used in the main signing protocol. A polynomial of maximum degree t+1
is represented as a list of t coefficients, where the constant term of the polynomial
is in the first position and the highest-degree coefficient is in the last position.
A point on the polynomial is a tuple (x, y), where y = f(x). For notational
convenience, we refer to the x-coordinate and y-coordinate of a
point p as p.x and p.y, respectively.Evaluation of a polynomialThis section describes a method for evaluating a polynomial f at a
particular input x, i.e., y = f(x) using Horner's method.Lagrange coefficientsThe function derive_lagrange_coefficient derives a Lagrange coefficient
to later perform polynomial interpolation, and is provided a list of x-coordinates
as input. Note that derive_lagrange_coefficient does not permit any x-coordinate
to equal 0. Lagrange coefficients are used in FROST to evaluate a polynomial f
at x-coordinate 0, i.e., f(0), given a list of t other x-coordinates. 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
L_i = numerator / denominator
return L_i
]]>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 bytestring for use in the FROST protocol.The following function is used to extract participant identifiers from a commitment
list.The following function is used to extract a binding factor from a list of binding factors.Binding Factors ComputationThis section describes the subroutine for computing binding factors based
on the participant commitment list and message to be signed.Group Commitment ComputationThis section describes the subroutine for creating the group commitment
from a commitment list.Signature Challenge ComputationThis section describes the subroutine for creating the per-message challenge.Two-Round FROST Signing ProtocolThis section describes the two-round variant of the FROST threshold signature
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 integer at least MIN_PARTICIPANTS but no larger than
MAX_PARTICIPANTS, where MIN_PARTICIPANTS < MAX_PARTICIPANTS, MIN_PARTICIPANTS is a positive
integer and MAX_PARTICIPANTS is 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 which participants will participate (at least MIN_PARTICIPANTS in number);
Coordinating rounds (receiving and forwarding inputs among participants); and
Aggregating signature shares output by each participant, and publishing the resulting signature.
FROST assumes that all participants, including the Coordinator and the set of participants,
are chosen externally to the protocol. Note that it is possible to deploy the protocol
without a distinguished Coordinator; see for more information.FROST produces signatures that are indistinguishable from those produced with a single
participant using a signing key s with corresponding public key PK, where s is a Scalar
value and PK = G.ScalarMultBase(s). As a threshold signing protocol, the group signing
key s is secret-shared amongst each participant and used to produce signatures. In particular,
FROST assumes each participant is configured with the following information:
An identifier, which is a Scalar value denoted i in the range [1, MAX_PARTICIPANTS]
and MUST be distinct from the identifier of every other participant.
A signing key share sk_i, which is a Scalar value representing the i-th secret share
of the group signing key s. The public key corresponding to this signing key share
is PK_i = G.ScalarMultBase(sk_i).
Each participant, including the Coordinator, is additionally configured
with common group information, denoted "group info," which consists of the following
information:
Group public key, which is an Element in G denoted PK.
Public keys PK_i for each signer, which are Element values in G denoted 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 possible configuration
mechanisms are possible: one that requires a single, trusted dealer, and the other
which requires performing a distributed key generation protocol. We highlight
key generation mechanism by a trusted dealer in for reference.The signing variant of FROST in this document requires participants to perform two network rounds:
1) generating and publishing commitments, and 2) signature share generation and
publication. The first round serves for each participant to issue a commitment to
a nonce. The second round receives commitments for all participants as well as the message,
and issues a signature share with respect to that message. The Coordinator performs
the coordination of each of these rounds. At the end of the second round, the
Coordinator then performs an aggregation step and outputs the final signature.
This complete interaction is shown in .Details for round one are described in , and details for round two
are described in . Note that each participant persists some state between
both rounds, and this state is deleted as described in . The final
Aggregation step is described in .FROST assumes that all inputs to each round, especially those of which are received
over the network, are validated before use. In particular, this means that any value
of type Element or Scalar is deserialized using DeserializeElement and DeserializeScalar,
respectively, as these functions perform the necessary input validation steps.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 any participant which has misbehaved (resulting in the protocol aborting)
to take actions such as excluding them from future signing operations, 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).The outputs nonce and comm from participant P_i should both be 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 reused in more than one invocation of FROST, and it 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
for choosing which participants will participate (of number 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 their identifier as
well as their commitment as from the first round appears in commitment_list.
Applications which require that participants not process arbitrary
input messages are also required to also 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.The output of this procedure is a signature share. Each participant then sends
these shares back to the Coordinator. Each participant MUST delete the nonce and
corresponding commitment after this round completes, and MUST use the nonce
to generate at most one signature share.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.Upon receipt from each Signer, the Coordinator MUST validate the input
signature share using DeserializeElement. If validation fails, the Coordinator MUST abort
the protocol. If validation succeeds, the Coordinator then verifies the set of
signature shares using the following procedure.Signature Share Verification and AggregationAfter participants perform round two and send their signature shares to the Coordinator,
the Coordinator verifies each signature share for correctness. In particular,
for each participant, the Coordinator uses commitment pairs generated during round
one and the signature share generated during round two, along with other group
parameters, to check that the signature share is valid using the following procedure.If any signature share fails to verify, i.e., if verify_signature_share returns False for
any participant share, the Coordinator MUST abort the protocol for correctness reasons
(this is true regardless of the size or makeup of the signing set selected by
the Coordinator).
