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Two-Round Threshold Schnorr Signatures with FROST
draft-irtf-cfrg-frost-04

Document Type Active Internet-Draft (cfrg RG)
Authors Deirdre Connolly , Chelsea Komlo , Ian Goldberg , Christopher A. Wood
Last updated 2022-03-29
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draft-irtf-cfrg-frost-04
CFRG                                                         D. Connolly
Internet-Draft                                          Zcash Foundation
Intended status: Informational                                  C. Komlo
Expires: 30 September 2022      University of Waterloo, Zcash Foundation
                                                             I. Goldberg
                                                  University of Waterloo
                                                              C. A. Wood
                                                              Cloudflare
                                                           29 March 2022

           Two-Round Threshold Schnorr Signatures with FROST
                        draft-irtf-cfrg-frost-04

Abstract

   In this draft, we present the two-round signing variant of FROST, a
   Flexible Round-Optimized Schnorr Threshold signature scheme.  FROST
   signatures can be issued after a threshold number of entities
   cooperate to issue a signature, allowing for improved distribution of
   trust and redundancy with respect to a secret key.  Further, this
   draft specifies signatures that are compatible with [RFC8032].
   However, unlike [RFC8032], the protocol for producing signatures in
   this draft is not deterministic, so as to ensure protection against a
   key-recovery attack that is possible when even only one participant
   is malicious.

Discussion Venues

   This note is to be removed before publishing as an RFC.

   Discussion of this document takes place on the Crypto Forum Research
   Group mailing list (cfrg@ietf.org), which is archived at
   https://mailarchive.ietf.org/arch/search/?email_list=cfrg.

   Source for this draft and an issue tracker can be found at
   https://github.com/cfrg/draft-irtf-cfrg-frost.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

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   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 30 September 2022.

Copyright Notice

   Copyright (c) 2022 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.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
     1.1.  Change Log  . . . . . . . . . . . . . . . . . . . . . . .   4
   2.  Conventions and Definitions . . . . . . . . . . . . . . . . .   5
   3.  Cryptographic Dependencies  . . . . . . . . . . . . . . . . .   6
     3.1.  Prime-Order Group . . . . . . . . . . . . . . . . . . . .   6
     3.2.  Cryptographic Hash Function . . . . . . . . . . . . . . .   7
   4.  Helper functions  . . . . . . . . . . . . . . . . . . . . . .   7
     4.1.  Schnorr Signature Operations  . . . . . . . . . . . . . .   8
     4.2.  Polynomial Operations . . . . . . . . . . . . . . . . . .   9
       4.2.1.  Evaluation of a polynomial  . . . . . . . . . . . . .   9
       4.2.2.  Lagrange coefficients . . . . . . . . . . . . . . . .  10
       4.2.3.  Deriving the constant term of a polynomial  . . . . .  11
     4.3.  Commitment List Encoding  . . . . . . . . . . . . . . . .  12
     4.4.  Binding Factor Computation  . . . . . . . . . . . . . . .  12
     4.5.  Group Commitment Computation  . . . . . . . . . . . . . .  13
     4.6.  Signature Challenge Computation . . . . . . . . . . . . .  13
   5.  Two-Round FROST Signing Protocol  . . . . . . . . . . . . . .  14
     5.1.  Round One - Commitment  . . . . . . . . . . . . . . . . .  17
     5.2.  Round Two - Signature Share Generation  . . . . . . . . .  17
     5.3.  Signature Share Verification and Aggregation  . . . . . .  19
   6.  Ciphersuites  . . . . . . . . . . . . . . . . . . . . . . . .  21
     6.1.  FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . .  22
     6.2.  FROST(ristretto255, SHA-512)  . . . . . . . . . . . . . .  22
     6.3.  FROST(Ed448, SHAKE256)  . . . . . . . . . . . . . . . . .  23
     6.4.  FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . .  24

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   7.  Security Considerations . . . . . . . . . . . . . . . . . . .  25
     7.1.  Nonce Reuse Attacks . . . . . . . . . . . . . . . . . . .  26
     7.2.  Protocol Failures . . . . . . . . . . . . . . . . . . . .  26
     7.3.  Removing the Coordinator Role . . . . . . . . . . . . . .  26
     7.4.  Input Message Validation  . . . . . . . . . . . . . . . .  27
   8.  Contributors  . . . . . . . . . . . . . . . . . . . . . . . .  27
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  27
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  27
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  28
   Appendix A.  Acknowledgments  . . . . . . . . . . . . . . . . . .  29
   Appendix B.  Trusted Dealer Key Generation  . . . . . . . . . . .  29
     B.1.  Shamir Secret Sharing . . . . . . . . . . . . . . . . . .  30
     B.2.  Verifiable Secret Sharing . . . . . . . . . . . . . . . .  32
   Appendix C.  Wire Format  . . . . . . . . . . . . . . . . . . . .  34
     C.1.  Signing Commitment  . . . . . . . . . . . . . . . . . . .  34
     C.2.  Signing Packages  . . . . . . . . . . . . . . . . . . . .  34
     C.3.  Signature Share . . . . . . . . . . . . . . . . . . . . .  35
   Appendix D.  Test Vectors . . . . . . . . . . . . . . . . . . . .  35
     D.1.  FROST(Ed25519, SHA-512) . . . . . . . . . . . . . . . . .  36
     D.2.  FROST(Ed448, SHAKE256)  . . . . . . . . . . . . . . . . .  37
     D.3.  FROST(ristretto255, SHA-512)  . . . . . . . . . . . . . .  39
     D.4.  FROST(P-256, SHA-256) . . . . . . . . . . . . . . . . . .  40
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  41

1.  Introduction

   DISCLAIMER: This is a work-in-progress draft of FROST.

   RFC EDITOR: PLEASE REMOVE THE FOLLOWING PARAGRAPH The source for this
   draft is maintained in GitHub.  Suggested changes should be submitted
   as pull requests at https://github.com/cfrg/draft-irtf-cfrg-frost.
   Instructions are on that page as well.

   Unlike signatures in a single-party setting, threshold signatures
   require cooperation among a threshold number of signers each holding
   a share of a common private key.  The security of threshold schemes
   in general assume that an adversary can corrupt strictly fewer than a
   threshold number of participants.

   This document presents a variant of a Flexible Round-Optimized
   Schnorr Threshold (FROST) signature scheme originally defined 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.  The variant of FROST presented in this document
   requires two rounds to compute a signature, and implements signing
   efficiency improvements described by [Schnorr21].  Single-round
   signing with FROST is out of scope.

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   For select ciphersuites, the signatures produced by this draft are
   compatible with [RFC8032].  However, unlike [RFC8032], signatures
   produced by FROST are not deterministic, since deriving nonces
   deterministically allows for a complete key-recovery attack in multi-
   party discrete logarithm-based signatures, such as FROST.

   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 B.

1.1.  Change Log

   draft-04

   *  Added methods to verify VSS commitments and derive group info
      (#126, #132).

   *  Changed check for participants to consider only nonnegative
      numbers (#133).

   *  Changed sampling for secrets and coefficients to allow the zero
      element (#130).

   *  Split test vectors into separate files (#129)

   *  Update wire structs to remove commitment shares where not
      necessary (#128)

   *  Add failure checks (#127)

   *  Update group info to include each participant's key and clarify
      how public key material is obtained (#120, #121).

