Internet-Draft FROST February 2022
Connolly, et al. Expires 16 August 2022 [Page]
Intended Status:
D. Connolly
Zcash Foundation
C. Komlo
University of Waterloo, Zcash Foundation
I. Goldberg
University of Waterloo
C. A. Wood

Two-Round Threshold Schnorr Signatures with FROST


In this draft, we present a 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.

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Table of Contents

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 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.

In this draft, we present a variant of FROST, a Flexible Round-Optimized Schnorr Threshold signature scheme. 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.

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, is insecure in a multi-party signature setting.

Further, this draft implements signing efficiency improvements for FROST described by Crites, Komlo, and Maller in [Schnorr21].

1.1. Change Log


  • 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.


  • Specify operations, notation and cryptographic dependencies.


  • 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.

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 depends on the following cryptographic constructs:

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 =.

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 non-zero element in GF(p).
  • SerializeElement(A): A member function of G that maps a group element A to a unique byte array buf of fixed length Ne. The output type of this function is SerializedElement.
  • DeserializeElement(buf): A member function of G that maps a byte array buf to a group element A, or fails if the input is not a valid byte representation of an element. This function can raise a DeserializeError if deserialization fails or A is the identity element of the group; see Section 3.1.1.

3.1.1. Input Validation

The DeserializeElement function recovers a group element from an arbitrary byte array. This function validates that the element is a proper member of the group and is not the identity element, and returns an error if either condition is not met.

For ristretto255, elements are deserialized by invoking the Decode function from [RISTRETTO], Section 4.3.1, which returns false if the element is invalid. If this function returns false, deserialization returns an error.

The DeserializeScalar function recovers a scalar field element from an arbitrary byte array. Like DeserializeElement, this function validates that the element is a member of the scalar field and returns an error if this condition is not met.

For ristretto255, this function ensures that the input, when treated as a little-endian integer, is a value greater than or equal to 0, and less than Order().

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:

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):

  - msg, message to be signed, an octet string
  - SK, private key, a scalar
  - PK, public key, a group element

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

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

    msg_hash = H3(msg)
    comm_enc = G.SerializeElement(R)
    pk_enc = G.SerializeElement(PK)
    challenge_input = comm_enc || pk_enc || msg_hash
    c = H2(challenge_input)

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

The corresponding verification operation is as follows.

  schnorr_signature_verify(msg, sig, PK):

  - 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):
    msg_hash = H3(msg)
    comm_enc = G.SerializeElement(R)
    pk_enc = G.SerializeElement(PK)
    challenge_input = comm_enc || pk_enc || msg_hash
    c = H2(challenge_input)

    l = ScalarBaseMult(z)
    r = R + (c * PK)
    if l == r:
      return 1
    return 0

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.

  polynomial_evaluate(x, coeffs):

  - 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
        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.

  derive_lagrange_coefficient(x_i, L):

  - 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

  def derive_lagrange_coefficient(x_i, L):
    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.

  - 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:

    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. Encoding Operations

This section describes various helper functions used for encoding data structures into values that can be processed with hash functions.

  - 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 = I2OSP(index, 2) ||
                           G.SerializeElement(hiding_nonce_commitment) ||
      encoded_group_commitment = encoded_group_commitment || encoded_commitment
    return encoded_group_commitment

5. Two-Round FROST

FROST is a two-round 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.

In FROST, all signers are assumed to have the group state and their corresponding signing key shares. In particular, FROST assumes that each signing participant P_i knows the following:

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.

There are two rounds in FROST: commitment and signature share generation. The first round serves for each participant to issue a commitment. The second round receives commitments for all signers as well as the message, and issues a signature share. The Coordinator performs the coordination of each of these rounds. At the end of the second round, the Coordinator then performs an aggregation step at the end and outputs the final signature. This complete interaction 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
      == 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.

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.RandomScalar()
    binding_nonce = G.RandomScalar()
    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. 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. Upon receipt, each Signer then runs the following procedure to produce its own signature share.

