path: root/doc/rfc/rfc9591.txt

Internet Research Task Force (IRTF)                          D. Connolly
Request for Comments: 9591                              Zcash Foundation
Category: Informational                                         C. Komlo
ISSN: 2070-1721                 University of Waterloo, Zcash Foundation
                                                             I. Goldberg
                                                  University of Waterloo
                                                              C. A. Wood
                                                              Cloudflare
                                                               June 2024


  The Flexible Round-Optimized Schnorr Threshold (FROST) Protocol for
                      Two-Round Schnorr Signatures

Abstract

   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.  This document is a product of the
   Crypto Forum Research Group (CFRG) in the IRTF.

Status of This Memo

   This document is not an Internet Standards Track specification; it is
   published for informational purposes.

   This document is a product of the Internet Research Task Force
   (IRTF).  The IRTF publishes the results of Internet-related research
   and development activities.  These results might not be suitable for
   deployment.  This RFC represents the consensus of the Crypto Forum
   Research Group of the Internet Research Task Force (IRTF).  Documents
   approved for publication by the IRSG are not candidates for any level
   of Internet Standard; see Section 2 of RFC 7841.

   Information about the current status of this document, any errata,
   and how to provide feedback on it may be obtained at
   https://www.rfc-editor.org/info/rfc9591.

Copyright Notice

   Copyright (c) 2024 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.

Table of Contents

   1.  Introduction
   2.  Conventions and Definitions
   3.  Cryptographic Dependencies
     3.1.  Prime-Order Group
     3.2.  Cryptographic Hash Function
   4.  Helper Functions
     4.1.  Nonce Generation
     4.2.  Polynomials
     4.3.  List Operations
     4.4.  Binding Factors Computation
     4.5.  Group Commitment Computation
     4.6.  Signature Challenge Computation
   5.  Two-Round FROST Signing Protocol
     5.1.  Round One - Commitment
     5.2.  Round Two - Signature Share Generation
     5.3.  Signature Share Aggregation
     5.4.  Identifiable Abort
   6.  Ciphersuites
     6.1.  FROST(Ed25519, SHA-512)
     6.2.  FROST(ristretto255, SHA-512)
     6.3.  FROST(Ed448, SHAKE256)
     6.4.  FROST(P-256, SHA-256)
     6.5.  FROST(secp256k1, SHA-256)
     6.6.  Ciphersuite Requirements
   7.  Security Considerations
     7.1.  Side-Channel Mitigations
     7.2.  Optimizations
     7.3.  Nonce Reuse Attacks
     7.4.  Protocol Failures
     7.5.  Removing the Coordinator Role
     7.6.  Input Message Hashing
     7.7.  Input Message Validation
   8.  IANA Considerations
   9.  References
     9.1.  Normative References
     9.2.  Informative References
   Appendix A.  Schnorr Signature Encoding
   Appendix B.  Schnorr Signature Generation and Verification for
           Prime-Order Groups
   Appendix C.  Trusted Dealer Key Generation
     C.1.  Shamir Secret Sharing
       C.1.1.  Additional Polynomial Operations
     C.2.  Verifiable Secret Sharing
   Appendix D.  Random Scalar Generation
     D.1.  Rejection Sampling
     D.2.  Wide Reduction
   Appendix E.  Test Vectors
     E.1.  FROST(Ed25519, SHA-512)
     E.2.  FROST(Ed448, SHAKE256)
     E.3.  FROST(ristretto255, SHA-512)
     E.4.  FROST(P-256, SHA-256)
     E.5.  FROST(secp256k1, SHA-256)
   Acknowledgments
   Authors' Addresses

1.  Introduction

   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
   [FROST20].  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 [FROST20] 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 [RFC8032], i.e., the
   signatures can be verified with a verifier that is compliant with
   [RFC8032].  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 Appendix C.

   This document represents the consensus of the Crypto Forum Research
   Group (CFRG).  It is not an IETF product and is not a standard.

2.  Conventions and Definitions

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in
   BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

   The following notation is used throughout the document.

   byte:  A sequence of eight bits.

   random_bytes(n):  Outputs n bytes, sampled uniformly at random using
      a cryptographically secure pseudorandom number generator (CSPRNG).

   count(i, L):  Outputs the number of times the element i is
      represented in the list L.

   len(l):  Outputs the length of list l, e.g., len([1,2,3]) = 3.

   reverse(l):  Outputs the list l in reverse order, e.g.,
      reverse([1,2,3]) = [3,2,1].

   range(a, b):  Outputs a list of integers from a to b-1 in ascending
      order, e.g., range(1, 4) = [1,2,3].

   pow(a, b):  Outputs the result, a Scalar, of a to the power of b,
      e.g., pow(2, 3) = 8 modulo the relevant group order p.

   ||:  Denotes concatenation of byte strings, i.e., x || y denotes the
      byte string x, immediately followed by the byte string y, with no
      extra separator, yielding xy.

   nil:  Denotes an empty byte string.

   Unless otherwise stated, we assume that secrets are sampled uniformly
   at random using a CSPRNG; see [RFC4086] for additional guidance on
   the generation of random numbers.

3.  Cryptographic Dependencies

   FROST signing depends on the following cryptographic constructs:

   *  Prime-order group (Section 3.1)

   *  Cryptographic hash function (Section 3.2)

   The following sections describe these constructs in more detail.

3.1.  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, 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 Section 6 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
      Section 6 for group-specific input validation steps.

3.2.  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 [FROST20] and [StrongerSec22]).
   For concrete recommendations on hash functions that SHOULD be used in
   practice, see Section 6.  Using H, we introduce distinct domain-
   separated hashes H1, H2, H3, H4, and H5:

   *  H1, H2, and H3 map arbitrary byte strings to Scalar elements
      associated with the prime-order group.

   *  H4 and H5 are aliases for H with distinct domain separators.

   The details of H1, H2, H3, H4, and H5 vary based on the ciphersuite
   used.  See Section 6 for more details about each.

4.  Helper Functions

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

   *  Nonce generation (Section 4.1);

   *  Polynomials (Section 4.2);

   *  List operations (Section 4.3);

   *  Binding factors computation (Section 4.4);

   *  Group commitment computation (Section 4.5); and

   *  Signature challenge computation (Section 4.6).

   The following sections describe these operations in more detail.

4.1.  Nonce Generation

   To hedge against a bad random number generator (RNG) that outputs
   predictable values, nonces are generated with the nonce_generate
   function by combining fresh randomness with the secret key as input
   to a domain-separated hash function built from the ciphersuite hash
   function H.  This domain-separated hash function is denoted as H3.
   This function always samples 32 bytes of fresh randomness to ensure
   that the probability of nonce reuse is at most 2^-128 as long as no
   more than 2^64 signatures are computed by a given signing
   participant.

   Inputs:
   - secret, a Scalar.

   Outputs:
   - nonce, a Scalar.

   def nonce_generate(secret):
     random_bytes = random_bytes(32)
     secret_enc = G.SerializeScalar(secret)
     return H3(random_bytes || secret_enc)

4.2.  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 three coefficients [3, 2, 1].  A
   point on the polynomial f is a tuple (x, y), where y = f(x).

   The function derive_interpolating_value derives a value that is used
   for polynomial interpolation.  It is provided a list of x-coordinates
   as input, each of which cannot equal 0.

   Inputs:
   - L, the list of x-coordinates, each a NonZeroScalar.
   - x_i, an x-coordinate contained in L, a NonZeroScalar.

   Outputs:
   - value, a Scalar.

   Errors:
   - "invalid parameters", if 1) x_i is not in L, or if 2) any
     x-coordinate is represented more than once in L.

   def derive_interpolating_value(L, x_i):
     if x_i not in L:
       raise "invalid parameters"
     for x_j in L:
       if count(x_j, L) > 1:
         raise "invalid parameters"

     numerator = Scalar(1)
     denominator = Scalar(1)
     for x_j in L:
       if x_j == x_i: continue
       numerator *= x_j
       denominator *= x_j - x_i

     value = numerator / denominator
     return value

4.3.  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.

   Inputs:
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.

   Outputs:
   - encoded_group_commitment, the serialized representation of
     commitment_list, a byte string.

   def encode_group_commitment_list(commitment_list):
     encoded_group_commitment = nil
     for (identifier, hiding_nonce_commitment,
          binding_nonce_commitment) in commitment_list:
       encoded_commitment = (
           G.SerializeScalar(identifier) ||
           G.SerializeElement(hiding_nonce_commitment) ||
           G.SerializeElement(binding_nonce_commitment))
       encoded_group_commitment = (
           encoded_group_commitment ||
           encoded_commitment)
     return encoded_group_commitment

   The following function is used to extract identifiers from a
   commitment list.

   Inputs:
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.

   Outputs:
   - identifiers, a list of NonZeroScalar values.

   def participants_from_commitment_list(commitment_list):
     identifiers = []
     for (identifier, _, _) in commitment_list:
       identifiers.append(identifier)
     return identifiers

   The following function is used to extract a binding factor from a
   list of binding factors.

   Inputs:
   - binding_factor_list = [(i, binding_factor), ...],
     a list of binding factors for each participant, where each element
     in the list indicates a NonZeroScalar identifier i and Scalar
     binding factor.
   - identifier, participant identifier, a NonZeroScalar.

   Outputs:
   - binding_factor, a Scalar.

   Errors:
   - "invalid participant", when the designated participant is
     not known.

   def binding_factor_for_participant(binding_factor_list, identifier):
     for (i, binding_factor) in binding_factor_list:
       if identifier == i:
         return binding_factor
     raise "invalid participant"

4.4.  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.

   Inputs:
   - group_public_key, the public key corresponding to the group signing
     key, an Element.
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.
   - msg, the message to be signed.

   Outputs:
   - binding_factor_list, a list of (NonZeroScalar, Scalar) tuples
     representing the binding factors.

   def compute_binding_factors(group_public_key, commitment_list, msg):
     group_public_key_enc = G.SerializeElement(group_public_key)
     // Hashed to a fixed length.
     msg_hash = H4(msg)
     // Hashed to a fixed length.
     encoded_commitment_hash =
         H5(encode_group_commitment_list(commitment_list))
     // The encoding of the group public key is a fixed length
     // within a ciphersuite.
     rho_input_prefix = group_public_key_enc || msg_hash ||
      encoded_commitment_hash

     binding_factor_list = []
     for (identifier, hiding_nonce_commitment,
          binding_nonce_commitment) in commitment_list:
       rho_input = rho_input_prefix || G.SerializeScalar(identifier)
       binding_factor = H1(rho_input)
       binding_factor_list.append((identifier, binding_factor))
     return binding_factor_list

4.5.  Group Commitment Computation

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

   Inputs:
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.
   - binding_factor_list = [(i, binding_factor), ...],
     a list of (NonZeroScalar, Scalar) tuples representing the binding
     factor Scalar for the given identifier.