   *  Define cofactor checks for verification (#118)

   *  Various editorial improvements and add contributors (#124, #123,
      #119, #116, #113, #109)

   draft-03

   *  Refactor the second round to use state from the first round (#94).

   *  Ensure that verification of signature shares from the second round
      uses commitments from the first round (#94).

   *  Clarify RFC8032 interoperability based on PureEdDSA (#86).

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   *  Specify signature serialization based on element and scalar
      serialization (#85).

   *  Fix hash function domain separation formatting (#83).

   *  Make trusted dealer key generation deterministic (#104).

   *  Add additional constraints on participant indexes and nonce usage
      (#105, #103, #98, #97).

   *  Apply various editorial improvements.

   draft-02

   *  Fully specify both rounds of FROST, as well as trusted dealer key
      generation.

   *  Add ciphersuites and corresponding test vectors, including suites
      for RFC8032 compatibility.

   *  Refactor document for editorial clarity.

   draft-01

   *  Specify operations, notation and cryptographic dependencies.

   draft-00

   *  Outline CFRG draft based on draft-komlo-frost.

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 and terminology are used throughout this
   document.

   *  A participant is an entity that is trusted to hold a secret share.

   *  NUM_SIGNERS denotes the number of participants, and the number of
      shares that s is split into.  This value MUST NOT exceed 2^16-1.

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   *  THRESHOLD_LIMIT denotes the threshold number of participants
      required to issue a signature.  More specifically, at least
      THRESHOLD_LIMIT shares must be combined to issue a valid
      signature.

   *  len(x) is the length of integer input x as an 8-byte, big-endian
      integer.

   *  encode_uint16(x): Convert two byte unsigned integer (uint16) x to
      a 2-byte, big-endian byte string.  For example, encode_uint16(310)
      = [0x01, 0x36].

   *  || denotes concatenation, i.e., x || y = xy.

   Unless otherwise stated, we assume that secrets are sampled uniformly
   at random using a cryptographically secure pseudorandom number
   generator (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;

   These are described in the following sections.

3.1.  Prime-Order Group

   FROST depends on an abelian group G of prime order p.  The
   fundamental group operation 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 GG, there exists an element -A such
   that A + (-A) = (-A) + A = I.  Scalar multiplication is equivalent to
   the repeated application of the group operation on an element A with
   itself r-1 times, this is denoted as r*A = A + ... + A.  For any
   element A, p * A = I.  We denote B as the fixed generator of the
   group.  Scalar base multiplication is equivalent to the repeated
   application of the group operation B with itself r-1 times, this is
   denoted as ScalarBaseMult(r).  The set of scalars corresponds to
   GF(p), which refer to as the scalar field.  This document uses types
   Element and Scalar to denote elements of the group G and its set of
   scalars, respectively.  We denote equality comparison as == and
   assignment of values by =.

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   We now detail a number of member functions that can be invoked on a
   prime-order group G.

   *  Order(): Outputs the order of G (i.e. p).

   *  Identity(): Outputs the identity element of the group (i.e.  I).

   *  RandomScalar(): A member function of G that chooses at random a
      Scalar element in GF(p).

   *  RandomNonzeroScalar(): A member function of G that chooses at
      random a non-zero Scalar element in GF(p).

   *  SerializeElement(A): A member function of G that maps an Element A
      to a unique byte array buf of fixed length Ne.

   *  DeserializeElement(buf): A member function of G that attempts to
      map a byte array buf to an Element A, and fails if the input is
      not a valid byte representation of an element of the group.  This
      function can raise a DeserializeError if deserialization fails or
      A is the identity element of the group; see Section 6 for group-
      specific input validation steps.

   *  SerializeScalar(s): A member function of G that maps a Scalar s to
      a unique byte array buf of fixed length Ns.

   *  DeserializeScalar(buf): A member function of G that attempts to
      map a byte array buf to a Scalar s.  This function can raise a
      DeserializeError 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 functions effectively as a random
   oracle.  For concrete recommendations on hash functions which SHOULD
   BE used in practice, see Section 6.  Using H, we introduce three
   separate domain-separated hashes, H1, H2, and H3, where H1 and H2 map
   arbitrary inputs to non-zero Scalar elements of the prime-order group
   scalar field, and H3 is an alias for H with domain separation
   applied.  The details of H1, H2, and H3 vary based on ciphersuite.
   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:

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   *  Schnorr signatures, Section 4.1;

   *  Polynomial operations, Section 4.2;

   *  Encoding operations, Section 4.3;

   *  Signature binding Section 4.4, group commitment Section 4.5, and
      challenge computation Section 4.6

   This sections describes these operations in more detail.

4.1.  Schnorr Signature Operations

   In the single-party setting, a Schnorr signature is generated with
   the following operation.

     schnorr_signature_generate(msg, SK):

     Inputs:
     - msg, message to be signed, an octet string
     - SK, private key, a scalar

     Outputs: signature (R, z), a pair of scalar values

     def schnorr_signature_generate(msg, SK):
       PK = G.ScalarBaseMult(SK)
       k = G.RandomScalar()
       R = G.ScalarBaseMult(k)

       comm_enc = G.SerializeElement(R)
       pk_enc = G.SerializeElement(PK)
       challenge_input = comm_enc || pk_enc || msg
       c = H2(challenge_input)

       z = k + (c * SK)
       return (R, z)

   The corresponding verification operation is as follows.  Here, h is
   the cofactor for the group being operated over, e.g. h=8 for the case
   of Curve25519, h=4 for Ed448, and h=1 for groups such as ristretto255
   and secp256k1, etc.  This final scalar multiplication MUST be
   performed when h>1.

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   schnorr_signature_verify(msg, sig, PK):

   Inputs:
   - msg, signed message, an octet string
   - sig, a tuple (R, z) output from schnorr_signature_generate or FROST
   - PK, public key, a group element

   Outputs: 1 if signature is valid, and 0 otherwise

   def schnorr_signature_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 + (c * PK)
     check = (l - r) * h
     return check == G.Identity()

4.2.  Polynomial Operations

   This section describes operations on and associated with polynomials
   that are used in the main signing protocol.  A polynomial of degree t
   is represented as a sorted list of t coefficients.  A point on the
   polynomial is a tuple (x, y), where y = f(x).  For notational
   convenience, we refer to the x-coordinate and y-coordinate of a point
   p as p.x and p.y, respectively.

4.2.1.  Evaluation of a polynomial

   This section describes a method for evaluating a polynomial f at a
   particular input x, i.e., y = f(x) using Horner's method.

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     polynomial_evaluate(x, coeffs):

     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 = 0
       for (counter, coeff) in coeffs.reverse():
         if counter == coeffs.len() - 1:
           value += coeff // add the constant term
         else:
           value += coeff
           value *= x

       return value

4.2.2.  Lagrange coefficients

   Lagrange coefficients are used in FROST to evaluate a polynomial f at
   f(0), given a set of t other points, where f is represented as a set
   of coefficients.

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    derive_lagrange_coefficient(x_i, L):

    Inputs:
    - x_i, an x-coordinate contained in L, a scalar
    - L, the set of x-coordinates, each a scalar

    Outputs: L_i, the i-th Lagrange coefficient

    Errors:
    - "invalid parameters", if any coordinate is less than or equal to 0

    def derive_lagrange_coefficient(x_i, L):
      if x_i = 0:
        raise "invalid parameters"
      for x_j in L:
        if x_j = 0:
          raise "invalid parameters"

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

      L_i = numerator / denominator
      return L_i

4.2.3.  Deriving the constant term of a polynomial

   Secret sharing requires "splitting" a secret, which is represented as
   a constant term of some polynomial f of degree t.  Recovering the
   constant term occurs with a set of t points using polynomial
   interpolation, defined as follows.