  - 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.
  - comm_i, pair of Element values (hiding_nonce_commitment, binding_nonce_commitment) 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 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 signature share sig_share and commitment share comm_share, which
           are Scalar and Element values respectively.

  def sign(index, sk_i, group_public_key, nonce_i, comm_i, msg, commitment_list, participant_list):
    # Compute the binding factor
    encoded_commitments = encode_group_commitment_list(commitment_list)
    msg_hash = H3(msg)
    binding_factor = H1(encoded_commitments || msg_hash)

    # Compute the group commitment
    R = G.Identity()
    for (_, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list:
      R = R + (hiding_nonce_commitment + (binding_nonce_commitment * binding_factor))

    lambda_i = derive_lagrange_coefficient(index, participant_list)

    # Compute the per-message challenge
    group_comm_enc = G.SerializeElement(R)
    group_public_key_enc = G.SerializeElement(group_public_key)
    challenge_input = group_comm_enc || group_public_key_enc || msg_hash
    c = H2(challenge_input)

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

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

    return sig_share, comm_share

The output of this procedure is a signature share and group commitment share. Each signer then sends these shares back to the collector; see Appendix C.3 for encoding recommendations.

The Coordinator MUST verify the set of signature shares using the following procedure.

  - index, Index `i` of the signer. Note index will never equal `0`.
  - PK, the public key for the group
  - PK_i, the public key for the ith signer, where PK_i = ScalarBaseMult(s[i])
  - sig_share, the signature share for the ith signer, computed from the signer
  - comm_share, the commitment for the ith signer, computed from the signer
  - R, the group commitment
  - msg, the message to be signed
  - participant_list, a set containing identifiers for each signer, similarly of length
    NUM_SIGNERS (sent by the Coordinator).

  Outputs: 1 if the signature share is valid, and 0 otherwise.

  def verify_signature_share(index, PK, PK_i, sig_share, comm_share, R, msg, participant_list):
    msg_hash = H3(msg)
    group_comm_enc = G.SerializeElement(R)
    group_public_key_enc = G.SerializeElement(group_public_key)
    challenge_input = group_comm_enc || group_public_key_enc || msg_hash
    c = H2(challenge_input)

    l = G.ScalarbaseMult(sig_share)

    lambda_i = derive_lagrange_coefficient(index, participant_list)
    r = comm_share + (sig_share * c * lambda_i)

    return l == r

5.3. Signature Share Aggregation

After signers perform round two and send their signature shares to the Coordinator, the Coordinator performs the aggregate operation and publishes the resulting signature. Note that here we do not specify the Coordinator as validating each signature schare, as if any signature share is invalid, the resulting joint signature will similarly be invalid. Deployments that wish to validate signature shares can do so using the verify_signature_share function in Section 5.2

  - R: the group commitment.
  - sig_shares: a set of signature shares z_i for each signer, of length NUM_SIGNERS,

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

  def frost_aggregate(R, sig_shares):
    z = 0
    for z_i in sig_shares:
      z = z + z_i
    return (R, 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].

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]

  • 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.

6.3. 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]

  • 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 - 13818066809895115352007386748515426880336692474882178609894547503885.
    • 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 - 13818066809895115352007386748515426880336692474882178609894547503885.
    • 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]

    • 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 of [KEYAGREEMENT].
    • Serialization: Elements are serialized as Ne = 33 byte string
  • 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]. The protocol as specified in this document assumes the following threat model.

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

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 determinstic 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 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.

8. Contributors

9. References

9.1. Normative References

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-13, , <>.
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, , <>.
Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <>.
Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch, "PKCS #1: RSA Cryptography Specifications Version 2.2", RFC 8017, DOI 10.17487/RFC8017, , <>.
Josefsson, S. and I. Liusvaara, "Edwards-Curve Digital Signature Algorithm (EdDSA)", RFC 8032, DOI 10.17487/RFC8032, , <>.
Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <>.
Valence, H. D., Grigg, J., Tankersley, G., Valsorda, F., Lovecruft, I., and M. Hamburg, "The ristretto255 and decaf448 Groups", Work in Progress, Internet-Draft, draft-irtf-cfrg-ristretto255-decaf448-01, , <>.
"Elliptic Curve Cryptography, Standards for Efficient Cryptography Group, ver. 2", , <>.
ANSI, "Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA)", ANSI X9.62-1998, .