Excluding one participant means that their nonce will not be included in the joint response z
and consequently the output signature will not verify. This is because the
group commitment will be with respect to a different signing set than the
the aggregated response.Otherwise, if all shares from participants that participated in Rounds 1 and 2 are valid, the Coordinator
performs the aggregate operation and publishes the resulting signature.The output signature (R, z) from the aggregation step MUST be encoded as follows
(using notation from ):Where Signature.R_encoded is G.SerializeElement(R) and Signature.z_encoded is
G.SerializeScalar(z).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 contextString,
which is an ASCII string literal (with no NULL terminating character).The RECOMMENDED ciphersuite is (ristretto255, SHA-512) .
The (Ed25519, SHA-512) ciphersuite is included for backwards compatibility
with .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 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.FROST(Ed25519, SHA-512)This ciphersuite uses edwards25519 for the Group and SHA-512 for the Hash function H
meant to produce signatures indistinguishable from Ed25519 as specified in .
The value of the contextString parameter is "FROST-ED25519-SHA512-v8".
Group: edwards25519
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 .
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. 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.
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
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, but for backwards compatibility
with , it is omitted.Signature verification is as specified in with the
constraint that implementations MUST check the group equation [8][S]B = [8]R + [8][k]A'.
The alternative check [S]B = R + [k]A' is not safe or interoperable in practice.
Note that optimizations for this check exist; see .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-v8".
Group: ristretto255
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 .
DeserializeElement(buf): Implemented using the 'Decode' function from .
Additionally, this function validates that the resulting element is not the group
identity element.
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
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 .FROST(Ed448, SHAKE256)This ciphersuite uses edwards448 for the Group and SHAKE256 for the Hash function H
meant to produce signatures indistinguishable from Ed448 as specified in .
The value of the contextString parameter is "FROST-ED448-SHAKE256-v8".
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 .
DeserializeElement(buf): Implemented as specified in .
Additionally, this function validates that the resulting element is not the group
identity element.
SerializeScalar(s): Implemented by outputting the little-endian 48-byte encoding of
the Scalar value.
DeserializeScalar(buf): Implemented by attempting to deserialize a Scalar from a
little-endian 48-byte string. This function can fail if the input does not
represent a Scalar in the range [0, G.Order() - 1].
Hash (H): SHAKE256
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, but for backwards compatibility
with , it is omitted.Signature verification is as specified in with the
constraint that implementations MUST check the group equation [4][S]B = [4]R + [4][k]A'.
The alternative check [S]B = R + [k]A' is not safe or interoperable in practice.
Note that optimizations for this check exist; see .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-v8".
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 compressed Elliptic-Curve-Point-to-Octet-String
method according to , yielding a 33 byte output.
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 performs partial public-key validation as defined in section 5.6.2.3.4 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. Additionally, this function
validates that the resulting element is not the group identity element.
If 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
H1(m): Implemented as hash_to_field(m, 1) from
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) from
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) from
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 .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-v8".
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 compressed Elliptic-Curve-Point-to-Octet-String
method according to .
DeserializeElement(buf): Implemented by attempting to deserialize a public key using
the compressed Octet-String-to-Elliptic-Curve-Point method according to ,
and then performs partial 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. Additionally, this function validates that the resulting
element is not the group identity element. If 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
H1(m): Implemented as hash_to_field(m, 1) from
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) from
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) from
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 .Security ConsiderationsA security analysis of FROST exists in and . The protocol as specified
in this document assumes the following threat model.
Trusted dealer. The dealer that performs key generation is trusted to follow
the protocol, although participants still are able to verify the consistency of their
shares via a VSS (verifiable secret sharing) step; see .
Unforgeability assuming at most (MIN_PARTICIPANTS-1) corrupted participants. So long as an adversary
corrupts fewer than MIN_PARTICIPANTS participants, the scheme remains secure against Existential
Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in ,
Definition 13.2.
Coordinator. We assume the Coordinator at the time of signing does not perform a
denial of service attack. A denial of service would include any action which either
prevents the protocol from completing or causing the resulting signature to be invalid.
Such actions for the latter include sending inconsistent values to participants,
such as messages or the set of individual commitments. Note that the Coordinator
is not trusted with any private information and communication at the time of signing
can be performed over a public but reliable channel.
The protocol as specified in this document does not target the following goals:
Post quantum security. FROST, like plain Schnorr signatures, requires the hardness of the Discrete Logarithm Problem.
Robustness. In the case of failure, FROST requires aborting the protocol.
Downgrade prevention. All participants in the protocol are assumed to agree on what algorithms to use.
Metadata protection. If protection for metadata is desired, a higher-level communication
channel can be used to facilitate key generation and signing.