   Outputs:
   - group_commitment, an Element.

   def compute_group_commitment(commitment_list, binding_factor_list):
     group_commitment = G.Identity()
     for (identifier, hiding_nonce_commitment,
          binding_nonce_commitment) in commitment_list:
       binding_factor = binding_factor_for_participant(
           binding_factor_list, identifier)
       binding_nonce = G.ScalarMult(
           binding_nonce_commitment,
           binding_factor)
       group_commitment = (
           group_commitment +
           hiding_nonce_commitment +
           binding_nonce)
     return group_commitment

   Note that the performance of this algorithm is defined naively and
   scales linearly relative to the number of signers.  For improved
   performance, the group commitment can be computed using multi-
   exponentiation techniques such as Pippinger's algorithm; see
   [MultExp] for more details.

4.6.  Signature Challenge Computation

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

   Inputs:
   - group_commitment, the group commitment, an Element.
   - group_public_key, the public key corresponding to the group signing
     key, an Element.
   - msg, the message to be signed, a byte string.

   Outputs:
   - challenge, a Scalar.

   def compute_challenge(group_commitment, group_public_key, msg):
     group_comm_enc = G.SerializeElement(group_commitment)
     group_public_key_enc = G.SerializeElement(group_public_key)
     challenge_input = group_comm_enc || group_public_key_enc || msg
     challenge = H2(challenge_input)
     return challenge

5.  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 and non-zero integer
   that MUST be at least MIN_PARTICIPANTS, but MUST NOT be larger than
   MAX_PARTICIPANTS, where MIN_PARTICIPANTS <= MAX_PARTICIPANTS and
   MIN_PARTICIPANTS is a positive and non-zero integer.  Additionally,
   MAX_PARTICIPANTS MUST be a positive integer less than the group
   order.  A signer participant, or simply "participant", is an entity
   that is trusted to hold and use a signing key share.  The Coordinator
   is an entity with the following responsibilities:

   1.  Determining the participants that will participate (at least
       MIN_PARTICIPANTS in number);

   2.  Coordinating rounds (receiving and forwarding inputs among
       participants);

   3.  Aggregating signature shares output by each participant; and

   4.  Publishing the resulting signature.

   FROST assumes that the Coordinator and the set of signer participants
   are chosen externally to the protocol.  Note that it is possible to
   deploy the protocol without designating a single Coordinator; see
   Section 7.5 for more information.

   FROST produces signatures that can be verified as if they were
   produced from a single signer using a signing key s with
   corresponding public key PK, where s is a Scalar value and PK =
   G.ScalarBaseMult(s).  As a threshold signing protocol, the group
   signing key s is Shamir secret-shared amongst each of the
   MAX_PARTICIPANTS participants and is used to produce signatures; see
   Appendix C.1 for more information about Shamir secret sharing.  In
   particular, FROST assumes each participant is configured with the
   following information:

   *  An identifier, which is a NonZeroScalar value denoted as i in the
      range [1, MAX_PARTICIPANTS] and MUST be distinct from the
      identifier of every other participant.

   *  A signing key sk_i, which is a Scalar value representing the i-th
      Shamir secret share of the group signing key s.  In particular,
      sk_i is the value f(i) on a secret polynomial f of degree
      (MIN_PARTICIPANTS - 1), where s is f(0).  The public key
      corresponding to this signing key share is PK_i =
      G.ScalarBaseMult(sk_i).

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

   *  Group public key, which is an Element in G denoted as PK.

   *  Public keys PK_i for each participant, which are Element values in
      G denoted as PK_i for each i in [1, MAX_PARTICIPANTS].

   This document does not specify how this information, including the
   signing key shares, are configured and distributed to participants.
   In general, two configuration mechanisms are possible: one that
   requires a single trusted dealer and one that requires performing a
   distributed key generation protocol.  We highlight the key generation
   mechanism by a trusted dealer in Appendix C for reference.

   FROST requires two rounds to complete.  In the first round,
   participants generate and publish one-time-use commitments to be used
   in the second round.  In the second round, each participant produces
   a share of the signature over the Coordinator-chosen message and the
   other participant commitments.  After the second round is completed,
   the Coordinator aggregates the signature shares to produce a final
   signature.  The Coordinator SHOULD abort the protocol if the
   signature is invalid; see Section 5.4 for more information about
   dealing with invalid signatures and misbehaving participants.  This
   complete interaction (without being aborted) is shown in Figure 1.

           (group info)            (group info,     (group info,
               |               signing key share)   signing key share)
               |                         |                |
               v                         v                v
           Coordinator               Signer-1   ...   Signer-n
       ------------------------------------------------------------
      signing request
   ------------>
               |
         == Round 1 (Commitment) ==
               | participant commitment |                 |
               |<-----------------------+                 |
               |          ...                             |
               | participant commitment            (commit state) ==\
               |<-----------------------------------------+         |
                                                                    |
         == Round 2 (Signature Share Generation) ==                 |
      message
   ------------>
               |                                                    |
               |   participant input    |                 |         |
               +------------------------>                 |         |
               |     signature share    |                 |         |
               |<-----------------------+                 |         |
               |          ...                             |         |
               |    participant input                     |         |
               +------------------------------------------>         /
               |     signature share                      |<=======/
               <------------------------------------------+
               |
         == Aggregation ==
               |
     signature |
   <-----------+

                     Figure 1: FROST Protocol Overview

   Details for round one are described in Section 5.1 and details for
   round two are described in Section 5.2.  Note that each participant
   persists some state between the two rounds; this state is deleted as
   described in Section 5.2.  The final Aggregation step is described in
   Section 5.3.

   FROST assumes that all inputs to each round, especially those that
   are received over the network, are validated before use.  In
   particular, this means that any value of type Element or Scalar
   received over the network MUST be deserialized using
   DeserializeElement and DeserializeScalar, respectively, as these
   functions perform the necessary input validation steps.
   Additionally, all messages sent over the wire MUST be encoded using
   their respective functions, e.g., Scalars and Elements are encoded
   using SerializeScalar and SerializeElement.

   FROST assumes reliable message delivery between the Coordinator and
   participants in order for the protocol to complete.  An attacker
   masquerading as another participant will result only in an invalid
   signature; see Section 7.  However, in order to identify misbehaving
   participants, we assume that the network channel is additionally
   authenticated; confidentiality is not required.

5.1.  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).

   Inputs:
   - sk_i, the secret key share, a Scalar.

   Outputs:
   - (nonce, comm), a tuple of nonce and nonce commitment pairs,
     where each value in the nonce pair is a Scalar and each value in
     the nonce commitment pair is an Element.

   def commit(sk_i):
     hiding_nonce = nonce_generate(sk_i)
     binding_nonce = nonce_generate(sk_i)
     hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
     binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
     nonces = (hiding_nonce, binding_nonce)
     comms = (hiding_nonce_commitment, binding_nonce_commitment)
     return (nonces, comms)

   The outputs nonce and comm from participant P_i are both stored
   locally and kept for use in the second round.  The nonce value is
   secret and MUST NOT be shared, whereas the public output comm is sent
   to the Coordinator.  The nonce values produced by this function MUST
   NOT be used in more than one invocation of sign, and the nonces MUST
   be generated from a source of secure randomness.

5.2.  Round Two - Signature Share Generation

   In round two, the Coordinator is responsible for sending the message
   to be signed and choosing the participants that will participate (a
   number of at least MIN_PARTICIPANTS).  Signers additionally require
   locally held data, specifically their private key and the nonces
   corresponding to their commitment issued in round one.

   The Coordinator begins by sending each participant the message to be
   signed along with the set of signing commitments for all participants
   in the participant list.  Each participant MUST validate the inputs
   before processing the Coordinator's request.  In particular, the
   signer MUST validate commitment_list, deserializing each group
   Element in the list using DeserializeElement from Section 3.1.  If
   deserialization fails, the signer MUST abort the protocol.  Moreover,
   each participant MUST ensure that its identifier and commitments
   (from the first round) appear in commitment_list.  Applications that
   restrict participants from processing arbitrary input messages are
   also required to perform relevant application-layer input validation
   checks; see Section 7.7 for more details.

   Upon receipt and successful input validation, each signer then runs
   the following procedure to produce its own signature share.

   Inputs:
   - identifier, identifier i of the participant, a NonZeroScalar.
   - sk_i, signer secret key share, a Scalar.
   - group_public_key, public key corresponding to the group signing
     key, an Element.
   - nonce_i, pair of Scalar values (hiding_nonce, binding_nonce)
     generated in round one.
   - msg, the message to be signed, a byte string.
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.


   Outputs:
   - sig_share, a signature share, a Scalar.

   def sign(identifier, sk_i, group_public_key,
            nonce_i, msg, commitment_list):
     # Compute the binding factor(s)
     binding_factor_list = compute_binding_factors(group_public_key,
      commitment_list, msg)
     binding_factor = binding_factor_for_participant(
         binding_factor_list, identifier)

     # Compute the group commitment
     group_commitment = compute_group_commitment(
         commitment_list, binding_factor_list)

     # Compute the interpolating value
     participant_list = participants_from_commitment_list(
         commitment_list)
     lambda_i = derive_interpolating_value(participant_list, identifier)

     # Compute the per-message challenge
     challenge = compute_challenge(
         group_commitment, group_public_key, msg)

     # Compute the signature share
     (hiding_nonce, binding_nonce) = nonce_i
     sig_share = hiding_nonce + (binding_nonce * binding_factor) +
         (lambda_i * sk_i * challenge)

     return sig_share

   The output of this procedure is a signature share.  Each participant
   sends these shares back to the Coordinator.  Each participant MUST
   delete the nonce and corresponding commitment after completing sign
   and MUST NOT use the nonce as input more than once to sign.

   Note that the lambda_i value derived during this procedure does not
   change across FROST signing operations for the same signing group.
   As such, participants can compute it once and store it for reuse
   across signing sessions.

5.3.  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.

