Two-Round Threshold Schnorr Signatures with FROST Zcash Foundation
durumcrustulum@gmail.com
University of Waterloo, Zcash Foundation
ckomlo@uwaterloo.ca
University of Waterloo
iang@uwaterloo.ca
Cloudflare
caw@heapingbits.net
General CFRG Internet-Draft 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 . However, unlike , the protocol for producing signatures in this draft is not deterministic, so as to ensure protection against a key-recovery attack that is possible when even only one participant is malicious. Discussion Venues Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at . Source for this draft and an issue tracker can be found at .
Introduction 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 . 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 . Single-round signing with FROST is out of scope. For select ciphersuites, the signatures produced by this draft are compatible with . However, unlike , signatures produced by FROST are not deterministic, since deriving nonces deterministically allows for a complete key-recovery attack in multi-party discrete logarithm-based signatures, such as FROST. Key generation for FROST signing is out of scope for this document. However, for completeness, key generation with a trusted dealer is specified in .
Change Log draft-05
  • Update test vectors to include version string (#202, #203)
  • Rename THRESHOLD_LIMIT to MIN_SIGNERS (#192)
  • Use non-contiguous signers for the test vectors (#187)
  • Add more reasoning why the coordinator MUST abort (#183)
  • Add a function to generate nonces (#182)
  • Add MUST that all participants have the same view of VSS commitment (#174)
  • Use THRESHOLD_LIMIT instead of t and MAX_SIGNERS instead of n (#171)
  • Specify what the dealer is trusted to do (#166)
  • Clarify types of NUM_SIGNERS and THRESHOLD_LIMIT (#165)
  • Assert that the network channel used for signing should be authenticated (#163)
  • Remove wire format section (#156)
  • Update group commitment derivation to have a single scalarmul (#150)
  • Use RandomNonzeroScalar for single-party Schnorr example (#148)
  • Fix group notation and clarify member functions (#145)
  • Update existing implementations table (#136)
  • Various editorial improvements (#135, #143, #147, #149, #153, #158, #162, #167, #168, #169, #170, #175, #176, #177, #178, #184, #186, #193, #198, #199)
draft-04
  • Added methods to verify VSS commitments and derive group info (#126, #132).
  • Changed check for participants to consider only nonnegative numbers (#133).
  • Changed sampling for secrets and coefficients to allow the zero element (#130).
  • Split test vectors into separate files (#129)
  • Update wire structs to remove commitment shares where not necessary (#128)
  • Add failure checks (#127)
  • Update group info to include each participant's key and clarify how public key material is obtained (#120, #121).
  • Define cofactor checks for verification (#118)
  • Various editorial improvements and add contributors (#124, #123, #119, #116, #113, #109)
draft-03
  • Refactor the second round to use state from the first round (#94).
  • Ensure that verification of signature shares from the second round uses commitments from the first round (#94).
  • Clarify RFC8032 interoperability based on PureEdDSA (#86).
  • Specify signature serialization based on element and scalar serialization (#85).
  • Fix hash function domain separation formatting (#83).
  • Make trusted dealer key generation deterministic (#104).
  • Add additional constraints on participant indexes and nonce usage (#105, #103, #98, #97).
  • Apply various editorial improvements.
draft-02
  • Fully specify both rounds of FROST, as well as trusted dealer key generation.
  • Add ciphersuites and corresponding test vectors, including suites for RFC8032 compatibility.
  • Refactor document for editorial clarity.
draft-01
  • Specify operations, notation and cryptographic dependencies.
draft-00
  • Outline CFRG draft based on draft-komlo-frost.
Conventions and Definitions The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 when, and only when, they appear in all capitals, as shown here. The following notation and terminology are used throughout this document.
  • A participant is an entity that is trusted to hold a secret share.
  • MAX_SIGNERS denotes the number of participants, and the number of shares that s is split into. This value MUST NOT exceed 2^16-1.
  • MIN_SIGNERS denotes the threshold number of participants required to issue a signature. More specifically, at least MIN_SIGNERS shares must be combined to issue a valid signature. This value MUST NOT exceed p.
  • NUM_SIGNERS denotes the number of signers that participate in an invocation of FROST signing, where MIN_SIGNERS <= NUM_SIGNERS <= MAX_SIGNERS. This value MUST NOT exceed p.