The rest of this section documents issues particular to implementations or deployments.Nonce Reuse Attacks describes the procedure that participants use to produce nonces during
the first round of singing. 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, which 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 that
the protocol abort. Examples of viable failure include when a verification check returns invalid or
if the underlying transport failed to deliver the required messages.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, but instead 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 as input from an external source the
message to be signed 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 instead will publish
this commitment to every other participant. Then, in the second round, participants will already have
sufficient information to perform signing. 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 messages that
are exchanged are public and so the channel simply must be reliable. However, in
the setting that a player attempts to split the view of all other players by
sending disjoint values to a subset of players, the signing operation will output
an invalid signature. To avoid this denial of service, implementations may wish
to define a mechanism where messages are authenticated, so that cheating players
can be identified and excluded.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 in inherent to the ciphersuite chosen. It is
RECOMMENDED that applications which choose to apply pre-hashing 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. One possible example is to construct this pre-hash over message m as
H(contextString \|\| "pre-hash" \|\| m).Input Message ValidationSome applications may require that participants only process messages of a certain
structure. For example, in digital currency applications wherein multiple
participants may collectively sign a transaction, it is reasonable to require that
each participant check the input message to be a syntactically valid transaction.As another example, use of threshold signatures in to produce
signatures of transcript hashes might require the participants receive the source
handshake messages themselves, and recompute the transcript hash which is used
as input message to the signature generation process, so that they can verify
that they are signing a proper TLS transcript hash and not some other data.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.ContributorsReferencesNormative ReferencesPublic Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)ANSElliptic Curve Cryptography, Standards for Efficient Cryptography Group, ver. 2Recommended Elliptic Curve Domain Parameters, Standards for Efficient Cryptography Group, ver. 2Edwards-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.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.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 Groups This 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.
Recommendation for pair-wise key-establishment schemes using discrete logarithm cryptographyHashing to Elliptic CurvesCloudflare, Inc.Cornell TechCloudflare, Inc.Stanford UniversityCloudflare, Inc. This 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.
Informative ReferencesTwo-Round Threshold Signatures with FROSTHow to Prove Schnorr Assuming SchnorrStronger Security for Non-Interactive Threshold Signatures: BLS and FROSTA Graduate Course in Applied CryptographyPoint-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.The 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.AcknowledgmentsThis document was improved based on input and contributions by the Zcash Foundation engineering team.
In addition, the authors of this document would like to thank
Isis Lovecruft,
Alden Torres,
T. Wilson-Brown,
and Conrado Gouvea
for their inputs and contributions.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.)The function prime_order_verify verifies Schnorr signatures with validated inputs.
Specifically, it assumes that signature R component and public key belong to the prime-order group.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 and to create secret
shares of s, denoted s_i for i = 0, ..., 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_SIGNER 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, and 2) delete secret values after distributing shares to each participant, and 3) keep secret values confidential.It is assumed 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.
Otherwise, 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 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. MAX_PARTICIPANTS or if MIN_PARTICIPANTS is less than 2
def secret_share_shard(s, coefficients, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
if MIN_PARTICIPANTS > MAX_PARTICIPANTS:
raise "invalid parameters"
if MIN_PARTICIPANTS < 2:
raise "invalid parameters"
# 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_share.append(secret_key_share_i)
return secret_key_shares, coefficients
]]>Let points be the output of this function. The i-th element in points is
the share for the i-th participant, which is the randomly generated polynomial
evaluated at coordinate i. We denote a secret share as the tuple (i, points[i]),
and the list of these shares as shares.
i MUST never equal 0; recall that f(0) = s, where f is the polynomial defined in a Shamir secret sharing operation.The procedure for combining a shares list of length MIN_PARTICIPANTS to recover the
secret s is as follows; the algorithm polynomial_interpolation is defined in {{dep-polynomial-interpolate}}.Deriving the constant term of a polynomialSecret sharing requires "splitting" a secret, which is represented as
a constant term of some polynomial f of degree t-1. Recovering the
constant term occurs with a set of t points using polynomial
interpolation, defined as follows.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 later reconstruct
the correct secret.The procedure for committing to a polynomial f of degree at most MIN_PARTICIPANTS-1 is as follows.The procedure for verification of a participant's share is as follows.
If vss_verify fails, the participant MUST abort the protocol, and failure should be investigated out of band.We now define how the Coordinator and participants can derive group info,
which is an input into the FROST signing protocol.Random Scalar GenerationTwo popular algorithms for generating a random integer uniformly distributed in
the range [0, G.Order() -1] are as follows: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 this in constant time can leak information
about the underlying corresponding Scalar.Note the that the Scalar size might be some bits smaller than the array size,
which can result in the loop iterating more times than required. In that case
it's acceptable to set the high-order bits to 0 before calling DeserializeScalar,
but care must be taken to not set to zero more bits than required. For example,
in the FROST(Ed25519, SHA-512) ciphersuite, the order has 253 bits while
the array has 256; thus the top 3 bits of the last byte can be set to zero.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 integer identifier, and 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 group binding factor as computed in ).
Round two parameters and outputs. This lists the NUM_PARTICIPANTS participants engaged
in the protocol, identified by their integer identifier, along with their corresponding
output signature share as produced in .
Final output. This lists the aggregate signature as produced in .