   Inputs:
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.
   - msg, the message to be signed, a byte string.
   - group_public_key, public key corresponding to the group signing
     key, an Element.
   - sig_shares, a set of signature shares z_i, Scalar values, for each
     participant, of length NUM_PARTICIPANTS, where
     MIN_PARTICIPANTS <= NUM_PARTICIPANTS <= MAX_PARTICIPANTS.

   Outputs:
   - (R, z), a Schnorr signature consisting of an Element R and
     Scalar z.

   def aggregate(commitment_list, msg, group_public_key, sig_shares):
     # Compute the binding factors
     binding_factor_list = compute_binding_factors(group_public_key,
      commitment_list, msg)

     # Compute the group commitment
     group_commitment = compute_group_commitment(
         commitment_list, binding_factor_list)

     # Compute aggregated signature
     z = Scalar(0)
     for z_i in sig_shares:
       z = z + z_i
     return (group_commitment, z)

   The output from the aggregation step is the output signature (R, z).
   The canonical encoding of this signature is specified in Section 6.

   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
   Section 6 for details.  The aggregate signature will verify
   successfully if all signature shares are valid.  Moreover, subsets of
   valid signature shares will not yield a valid aggregate signature
   themselves.

   If the aggregate signature verification fails, the Coordinator MAY
   verify each signature share individually to identify and act on
   misbehaving participants.  The mechanism for acting on a misbehaving
   participant is out of scope for this specification; see Section 5.4
   for more information about dealing with invalid signatures and
   misbehaving participants.

   The function for verifying a signature share, denoted as
   verify_signature_share, is described below.  Recall that the
   Coordinator is configured with "group info" that contains the group
   public key PK and public keys PK_i for each participant.  The
   group_public_key and PK_i function arguments MUST come from that
   previously stored group info.

   Inputs:
   - identifier, identifier i of the participant, a NonZeroScalar.
   - PK_i, the public key for the i-th participant, where
     PK_i = G.ScalarBaseMult(sk_i), an Element.
   - comm_i, pair of Element values in G
     (hiding_nonce_commitment, binding_nonce_commitment) generated in
     round one from the i-th participant.
   - sig_share_i, a Scalar value indicating the signature share as
     produced in round two from the i-th participant.
   - commitment_list = [(i, hiding_nonce_commitment_i,
     binding_nonce_commitment_i), ...], a list of commitments issued by
     each participant, where each element in the list indicates a
     NonZeroScalar identifier i and two commitment Element values
     (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list
     MUST be sorted in ascending order by identifier.
   - group_public_key, public key corresponding to the group signing
     key, an Element.
   - msg, the message to be signed, a byte string.

   Outputs:
   - True if the signature share is valid, and False otherwise.

   def verify_signature_share(
           identifier, PK_i, comm_i, sig_share_i, commitment_list,
           group_public_key, msg):
     # Compute the binding factors
     binding_factor_list = compute_binding_factors(group_public_key,
      commitment_list, msg)
     binding_factor = binding_factor_for_participant(
         binding_factor_list, identifier)

     # Compute the group commitment
     group_commitment = compute_group_commitment(
         commitment_list, binding_factor_list)

     # Compute the commitment share
     (hiding_nonce_commitment, binding_nonce_commitment) = comm_i
     comm_share = hiding_nonce_commitment + G.ScalarMult(
         binding_nonce_commitment, binding_factor)

     # Compute the challenge
     challenge = compute_challenge(
         group_commitment, group_public_key, msg)

     # Compute the interpolating value
     participant_list = participants_from_commitment_list(
         commitment_list)
     lambda_i = derive_interpolating_value(participant_list, identifier)

     # Compute relation values
     l = G.ScalarBaseMult(sig_share_i)
     r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i)

     return l == r

   The Coordinator can verify each signature share before aggregating
   and verifying the signature under the group public key.  However,
   since the aggregate signature is valid if all signature shares are
   valid, this order of operations is more expensive if the signature is
   valid.

5.4.  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 Section 7 for more
   information about FROST's security properties and the threat model.

   As a result of this property, a misbehaving participant can cause a
   denial of service (DoS) on the signing protocol by contributing
   malformed signature shares or refusing to participate.  Identifying
   misbehaving participants that produce invalid shares can be done by
   checking signature shares from each participant using
   verify_signature_share as described in Section 5.3.  FROST assumes
   the network channel is authenticated to identify the signer that
   misbehaved.  FROST allows for identifying misbehaving participants
   that produce invalid signature shares as described in Section 5.3.
   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.

6.  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 as contextString, which is an ASCII string literal
   (with no terminating NUL character).

   The RECOMMENDED ciphersuite is (ristretto255, SHA-512) as described
   in Section 6.2.  The (Ed25519, SHA-512) and (Ed448, SHAKE256)
   ciphersuites are included for compatibility with Ed25519 and Ed448 as
   defined in [RFC8032].

   The DeserializeElement and DeserializeScalar functions instantiated
   for a particular prime-order group corresponding to a ciphersuite
   MUST adhere to the description in Section 3.1.  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 Section 6.6.  Future
   ciphersuites MUST also adhere to these requirements.

6.1.  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 Section 5.1 of [RFC8032].  The value of the
   contextString parameter is "FROST-ED25519-SHA512-v1".

   Group:  edwards25519 [RFC8032], where Ne = 32 and Ns = 32.

      Order():  Return 2^252 + 27742317777372353535851937790883648493
         (see [RFC7748]).

      Identity():  As defined in [RFC7748].

      RandomScalar():  Implemented by returning a uniformly random
         Scalar in the range [0, G.Order() - 1].  Refer to Appendix D
         for implementation guidance.

      SerializeElement(A):  Implemented as specified in [RFC8032],
         Section 5.1.2.  Additionally, this function validates that the
         input element is not the group identity element.

      DeserializeElement(buf):  Implemented as specified in [RFC8032],
         Section 5.1.3.  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 [Pornin22].

      SerializeScalar(s):  Implemented by outputting the little-endian
         32-byte encoding of the Scalar value with the top three bits
         set to zero.

      DeserializeScalar(buf):  Implemented by attempting to deserialize
         a Scalar from a little-endian 32-byte string.  This function
         can fail if the input does not represent a Scalar in the range
         [0, G.Order() - 1].  Note that this means the top three bits of
         the input MUST be zero.

   Hash (H):  SHA-512, which has an output of 64 bytes.

      H1(m):  Implemented by computing H(contextString || "rho" || m),
         interpreting the 64-byte digest as a little-endian integer, and
         reducing the resulting integer modulo 2^252 +
         27742317777372353535851937790883648493.

      H2(m):  Implemented by computing H(m), interpreting the 64-byte
         digest as a little-endian integer, and reducing the resulting
         integer modulo 2^252 + 27742317777372353535851937790883648493.

      H3(m):  Implemented by computing H(contextString || "nonce" || m),
         interpreting the 64-byte digest as a little-endian integer, and
         reducing the resulting integer modulo 2^252 +
         27742317777372353535851937790883648493.

      H4(m):  Implemented by computing H(contextString || "msg" || m).

      H5(m):  Implemented by computing H(contextString || "com" || m).

   Normally, H2 would also include a domain separator; however, for
   compatibility with [RFC8032], it is omitted.

   Signature verification is as specified in Section 5.1.7 of [RFC8032]
   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 Appendix A.

6.2.  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 [RISTRETTO], where Ne = 32 and Ns = 32.

      Order():  Return 2^252 + 27742317777372353535851937790883648493
         (see [RISTRETTO]).

      Identity():  As defined in [RISTRETTO].

      RandomScalar():  Implemented by returning a uniformly random
         Scalar in the range [0, G.Order() - 1].  Refer to Appendix D
         for implementation guidance.

      SerializeElement(A):  Implemented using the "Encode" function from
         [RISTRETTO].  Additionally, this function validates that the
         input element is not the group identity element.

      DeserializeElement(buf):  Implemented using the "Decode" function
         from [RISTRETTO].  Additionally, this function validates that
         the resulting element is not the group identity element.  If
         either the "Decode" function or the check fails,
         deserialization returns an error.

      SerializeScalar(s):  Implemented by outputting the little-endian
         32-byte encoding of the Scalar value with the top three bits
         set to zero.

      DeserializeScalar(buf):  Implemented by attempting to deserialize
         a Scalar from a little-endian 32-byte string.  This function
         can fail if the input does not represent a Scalar in the range
         [0, G.Order() - 1].  Note that this means the top three bits of
         the input MUST be zero.

   Hash (H):  SHA-512, which has 64 bytes of output.

      H1(m):  Implemented by computing H(contextString || "rho" || m)
         and mapping the output to a Scalar as described in [RISTRETTO],
         Section 4.4.

      H2(m):  Implemented by computing H(contextString || "chal" || m)
         and mapping the output to a Scalar as described in [RISTRETTO],
         Section 4.4.

      H3(m):  Implemented by computing H(contextString || "nonce" || m)
         and mapping the output to a Scalar as described in [RISTRETTO],
         Section 4.4.

      H4(m):  Implemented by computing H(contextString || "msg" || m).

      H5(m):  Implemented by computing H(contextString || "com" || m).

   Signature verification is as specified in Appendix B.

   Canonical signature encoding is as specified in Appendix A.

6.3.  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 Section 5.2 of [RFC8032].  Unlike Ed448 in [RFC8032],
   this ciphersuite does not allow applications to specify a context
   string and always sets the context of [RFC8032] to the empty string.
   Note that this ciphersuite does not allow applications to specify a
   context string as is allowed for Ed448 in [RFC8032], and always sets
   the [RFC8032] context string to the empty string.  The value of the
   (internal to FROST) contextString parameter is "FROST-
   ED448-SHAKE256-v1".

   Group:  edwards448 [RFC8032], where Ne = 57 and Ns = 57.

      Order():  Return 2^446 - 13818066809895115352007386748515426880336
         692474882178609894547503885.

      Identity():  As defined in [RFC7748].

      RandomScalar():  Implemented by returning a uniformly random
         Scalar in the range [0, G.Order() - 1].  Refer to Appendix D
         for implementation guidance.

      SerializeElement(A):  Implemented as specified in [RFC8032],
         Section 5.2.2.  Additionally, this function validates that the
         input element is not the group identity element.

      DeserializeElement(buf):  Implemented as specified in [RFC8032],
         Section 5.2.3.  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 [Pornin22].

      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 - 1381806680989
         5115352007386748515426880336692474882178609894547503885.