9.2. Informative References

Komlo, C. and I. Goldberg, "Two-Round Threshold Signatures with FROST", , <>.
Eastlake 3rd, D., Schiller, J., and S. Crocker, "Randomness Requirements for Security", BCP 106, RFC 4086, DOI 10.17487/RFC4086, , <>.
Crites, E., Komlo, C., and M. Maller, "How to Prove Schnorr Assuming Schnorr", , <>.

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.

  - n, the number of shares to generate, an integer
  - t, the threshold of the secret sharing scheme, an integer

  Outputs: a secret key Scalar, public key Element, along with `n`
  shares of the secret key, each a Scalar value.

  def trusted_dealer_keygen(n, t):
    secret_key = G.RandomScalar()
    secret_key_shares = secret_share_split(secret_key, n, t)
    public_key = G.ScalarBaseMult(s)
    return secret_key, public_key, secret_key_shares

It is assumed the dealer then sends one secret key to each of the NUM_SIGNERS participants, and afterwards deletes the secrets from their local device.

Use of this method for key generation requires a mutually authenticated secure channel between Coordinator and participants, 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.

  secret_share_split(s, n, t):

  - 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: A list of n secret shares, each of which is an element of F

  - "invalid parameters", if t > n

  def secret_share(s, n, t):
    if t > n:
      raise "invalid parameters"

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

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

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.

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


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

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

  - "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. If the validation fails, the participant can issue a complaint against the dealer, and take actions such as broadcasting this complaint to all other participants. We do not specify the complaint procedure in this draft, as it will be implementation-specific.

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


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

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

  def vss_commit(coeffs):
    C = []
    for coeff in coeffs:
      A_i = ScalarBaseMult(coeff)
    return C

The procedure for verification of a participant's share is as follows.

  vss_verify(sk_i, C):

  - sk_i: A participant's secret key, the tuple sk_i = (i, s[i]),
  where s[i] is a secret share of the constant term of f.
  - C: A VSS commitment to a secret polynomial f.

  Outputs: 1 if s[i] is valid, and 0 otherwise

  vss_verify(sk_i, commitment)
    S_i = ScalarBaseMult(s[i])
    S_i' = SUM(commitment[0], commitment[t-1]){A_j}: A_j*(i^j)
    if S_i == S_i':
      return 1
    return 0

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;

The SignerID.


The commitment hiding factor encoded as a serialized group element.


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;

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


The message to be signed.

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];
    opaque commitment_share[Ne];
  } SignatureShare;

The SignerID.


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.

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

// Configuration information

// Group input parameters
group_secret_key: 7c1c33d3f5291d85de664833beb1ad469f7fb6025a0ec78b3a7
group_public_key: 377a6acb3b9b5f642c5ce355d23cac0568aad0da63c633d59d4
message: 74657374

// Signer input parameters
S1 signer_share: 949dcc590407aae7d388761cddb0c0db6f5627aea8e217f4a033
S2 signer_share: ac1e66e012e4364ac9aaa405fcafd370402d9859f7b6685c07ee
S3 signer_share: d7cb090a075eb154e82fdb4b3cb507f110040905468bb9c46da8

// Round one parameters
participants: 1,2
group_binding_factor_input: 000178e175d15cb5cec1257e0d84d797ba8c3dd9b
group_binding_factor: c4d7668d793ff4c6ec424fb493cdab3ef5b625eefffe775

// Signer round one outputs
S1 hiding_nonce: 570f27bfd808ade115a701eeee997a488662bca8c2a073143e66
S1 binding_nonce: 6720f0436bd135fe8dddc3fadd6e0d13dbd58a1981e587d377d
S1 hiding_nonce_commitment: 78e175d15cb5cec1257e0d84d797ba8c3dd9b4c7b
S1 binding_nonce_commitment: 4cdad16ae67ac5919159d655b681bd038574383b
S2 hiding_nonce: 2a67c5e85884d0275a7a740ba8f53617527148418797345071dd
S2 binding_nonce: a0609158eeb448abe5b0df27f5ece96196df5722c01a999e8a4
S2 hiding_nonce_commitment: 88a4e6c3d8353dc3f4aca2e10d10a75fb98d9fbea
S2 binding_nonce_commitment: 32bbf10c41feb17d41cc6433e69f16cceccc42a0