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    Inputs:
    - points, a set of `t` points on a polynomial f, each a tuple of two
      scalar values representing the x and y coordinates

    Outputs: The constant term of f, i.e., f(0)

    def polynomial_interpolation(points):
      L = []
      for point in points:
        L.append(point.x)

      f_zero = F(0)
      for point in points:
        delta = point.y * derive_lagrange_coefficient(point.x, L)
        f_zero = f_zero + delta

      return f_zero

4.3.  Commitment List Encoding

   This section describes the subroutine used for encoding a list of
   signer commitments into a bytestring that is used in the FROST
   protocol.

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

  Outputs: A byte string containing the serialized representation of commitment_list.

  def encode_group_commitment_list(commitment_list):
    encoded_group_commitment = nil
    for (index, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
      encoded_commitment = encode_uint16(index) ||
                           G.SerializeElement(hiding_nonce_commitment) ||
                           G.SerializeElement(binding_nonce_commitment)
      encoded_group_commitment = encoded_group_commitment || encoded_commitment
    return encoded_group_commitment

4.4.  Binding Factor Computation

   This section describes the subroutine for computing the binding
   factor based on the signer commitment list and message to be signed.

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     Inputs:
     - encoded_commitment_list, an encoded commitment list (as computed
       by encode_group_commitment_list)
     - msg, the message to be signed (sent by the Coordinator).

     Outputs: binding_factor, a Scalar representing the binding factor

     def compute_binding_factor(encoded_commitment_list, msg):
       msg_hash = H3(msg)
       rho_input = encoded_commitment_list || msg_hash
       binding_factor = H1(rho_input)
       return binding_factor

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 signer, where each element in the list indicates the signer index i and their
    two commitment Element values (hiding_nonce_commitment_i, binding_nonce_commitment_i).
    This list MUST be sorted in ascending order by signer index.
  - binding_factor, a Scalar

  Outputs: An Element representing the group commitment

  def compute_group_commitment(commitment_list, binding_factor):
    group_commitment = G.Identity()
    for (_, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
      group_commitment = group_commitment + (hiding_nonce_commitment + (binding_nonce_commitment * binding_factor))
    return group_commitment

4.6.  Signature Challenge Computation

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

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  Inputs:
  - group_commitment, an Element representing the group commitment
  - group_public_key, public key corresponding to the signer secret key share.
  - msg, the message to be signed (sent by the Coordinator).

  Outputs: a challenge Scalar value

  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

   We now present the two-round variant of the FROST threshold signature
   protocol for producing Schnorr signatures.  It involves signer
   participants and a coordinator.  Signing participants are entities
   with signing key shares that participate in the threshold signing
   protocol.  The coordinator is a distinguished signer with the
   following responsibilities:

   1.  Determining which signers will participate (at least
       THRESHOLD_LIMIT in number);

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

   3.  Aggregating signature shares output by each participant, and
       publishing the resulting signature.

   FROST assumes the selection of all participants, including the
   dealer, signer, and Coordinator are all chosen external to the
   protocol.  Note that it is possible to deploy the protocol without a
   distinguished Coordinator; see Section 7.3 for more information.

   Because key generation is not specified, all signers are assumed to
   have the (public) group state that we refer to as "group info" below,
   and their corresponding signing key shares.

   In particular, it is assumed that the coordinator and each signing
   participant P_i knows the following group info:

   *  Group public key, denoted PK = G.ScalarMultBase(s), corresponding
      to the group secret key s.  PK is an output from the group's key
      generation protocol, such as trusted_dealer_keygenor a DKG.

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   *  Public keys for each signer, denoted PK_i =
      G.ScalarMultBase(sk_i), which are similarly outputs from the
      group's key generation protocol.

   And that each participant with identifier i additionally knows the
   following:

   *  Participant is signing key share sk_i, which is the i-th secret
      share of s.

   The exact key generation mechanism is out of scope for this
   specification.  In general, key generation is a protocol that outputs
   (1) a shared, group public key PK owned by each Signer, and (2)
   individual shares of the signing key owned by each Signer.  In
   general, two possible key generation mechanisms are possible, one
   that requires a single, trusted dealer, and the other which requires
   performing a distributed key generation protocol.  We highlight key
   generation mechanism by a trusted dealer in Appendix B, for
   reference.

   This signing variant of FROST requires signers to perform two network
   rounds: 1) generating and publishing commitments, and 2) signature
   share generation and publication.  The first round serves for each
   participant to issue a commitment to a nonce.  The second round
   receives commitments for all signers as well as the message, and
   issues a signature share with respect to that message.  The
   Coordinator performs the coordination of each of these rounds.  At
   the end of the second round, the Coordinator then performs an
   aggregation step and outputs the final signature.  This complete
   interaction is shown in Figure 1.

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           (group info)            (group info,     (group info,
               |               signing key share)   signing key share)
               |                         |                |
               v                         v                v
           Coordinator               Signer-1   ...   Signer-n
       ------------------------------------------------------------
      message
   ------------>
               |
         == Round 1 (Commitment) ==
               |    signer commitment   |                 |
               |<-----------------------+                 |
               |          ...                             |
               |    signer commitment                     |
               |<-----------------------------------------+

         == Round 2 (Signature Share Generation) ==
               |
               |     signer input       |                 |
               +------------------------>                 |
               |     signature share    |                 |
               |<-----------------------+                 |
               |          ...                             |
               |     signer input                         |
               +------------------------------------------>
               |     signature share                      |
               <------------------------------------------+
               |
         == Aggregation ==
               |
     signature |
   <-----------+

                     Figure 1: FROST signature overview

   Details for round one are described in Section 5.1, and details for
   round two are described in Section 5.2.  The final Aggregation step
   is described in Section 5.3.

   FROST assumes reliable message delivery between Coordinator and
   signing participants in order for the protocol to complete.  Messages
   exchanged during signing operations are all within the public domain.
   An attacker masquerading as another participant will result only in
   an invalid signature; see Section 7.

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5.1.  Round One - Commitment

   Round one involves each signer generating a pair of nonces and their
   corresponding public commitments.  A nonce is a pair of Scalar
   values, and a commitment is a pair of Element values.

   Each signer in round one generates a nonce nonce = (hiding_nonce,
   binding_nonce) and commitment comm = (hiding_nonce_commitment,
   binding_nonce_commitment).

    Inputs: None

    Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs.

    def commit():
      hiding_nonce = G.RandomNonzeroScalar()
      binding_nonce = G.RandomNonzeroScalar()
      hiding_nonce_commitment = G.ScalarBaseMult(hiding_nonce)
      binding_nonce_commitment = G.ScalarBaseMult(binding_nonce)
      nonce = (hiding_nonce, binding_nonce)
      comm = (hiding_nonce_commitment, binding_nonce_commitment)
      return (nonce, comm)

   The private output nonce from Participant P_i is stored locally and
   kept private for use in the second round.  This nonce MUST NOT be
   reused in more than one invocation of FROST, and it MUST be generated
   from a source of secure randomness.  The public output comm from
   Participant P_i is sent to the Coordinator; see Appendix C.1 for
   encoding recommendations.