      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 - 1381806680989
         5115352007386748515426880336692474882178609894547503885.

      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 - 1381806680989
         5115352007386748515426880336692474882178609894547503885.

      H4(m):  Implemented by computing H(contextString || "msg" || m).

      H5(m):  Implemented by computing H(contextString || "com" || m).

   Normally, H2 would also include a domain separator.  However, it is
   omitted for compatibility with [RFC8032].

   Signature verification is as specified in Section 5.2.7 of [RFC8032]
   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 Appendix A.

6.4.  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) [x9.62], where Ne = 33 and Ns = 32.

      Order():  Return 0xffffffff00000000ffffffffffffffffbce6faada7179e8
         4f3b9cac2fc632551.

      Identity():  As defined in [x9.62].

      RandomScalar():  Implemented by returning a uniformly random
         Scalar in the range [0, G.Order() - 1].  Refer to Appendix D
         for implementation guidance.

      SerializeElement(A):  Implemented using the compressed Elliptic-
         Curve-Point-to-Octet-String method according to [SEC1],
         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 [SEC1]
         and then performing public key validation as defined in
         Section 3.2.2.1 of [SEC1].  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 [SEC1].

      DeserializeScalar(buf):  Implemented by attempting to deserialize
         a Scalar from a 32-byte string using Octet-String-to-Field-
         Element from [SEC1].  This function can fail if the input does
         not represent a Scalar in the range [0, G.Order() - 1].

   Hash (H):  SHA-256, which has 32 bytes of output.

      H1(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) using expand_message_xmd with SHA-256 with
         parameters DST = contextString || "rho", F set to the Scalar
         field, p set to G.Order(), m = 1, and L = 48.

      H2(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) using expand_message_xmd with SHA-256 with
         parameters DST = contextString || "chal", F set to the Scalar
         field, p set to G.Order(), m = 1, and L = 48.

      H3(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) 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 Appendix B.

   Canonical signature encoding is as specified in Appendix A.

6.5.  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 [SEC2], where Ne = 33 and Ns = 32.

      Order():  Return 0xfffffffffffffffffffffffffffffffebaaedce6af48a03
         bbfd25e8cd0364141.

      Identity():  As defined in [SEC2].

      RandomScalar():  Implemented by returning a uniformly random
         Scalar in the range [0, G.Order() - 1].  Refer to Appendix D
         for implementation guidance.

      SerializeElement(A):  Implemented using the compressed Elliptic-
         Curve-Point-to-Octet-String method according to [SEC1],
         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 [SEC1]
         and then performing public key validation as defined in
         Section 3.2.2.1 of [SEC1].  This includes checking that the
         coordinates of the resulting point are in the correct range,
         the point is on the curve, and the point is not the point at
         infinity.  (As noted in the specification, validation of the
         point order is not required since the cofactor is 1.)  If any
         of these checks fail, deserialization returns an error.

      SerializeScalar(s):  Implemented using the Field-Element-to-Octet-
         String conversion according to [SEC1].

      DeserializeScalar(buf):  Implemented by attempting to deserialize
         a Scalar from a 32-byte string using Octet-String-to-Field-
         Element from [SEC1].  This function can fail if the input does
         not represent a Scalar in the range [0, G.Order() - 1].

   Hash (H):  SHA-256, which has 32 bytes of output.

      H1(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) using expand_message_xmd with SHA-256 with
         parameters DST = contextString || "rho", F set to the Scalar
         field, p set to G.Order(), m = 1, and L = 48.

      H2(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) using expand_message_xmd with SHA-256 with
         parameters DST = contextString || "chal", F set to the Scalar
         field, p set to G.Order(), m = 1, and L = 48.

      H3(m):  Implemented as hash_to_field(m, 1) (see [HASH-TO-CURVE],
         Section 5.2) 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 Appendix B.

   Canonical signature encoding is as specified in Appendix A.

6.6.  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 in
       order to maintain compatibility with [RFC8032].

   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 else fail.

   4.  The canonical signature encoding details are clearly specified.

7.  Security Considerations

   A security analysis of FROST is documented in [FROST20] and
   [StrongerSec22].  At a high level, FROST provides security against
   Existential Unforgeability Under Chosen Message Attacks (EUF-CMA) as
   defined in [StrongerSec22].  To satisfy this requirement, the
   ciphersuite needs to adhere to the requirements in Section 6.6 and
   the following assumptions must hold.

   *  The signer key shares are generated and distributed securely,
      e.g., via a trusted dealer that performs key generation (see
      Appendix C.2) or through a distributed key generation protocol.

   *  The Coordinator and at most (MIN_PARTICIPANTS-1) participants may
      be corrupted.

   Note that the Coordinator is not trusted with any private
   information, and communication at the time of signing can be
   performed over a public channel as long as it is authenticated and
   reliable.

   FROST provides security against DoS attacks under the following
   assumptions:

   *  The Coordinator does not perform a DoS attack.

   *  The Coordinator identifies misbehaving participants such that they
      can be removed from future invocations of FROST.  The Coordinator
      may also abort upon detecting a misbehaving participant to ensure
      that invalid signatures are not produced.

   FROST does not aim to achieve the following goals:

   *  Post-quantum security.  FROST, like plain Schnorr signatures,
      requires the hardness of the Discrete Logarithm Problem.

   *  Robustness.  Preventing DoS attacks against misbehaving
      participants requires the Coordinator to identify and act on
      misbehaving participants; see Section 5.4 for more information.
      While FROST does not provide robustness, [ROAST] is a wrapper
      protocol around FROST that does.

   *  Downgrade prevention.  All participants in the protocol are
      assumed to agree on which algorithms to use.

   *  Metadata protection.  If protection for metadata is desired, a
      higher-level communication channel can be used to facilitate key
      generation and signing.

   The rest of this section documents issues particular to
   implementations or deployments.

7.1.  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.

7.2.  Optimizations

   [StrongerSec22] 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
   [StrongerSec22], this optimization removes the guarantee that the set
   of signer participants that started round one of the protocol is the
   same set of signing participants that produced the signature output
   by round two.  As such, the optimization is NOT RECOMMENDED and is
   not covered in this document.

7.3.  Nonce Reuse Attacks

   Section 4.1 describes the procedure that participants use to produce
   nonces during the first round of signing.  The randomness produced in
   this procedure MUST be sampled uniformly at random.  The resulting
   nonces produced via nonce_generate are indistinguishable from values
   sampled uniformly at random.  This requirement is necessary to avoid
   replay attacks initiated by other participants that allow for a
   complete key-recovery attack.  The Coordinator MAY further hedge
   against nonce reuse attacks by tracking participant nonce commitments
   used for a given group key at the cost of additional state.

7.4.  Protocol Failures

   We do not specify what implementations should do when the protocol
   fails other than requiring the protocol to abort.  Examples of viable
   failures include when a verification check returns invalid or the
   underlying transport failed to deliver the required messages.

7.5.  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; instead, it simply requires each participant
   to communicate with all other participants.  We loosely describe how
   to perform FROST signing among participants without this coordinator
   role.  We assume that every participant receives a message to be
   signed from an external source as input prior to performing the
   protocol.

   Every participant begins by performing commit() as is done in the
   setting where a Coordinator is used.  However, instead of sending the
   commitment to the Coordinator, every participant will publish this
   commitment to every other participant.  In the second round,
   participants will already have sufficient information to perform
   signing, and they will directly perform sign().  All participants
   will then publish their signature shares to one another.  After
   having received all signature shares from all other participants,
   each participant will then perform verify_signature_share and then
   aggregate directly.

   The requirements for the underlying network channel remain the same
   in the setting where all participants play the role of the
   Coordinator, in that all exchanged messages are public and the
   channel must be reliable.  However, in the setting where a player
   attempts to split the view of all other players by sending disjoint
   values to a subset of players, the signing operation will output an
   invalid signature.  To avoid this DoS, implementations may wish to
   define a mechanism where messages are authenticated so that cheating
   players can be identified and excluded.

7.6.  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.  For applications that choose to apply pre-
   hashing, it is RECOMMENDED that they use the hash function (H)
   associated with the chosen ciphersuite in a manner similar to how H4
   is defined.  In particular, a different prefix SHOULD be used to
   differentiate this pre-hash from H4.  For example, if a fictional
   protocol Quux decided to pre-hash its input messages, one possible
   way to do so is via H(contextString || "Quux-pre-hash" || m).

7.7.  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 each participant to check that the input
   message is a syntactically valid transaction.

   As another example, some applications may require that participants
   only process messages with permitted content according to some
   policy.  In digital currency applications, this might mean that a
   transaction being signed is allowed and intended by the relevant
   stakeholders.  Another instance of this type of message validation is
   in the context of [TLS], 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.

8.  IANA Considerations

   This document has no IANA actions.

9.  References

9.1.  Normative References

   [HASH-TO-CURVE]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
              and C. A. Wood, "Hashing to Elliptic Curves", RFC 9380,
              DOI 10.17487/RFC9380, August 2023,
              <https://www.rfc-editor.org/info/rfc9380>.

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/info/rfc2119>.

   [RFC8032]  Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital
              Signature Algorithm (EdDSA)", RFC 8032,
              DOI 10.17487/RFC8032, January 2017,
              <https://www.rfc-editor.org/info/rfc8032>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/info/rfc8174>.

   [RISTRETTO]
              de Valence, H., Grigg, J., Hamburg, M., Lovecruft, I.,
              Tankersley, G., and F. Valsorda, "The ristretto255 and
              decaf448 Groups", RFC 9496, DOI 10.17487/RFC9496, December
              2023, <https://www.rfc-editor.org/info/rfc9496>.

   [SEC1]     Standards for Efficient Cryptography, "SEC 1: Elliptic
              Curve Cryptography", Version 2.0, May 2009,
              <https://secg.org/sec1-v2.pdf>.

   [SEC2]     Standards for Efficient Cryptography, "SEC 2: Recommended
              Elliptic Curve Domain Parameters", Version 2.0, January
              2010, <https://secg.org/sec2-v2.pdf>.

   [x9.62]    American National Standards Institute, "Public Key
              Cryptography for the Financial Services Industry: the
              Elliptic Curve Digital Signature Algorithm (ECDSA)",
              ANSI X9.62-2005, November 2005.