// Round two parameters
participants: 1,2

// Signer round two outputs
S1 sig_share: f8bbaf924e1c90e11dec1eb679194aade084f92fbf52fdd436ba0a0
S1 group_commitment_share: 11bb9777aa393b92e814e415039adf62687a0be543
S2 sig_share: 80f589405714ca0e6adc87c2c0186a0ae4d6e352f7b248b23149a5d
S2 group_commitment_share: 4af85179d17ed031b767ab579e59c7018dac09ae40

sig: ebe7efbb42c4b1c55106b5536fb5e9ac7a6d0803ea4ae9c8c629ca51e05c230e

D.2. FROST(Ed448, SHAKE256)

// Configuration information

// Group input parameters
group_secret_key: cef4a803a21d82fa90692e86541e08d878c9f688e5d71a2bd35
group_public_key: 005f23508a78131aee4d6cb027f967d89557ec5f24dc3ebeede
message: 74657374

// Signer input parameters
S1 signer_share: d408a2f1d9ead0cc4b4b9b2e84a22f8e2aa2ab4ee715febe7a08
S2 signer_share: da1c9bdf11b81f9f062d08d7b3265744dc7a6014e953e15222bc
S3 signer_share: e03094cd49856e71c10e757fe3aa7efa8d5315daea91c4e6c96f

// Round one parameters
participants: 1,2
group_binding_factor_input: 00016d8ef55145bab18c129311f1d07bef2110d0b
group_binding_factor: 2716e157c3da80b65149b1c2cb546723516272ccf75e111

// Signer round one outputs
S1 hiding_nonce: 04eccfe12348a5a2e4b30e95efcf4e494ce64b89f6504de46b3d
S1 binding_nonce: 80bcd1b09e82d7d2ff6dd433b0f81e012cadd4661011c44d929
S1 hiding_nonce_commitment: 6d8ef55145bab18c129311f1d07bef2110d0b6841
S1 binding_nonce_commitment: a0c29f750605b10c52e347fc538af0d4ebddd23a
S2 hiding_nonce: 3b3bbe82babf2a67ded81b308ba45f73b88f6cf3f6aaa4442256
S2 binding_nonce: bcbd112a88bebf463e3509076c5ef280304cb4f1b3a7499cca1
S2 hiding_nonce_commitment: 42c2fdc11e5f726d4c897ed118f668a27bfb0d594
S2 binding_nonce_commitment: 4fa4ecba88458bcf7677a3952f540e20556d5e90

// Round two parameters
participants: 1,2

// Signer round two outputs
S1 sig_share: ad41cd3320c82edd20c344769bd7b250105d9d0516109b7f774c297
S1 group_commitment_share: 086d4d2ff2555fab65afc8eb473cc708f37cdb9c5d
S2 sig_share: 5dcc0aec7d0a71eddd5ba2dd0f185ba7990bcd39b6fc0e4b0470c35
S2 group_commitment_share: 7e91f66097b6450c52c89c14400a506ee1d37f5e52

sig: 4d9883057726b029d042418600abe88ad3fec06d6a48dca289482e9d51c10353

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

// Configuration information

// Group input parameters
group_secret_key: b120be204b5e758960458ca9c4675b56b12a8faff2be9c94891
group_public_key: 563b80013f337deaa2a282af7b281bd70d2f501928a89c1aa48
message: 74657374

// Signer input parameters
S1 signer_share: 94ae65bb90030a89507fa00fff08dfed841cf996de5a0c574f1f
S2 signer_share: 641003b3f00bb1e01656ac1818a4419a580e637ecaf67b191521
S3 signer_share: 479eaa4d36b145e00690c07e5245c5312c00cd65b692ebdbda22

// Round one parameters
participants: 1,2
group_binding_factor_input: 0001824e9eddddf02b2a9caf5859825e999d791ca
group_binding_factor: f49fbf1a092173b9338394b5818966480c0413c5f90e2a7