  • 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].
  • random_bytes(n): Outputs n bytes, sampled uniformly at random using a cryptographically secure pseudorandom number generator (CSPRNG).
  • || denotes concatenation, i.e., x || y = xy.
  • nil denotes an empty byte string.
Unless otherwise stated, we assume that secrets are sampled uniformly at random using a cryptographically secure pseudorandom number generator (CSPRNG); see for additional guidance on the generation of random numbers.
Cryptographic Dependencies FROST signing depends on the following cryptographic constructs:
  • Prime-order Group, ;
  • Cryptographic hash function, ;
These are described in the following sections.
Prime-Order Group FROST depends on an abelian group of prime order p. We represent this group as the object G that additionally defines helper functions described below. The group operation for G is addition + with identity element I. For any elements A and B of the group G, A + B = B + A is also a member of G. Also, for any A in G, there exists an element -A such that A + (-A) = (-A) + A = I. For convenience, we use - to denote the subtraction, e.g., A - B = A + (-B). 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 a fixed generator of the group. Scalar base multiplication is equivalent to the repeated application of the group operation B with itself r-1 times, this is denoted as ScalarBaseMult(r). The set of scalars corresponds to GF(p), which 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 G.
  • Order(): Outputs the order of G (i.e. p).
  • Identity(): Outputs the identity Element of the group (i.e. I).
  • RandomScalar(): Outputs a random Scalar element in GF(p).
  • RandomNonzeroScalar(): Outputs a random non-zero Scalar element in GF(p).
  • SerializeElement(A): Maps an Element A to a unique byte array buf of fixed length Ne.
  • DeserializeElement(buf): 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 for group-specific input validation steps.
  • SerializeScalar(s): Maps a Scalar s to a unique byte array buf of fixed length Ns.
  • DeserializeScalar(buf): Attempts to map a byte array buf to a Scalar s. This function can raise a DeserializeError if deserialization fails; see for group-specific input validation steps.
Cryptographic Hash Function FROST requires the use of a cryptographically secure hash function, generically written as H, which functions effectively as a random oracle. For concrete recommendations on hash functions which SHOULD be used in practice, see . Using H, we introduce three separate domain-separated hashes, H1, H2, H3, and H4, where H1, H2, and H4 map arbitrary byte strings to Scalar elements of the prime-order group scalar field, and H3 is an alias for H with a domain separator. The details of H1, H2, H3, and H4 vary based on ciphersuite. See for more details about each.
Helper functions Beyond the core dependencies, the protocol in this document depends on the following helper operations:
  • Nonce generation,
  • Schnorr signatures, ;
  • Polynomial operations, ;
  • Encoding operations, ;
  • Signature binding , group commitment , and challenge computation
These sections describes these operations in more detail.
Nonce generation To hedge against a bad RNG, we generate nonces by sourcing fresh randomness and combine with the secret key, to create a domain-separated hash function from the ciphersuite hash function H, H4:
Schnorr Signature Operations In the single-party setting, a Schnorr signature is generated with the following operation. The corresponding verification operation is as follows. By definition, this function assumes that all group Elements passed as input, including the signature R component and public key, belong to prime-order group G. Some ciphersuites defined in operate in settings where some inputs may not belong to this prime-order group, e.g., because they operate over an elliptic curve with a cofactor larger than 1. In these settings, input validation is required before invoking this verification function.
Polynomial Operations This section describes operations on and associated with polynomials over Scalars that are used in the main signing protocol. A polynomial of degree t is represented as a list of t coefficients, where the constant term of the polynomial is in the first position and the highest-degree coefficient is in the last position. A point on the polynomial is a tuple (x, y), where y = f(x). For notational convenience, we refer to the x-coordinate and y-coordinate of a point p as p.x and p.y, respectively.
Evaluation of a 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.
Lagrange coefficients The function derive_lagrange_coefficient derives a Lagrange coefficient to later perform polynomial interpolation, and is provided a set of x-coordinates as input. Lagrange coefficients are used in FROST to evaluate a polynomial f at point 0, i.e., f(0), given a set of t other points.
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-1. Recovering the constant term occurs with a set of t points using polynomial interpolation, defined as follows. Note that these coefficients can be computed once and then stored, as these values remain constant across FROST signing sessions.