5.2.  Round Two - Signature Share Generation

   In round two, the Coordinator is responsible for sending the message
   to be signed, and for choosing which signers will participate (of
   number at least THRESHOLD_LIMIT).  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 signer the message to be
   signed along with the set of signing commitments for other signers in
   the participant list.  Each signer 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.  Applications which
   require that signers not process arbitrary input messages are also
   required to also perform relevant application-layer input validation
   checks; see Section 7.4 for more details.

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

  Inputs:
  - index, Index `i` of the signer. Note index will never equal `0`.
  - sk_i, Signer secret key share.
  - group_public_key, public key corresponding to the signer secret key share.
  - nonce_i, pair of Scalar values (hiding_nonce, binding_nonce) generated in round one.
  - msg, the message to be signed (sent by the Coordinator).
  - commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a
    list of commitments issued in Round 1 by each signer, where each element in the list indicates the signer index j and their
    two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
    This list MUST be sorted in ascending order by signer index.
  - participant_list, a set containing identifiers for each signer, similarly of length
    NUM_SIGNERS (sent by the Coordinator).

  Outputs: a Scalar value representing the signature share

  def sign(index, sk_i, group_public_key, nonce_i, msg, commitment_list, participant_list):
    # Encode the commitment list
    encoded_commitments = encode_group_commitment_list(commitment_list)

    # Compute the binding factor
    binding_factor = compute_binding_factor(encoded_commitments, msg)

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

    # Compute Lagrange coefficient
    lambda_i = derive_lagrange_coefficient(index, participant_list)

    # 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 signer then
   sends these shares back to the collector; see Appendix C.3 for
   encoding recommendations.  Each signer MUST delete the nonce and
   corresponding commitment after this round completes.

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   Upon receipt from each Signer, the Coordinator MUST validate the
   input signature using DeserializeElement.  If validation fails, the
   Coordinator MUST abort the protocol.  If validation succeeds, the
   Coordinator then verifies the set of signature shares using the
   following procedure.

5.3.  Signature Share Verification and Aggregation

   After signers perform round two and send their signature shares to
   the Coordinator, the Coordinator verifies each signature share for
   correctness.  In particular, for each signer, the Coordinator uses
   commitment pairs generated during round one and the signature share
   generated during round two, along with other group parameters, to
   check that the signature share is valid using the following
   procedure.

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  Inputs:
  - index, Index `i` of the signer. Note index will never equal `0`.
  - PK_i, the public key for the ith signer, where `PK_i = G.ScalarBaseMult(sk_i)`
  - comm_i, pair of Element values (hiding_nonce_commitment, binding_nonce_commitment) generated
    in round one from the ith signer.
  - sig_share_i, a Scalar value indicating the signature share as produced in round two from the ith signer.
  - commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a list of commitments
    issued in Round 1 by each signer, where each element in the list indicates the signer index j and their
    two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j).
    This list MUST be sorted in ascending order by signer index.
  - participant_list, a set containing identifiers for each signer, similarly of length
    NUM_SIGNERS (sent by the Coordinator).
  - group_public_key, the public key for the group
  - msg, the message to be signed

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

  def verify_signature_share(index, PK_i, comm_i, sig_share_i, commitment_list,
                             participant_list, group_public_key, msg):
    # Encode the commitment list
    encoded_commitments = encode_group_commitment_list(commitment_list)

    # Compute the binding factor
    binding_factor = compute_binding_factor(encoded_commitments, msg)

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

    # Compute the commitment share
    (hiding_nonce_commitment, binding_nonce_commitment) = comm_i
    comm_share = hiding_nonce_commitment + (binding_nonce_commitment * binding_factor)

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

    # Compute Lagrange coefficient
    lambda_i = derive_lagrange_coefficient(index, participant_list)

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

    return l == r

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   If any signature share fails to verify, i.e., if
   verify_signature_share returns False for any signer share, the
   Coordinator MUST abort the protocol.  Otherwise, if all signer shares
   are valid, the Coordinator performs the aggregate operation and
   publishes the resulting signature.

  Inputs:
  - group_commitment, the group commitment returned by compute_group_commitment
  - sig_shares, a set of signature shares z_i for each signer, of length NUM_SIGNERS,
  where THRESHOLD_LIMIT <= NUM_SIGNERS <= MAX_SIGNERS.

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

  def frost_aggregate(group_commitment, sig_shares):
    z = 0
    for z_i in sig_shares:
      z = z + z_i
    return (group_commitment, z)

   The output signature (R, z) from the aggregation step MUST be encoded
   as follows:

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

   Where Signature.R_encoded is G.SerializeElement(R) and
   Signature.z_encoded is G.SerializeScalar(z).

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.

   The RECOMMENDED ciphersuite is (ristretto255, SHA-512) Section 6.2.
   The (Ed25519, SHA-512) ciphersuite is included for backwards
   compatibility with [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 the ciphersuites below.
   Future ciphersuites MUST describe how input validation is done for
   DeserializeElement and DeserializeScalar.

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6.1.  FROST(Ed25519, SHA-512)

   This ciphersuite uses edwards25519 for the Group and SHA-512 for the
   Hash function H meant to produce signatures indistinguishable from
   Ed25519 as specified in [RFC8032].  The value of the contextString
   parameter is empty.

   *  Group: edwards25519 [RFC8032]

      -  Cofactor (h): 8

      -  SerializeElement: Implemented as specified in [RFC8032],
         Section 5.1.2.

      -  DeserializeElement: Implemented as specified in [RFC8032],
         Section 5.1.3.  Additionally, this function validates that the
         resulting element is not the group identity element.

      -  SerializeScalar: Implemented by outputting the little-endian
         32-byte encoding of the Scalar value.

      -  DeserializeScalar: Implemented by attempting to deserialize a
         Scalar from a 32-byte string.  This function can fail if the
         input does not represent a Scalar between the value 0 and
         G.Order() - 1.

   *  Hash (H): SHA-512, and Nh = 64.

      -  H1(m): Implemented by computing H("rho" || m), interpreting the
         lower 32 bytes as a little-endian integer, and reducing the
         resulting integer modulo L =
         2^252+27742317777372353535851937790883648493.

      -  H2(m): Implemented by computing H(m), interpreting the lower 32
         bytes as a little-endian integer, and reducing the resulting
         integer modulo L =
         2^252+27742317777372353535851937790883648493.

      -  H3(m): Implemented as an alias for H, i.e., H(m).

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

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".

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   *  Group: ristretto255 [RISTRETTO]

      -  Cofactor (h): 1

      -  SerializeElement: Implemented using the 'Encode' function from
         [RISTRETTO].

      -  DeserializeElement: Implemented using the 'Decode' function
         from [RISTRETTO].

      -  SerializeScalar: Implemented by outputting the little-endian
         32-byte encoding of the Scalar value.

      -  DeserializeScalar: Implemented by attempting to deserialize a
         Scalar from a 32-byte string.  This function can fail if the
         input does not represent a Scalar between the value 0 and
         G.Order() - 1.

   *  Hash (H): SHA-512, and Nh = 64.

      -  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 || "digest" ||
         m).

6.3.  FROST(Ed448, SHAKE256)

   This ciphersuite uses edwards448 for the Group and SHA256 for the
   Hash function H meant to produce signatures indistinguishable from
   Ed448 as specified in [RFC8032].  The value of the contextString
   parameter is empty.

   *  Group: edwards448 [RFC8032]

      -  Cofactor (h): 4

      -  SerializeElement: Implemented as specified in [RFC8032],
         Section 5.2.2.