9.2.  Informative References

   [FeldmanSecretSharing]
              Feldman, P., "A practical scheme for non-interactive
              verifiable secret sharing", IEEE, 28th Annual Symposium on
              Foundations of Computer Science (sfcs 1987),
              DOI 10.1109/sfcs.1987.4, October 1987,
              <https://doi.org/10.1109/sfcs.1987.4>.

   [FROST20]  Komlo, C. and I. Goldberg, "FROST: Flexible Round-
              Optimized Schnorr Threshold Signatures", December 2020,
              <https://eprint.iacr.org/2020/852.pdf>.

   [MultExp]  Connolly, D. and C. Gouvea, "Speeding up FROST with multi-
              scalar multiplication", June 2023, <https://zfnd.org/
              speeding-up-frost-with-multi-scalar-multiplication/>.

   [Pornin22] Pornin, T., "Point-Halving and Subgroup Membership in
              Twisted Edwards Curves", September 2022,
              <https://eprint.iacr.org/2022/1164.pdf>.

   [RFC4086]  Eastlake 3rd, D., Schiller, J., and S. Crocker,
              "Randomness Requirements for Security", BCP 106, RFC 4086,
              DOI 10.17487/RFC4086, June 2005,
              <https://www.rfc-editor.org/info/rfc4086>.

   [RFC7748]  Langley, A., Hamburg, M., and S. Turner, "Elliptic Curves
              for Security", RFC 7748, DOI 10.17487/RFC7748, January
              2016, <https://www.rfc-editor.org/info/rfc7748>.

   [ROAST]    Ruffing, T., Ronge, V., Jin, E., Schneider-Bensch, J., and
              D. Schröder, "ROAST: Robust Asynchronous Schnorr Threshold
              Signatures", Paper 2022/550, DOI 10.1145/3548606, November
              2022, <https://eprint.iacr.org/2022/550>.

   [ShamirSecretSharing]
              Shamir, A., "How to share a secret", Association for
              Computing Machinery (ACM), Communications of the ACM, Vol.
              22, Issue 11, pp. 612-613, DOI 10.1145/359168.359176,
              November 1979, <https://doi.org/10.1145/359168.359176>.

   [StrongerSec22]
              Bellare, M., Crites, E., Komlo, C., Maller, M., Tessaro,
              S., and C. Zhu, "Better than Advertised Security for Non-
              interactive Threshold Signatures",
              DOI 10.1007/978-3-031-15985-5_18, August 2022,
              <https://crypto.iacr.org/2022/
              papers/538806_1_En_18_Chapter_OnlinePDF.pdf>.

   [TLS]      Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/info/rfc8446>.

Appendix A.  Schnorr Signature Encoding

   This section describes one possible canonical encoding of FROST
   signatures.  Using notation from Section 3 of [TLS], the encoding of
   a FROST signature (R, z) is as follows:

     struct {
       opaque R_encoded[Ne];
       opaque z_encoded[Ns];
     } Signature;

   Where Signature.R_encoded is G.SerializeElement(R),
   Signature.z_encoded is G.SerializeScalar(z), and G is determined by
   ciphersuite.

Appendix B.  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 [RFC8032] for prime-order groups, including
   ristretto255, P-256, and secp256k1.  The functions for generating and
   verifying signatures are prime_order_sign and prime_order_verify,
   respectively.

   The function prime_order_sign produces a Schnorr signature over a
   message given a full secret signing key as input (as opposed to a key
   share).

   Inputs:
   - msg, message to sign, a byte string.
   - sk, secret key, a Scalar.

   Outputs:
   - (R, z), a Schnorr signature consisting of an Element R and
     Scalar z.

   def prime_order_sign(msg, sk):
     r = G.RandomScalar()
     R = G.ScalarBaseMult(r)
     PK = G.ScalarBaseMult(sk)
     comm_enc = G.SerializeElement(R)
     pk_enc = G.SerializeElement(PK)
     challenge_input = comm_enc || pk_enc || msg
     c = H2(challenge_input)
     z = r + (c * sk) // Scalar addition and multiplication
     return (R, z)

   The function prime_order_verify verifies Schnorr signatures with
   validated inputs.  Specifically, it assumes that the signature R
   component and public key belong to the prime-order group.

   Inputs:
   - msg, signed message, a byte string.
   - sig, a tuple (R, z) output from signature generation.
   - PK, public key, an Element.

   Outputs:
   - True if signature is valid, and False otherwise.

   def prime_order_verify(msg, sig = (R, z), PK):
     comm_enc = G.SerializeElement(R)
     pk_enc = G.SerializeElement(PK)
     challenge_input = comm_enc || pk_enc || msg
     c = H2(challenge_input)

     l = G.ScalarBaseMult(z)
     r = R + G.ScalarMult(PK, c)
     return l == r

Appendix C.  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
   [ShamirSecretSharing] as described in Appendices C.1 and C.2 to
   create secret shares of s, denoted as s_i for i = 1, ...,
   MAX_PARTICIPANTS, to be sent to all MAX_PARTICIPANTS participants.
   This operation is specified in the trusted_dealer_keygen algorithm.
   The mathematical relation between the secret key s and the
   MAX_PARTICIPANTS secret shares is formalized in the
   secret_share_combine(shares) algorithm, defined in Appendix C.1.

   The dealer that performs trusted_dealer_keygen is trusted to 1)
   generate good randomness, 2) delete secret values after distributing
   shares to each participant, and 3) keep secret values confidential.

   Inputs:
   - secret_key, a group secret, a Scalar, that MUST be derived from at
     least Ns bytes of entropy.
   - MAX_PARTICIPANTS, the number of shares to generate, an integer.
   - MIN_PARTICIPANTS, the threshold of the secret sharing scheme,
     an integer.

   Outputs:
   - participant_private_keys, MAX_PARTICIPANTS shares of the secret
     key s, each a tuple consisting of the participant identifier
     (a NonZeroScalar) and the key share (a Scalar).
   - group_public_key, public key corresponding to the group signing
     key, an Element.
   - vss_commitment, a vector commitment of Elements in G, to each of
     the coefficients in the polynomial defined by secret_key_shares and
     whose first element is G.ScalarBaseMult(s).

   def trusted_dealer_keygen(
           secret_key, MAX_PARTICIPANTS, MIN_PARTICIPANTS):
     # Generate random coefficients for the polynomial
     coefficients = []
     for i in range(0, MIN_PARTICIPANTS - 1):
       coefficients.append(G.RandomScalar())
     participant_private_keys, coefficients = secret_share_shard(
         secret_key, coefficients, MAX_PARTICIPANTS)
     vss_commitment = vss_commit(coefficients):
     return participant_private_keys, vss_commitment[0], vss_commitment

   It is assumed that the dealer then sends one secret key share to each
   of the NUM_PARTICIPANTS participants, along with vss_commitment.
   After receiving their secret key share and vss_commitment,
   participants MUST abort if they do not have the same view of
   vss_commitment.  The dealer can use a secure broadcast channel to
   ensure each participant has a consistent view of this commitment.
   Furthermore, each participant MUST perform
   vss_verify(secret_key_share_i, vss_commitment) and abort if the check
   fails.  The trusted dealer MUST delete the secret_key and
   secret_key_shares upon completion.

   Use of this method for key generation requires a mutually
   authenticated secure channel between the dealer and participants to
   send secret key shares, wherein the channel provides confidentiality
   and integrity.  Mutually authenticated TLS is one possible deployment
   option.

C.1.  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 Appendix C.1.1.

   Inputs:
   - s, secret value to be shared, a Scalar.
   - coefficients, an array of size MIN_PARTICIPANTS - 1 with randomly
     generated Scalars, not including the 0th coefficient of the
     polynomial.
   - MAX_PARTICIPANTS, the number of shares to generate, an integer less
     than the group order.

   Outputs:
   - secret_key_shares, A list of MAX_PARTICIPANTS number of secret
     shares, each a tuple consisting of the participant identifier
     (a NonZeroScalar) and the key share (a Scalar).
   - coefficients, a vector of MIN_PARTICIPANTS coefficients which
     uniquely determine a polynomial f.

   def secret_share_shard(s, coefficients, MAX_PARTICIPANTS):
     # Prepend the secret to the coefficients
     coefficients = [s] + coefficients

     # Evaluate the polynomial for each point x=1,...,n
     secret_key_shares = []
     for x_i in range(1, MAX_PARTICIPANTS + 1):
       y_i = polynomial_evaluate(Scalar(x_i), coefficients)
       secret_key_share_i = (x_i, y_i)
       secret_key_shares.append(secret_key_share_i)
     return secret_key_shares, coefficients

   Let points be the output of this function.  The i-th element in
   points is the share for the i-th participant, which is the randomly
   generated polynomial evaluated at coordinate i.  We denote a secret
   share as the tuple (i, points[i]) and the list of these shares as
   shares. 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 Appendix C.1.1.

   Inputs:
   - shares, a list of at minimum MIN_PARTICIPANTS secret shares, each a
     tuple (i, f(i)) where i and f(i) are Scalars.

   Outputs:
   - s, the resulting secret that was previously split into shares,
     a Scalar.

   Errors:
   - "invalid parameters", if fewer than MIN_PARTICIPANTS input shares
     are provided.

   def secret_share_combine(shares):
     if len(shares) < MIN_PARTICIPANTS:
       raise "invalid parameters"
     s = polynomial_interpolate_constant(shares)
     return s

C.1.1.  Additional Polynomial Operations

   This section describes two functions.  One function, denoted as
   polynomial_evaluate, is for evaluating a polynomial f(x) at a
   particular point x using Horner's method, i.e., computing y = f(x).
   The other function, polynomial_interpolate_constant, is for
   recovering the constant term of an interpolating polynomial defined
   by a set of points.

   The function polynomial_evaluate is defined as follows.

   Inputs:
   - x, input at which to evaluate the polynomial, a Scalar
   - coeffs, the polynomial coefficients, a list of Scalars

   Outputs: Scalar result of the polynomial evaluated at input x

   def polynomial_evaluate(x, coeffs):
     value = Scalar(0)
     for coeff in reverse(coeffs):
       value *= x
       value += coeff
     return value

   The function polynomial_interpolate_constant is defined as follows.

   Inputs:
   - points, a set of t points with distinct x coordinates on
     a polynomial f, each a tuple of two Scalar values representing the
     x and y coordinates.