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.
Binding Factor Computation This section describes the subroutine for computing the binding factor based on the signer commitment list and message to be signed.
Group Commitment Computation This section describes the subroutine for creating the group commitment from a commitment list.
Signature Challenge Computation This section describes the subroutine for creating the per-message challenge.
Two-Round FROST Signing Protocol 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 an entity with the following responsibilities:
  1. Determining which signers will participate (at least MIN_SIGNERS 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 Coordinator and set of signers, are all chosen external to the protocol. Note that it is possible to deploy the protocol without a distinguished Coordinator; see 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, an Element in G, denoted PK = G.ScalarMultBase(s), corresponding to the group secret key s, which is a Scalar. PK is an output from the group's key generation protocol, such as trusted_dealer_keygenor a DKG.
  • Public keys for each signer, denoted PK_i = G.ScalarMultBase(), which are similarly outputs from the group's key generation protocol, Elements in G.
And that each participant with identifier i where i is an integer in the range between 1 and MAX_SIGNERS additionally knows the following:
  • Participant is signing key share sk_i, which is the i-th secret share of s, a Scalar.
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 , 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 .
FROST signature overview | == Round 1 (Commitment) == | signer commitment | | |<-----------------------+ | | ... | | signer commitment | |<-----------------------------------------+ == Round 2 (Signature Share Generation) == | | signer input | | +------------------------> | | signature share | | |<-----------------------+ | | ... | | signer input | +------------------------------------------> | signature share | <------------------------------------------+ | == Aggregation == | signature | <-----------+ ]]>
Details for round one are described in , and details for round two are described in . The final Aggregation step is described in . FROST assumes that all inputs to each round, especially those of which are received over the network, are validated before use. In particular, this means that any value of type Element or Scalar is deserialized using DeserializeElement and DeserializeScalar, respectively, as these functions perform the necessary input validation steps. FROST assumes reliable message delivery between the Coordinator and signing participants in order for the protocol to complete. An attacker masquerading as another participant will result only in an invalid signature; see . However, in order to identify any participant which has misbehaved (resulting in the protocol aborting) to take actions such as excluding them from future signing operations, we assume that the network channel is additionally authenticated; confidentiality is not required.
Round One - Commitment Round one involves each signer generating nonces and their corresponding public commitments. A nonce is a pair of Scalar values, and a commitment is a pair of Element values. Each signer in round one generates a nonce nonce = (hiding_nonce, binding_nonce) and commitment comm = (hiding_nonce_commitment, binding_nonce_commitment). 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.
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 MIN_SIGNERS). 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 all 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 . 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 for more details. Upon receipt and successful input validation, each Signer then runs the following procedure to produce its own signature share. The output of this procedure is a signature share. Each signer then sends these shares back to the Coordinator. Each signer MUST delete the nonce and corresponding commitment after this round completes. Upon receipt from each Signer, the Coordinator MUST validate the input signature share using DeserializeElement. If validation fails, the Coordinator MUST abort the protocol. If validation succeeds, the Coordinator then verifies the set of signature shares using the following procedure.
Signature Share Verification and 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. 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 for correctness reasons. Excluding one signer means that their nonce will not be included in the joint response z and consequently the output signature will not verify. Otherwise, if all signer shares are valid, the Coordinator performs the aggregate operation and publishes the resulting signature. The output signature (R, z) from the aggregation step MUST be encoded as follows (using notation from ): Where Signature.R_encoded is G.SerializeElement(R) and Signature.z_encoded is G.SerializeScalar(z).
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) . The (Ed25519, SHA-512) ciphersuite is included for backwards compatibility with . The DeserializeElement and DeserializeScalar functions instantiated for a particular prime-order group corresponding to a ciphersuite MUST adhere to the description in . Validation steps for these functions are described for each the ciphersuites below. Future ciphersuites MUST describe how input validation is done for DeserializeElement and DeserializeScalar.
FROST(Ed25519, SHA-512) This ciphersuite uses edwards25519 for the Group and SHA-512 for the Hash function H meant to produce signatures indistinguishable from Ed25519 as specified in . The value of the contextString parameter is empty.