      -  DeserializeElement: Implemented as specified in [RFC8032],
         Section 5.2.3.  Additionally, this function validates that the
         resulting element is not the group identity element.

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      -  SerializeScalar: Implemented by outputting the little-endian
         48-byte encoding of the Scalar value.

      -  DeserializeScalar: Implemented by attempting to deserialize a
         Scalar from a 48-byte string.  This function can fail if the
         input does not represent a Scalar between the value 0 and
         G.Order() - 1.

   *  Hash (H): SHAKE256, and Nh = 117.

      -  H1(m): Implemented by computing H("rho" || m), interpreting the
         lower 57 bytes as a little-endian integer, and reducing the
         resulting integer modulo L = 2^446 - 13818066809895115352007386
         748515426880336692474882178609894547503885.

      -  H2(m): Implemented by computing H(m), interpreting the lower 57
         bytes as a little-endian integer, and reducing the resulting
         integer modulo L = 2^446 - 138180668098951153520073867485154268
         80336692474882178609894547503885.

      -  H3(m): Implemented as an alias for H, i.e., H(m).

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

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".

   *  Group: P-256 (secp256r1) [x9.62]

      -  Cofactor (h): 1

      -  SerializeElement: Implemented using the compressed Elliptic-
         Curve-Point-to-Octet-String method according to [SECG].

      -  DeserializeElement: Implemented by attempting to deserialize a
         public key using the compressed Octet-String-to-Elliptic-Curve-
         Point method according to [SECG], and then performs partial
         public-key validation as defined in section 5.6.2.3.4 of
         [KEYAGREEMENT].  This includes checking that the coordinates of
         the resulting point are in the correct range, that the point is
         on the curve, and that the point is not the point at infinity.
         Additionally, this function validates that the resulting
         element is not the group identity element.  If these checks
         fail, deserialization returns an error.

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      -  SerializeScalar: Implemented using the Field-Element-to-Octet-
         String conversion according to [SECG].

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

   *  Hash (H): SHA-256, and Nh = 32.

      -  H1(m): Implemented using hash_to_field from [HASH-TO-CURVE],
         Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST
         = contextString || "rho", and prime modulus equal to Order().

      -  H2(m): Implemented using hash_to_field from [HASH-TO-CURVE],
         Section 5.3 using L = 48, expand_message_xmd with SHA-256, DST
         = contextString || "chal", and prime modulus equal to Order().

      -  H3(m): Implemented by computing H(contextString || "digest" ||
         m).

7.  Security Considerations

   A security analysis of FROST exists in [FROST20] and [Schnorr21].
   The protocol as specified in this document assumes the following
   threat model.

   *  Trusted dealer.  The dealer that performs key generation is
      trusted to follow the protocol, although participants still are
      able to verify the consistency of their shares via a VSS
      (verifiable secret sharing) step; see Appendix B.2.

   *  Unforgeability assuming less than (t-1) corrupted signers.  So
      long as an adverary corrupts fewer than (t-1) participants, the
      scheme remains secure against EUF-CMA attacks.

   *  Coordinator.  We assume the Coordinator at the time of signing
      does not perform a denial of service attack.  A denial of service
      would include any action which either prevents the protocol from
      completing or causing the resulting signature to be invalid.  Such
      actions for the latter include sending inconsistent values to
      signing participants, such as messages or the set of individual
      commitments.  Note that the Coordinator is _not_ trusted with any
      private information and communication at the time of signing can
      be performed over a public but reliable channel.

   The protocol as specified in this document does not target the
   following goals:

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

   *  Robustness.  In the case of failure, FROST requires aborting the
      protocol.

   *  Downgrade prevention.  The sender and receiver are assumed to
      agree on what algorithms to use.

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

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

7.1.  Nonce Reuse Attacks

   Nonces generated by each participant in the first round of signing
   must be sampled uniformly at random and cannot be derived from some
   deterministic function.  This is to avoid replay attacks initiated by
   other signers, which allows for a complete key-recovery attack.
   Coordinates MAY further hedge against nonce reuse attacks by tracking
   signer nonce commitments used for a given group key, at the cost of
   additional state.

7.2.  Protocol Failures

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

7.3.  Removing the Coordinator Role

   In some settings, it may be desirable to omit the role of the
   coordinator entirely.  Doing so does not change the security
   implications of FROST, but instead simply requires each participant
   to communicate with all other participants.  We loosely describe how
   to perform FROST signing among signers without this coordinator role.
   We assume that every participant receives as input from an external
   source the message to be signed prior to performing the protocol.

   Every participant begins by performing frost_commit() as is done in
   the setting where a coordinator is used.  However, instead of sending
   the commitment SigningCommitment to the coordinator, every
   participant instead will publish this commitment to every other
   participant.  Then, in the second round, instead of receiving a

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   SigningPackage from the coordinator, signers will already have
   sufficient information to perform signing.  They will directly
   perform frost_sign.  All participants will then publish a
   SignatureShare to one another.  After having received all signature
   shares from all other signers, each signer will then perform
   frost_verify and then frost_aggregate directly.

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

7.4.  Input Message Validation

   Some applications may require that signers only process messages of a
   certain structure.  For example, in digital currency applications
   wherein multiple signers may collectively sign a transaction, it is
   reasonable to require that each signer check the input message to be
   a syntactically valid transaction.  As another example, use of
   threshold signatures in TLS [TLS] to produce signatures of transcript
   hashes might require that signers check that the input message is a
   valid TLS transcript from which the corresponding transcript hash can
   be derived.

   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 signers do not operate as
   signing oracles for arbitrary messages.

8.  Contributors

   *  Isis Lovecruft

   *  T.  Wilson-Brown

   *  Alden Torres

9.  References

9.1.  Normative References

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   [HASH-TO-CURVE]
              Faz-Hernandez, A., Scott, S., Sullivan, N., Wahby, R. S.,
              and C. A. Wood, "Hashing to Elliptic Curves", Work in
              Progress, Internet-Draft, draft-irtf-cfrg-hash-to-curve-
              14, 18 February 2022,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              hash-to-curve-14>.

   [KEYAGREEMENT]
              Barker, E., Chen, L., Roginsky, A., Vassilev, A., and R.
              Davis, "Recommendation for pair-wise key-establishment
              schemes using discrete logarithm cryptography", National
              Institute of Standards and Technology report,
              DOI 10.6028/nist.sp.800-56ar3, April 2018,
              <https://doi.org/10.6028/nist.sp.800-56ar3>.

   [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/rfc/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/rfc/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/rfc/rfc8174>.

   [RISTRETTO]
              Valence, H. D., Grigg, J., Hamburg, M., Lovecruft, I.,
              Tankersley, G., and F. Valsorda, "The ristretto255 and
              decaf448 Groups", Work in Progress, Internet-Draft, draft-
              irtf-cfrg-ristretto255-decaf448-03, 25 February 2022,
              <https://datatracker.ietf.org/doc/html/draft-irtf-cfrg-
              ristretto255-decaf448-03>.

   [SECG]     "Elliptic Curve Cryptography, Standards for Efficient
              Cryptography Group, ver. 2", 2009,
              <https://secg.org/sec1-v2.pdf>.

   [x9.62]    ANSI, "Public Key Cryptography for the Financial Services
              Industry: the Elliptic Curve Digital Signature Algorithm
              (ECDSA)", ANSI X9.62-1998, September 1998.

