]> Zcash Foundation

durumcrustulum@gmail.com

University of Waterloo, Zcash Foundation

ckomlo@uwaterloo.ca

University of Waterloo

iang@uwaterloo.ca

Cloudflare

caw@heapingbits.net

General CFRG Internet-Draft This 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. One such ciphersuite can be used to produce signatures that can be verified with an Edwards-Curve Digital Signature Algorithm (EdDSA, as defined in RFC8032) compliant verifier. However, unlike EdDSA, the signatures produced by FROST are not deterministic. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. Discussion Venues Discussion 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 .

Introduction 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 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 an compliant verifier. 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.

Change Log draft-13 and draft-14

• Hash group public key into binding computation (#439)
• Finalize test vectors (#442)

draft-12

• Address RGLC feedback (#399, #396, #395, #394, #393, #384, #382, #397, #378, #376, #375, #374, #373, #371, #370, #369, #368, #367, #366, #364, #363, #362, #361, #359, #358, #357, #356, #354, #353, #352, #350, #349, #348, #347, #314)
• Fix bug in signature share serialization (#397)
• Fix various editorial issues (#385)

draft-11

• Update version string constant (#307)
• Make SerializeElement reject the identity element (#306)
• Make ciphersuite requirements explicit (#302)
• Fix various editorial issues (#303, #301, #299, #297)

draft-10

• Update version string constant (#296)
• Fix some editorial issues from Ian Goldberg (#295)

draft-09

• Add single-signer signature generation to complement RFC8032 functions (#293)
• Address Thomas Pornin review comments from https://mailarchive.ietf.org/arch/msg/crypto-panel/bPyYzwtHlCj00g8YF1tjj-iYP2c/ (#292, #291, #290, #289, #287, #286, #285, #282, #281, #280, #279, #278, #277, #276, #275, #273, #272, #267)
• Correct Ed448 ciphersuite (#246)
• Various editorial changes (#241, #240)

draft-08

• Add notation for Scalar multiplication (#237)
• Add secp2561k1 ciphersuite (#223)
• Remove RandomScalar implementation details (#231)
• Add domain separation for message and commitment digests (#228)

draft-07

• Fix bug in per-rho signer computation (#222)

draft-06

• Make verification a per-ciphersuite functionality (#219)
• Use per-signer values of rho to mitigate protocol malleability (#217)
• Correct prime-order subgroup checks (#215, #211)
• Fix bug in ed25519 ciphersuite description (#205)
• Various editorial improvements (#208, #209, #210, #218)

draft-05

• Update test vectors to include version string (#202, #203)
• Rename THRESHOLD_LIMIT to MIN_PARTICIPANTS (#192)
• Use non-contiguous signers for the test vectors (#187)
• Add more reasoning why the coordinator MUST abort (#183)
• Add a function to generate nonces (#182)
• Add MUST that all participants have the same view of VSS commitment (#174)
• Use THRESHOLD_LIMIT instead of t and MAX_PARTICIPANTS instead of n (#171)
• Specify what the dealer is trusted to do (#166)
• Clarify types of NUM_PARTICIPANTS and THRESHOLD_LIMIT (#165)
• Assert that the network channel used for signing should be authenticated (#163)
• Remove wire format section (#156)
• Update group commitment derivation to have a single scalarmul (#150)
• Use RandomNonzeroScalar for single-party Schnorr example (#148)
• Fix group notation and clarify member functions (#145)
• Update existing implementations table (#136)
• Various editorial improvements (#135, #143, #147, #149, #153, #158, #162, #167, #168, #169, #170, #175, #176, #177, #178, #184, #186, #193, #198, #199)

draft-04

• Added methods to verify VSS commitments and derive group info (#126, #132).
• Changed check for participants to consider only nonnegative numbers (#133).
• Changed sampling for secrets and coefficients to allow the zero element (#130).
• Split test vectors into separate files (#129)
• Update wire structs to remove commitment shares where not necessary (#128)
• Add failure checks (#127)
• Update group info to include each participant's key and clarify how public key material is obtained (#120, #121).
• Define cofactor checks for verification (#118)
• Various editorial improvements and add contributors (#124, #123, #119, #116, #113, #109)

draft-03

• 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 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 cryptographically secure pseudorandom number generator (CSPRNG); see for additional guidance on the generation of random numbers.