  • Group: edwards25519
    • Order: 2^252 + 27742317777372353535851937790883648493 (see )
    • Identity: As defined in .
    • RandomScalar: Implemented by generating a random 32-byte string and invoking DeserializeScalar on the result.
    • RandomNonZeroScalar: Implemented by generating a random 32-byte string that is not equal to the all-zero string and invoking DeserializeScalar on the result.
    • SerializeElement: Implemented as specified in .
    • DeserializeElement: Implemented as specified in . 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).
    • H4(m): Implemented by computing H("nonce" || m), interpreting the lower 32 bytes as a little-endian integer, and reducing the resulting integer modulo L = 2^252+27742317777372353535851937790883648493.
Normally H2 would also include a domain separator, but for backwards compatibility with , it is omitted.
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-v5".
  • Group: ristretto255
    • Order: 2^252 + 27742317777372353535851937790883648493 (see )
    • Identity: As defined in .
    • RandomScalar: Implemented by generating a random 32-byte string and invoking DeserializeScalar on the result.
    • RandomNonZeroScalar: Implemented by generating a random 32-byte string that is not equal to the all-zero string and invoking DeserializeScalar on the result.
    • SerializeElement: Implemented using the 'Encode' function from .
    • DeserializeElement: Implemented using the 'Decode' function from .
    • 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 .
    • H2(m): Implemented by computing H(contextString || "chal" || m) and mapping the output to a Scalar as described in .
    • H3(m): Implemented by computing H(contextString || "digest" || m).
    • H4(m): Implemented by computing H(contextString || "nonce" || m) and mapping the output to a Scalar as described in .
FROST(Ed448, SHAKE256) This ciphersuite uses edwards448 for the Group and SHAKE256 for the Hash function H meant to produce signatures indistinguishable from Ed448 as specified in . The value of the contextString parameter is empty.
  • Group: edwards448
    • Order: 2^446 - 13818066809895115352007386748515426880336692474882178609894547503885
    • Identity: As defined in .
    • RandomScalar: Implemented by generating a random 48-byte string and invoking DeserializeScalar on the result.
    • RandomNonZeroScalar: Implemented by generating a random 48-byte string that is not equal to the all-zero string and invoking DeserializeScalar on the result.
    • SerializeElement: Implemented as specified in .
    • DeserializeElement: Implemented as specified in . Additionally, this function validates that the resulting element is not the group identity element.
    • 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 - 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).
    • H4(m): Implemented by computing H("nonce" || m), interpreting the lower 57 bytes as a little-endian integer, and reducing the resulting integer modulo L = 2^446 - 13818066809895115352007386748515426880336692474882178609894547503885.
Normally H2 would also include a domain separator, but for backwards compatibility with , it is omitted.
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-v5".
  • Group: P-256 (secp256r1)
    • Order: 0xffffffff00000000ffffffffffffffffbce6faada7179e84f3b9cac2fc632551
    • Identity: As defined in .
    • RandomScalar: Implemented by generating a random 32-byte string and invoking DeserializeScalar on the result.
    • RandomNonZeroScalar: Implemented by generating a random 32-byte string that is not equal to the all-zero string and invoking DeserializeScalar on the result.
    • SerializeElement: Implemented using the compressed Elliptic-Curve-Point-to-Octet-String method according to .
    • DeserializeElement: Implemented by attempting to deserialize a public key using the compressed Octet-String-to-Elliptic-Curve-Point method according to , and then performs partial public-key validation as defined in section 5.6.2.3.4 of . This includes checking that the coordinates of the resulting point are in the correct range, that the point is on the curve, and that the point is not the point at infinity. Additionally, this function validates that the resulting element is not the group identity element. If these checks fail, deserialization returns an error.
    • SerializeScalar: Implemented using the Field-Element-to-Octet-String conversion according to .
    • DeserializeScalar: Implemented by attempting to deserialize a Scalar from a 32-byte string using Octet-String-to-Field-Element from . This function can fail if the input does not represent a Scalar between the value 0 and G.Order() - 1.
  • Hash (H): SHA-256, and Nh = 32.
    • H1(m): Implemented using hash_to_field from 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 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).
    • H4(m): Implemented using hash_to_field from using L = 48, expand_message_xmd with SHA-256, DST = contextString || "nonce", and prime modulus equal to Order().