9.2.  Informative References

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   [FROST20]  Komlo, C. and I. Goldberg, "Two-Round Threshold Signatures
              with FROST", 22 December 2020,
              <https://eprint.iacr.org/2020/852.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/rfc/rfc4086>.

   [Schnorr21]
              Crites, E., Komlo, C., and M. Maller, "How to Prove
              Schnorr Assuming Schnorr", 11 October 2021,
              <https://eprint.iacr.org/2021/1375>.

   [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/rfc/rfc8446>.

Appendix A.  Acknowledgments

   The Zcash Foundation engineering team designed a serialization format
   for FROST messages which we employ a slightly adapted version here.

Appendix B.  Trusted Dealer Key Generation

   One possible key generation mechanism is to depend on a trusted
   dealer, wherein the dealer generates a group secret s uniformly at
   random and uses Shamir and Verifiable Secret Sharing as described in
   Sections Appendix B.1 and Appendix B.2 to create secret shares of s
   to be sent to all other participants.  We highlight at a high level
   how this operation can be performed.

  Inputs:
  - s, a group secret that MUST be derived from at least `Ns` bytes of entropy
  - n, the number of shares to generate, an integer
  - t, the threshold of the secret sharing scheme, an integer

  Outputs:
  - signer_private_keys, `n` shares of the secret key `s`, each a Scalar value.
  - vss_commitment, a vector commitment to each of the coefficients in the polynomial defined by secret_key_shares and whose constant term is s.

  def trusted_dealer_keygen(s, n, t):
    signer_private_keys, coefficients = secret_share_shard(secret_key, n, t)
    vss_commitment = vss_commit(coefficients):
    PK = G.ScalarBaseMult(secret_key)
    return signer_private_keys, vss_commitment

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   It is assumed the dealer then sends one secret key share to each of
   the NUM_SIGNERS participants, along with C.  After receiving their
   secret key share and C each participant MUST perform
   vss_verify(secret_key_share_i, C).  It is assumed that all
   participant have the same view of C.  The trusted dealer MUST delete
   the secret_key and secret_key_shares upon completion.

   Use of this method for key generation requires a mutually
   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.

B.1.  Shamir Secret Sharing

   In Shamir secret sharing, a dealer distributes a secret s to n
   participants in such a way that any cooperating subset of t
   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.

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  secret_share_shard(s, n, t):

  Inputs:
  - s, secret to be shared, an element of F
  - n, the number of shares to generate, an integer
  - t, the threshold of the secret sharing scheme, an integer

  Outputs:
  - secret_key_shares, A list of n secret shares, which is a tuple
  consisting of the participant identifier and the key share, each of
  which is an element of F
  - coefficients, a vector of the t coefficients which uniquely determine
  a polynomial f.

  Errors:
  - "invalid parameters", if t > n or if t is less than 2

  def secret_share_shard(s, n, t):
    if t > n:
      raise "invalid parameters"
    if t < 2:
      raise "invalid parameters"

    # Generate random coefficients for the polynomial, yielding
    # a polynomial of degree (t - 1)
    coefficients = [s]
    for i in range(t - 1):
      coefficients.append(G.RandomScalar())

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

   Let points be the output of this function.  The i-th element in
   points is the share for the i-th participant, which is the randomly
   generated polynomial evaluated at coordinate i.  We denote a secret
   share as the tuple (i, points[i]), and the list of these shares as
   shares. i MUST never equal 0; recall that f(0) = s, where f is the
   polynomial defined in a Shamir secret sharing operation.

   The procedure for combining a shares list of length t to recover the
   secret s is as follows.

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  secret_share_combine(shares):

  Inputs:
  - shares, a list of t secret shares, each a tuple (i, f(i))

  Outputs: The resulting secret s, that was previously split into shares

  Errors:
  - "invalid parameters", if less than t input shares are provided

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

B.2.  Verifiable Secret Sharing

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

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

    vss_commit(coeffs):

    Inputs:
    - coeffs, a vector of the t coefficients which uniquely determine
    a polynomial f.

    Outputs: a commitment vss_commitment, which is a vector commitment to each of the
    coefficients in coeffs.

    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 failure should be investigated out of band.

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    vss_verify(share_i, vss_commitment):

    Inputs:
    - share_i: A tuple of the form (i, sk_i), where i indicates the participant
    identifier, and sk_i the participant's secret key, where sk_i is a secret share of
    the constant term of f.
    - vss_commitment: A VSS commitment to a secret polynomial f.

    Outputs: 1 if sk_i is valid, and 0 otherwise

    vss_verify(share_i, commitment)
      (i, sk_i) = share_i
      S_i = ScalarBaseMult(sk_i)
      S_i' = G.Identity()
      for j in range(0, THRESHOLD_LIMIT-1):
        S_i' += vss_commitment_j * i^j
      if S_i == S_i':
        return 1
      return 0

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

    derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment):

    Inputs:
    - MAX_SIGNERS, the number of shares to generate, an integer
    - THRESHOLD_LIMIT, the threshold of the secret sharing scheme, an integer
    - vss_commitment: A VSS commitment to a secret polynomial f.

    Outputs:
    - PK, the public key representing the group
    - signer_public_keys, a list of MAX_SIGNERS public keys PK_i for i=1,...,MAX_SIGNERS, where PK_i is the public key for participant i.

    derive_group_info(MAX_SIGNERS, THRESHOLD_LIMIT, vss_commitment)
      PK = vss_commitment[0]
      signer_public_keys = []
      for i in range(1, MAX_SIGNERS):
        PK_i = G.Identity()
        for j in range(0, THRESHOLD_LIMIT-1):
          PK_i += vss_commitment_j * i^j
        signer_public_keys.append(PK_i)
      return PK, signer_public_keys

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Appendix C.  Wire Format

   Applications are responsible for encoding protocol messages between
   peers.  This section contains RECOMMENDED encodings for different
   protocol messages as described in Section 5.

C.1.  Signing Commitment

   A commitment from a signer is a pair of Element values.  It can be
   encoded in the following manner.

     SignerID uint64;

     struct {
       SignerID id;
       opaque D[Ne];
       opaque E[Ne];
     } SigningCommitment;

   id  The SignerID.

   D  The commitment hiding factor encoded as a serialized group
      element.

   E  The commitment binding factor encoded as a serialized group
      element.

C.2.  Signing Packages

   The Coordinator sends "signing packages" to each Signer in Round two.
   Each package contains the list of signing commitments generated
   during round one along with the message to sign.  This package can be
   encoded in the following manner.

   struct {
     SigningCommitment signing_commitments<1..2^16-1>;
     opaque msg<0..2^16-1>;
   } SigningPackage;

   signing_commitments  An list of SIGNING_COUNT SigningCommitment
      values, where THRESHOLD_LIMIT <= SIGNING_COUNT <= NUM_SIGNERS,
      ordered in ascending order by SigningCommitment.id.  This list
      MUST NOT contain more than one SigningCommitment value
      corresponding to each signer.  Signers MUST ignore SigningPackage
      values with duplicate SignerIDs.

   msg  The message to be signed.

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C.3.  Signature Share

   The output of each signer is a signature share which is sent to the
   Coordinator.  This can be constructed as follows.

     struct {
       SignerID id;
       opaque signature_share[Ns];
     } SignatureShare;

   id  The SignerID.

   signature_share  The signature share from this signer encoded as a
      serialized scalar.

Appendix D.  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_SIGNERS, THRESHOLD_LIMIT,
      and NUM_SIGNERS.