   Outputs:
   - f_zero, the constant term of f, i.e., f(0), a Scalar.

   def polynomial_interpolate_constant(points):
     x_coords = []
     for (x, y) in points:
       x_coords.append(x)

     f_zero = Scalar(0)
     for (x, y) in points:
       delta = y * derive_interpolating_value(x_coords, x)
       f_zero += delta

     return f_zero

C.2.  Verifiable Secret Sharing

   Feldman's Verifiable Secret Sharing (VSS) [FeldmanSecretSharing]
   builds upon Shamir secret sharing, adding a verification step to
   demonstrate the consistency of a participant's share with a public
   commitment to the polynomial f for which the secret s is the constant
   term.  This check ensures that all participants have a point (their
   share) on the same polynomial, ensuring that they can reconstruct the
   correct secret later.

   The procedure for committing to a polynomial f of degree at most
   MIN_PARTICIPANTS-1 is as follows.

   Inputs:
   - coeffs, a vector of the MIN_PARTICIPANTS coefficients that
     uniquely determine a polynomial f.

   Outputs:
   - vss_commitment, a vector commitment to each of the coefficients in
     coeffs, where each item of the vector commitment is an Element.

   def vss_commit(coeffs):
     vss_commitment = []
     for coeff in coeffs:
       A_i = G.ScalarBaseMult(coeff)
       vss_commitment.append(A_i)
     return vss_commitment

   The procedure for verification of a participant's share is as
   follows.  If vss_verify fails, the participant MUST abort the
   protocol, and the failure should be investigated out of band.

   Inputs:
   - share_i: A tuple of the form (i, sk_i), where i indicates the
     participant identifier (a NonZeroScalar), and sk_i the
     participant's secret key, a secret share of the constant term of f,
     where sk_i is a Scalar.
   - vss_commitment, a VSS commitment to a secret polynomial f, a vector
     commitment to each of the coefficients in coeffs, where each
     element of the vector commitment is an Element.

   Outputs:
   - True if sk_i is valid, and False otherwise.

   def vss_verify(share_i, vss_commitment)
     (i, sk_i) = share_i
     S_i = G.ScalarBaseMult(sk_i)
     S_i' = G.Identity()
     for j in range(0, MIN_PARTICIPANTS):
       S_i' += G.ScalarMult(vss_commitment[j], pow(i, j))
     return S_i == S_i'

   We now define how the Coordinator and participants can derive group
   info, which is an input into the FROST signing protocol.

   Inputs:
   - MAX_PARTICIPANTS, the number of shares to generate, an integer.
   - MIN_PARTICIPANTS, the threshold of the secret sharing scheme,
     an integer.
   - vss_commitment, a VSS commitment to a secret polynomial f, a vector
     commitment to each of the coefficients in coeffs, where each
     element of the vector commitment is an Element.

   Outputs:
   - PK, the public key representing the group, an Element.
   - participant_public_keys, a list of MAX_PARTICIPANTS public keys
     PK_i for i=1,...,MAX_PARTICIPANTS, where each PK_i is the public
     key, an Element, for participant i.

   def derive_group_info(MAX_PARTICIPANTS, MIN_PARTICIPANTS,
    vss_commitment):
     PK = vss_commitment[0]
     participant_public_keys = []
     for i in range(1, MAX_PARTICIPANTS+1):
       PK_i = G.Identity()
       for j in range(0, MIN_PARTICIPANTS):
         PK_i += G.ScalarMult(vss_commitment[j], pow(i, j))
       participant_public_keys.append(PK_i)
     return PK, participant_public_keys

Appendix D.  Random Scalar Generation

   Two popular algorithms for generating a random integer uniformly
   distributed in the range [0, G.Order() -1] are described in the
   sections that follow.

D.1.  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.

D.2.  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 Section 5 of [HASH-TO-CURVE] for the underlying derivation of l.

Appendix E.  Test Vectors

   This section contains test vectors for all ciphersuites listed in
   Section 6.  All Element and Scalar values are represented in
   serialized form and encoded in hexadecimal strings.  Signatures are
   represented as the concatenation of their constituent parts.  The
   input message to be signed is also encoded as a hexadecimal string.

   Each test vector consists of the following information.

   *  Configuration.  This lists the fixed parameters for the particular
      instantiation of FROST, including MAX_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
      Appendix C, as well as the input message to be signed.  The
      randomly generated coefficients produced by the trusted dealer to
      share the group signing secret are also listed.  Each coefficient
      is identified by its index, e.g., share_polynomial_coefficients[1]
      is the coefficient of the first term in the polynomial.  Note that
      the 0-th coefficient is omitted, as this is equal to the group
      secret key.  All values are encoded as hexadecimal strings.

   *  Signer input parameters.  This lists the signing key share for
      each of the NUM_PARTICIPANTS participants.

   *  Round one parameters and outputs.  This lists the NUM_PARTICIPANTS
      participants engaged in the protocol, identified by their
      NonZeroScalar identifier, and the following for each participant:
      the hiding and binding commitment values produced in Section 5.1;
      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
      Section 4.3; and the group binding factor as computed in
      Section 5.2.

   *  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 Section 5.2.

   *  Final output.  This lists the aggregate signature as produced in
      Section 5.3.

E.1.  FROST(Ed25519, SHA-512)

   // Configuration information
   MAX_PARTICIPANTS: 3
   MIN_PARTICIPANTS: 2
   NUM_PARTICIPANTS: 2

   // Group input parameters
   participant_list: 1,3
   group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
   90c6e13a98304
   group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040
   d380fb9738673
   message: 74657374
   share_polynomial_coefficients[1]: 178199860edd8c62f5212ee91eff1295d0d
   670ab4ed4506866bae57e7030b204

   // Signer input parameters
   P1 participant_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f
   4a033f2ec83d93509
   P2 participant_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685
   c07eed76bf409e80d
   P3 participant_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c
   46da8bdea643a9a02

   // Signer round one outputs
   P1 hiding_nonce_randomness: 0fd2e39e111cdc266f6c0f4d0fd45c947761f1f5d
   3cb583dfcb9bbaf8d4c9fec
   P1 binding_nonce_randomness: 69cd85f631d5f7f2721ed5e40519b1366f340a87
   c2f6856363dbdcda348a7501
   P1 hiding_nonce: 812d6104142944d5a55924de6d49940956206909f2acaeedecda
   2b726e630407
   P1 binding_nonce: b1110165fc2334149750b28dd813a39244f315cff14d4e89e61
   42f262ed83301
   P1 hiding_nonce_commitment: b5aa8ab305882a6fc69cbee9327e5a45e54c08af6
   1ae77cb8207be3d2ce13de3
   P1 binding_nonce_commitment: 67e98ab55aa310c3120418e5050c9cf76cf387cb
   20ac9e4b6fdb6f82a469f932
   P1 binding_factor_input: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3f
   af66040d380fb9738673504df914fa965023fb75c25ded4bb260f417de6d32e5c442c
   6ba313791cc9a4948d6273e8d3511f93348ea7a708a9b862bc73ba2a79cfdfe07729a
   193751cbc973af46d8ac3440e518d4ce440a0e7d4ad5f62ca8940f32de6d8dc00fc12
   c660b817d587d82f856d277ce6473cae6d2f5763f7da2e8b4d799a3f3e725d4522ec7
   0100000000000000000000000000000000000000000000000000000000000000
   P1 binding_factor: f2cb9d7dd9beff688da6fcc83fa89046b3479417f47f55600b
   106760eb3b5603
   P3 hiding_nonce_randomness: 86d64a260059e495d0fb4fcc17ea3da7452391baa
   494d4b00321098ed2a0062f
   P3 binding_nonce_randomness: 13e6b25afb2eba51716a9a7d44130c0dbae0004a
   9ef8d7b5550c8a0e07c61775
   P3 hiding_nonce: c256de65476204095ebdc01bd11dc10e57b36bc96284595b8215
   222374f99c0e
   P3 binding_nonce: 243d71944d929063bc51205714ae3c2218bd3451d0214dfb5ae
   ec2a90c35180d
   P3 hiding_nonce_commitment: cfbdb165bd8aad6eb79deb8d287bcc0ab6658ae57
   fdcc98ed12c0669e90aec91
   P3 binding_nonce_commitment: 7487bc41a6e712eea2f2af24681b58b1cf1da278
   ea11fe4e8b78398965f13552
   P3 binding_factor_input: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3f
   af66040d380fb9738673504df914fa965023fb75c25ded4bb260f417de6d32e5c442c
   6ba313791cc9a4948d6273e8d3511f93348ea7a708a9b862bc73ba2a79cfdfe07729a
   193751cbc973af46d8ac3440e518d4ce440a0e7d4ad5f62ca8940f32de6d8dc00fc12
   c660b817d587d82f856d277ce6473cae6d2f5763f7da2e8b4d799a3f3e725d4522ec7
   0300000000000000000000000000000000000000000000000000000000000000
   P3 binding_factor: b087686bf35a13f3dc78e780a34b0fe8a77fef1b9938c563f5
   573d71d8d7890f

   // Signer round two outputs
   P1 sig_share: 001719ab5a53ee1a12095cd088fd149702c0720ce5fd2f29dbecf24
   b7281b603
   P3 sig_share: bd86125de990acc5e1f13781d8e32c03a9bbd4c53539bbc106058bf
   d14326007

   sig: 36282629c383bb820a88b71cae937d41f2f2adfcc3d02e55507e2fb9e2dd3cbe
   bd9d2b0844e49ae0f3fa935161e1419aab7b47d21a37ebeae1f17d4987b3160b

E.2.  FROST(Ed448, SHAKE256)

   // Configuration information
   MAX_PARTICIPANTS: 3
   MIN_PARTICIPANTS: 2
   NUM_PARTICIPANTS: 2

   // Group input parameters
   participant_list: 1,3
   group_secret_key: 6298e1eef3c379392caaed061ed8a31033c9e9e3420726f23b4
   04158a401cd9df24632adfe6b418dc942d8a091817dd8bd70e1c72ba52f3c00
   group_public_key: 3832f82fda00ff5365b0376df705675b63d2a93c24c6e81d408
   01ba265632be10f443f95968fadb70d10786827f30dc001c8d0f9b7c1d1b000
   message: 74657374
   share_polynomial_coefficients[1]: dbd7a514f7a731976620f0436bd135fe8dd
   dc3fadd6e0d13dbd58a1981e587d377d48e0b7ce4e0092967c5e85884d0275a7a740b
   6abdcd0500