Cryptographic Dependencies FROST signing depends on the following cryptographic constructs:

• Prime-order Group, ;
• Cryptographic hash function, ;

These are described in the following sections.

Prime-Order Group FROST 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, this is 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 Element A 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 Element A, 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 Scalar s. This function raises an error if deserialization fails; see for group-specific input validation steps.

Cryptographic Hash Function FROST 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 which 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 ciphersuite. See for more details about each.

Helper Functions Beyond the core dependencies, the protocol in this document depends on the following helper operations:

• Nonce generation, ;
• Polynomials, ;
• Encoding operations, ;
• Signature binding computation ;
• Group commitment computation ; and
• Signature challenge computation .

The following sections describe these operations in more detail.

Nonce generation To hedge against a bad 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 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 2⁶⁴ signatures are computed by a given signing participant.

Polynomials This 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 3 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 used for polynomial interpolation. It is provided a list of x-coordinates as input, each of which cannot equal 0. 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 Operations This 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. The following function is used to extract identifiers from a commitment list. The following function is used to extract a binding factor from a list of binding factors.

Binding Factors Computation This section describes the subroutine for computing binding factors based on the participant commitment list, message to be signed, and group public key.

Group Commitment Computation This section describes the subroutine for creating the group commitment from a commitment list.

Signature Challenge Computation This section describes the subroutine for creating the per-message challenge.

Two-Round FROST Signing Protocol This 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 integer at least MIN_PARTICIPANTS but no larger than MAX_PARTICIPANTS, where MIN_PARTICIPANTS <= MAX_PARTICIPANTS, MIN_PARTICIPANTS is a positive non-zero 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:

1. Determining which participants will participate (at least MIN_PARTICIPANTS in number);
2. Coordinating rounds (receiving and forwarding inputs among participants); and
3. Aggregating signature shares output by each participant, and publishing the resulting signature.

FROST assumes 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 a distinguished 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 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 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).

The Coordinator and each participant are additionally configured with common group information, denoted "group info," which consists of the following:

• Group public key, which is an Element in G denoted PK.
• Public keys PK_i for each participant, 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. 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 completes, the Coordinator aggregates the signature shares to produce a final signature. The Coordinator SHOULD abort if the signature is invalid; see for more information about dealing with invalid signatures and misbehaving participants. This complete interaction, without abort, is shown in .

FROST protocol overview | == Round 1 (Commitment) == | participant commitment | | |<-----------------------+ | | ... | | participant commitment (commit state) ==\ |<-----------------------------------------+ | | == Round 2 (Signature Share Generation) == | | | | participant input | | | +------------------------> | | | signature share | | | |<-----------------------+ | | | ... | | | participant input | | +------------------------------------------> / | signature share |<=======/ <------------------------------------------+ | == Aggregation == | signature | <-----------+ ]]>

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, 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, and that all messages sent over the wire are encoded appropriately, e.g., that Scalars and Elements are encoded using their respective functions. 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 - Commitment Round 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 used in more than one invocation of sign, and the nonces MUST be generated from a source of secure randomness.

Round Two - Signature Share Generation In 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 its identifier and commitments (from the first round) appear in commitment_list. Applications which require that participants not process 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. 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 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 Aggregation After 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. 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 themselves not yield a valid aggregate signature. If the aggregate signature verification fails, the Coordinator can 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 verify_signature_share, is described below. Recall that the Coordinator is configured with "group info" which contains the group public key PK and public keys PK_i for each participant, so the group_public_key and PK_i function arguments should come from that previously stored group info. The Coordinator can verify each signature share before first 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 Abort FROST 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 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 which signer 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.