   *  Group input parameters: This lists the group secret key and shared
      public key, generated by a trusted dealer as described in
      Appendix B, as well as the input message to be signed.  All values
      are encoded as hexadecimal strings.

   *  Signer input parameters: This lists the signing key share for each
      of the NUM_SIGNERS signers.

   *  Round one parameters and outputs: This lists the NUM_SIGNERS
      participants engaged in the protocol, identified by their integer
      index, the hiding and binding commitment values produced in
      Section 5.1, as well as the resulting group binding factor input,
      computed in part from the group commitment list encoded as
      described in Section 4.3, and group binding factor as computed in
      Section 5.2).

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   *  Round two parameters and outputs: This lists the NUM_SIGNERS
      participants engaged in the protocol, identified by their integer
      index, 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.

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

   // Configuration information
   MAX_SIGNERS: 3
   THRESHOLD_LIMIT: 2
   NUM_SIGNERS: 2

   // Group input parameters
   group_secret_key: 7b1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
   90c6e13a98304
   group_public_key: 15d21ccd7ee42959562fc8aa63224c8851fb3ec85a3faf66040
   d380fb9738673
   message: 74657374

   // Signer input parameters
   S1 signer_share: 929dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033
   f2ec83d93509
   S2 signer_share: a91e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee
   d76bf409e80d
   S3 signer_share: d3cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8
   bdea643a9a02

   // Round one parameters
   participants: 1,2
   group_binding_factor_input: 000178e175d15cb5cec1257e0d84d797ba8c3dd9b
   4c7bc50f3fa527c200bcc6c4a954cdad16ae67ac5919159d655b681bd038574383bab
   423614f8967396ee12ca62000288a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea98
   981bfb25375996c5767c932bbf10c41feb17d41cc6433e69f16cceccc42a00aedf72f
   eb5f44929fdf2e2fee26b0dd4af7e749aa1a8ee3c10ae9923f618980772e473f8819a
   5d4940e0db27ac185f8a0e1d5f84f88bc887fd67b143732c304cc5fa9ad8e6f57f500
   28a8ff
   group_binding_factor: c4d7668d793ff4c6ec424fb493cdab3ef5b625eefffe775
   71ff28a345e5f700a

   // Signer round one outputs
   S1 hiding_nonce: 570f27bfd808ade115a701eeee997a488662bca8c2a073143e66
   2318f1ed8308
   S1 binding_nonce: 6720f0436bd135fe8dddc3fadd6e0d13dbd58a1981e587d377d
   48e0b8f1c3c01
   S1 hiding_nonce_commitment: 78e175d15cb5cec1257e0d84d797ba8c3dd9b4c7b

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   c50f3fa527c200bcc6c4a95
   S1 binding_nonce_commitment: 4cdad16ae67ac5919159d655b681bd038574383b
   ab423614f8967396ee12ca62
   S2 hiding_nonce: 2a67c5e85884d0275a7a740ba8f53617527148418797345071dd
   cf1a1bd37206
   S2 binding_nonce: a0609158eeb448abe5b0df27f5ece96196df5722c01a999e8a4
   5d2d5dfc5620c
   S2 hiding_nonce_commitment: 88a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea
   98981bfb25375996c5767c9
   S2 binding_nonce_commitment: 32bbf10c41feb17d41cc6433e69f16cceccc42a0
   0aedf72feb5f44929fdf2e2f

   // Round two parameters
   participants: 1,2

   // Signer round two outputs
   S1 sig_share: b7e8f03a1a1149adacb96f952dbc39b6034facceafe4a70d6963592
   fce75570c
   S2 sig_share: cd388f9aff4376397c5ad231713fe6b167bed9cc88a1cc97b0b6bbe
   0316a7909

   sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e
   974d8a78fff1ac8e52774a24c00141536b0d869b388674a5191a151000e0d005

D.2.  FROST(Ed448, SHAKE256)

   // Configuration information
   MAX_SIGNERS: 3
   THRESHOLD_LIMIT: 2
   NUM_SIGNERS: 2

   // Group input parameters
   group_secret_key: cdf4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35
   4a9a3af62b8c7c89753055949cab8fd044c17c94211f167672b053659420b00
   group_public_key: 800e9b495543b04aaebdba2813de65d1aefe78e8b219d38966b
   c0afa1d5d9d685c740c8ab720bff3c84cd9f4a701c1588e40d981f4abb19600
   message: 74657374

   // Signer input parameters
   S1 signer_share: d208a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08
   175d4298dd6bbe2e1c0b29aaa972c78555ea3b3d7308b248994780219e0800
   S2 signer_share: d71c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc
   8416d5cd0210b4c5e410f90a892c91065fbdae37d51ffc29078acae9f90500
   S3 signer_share: dc3094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f
   f2cf670328b4a95cad16c96b68e65a87689021323737460b75cc14b2550300

   // Round one parameters
   participants: 1,2

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   group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b
   6841aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c8
   9a870400a0c29f750605b10c52e347fc538af0d4ebddd23a1e0300482a7d98a39d408
   356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c1753a80000242c2fdc11e5f
   726d4c897ed118f668a27bfb0d5946b5f513e975638b7c4b0a46cf5184d4a9c1f6310
   fd3c10f84d9de704a33aab2af976d60804fa4ecba88458bcf7677a3952f540e20556d
   5e90d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e
   1b0b28a80b54ff7255705a71ee2925e4a3e30e41aed489a579d5595e0df13e32e1e4d
   d202a7c7f68b31d6418d9845eb4d757adda6ab189e1bb340db818e5b3bc725d992faf
   63e9b0500db10517fe09d3f566fba3a80e46a403e0c7d41548fbf75cf2662b00225b5
   02961f98d8c9ff937de0b24c231845
   group_binding_factor: 2716e157c3da80b65149b1c2cb546723516272ccf75e111
   334533e2840a9bf85f3c71478ade11be26d26d8e4b9a1667af88f7df61670f60a00

   // Signer round one outputs
   S1 hiding_nonce: 04eccfe12348a5a2e4b30e95efcf4e494ce64b89f6504de46b3d
   67a5341baaa931e455c57c6c5c81f4895e333da9d71f7d119fcfbd0d7d2000
   S1 binding_nonce: 80bcd1b09e82d7d2ff6dd433b0f81e012cadd4661011c44d929
   1269cf24820f5c5086d4363dc67450f24ebe560eb4c2059883545d54aa43a00
   S1 hiding_nonce_commitment: 6d8ef55145bab18c129311f1d07bef2110d0b6841
   aae919eb6abf5e523d26f819d3695d78f8aa246c6b6d6fd6c2b8a63dd1cf8e8c89a87
   0400
   S1 binding_nonce_commitment: a0c29f750605b10c52e347fc538af0d4ebddd23a
   1e0300482a7d98a39d408356b9041d5fbaa274c2dc3f248601f21cee912e2f5700c17
   53a80
   S2 hiding_nonce: 3b3bbe82babf2a67ded81b308ba45f73b88f6cf3f6aaa4442256
   b7a0354d1567478cfde0a2bba98ba4c3e65645e1b77386eb4063f925e00700
   S2 binding_nonce: bcbd112a88bebf463e3509076c5ef280304cb4f1b3a7499cca1
   d5e282cc2010a92ff56a3bdcf5ba352e0f4241ba2e54c1431a895c19fff0600
   S2 hiding_nonce_commitment: 42c2fdc11e5f726d4c897ed118f668a27bfb0d594
   6b5f513e975638b7c4b0a46cf5184d4a9c1f6310fd3c10f84d9de704a33aab2af976d
   6080
   S2 binding_nonce_commitment: 4fa4ecba88458bcf7677a3952f540e20556d5e90
   d5aa7e8f226d303ef7b88fb33a63f6cac6a9d638089b1739a5d2564d15fb3e43e1b0b
   28a80