   // Signer input parameters
   P1 participant_share: 4a2b2f5858a932ad3d3b18bd16e76ced3070d72fd79ae44
   02df201f525e754716a1bc1b87a502297f2a99d89ea054e0018eb55d39562fd0100
   P2 participant_share: 2503d56c4f516444a45b080182b8a2ebbe4d9b2ab509f25
   308c88c0ea7ccdc44e2ef4fc4f63403a11b116372438a1e287265cadeff1fcb0700
   P3 participant_share: 00db7a8146f995db0a7cf844ed89d8e94c2b5f259378ff6
   6e39d172828b264185ac4decf7219e4aa4478285b9c0eef4fccdf3eea69dd980d00

   // Signer round one outputs
   P1 hiding_nonce_randomness: 9cda90c98863ef3141b75f09375757286b4bc323d
   d61aeb45c07de45e4937bbd
   P1 binding_nonce_randomness: 781bf4881ffe1aa06f9341a747179f07a49745f8
   cd37d4696f226aa065683c0a
   P1 hiding_nonce: f922beb51a5ac88d1e862278d89e12c05263b945147db04b9566
   acb2b5b0f7422ccea4f9286f4f80e6b646e72143eeaecc0e5988f8b2b93100
   P1 binding_nonce: 1890f16a120cdeac092df29955a29c7cf29c13f6f7be60e63d6
   3f3824f2d37e9c3a002dfefc232972dc08658a8c37c3ec06a0c5dc146150500
   P1 hiding_nonce_commitment: 3518c2246c874569e54ab254cb1da666ca30f7879
   605cc43b4d2c47a521f8b5716080ab723d3a0cd04b7e41f3cc1d3031c94ccf3829b23
   fe80
   P1 binding_nonce_commitment: 11b3d5220c57d02057497de3c4eebab384900206
   592d877059b0a5f1d5250d002682f0e22dff096c46bb81b46d60fcfe7752ed47cea76
   c3900
   P1 binding_factor_input: 3832f82fda00ff5365b0376df705675b63d2a93c24c6
   e81d40801ba265632be10f443f95968fadb70d10786827f30dc001c8d0f9b7c1d1b00
   0e9a0f30b97fe77ef751b08d4e252a3719ae9135e7f7926f7e3b7dd6656b27089ca35
   4997fe5a633aa0946c89f022462e7e9d50fd6ef313f72d956ea4571089427daa1862f
   623a41625177d91e4a8f350ce9c8bd3bc7c766515dc1dd3a0eab93777526b616cccb1
   48fe1e5992dc1ae705c8ba2f97ca8983328d41d375ed1e5fde5c9d672121c9e8f177f
   4a1a9b2575961531b33f054451363c8f27618382cd66ce14ad93b68dac6a09f5edcbc
   cc813906b3fc50b8fef1cc09757b06646f38ceed1674cd6ced28a59c93851b325c6a9
   ef6a4b3b88860b7138ee246034561c7460db0b3fae501000000000000000000000000
   000000000000000000000000000000000000000000000000000000000000000000000
   0000000000000000000
   P1 binding_factor: 71966390dfdbed73cf9b79486f3b70e23b243e6c40638fb559
   98642a60109daecbfcb879eed9fe7dbbed8d9e47317715a5740f772173342e00
   P3 hiding_nonce_randomness: b3adf97ceea770e703ab295babf311d77e956a20d
   3452b4b3344aa89a828e6df
   P3 binding_nonce_randomness: 81dbe7742b0920930299197322b255734e52bbb9
   1f50cfe8ce689f56fadbce31
   P3 hiding_nonce: ccb5c1e82f23e0a4b966b824dbc7b0ef1cc5f56eeac2a4126e2b
   2143c5f3a4d890c52d27803abcf94927faf3fc405c0b2123a57a93cefa3b00
   P3 binding_nonce: e089df9bf311cf711e2a24ea27af53e07b846d09692fe11035a
   1112f04d8b7462a62f34d8c01493a22b57a1cbf1f0a46c77d64d46449a90100
   P3 hiding_nonce_commitment: 1254546d7d104c04e4fbcf29e05747e2edd392f67
   87d05a6216f3713ef859efe573d180d291e48411e5e3006e9f90ee986ccc26b7a4249
   0b80
   P3 binding_nonce_commitment: 3ef0cec20be15e56b3ddcb6f7b956fca0c8f7199
   0f45316b537b4f64c5e8763e6629d7262ff7cd0235d0781f23be97bf8fa8817643ea1
   9cd00
   P3 binding_factor_input: 3832f82fda00ff5365b0376df705675b63d2a93c24c6
   e81d40801ba265632be10f443f95968fadb70d10786827f30dc001c8d0f9b7c1d1b00
   0e9a0f30b97fe77ef751b08d4e252a3719ae9135e7f7926f7e3b7dd6656b27089ca35
   4997fe5a633aa0946c89f022462e7e9d50fd6ef313f72d956ea4571089427daa1862f
   623a41625177d91e4a8f350ce9c8bd3bc7c766515dc1dd3a0eab93777526b616cccb1
   48fe1e5992dc1ae705c8ba2f97ca8983328d41d375ed1e5fde5c9d672121c9e8f177f
   4a1a9b2575961531b33f054451363c8f27618382cd66ce14ad93b68dac6a09f5edcbc
   cc813906b3fc50b8fef1cc09757b06646f38ceed1674cd6ced28a59c93851b325c6a9
   ef6a4b3b88860b7138ee246034561c7460db0b3fae503000000000000000000000000
   000000000000000000000000000000000000000000000000000000000000000000000
   0000000000000000000
   P3 binding_factor: 236a6f7239ac2019334bad21323ec93bef2fead37bd5511435
   6419f3fc1fb59f797f44079f28b1a64f51dd0a113f90f2c3a1c27d2faa4f1300

   // Signer round two outputs
   P1 sig_share: e1eb9bfbef792776b7103891032788406c070c5c315e3bf5d64acd4
   6ea8855e85b53146150a09149665cbfec71626810b575e6f4dbe9ba3700
   P3 sig_share: 815434eb0b9f9242d54b8baf2141fe28976cabe5f441ccfcd5ee7cd
   b4b52185b02b99e6de28e2ab086c7764068c5a01b5300986b9f084f3e00

   sig: cd642cba59c449dad8e896a78a60e8edfcbd9040df524370891ff8077d47ce72
   1d683874483795f0d85efcbd642c4510614328605a19c6ed806ffb773b6956419537c
   dfdb2b2a51948733de192dcc4b82dc31580a536db6d435e0cb3ce322fbcf9ec23362d
   da27092c08767e607bf2093600

E.3.  FROST(ristretto255, SHA-512)

   // Configuration information
   MAX_PARTICIPANTS: 3
   MIN_PARTICIPANTS: 2
   NUM_PARTICIPANTS: 2

   // Group input parameters
   participant_list: 1,3
   group_secret_key: 1b25a55e463cfd15cf14a5d3acc3d15053f08da49c8afcf3ab2
   65f2ebc4f970b
   group_public_key: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3dd9c3c0d
   05083c7254f57
   message: 74657374
   share_polynomial_coefficients[1]: 410f8b744b19325891d73736923525a4f59
   6c805d060dfb9c98009d34e3fec02

   // Signer input parameters
   P1 participant_share: 5c3430d391552f6e60ecdc093ff9f6f4488756aa6cebdba
   d75a768010b8f830e
   P2 participant_share: b06fc5eac20b4f6e1b271d9df2343d843e1e1fb03c4cbb6
   73f2872d459ce6f01
   P3 participant_share: f17e505f0e2581c6acfe54d3846a622834b5e7b50cad9a2
   109a97ba7a80d5c04

   // Signer round one outputs
   P1 hiding_nonce_randomness: f595a133b4d95c6e1f79887220c8b275ce6277e7f
   68a6640e1e7140f9be2fb5c
   P1 binding_nonce_randomness: 34dd1001360e3513cb37bebfabe7be4a32c5bb91
   ba19fbd4360d039111f0fbdc
   P1 hiding_nonce: 214f2cabb86ed71427ea7ad4283b0fae26b6746c801ce824b83c
   eb2b99278c03
   P1 binding_nonce: c9b8f5e16770d15603f744f8694c44e335e8faef00dad182b8d
   7a34a62552f0c
   P1 hiding_nonce_commitment: 965def4d0958398391fc06d8c2d72932608b1e625
   5226de4fb8d972dac15fd57
   P1 binding_nonce_commitment: ec5170920660820007ae9e1d363936659ef622f9
   9879898db86e5bf1d5bf2a14
   P1 binding_factor_input: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3d
   d9c3c0d05083c7254f572889dde2854e26377a16caf77dfee5f6be8fe5b4c80318da8
   4698a4161021b033911db5ef8205362701bc9ecd983027814abee94f46d094943a2f4
   b79a6e4d4603e52c435d8344554942a0a472d8ad84320585b8da3ae5b9ce31cd1903f
   795c1af66de22af1a45f652cd05ee446b1b4091aaccc91e2471cd18a85a659cecd11f
   0100000000000000000000000000000000000000000000000000000000000000
   P1 binding_factor: 8967fd70fa06a58e5912603317fa94c77626395a695a0e4e4e
   fc4476662eba0c
   P3 hiding_nonce_randomness: daa0cf42a32617786d390e0c7edfbf2efbd428037
   069357b5173ae61d6dd5d5e
   P3 binding_nonce_randomness: b4387e72b2e4108ce4168931cc2c7fcce5f345a5
   297368952c18b5fc8473f050
   P3 hiding_nonce: 3f7927872b0f9051dd98dd73eb2b91494173bbe0feb65a3e7e58
   d3e2318fa40f
   P3 binding_nonce: ffd79445fb8030f0a3ddd3861aa4b42b618759282bfe24f1f93
   04c7009728305
   P3 hiding_nonce_commitment: 480e06e3de182bf83489c45d7441879932fd7b434
   a26af41455756264fbd5d6e
   P3 binding_nonce_commitment: 3064746dfd3c1862ef58fc68c706da287dd92506
   6865ceacc816b3a28c7b363b
   P3 binding_factor_input: e2a62f39eede11269e3bd5a7d97554f5ca384f9f6d3d
   d9c3c0d05083c7254f572889dde2854e26377a16caf77dfee5f6be8fe5b4c80318da8
   4698a4161021b033911db5ef8205362701bc9ecd983027814abee94f46d094943a2f4
   b79a6e4d4603e52c435d8344554942a0a472d8ad84320585b8da3ae5b9ce31cd1903f
   795c1af66de22af1a45f652cd05ee446b1b4091aaccc91e2471cd18a85a659cecd11f
   0300000000000000000000000000000000000000000000000000000000000000
   P3 binding_factor: f2c1bb7c33a10511158c2f1766a4a5fadf9f86f2a92692ed33
   3128277cc31006