Ciphersuites A FROST ciphersuite must specify the underlying prime-order group details and cryptographic hash function. Each ciphersuite is denoted as (Group, Hash), e.g., (ristretto255, SHA-512). This section contains some ciphersuites. 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) 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 64 bytes of output
  • 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 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 'Decode' or that 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 . 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".

• Group: edwards448 , where Ne = 57 and Ns = 57.
  • Order(): Return 2^446 - 13818066809895115352007386748515426880336692474882178609894547503885.
  • 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 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, but for compatibility with , it is omitted. 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".

• Group: P-256 (secp256r1) , where Ne = 33 and Ns = 32.
  • Order(): Return 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551.
  • 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 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 performs 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) 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 . 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".

• Group: secp256k1 , where Ne = 33 and Ns = 32.
  • Order(): Return 0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141.
  • 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 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 performs 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) 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 . Canonical signature encoding is as specified in .

Ciphersuite Requirements Future documents that introduce new ciphersuites MUST adhere to the following requirements.

1. H1, H2, and H3 all have output distributions that are close to (indistinguishable from) the uniform distribution.
2. 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 to maintain compatibility with .
3. 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 failure when deserialization fails.
4. The canonical signature encoding details are clearly specified.

Security Considerations A security analysis of FROST exists in and . At a high level, FROST provides security against Existential Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in . Satisfying this requirement requires the ciphersuite to adhere to the requirements in , as well as the following assumptions to 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 denial of service attacks under the following assumptions:

• The Coordinator does not perform a denial of service 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 denial-of-service 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 as a wrapper protocol around FROST that does.
• 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.

Side-channel mitigations Several 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 RECOMENDED, and it 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, 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 Failures We do not specify what implementations should do when the protocol fails, other than requiring that the protocol abort. Examples of viable failure include when a verification check returns invalid or if the underlying transport failed to deliver the required messages.

Removing the Coordinator Role In some settings, it may be desirable to omit the role of the Coordinator entirely. Doing so does not change the security implications of FROST, but instead simply requires each participant to communicate with all other participants. We loosely describe how to perform FROST signing among 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 Hashing FROST 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. 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. 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 Validation Message 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 that each participant check the input message to be 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 Considerations This document makes no IANA requests.

References Normative References ANS 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. In 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. RFC 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. 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. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. Cloudflare, Inc. Cornell Tech Cloudflare, Inc. Stanford University Cloudflare, 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 References Security 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. This 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. This 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. Massachusetts Institute of Technology, Cambridge

Acknowledgments This 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 Encoding This section describes one possible canonical encoding of FROST signatures. Using notation from , the encoding of a FROST signature (R, z) is as follows: Where Signature.R_encoded is G.SerializeElement(R) and Signature.z_encoded is G.SerializeScalar(z) and G is determined by ciphersuite.

Schnorr Signature Generation and Verification for Prime-Order Groups This 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 Generation One possible key generation mechanism is to depend on a trusted dealer, wherein the dealer generates a group secret s uniformly at random and uses Shamir and Verifiable Secret Sharing , as described in and to create secret shares of s, denoted 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, 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. 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 Sharing In Shamir secret sharing, a dealer distributes a secret Scalar s 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 . 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_interpolate_constant is defined in .

Additional polynomial operations This section describes two functions. One function, denoted 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. The function polynomial_interpolate_constant is defined as follows.

Verifiable Secret Sharing Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret sharing, adding a verification step to demonstrate the consistency of a participant's share with a public commitment to the polynomial f for which the secret s is the constant term. This check 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 Generation Two popular algorithms for generating a random integer uniformly distributed in the range [0, G.Order() -1] are as follows:

Rejection Sampling Generate 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 Reduction Generate 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 Vectors This 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 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 NonZeroScalar identifier, along with their corresponding output signature share as produced in .
• Final output. This lists the aggregate signature as produced in .

FROST(Ed25519, SHA-512)

FROST(Ed448, SHAKE256)

FROST(ristretto255, SHA-512)

FROST(P-256, SHA-256)

FROST(secp256k1, SHA-256)