   // Round two parameters
   participants: 1,2

   // Signer round two outputs
   S1 sig_share: c5ab0a80c561d1a616ac70f4f13d993156f65f2b44a4a90f37f0640
   7a1b62e3940bf14199301d128358b812bef32cb4bffaf03030238772000
   S2 sig_share: 15211cb96d6aa73de803d46caf2043859fd796a6282f9adb00033f1
   4f4827f23f8cc792c2e322a1f30631ec7690ac587e5eb9c2afd323e3300

   sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353
   37e4d1aae5fd1c73a55701133238602f423886fc134a3c6580e787ce8da00900c1a92
   07fd32e9c6f956597202323f8f4264ecfd99e9539ae5c388c8e45c133fb4765ee9ff2

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   583d90d3e49ba02dff6ab51300

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

   // Configuration information
   MAX_SIGNERS: 3
   THRESHOLD_LIMIT: 2
   NUM_SIGNERS: 2

   // Group input parameters
   group_secret_key: b020be204b5e758960458ca9c4675b56b12a8faff2be9c94891
   d5e1cd75c880e
   group_public_key: e22ac4850672021eac8e0a36dfc4811466fb01108c3427d2347
   827467ba02a34
   message: 74657374

   // Signer input parameters
   S1 signer_share: 92ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f
   4693ddcb6705
   S2 signer_share: 611003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521
   2e0ae43a470c
   S3 signer_share: 439eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22
   1681eaa92603

   // Round one parameters
   participants: 1,2
   group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca
   094f65b814a8bca6013d9cc312774c7e1271d2939a84a9a867e3a06579b4d25659b42
   7439ccf0d745b43f75b76600028013834ff4d48e7d6b76c2e732bc611f54720ef8933
   c4ca4de7eaaa77ff5cd125e056ecc4f7c4657d3a742354430d768f945db229c335d25
   8e9622ad99f3e7582d07b35bd9849ce4af6ad403090d69a7d0eb88bba669a9f985175
   d70cd15ad5f1ef5b734c98a32b4aab7b43a57e93fc09281f2e7a207076b31e416ba63
   f53d9d
   group_binding_factor: f00ae6007f2d74a1507c962cf30006be77596106db28f2d
   5443fd66d755e780c

   // Signer round one outputs
   S1 hiding_nonce: 349b3bb8464a1d87f7d6b56f4559a3f9a6335261a3266089a9b1
   2d9d6f6ce209
   S1 binding_nonce: ce7406016a854be4291f03e7d24fe30e77994c3465de031515a
   4c116f22ca901
   S1 hiding_nonce_commitment: 824e9eddddf02b2a9caf5859825e999d791ca094f
   65b814a8bca6013d9cc3127
   S1 binding_nonce_commitment: 74c7e1271d2939a84a9a867e3a06579b4d25659b
   427439ccf0d745b43f75b766
   S2 hiding_nonce: 4d66d319f20a728ec3d491cbf260cc6be687bd87cc2b5fdb4d5f
   528f65fd650d
   S2 binding_nonce: 278b9b1e04632e6af3f1a3c144d07922ffcf5efd3a341b47abc

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   19c43f48ce306
   S2 hiding_nonce_commitment: 8013834ff4d48e7d6b76c2e732bc611f54720ef89
   33c4ca4de7eaaa77ff5cd12
   S2 binding_nonce_commitment: 5e056ecc4f7c4657d3a742354430d768f945db22
   9c335d258e9622ad99f3e758

   // Round two parameters
   participants: 1,2

   // Signer round two outputs
   S1 sig_share: 6a539c3a4ee281879a6fb350d20d53e17473f28cd3409ffc238dafe
   8d9330605
   S2 sig_share: 1d4e59636ee089bfaf548834b07658216649a37f87f0818d5190aa9
   b90957505

   sig: 7e92309bf40993141acd5f2c7680a302cc5aa5dd291a833906da8e35bc39b03e
   87a1f59dbcc20b474ac43b858284ab02dbbc950c5b31218a751d5a846ac97b0a

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

   // Configuration information
   MAX_SIGNERS: 3
   THRESHOLD_LIMIT: 2
   NUM_SIGNERS: 2

   // Group input parameters
   group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0
   3a6a51a5a5928
   group_public_key: 033a2a83f9c9fdfdab7d620f48238a5e6157a8eb1d6c382c7b0
   ba95b7c9f69679c
   message: 74657374

   // Signer input parameters
   S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678
   6f1e410dcecd
   S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed
   379767c14472
   S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62
   00108e74ba17

   // Round one parameters
   participants: 1,2
   group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7
   f702906930766b86c462bb8ff7f3402b724640ea9e262469f401c9006991ba3247c2c
   91b97cdb1f0eeab1a777e24e1e0002037f8a998dfc2e60a7ad63bc987cb27b8abf78a
   68bd924ec6adb9f251850cbe711024a4e90422a19dd8463214e997042206c39d3df56
   168b458592462090c89dbcf84efca0c54f70a585d6aae28679482b4aed03ae5d38297
   b9092ab3376d46fdf55

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   group_binding_factor: 9df349a9f34bf01627f6b4f8b376e8c8261d55508d1cac2
   919cdaf7f9cb20e70

   // Signer round one outputs
   S1 hiding_nonce: 3da92a503cf7e3f72f62dabedbb3ffcc9f555f1c1e78527940fe
   3fed6d45e56f
   S1 binding_nonce: ec97c41fc77ae7e795067976b2edd8b679f792abb062e4d0c33
   f0f37d2e363eb
   S1 hiding_nonce_commitment: 02f34caab210d59324e12ba41f0802d9545f7f702
   906930766b86c462bb8ff7f34
   S1 binding_nonce_commitment: 02b724640ea9e262469f401c9006991ba3247c2c
   91b97cdb1f0eeab1a777e24e1e
   S2 hiding_nonce: 06cb4425031e695d1f8ac61320717d63918d3edc7a02fcd3f23a
   de47532b1fd9
   S2 binding_nonce: 2d965a4ea73115b8065c98c1d95c7085db247168012a834d828
   5a7c02f11e3e0
   S2 hiding_nonce_commitment: 037f8a998dfc2e60a7ad63bc987cb27b8abf78a68
   bd924ec6adb9f251850cbe711
   S2 binding_nonce_commitment: 024a4e90422a19dd8463214e997042206c39d3df
   56168b458592462090c89dbcf8

   // Round two parameters
   participants: 1,2

   // Signer round two outputs
   S1 sig_share: 0a658fe198caddf5ddc407ad58c4615458f02a58d0c1f7a38e25692
   98dc41df0
   S2 sig_share: e84d948cfec74b5e7540ad09fd69dcd1570f708f2d8573dbbf08cb0
   2bc872c75

   sig: 035cfbd148da711bbc823455b682ed01a1be3c5415cf692f4a91b7fe22d1dec3
   45f2b3246e979229545304b4b7562e3e25afff9ae7fe476b7f4d2e342c4a4b4a65

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

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

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