// Signer round one outputs
S1 hiding_nonce: 349b3bb8464a1d87f7d6b56f4559a3f9a6335261a3266089a9b1
S1 binding_nonce: ce7406016a854be4291f03e7d24fe30e77994c3465de031515a
S1 hiding_nonce_commitment: 824e9eddddf02b2a9caf5859825e999d791ca094f
S1 binding_nonce_commitment: 74c7e1271d2939a84a9a867e3a06579b4d25659b
S2 hiding_nonce: 4d66d319f20a728ec3d491cbf260cc6be687bd87cc2b5fdb4d5f
S2 binding_nonce: 278b9b1e04632e6af3f1a3c144d07922ffcf5efd3a341b47abc
S2 hiding_nonce_commitment: 8013834ff4d48e7d6b76c2e732bc611f54720ef89
S2 binding_nonce_commitment: 5e056ecc4f7c4657d3a742354430d768f945db22

// Round two parameters
participants: 1,2

// Signer round two outputs
S1 sig_share: 503c943f6af38f3ad2271de3d792c4756c5d9c483989e2005add0e8
S1 group_commitment_share: 40587ff15e9cba4fe601c4378b40ae4babe2ea6f0b
S2 sig_share: eee969c17354ec84933723b45ac9965d2874e21c3a50cd31648f708
S2 group_commitment_share: 70661957faf398410a9da3e20cfd6f1233b368548e

sig: 90ac875023a311624948b2660bc5524f690ae9e14fcd6541959bee2e868d8c32

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

// Configuration information

// Group input parameters
group_secret_key: 6f090d1393ff53bbcbba036c00b8830ab4546c251dece199eb0
group_public_key: 03db0945167b62e6472ad46373b6cbbca88e2a9a4883071f0b3
message: 74657374

// Signer input parameters
S1 signer_share: 738552e18ea4f2090597aca6c23c1666845c21c676813f9e2678
S2 signer_share: 780198af894a90563f7555e183bfa9c25463d767cf159da261ed
S3 signer_share: 7c7dde7d83f02ea37952ff1c45433d1e246b8d0927a9fba69d62

// Round one parameters
participants: 1,2
group_binding_factor_input: 000102f34caab210d59324e12ba41f0802d9545f7
group_binding_factor: 9c649ba6084d89db49dafb28f4b50fda14fd8263b3fe907

// Signer round one outputs
S1 hiding_nonce: 3da92a503cf7e3f72f62dabedbb3ffcc9f555f1c1e78527940fe
S1 binding_nonce: ec97c41fc77ae7e795067976b2edd8b679f792abb062e4d0c33
S1 hiding_nonce_commitment: 02f34caab210d59324e12ba41f0802d9545f7f702
S1 binding_nonce_commitment: 02b724640ea9e262469f401c9006991ba3247c2c
S2 hiding_nonce: 06cb4425031e695d1f8ac61320717d63918d3edc7a02fcd3f23a
S2 binding_nonce: 2d965a4ea73115b8065c98c1d95c7085db247168012a834d828
S2 hiding_nonce_commitment: 037f8a998dfc2e60a7ad63bc987cb27b8abf78a68
S2 binding_nonce_commitment: 024a4e90422a19dd8463214e997042206c39d3df

// Round two parameters
participants: 1,2

// Signer round two outputs
S1 sig_share: 0400c1ba4351343192777323e20847b4fd9067af35e0261f6af0413
S1 group_commitment_share: 03208d89db28e4946b02c9942a5245140014c930d8
S2 sig_share: 54d1dbea644a643ca91948398c40f20f12a00f15075b9614095ecfb
S2 group_commitment_share: 026842e4516527c3f47b9c5300d231ae0b61a13ede

sig: 03f8bda19543758dab107009bd44f2bfb0f6192c8fd1d0eded6bdbee3169a78e

Authors' Addresses

Deirdre Connolly
Zcash Foundation
Chelsea Komlo
University of Waterloo, Zcash Foundation
Ian Goldberg
University of Waterloo
Christopher A. Wood