   // Signer round two outputs
   P1 sig_share: 9285f875923ce7e0c491a592e9ea1865ec1b823ead4854b48c8a462
   87749ee09
   P3 sig_share: 7cb211fe0e3d59d25db6e36b3fb32344794139602a7b24f1ae0dc4e
   26ad7b908

   sig: fc45655fbc66bbffad654ea4ce5fdae253a49a64ace25d9adb62010dd9fb2555
   2164141787162e5b4cab915b4aa45d94655dbb9ed7c378a53b980a0be220a802

E.4.  FROST(P-256, SHA-256)

   // Configuration information
   MAX_PARTICIPANTS: 3
   MIN_PARTICIPANTS: 2
   NUM_PARTICIPANTS: 2

   // Group input parameters
   participant_list: 1,3
   group_secret_key: 8ba9bba2e0fd8c4767154d35a0b7562244a4aaf6f36c8fb8735
   fa48b301bd8de
   group_public_key: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99cfbd02b4
   dc00585ec45ab70
   message: 74657374
   share_polynomial_coefficients[1]: 80f25e6c0709353e46bfbe882a11bdbb1f8
   097e46340eb8673b7e14556e6c3a4

   // Signer input parameters
   P1 participant_share: 0c9c1a0fe806c184add50bbdcac913dda73e482daf95dcb
   9f35dbb0d8a9f7731
   P2 participant_share: 8d8e787bef0ff6c2f494ca45f4dad198c6bee01212d6c84
   067159c52e1863ad5
   P3 participant_share: 0e80d6e8f6192c003b5488ce1eec8f5429587d48cf00154
   1e713b2d53c09d928

   // Signer round one outputs
   P1 hiding_nonce_randomness: ec4c891c85fee802a9d757a67d1252e7f4e5efb8a
   538991ac18fbd0e06fb6fd3
   P1 binding_nonce_randomness: 9334e29d09061223f69a09421715a347e4e6deba
   77444c8f42b0c833f80f4ef9
   P1 hiding_nonce: 9f0542a5ba879a58f255c09f06da7102ef6a2dec6279700c656d
   58394d8facd4
   P1 binding_nonce: 6513dfe7429aa2fc972c69bb495b27118c45bbc6e654bb9dc9b
   e55385b55c0d7
   P1 hiding_nonce_commitment: 0213b3e6298bf8ad46fd5e9389519a8665d63d98f
   4ec6a1fcca434e809d2d8070e
   P1 binding_nonce_commitment: 02188ff1390bf69374d7b272e454b1878ef10a6b
   6ea3ff36f114b300b4dbd5233b
   P1 binding_factor_input: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99c
   fbd02b4dc00585ec45ab70825371853e974bc30ac5b947b216d70461919666584c70c
   51f9f56f117736c5d178dd0b521ad9c1abe98048419cbdec81504c85e12eb40e3bcb6
   ec73d3fc4afd000000000000000000000000000000000000000000000000000000000
   0000001
   P1 binding_factor: 7925f0d4693f204e6e59233e92227c7124664a99739d2c06b8
   1cf64ddf90559e
   P3 hiding_nonce_randomness: c0451c5a0a5480d6c1f860e5db7d655233dca2669
   fd90ff048454b8ce983367b
   P3 binding_nonce_randomness: 2ba5f7793ae700e40e78937a82f407dd35e847e3
   3d1e607b5c7eb6ed2a8ed799
   P3 hiding_nonce: f73444a8972bcda9e506bbca3d2b1c083c10facdf4bb5d47fef7
   c2dc1d9f2a0d
   P3 binding_nonce: 44c6a29075d6e7e4f8b97796205f9e22062e7835141470afe94
   17fd317c1c303
   P3 hiding_nonce_commitment: 033ac9a5fe4a8b57316ba1c34e8a6de453033b750
   e8984924a984eb67a11e73a3f
   P3 binding_nonce_commitment: 03a7a2480ee16199262e648aea3acab628a53e9b
   8c1945078f2ddfbdc98b7df369
   P3 binding_factor_input: 023a309ad94e9fe8a7ba45dfc58f38bf091959d3c99c
   fbd02b4dc00585ec45ab70825371853e974bc30ac5b947b216d70461919666584c70c
   51f9f56f117736c5d178dd0b521ad9c1abe98048419cbdec81504c85e12eb40e3bcb6
   ec73d3fc4afd000000000000000000000000000000000000000000000000000000000
   0000003
   P3 binding_factor: e10d24a8a403723bcb6f9bb4c537f316593683b472f7a89f16
   6630dde11822c4

   // Signer round two outputs
   P1 sig_share: 400308eaed7a2ddee02a265abe6a1cfe04d946ee8720768899619cf
   abe7a3aeb
   P3 sig_share: 561da3c179edbb0502d941bb3e3ace3c37d122aaa46fb54499f15f3
   a3331de44

   sig: 026d8d434874f87bdb7bc0dfd239b2c00639044f9dcb195e9a04426f70bfa4b7
   0d9620acac6767e8e3e3036815fca4eb3a3caa69992b902bcd3352fc34f1ac192f

E.5.  FROST(secp256k1, SHA-256)

   // Configuration information
   MAX_PARTICIPANTS: 3
   MIN_PARTICIPANTS: 2
   NUM_PARTICIPANTS: 2

   // Group input parameters
   participant_list: 1,3
   group_secret_key: 0d004150d27c3bf2a42f312683d35fac7394b1e9e318249c1bf
   e7f0795a83114
   group_public_key: 02f37c34b66ced1fb51c34a90bdae006901f10625cc06c4f646
   63b0eae87d87b4f
   message: 74657374
   share_polynomial_coefficients[1]: fbf85eadae3058ea14f19148bb72b45e439
   9c0b16028acaf0395c9b03c823579

   // Signer input parameters
   P1 participant_share: 08f89ffe80ac94dcb920c26f3f46140bfc7f95b493f8310
   f5fc1ea2b01f4254c
   P2 participant_share: 04f0feac2edcedc6ce1253b7fab8c86b856a797f44d83d8
   2a385554e6e401984
   P3 participant_share: 00e95d59dd0d46b0e303e500b62b7ccb0e555d49f5b849f
   5e748c071da8c0dbc

   // Signer round one outputs
   P1 hiding_nonce_randomness: 7ea5ed09af19f6ff21040c07ec2d2adbd35b759da
   5a401d4c99dd26b82391cb2
   P1 binding_nonce_randomness: 47acab018f116020c10cb9b9abdc7ac10aae1b48
   ca6e36dc15acb6ec9be5cdc5
   P1 hiding_nonce: 841d3a6450d7580b4da83c8e618414d0f024391f2aeb511d7579
   224420aa81f0
   P1 binding_nonce: 8d2624f532af631377f33cf44b5ac5f849067cae2eacb88680a
   31e77c79b5a80
   P1 hiding_nonce_commitment: 03c699af97d26bb4d3f05232ec5e1938c12f1e6ae
   97643c8f8f11c9820303f1904
   P1 binding_nonce_commitment: 02fa2aaccd51b948c9dc1a325d77226e98a5a3fe
   65fe9ba213761a60123040a45e
   P1 binding_factor_input: 02f37c34b66ced1fb51c34a90bdae006901f10625cc0
   6c4f64663b0eae87d87b4fff9b5210ffbb3c07a73a7c8935be4a8c62cf015f6cf7ade
   6efac09a6513540fc3f5a816aaebc2114a811a415d7a55db7c5cbc1cf27183e79dd9d
   ef941b5d4801000000000000000000000000000000000000000000000000000000000
   0000001
   P1 binding_factor: 3e08fe561e075c653cbfd46908a10e7637c70c74f0a77d5fd4
   5d1a750c739ec6
   P3 hiding_nonce_randomness: e6cc56ccbd0502b3f6f831d91e2ebd01c4de0479e
   0191b66895a4ffd9b68d544
   P3 binding_nonce_randomness: 7203d55eb82a5ca0d7d83674541ab55f6e76f1b8
   5391d2c13706a89a064fd5b9
   P3 hiding_nonce: 2b19b13f193f4ce83a399362a90cdc1e0ddcd83e57089a7af0bd
   ca71d47869b2
   P3 binding_nonce: 7a443bde83dc63ef52dda354005225ba0e553243402a4705ce2
   8ffaafe0f5b98
   P3 hiding_nonce_commitment: 03077507ba327fc074d2793955ef3410ee3f03b82
   b4cdc2370f71d865beb926ef6
   P3 binding_nonce_commitment: 02ad53031ddfbbacfc5fbda3d3b0c2445c8e3e99
   cbc4ca2db2aa283fa68525b135
   P3 binding_factor_input: 02f37c34b66ced1fb51c34a90bdae006901f10625cc0
   6c4f64663b0eae87d87b4fff9b5210ffbb3c07a73a7c8935be4a8c62cf015f6cf7ade
   6efac09a6513540fc3f5a816aaebc2114a811a415d7a55db7c5cbc1cf27183e79dd9d
   ef941b5d4801000000000000000000000000000000000000000000000000000000000
   0000003
   P3 binding_factor: 93f79041bb3fd266105be251adaeb5fd7f8b104fb554a4ba9a
   0becea48ddbfd7

   // Signer round two outputs
   P1 sig_share: c4fce1775a1e141fb579944166eab0d65eefe7b98d480a569bbbfcb
   14f91c197
   P3 sig_share: 0160fd0d388932f4826d2ebcd6b9eaba734f7c71cf25b4279a4ca25
   81e47b18d

   sig: 0205b6d04d3774c8929413e3c76024d54149c372d57aae62574ed74319b5ea14
   d0c65dde8492a7471437e6c2fe3da49b90d23f642b5c6dbe7e36089f096dd97324

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 input and contributions.

Authors' Addresses

   Deirdre Connolly
   Zcash Foundation
   Email: durumcrustulum@gmail.com


   Chelsea Komlo
   University of Waterloo, Zcash Foundation
   Email: ckomlo@uwaterloo.ca


   Ian Goldberg
   University of Waterloo
   Email: iang@uwaterloo.ca


   Christopher A. Wood
   Cloudflare
   Email: caw@heapingbits.net