Security Considerations A security analysis of FROST exists in and . The protocol as specified in this document assumes the following threat model.
  • Trusted dealer. The dealer that performs key generation is trusted to follow the protocol, although participants still are able to verify the consistency of their shares via a VSS (verifiable secret sharing) step; see .
  • Unforgeability assuming at most (MIN_SIGNERS-1) corrupted signers. So long as an adversary corrupts fewer than MIN_SIGNERS participants, the scheme remains secure against Existential Unforgeability Under Chosen Message Attack (EUF-CMA) attacks, as defined in , Definition 13.2.
  • Coordinator. We assume the Coordinator at the time of signing does not perform a denial of service attack. A denial of service would include any action which either prevents the protocol from completing or causing the resulting signature to be invalid. Such actions for the latter include sending inconsistent values to 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:
  • Post quantum security. FROST, like plain Schnorr signatures, requires the hardness of the Discrete Logarithm Problem.
  • Robustness. In the case of failure, FROST requires aborting the protocol.
  • Downgrade prevention. All participants in the protocol are assumed to agree on what algorithms to use.
  • Metadata protection. If protection for metadata is desired, a higher-level communication channel can be used to facilitate key generation and signing.
The rest of this section documents issues particular to implementations or deployments.
Nonce Reuse Attacks 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. The Coordinator MAY further hedge against nonce reuse attacks by tracking signer nonce commitments used for a given group key, at the cost of additional state.
Protocol Failures We do not specify what implementations should do when the protocol fails, other than requiring that the protocol abort. Examples of viable failure include when a verification check returns invalid or if the underlying transport failed to deliver the required messages.
Removing the Coordinator Role In some settings, it may be desirable to omit the role of the Coordinator entirely. Doing so does not change the security implications of FROST, but instead simply requires each participant to communicate with all other participants. We loosely describe how to perform FROST signing among 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 commit() as is done in the setting where a Coordinator is used. However, instead of sending the commitment to the Coordinator, every participant instead will publish this commitment to every other participant. Then, in the second round, signers will already have sufficient information to perform signing. They will directly perform sign. All participants will then publish their signature shares to one another. After having received all signature shares from all other signers, each signer will then perform verify_signature_share and then aggregate directly. The requirements for the underlying network channel remain the same in the setting where all participants play the role of the Coordinator, in that all messages that are exchanged are public and so the channel simply must be reliable. However, in the setting that a player attempts to split the view of all other players by sending disjoint values to a subset of players, the signing operation will output an invalid signature. To avoid this denial of service, implementations may wish to define a mechanism where messages are authenticated, so that cheating players can be identified and excluded.
Input Message 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 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.
Contributors
  • Isis Lovecruft
  • T. Wilson-Brown
  • Alden Torres
 References Normative References Public Key Cryptography for the Financial Services Industry: the Elliptic Curve Digital Signature Algorithm (ECDSA) ANSI Elliptic Curve Cryptography, Standards for Efficient Cryptography Group, ver. 2 Edwards-Curve Digital Signature Algorithm (EdDSA) This document describes elliptic curve signature scheme Edwards-curve Digital Signature Algorithm (EdDSA). The algorithm is instantiated with recommended parameters for the edwards25519 and edwards448 curves. An example implementation and test vectors are provided. Key words for use in RFCs to Indicate Requirement Levels In many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words RFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings. The ristretto255 and decaf448 Groups This memo specifies two prime-order groups, ristretto255 and decaf448, suitable for safely implementing higher-level and complex cryptographic protocols. The ristretto255 group can be implemented using Curve25519, allowing existing Curve25519 implementations to be reused and extended to provide a prime-order group. Likewise, the decaf448 group can be implemented using edwards448. Recommendation for pair-wise key-establishment schemes using discrete logarithm cryptography Hashing to Elliptic Curves Cloudflare, Inc. Cornell Tech Cloudflare, Inc. Stanford University Cloudflare, Inc. This document specifies a number of algorithms for encoding or hashing an arbitrary string to a point on an elliptic curve. This document is a product of the Crypto Forum Research Group (CFRG) in the IRTF. Informative References Two-Round Threshold Signatures with FROST How to Prove Schnorr Assuming Schnorr A Graduate Course in Applied Cryptography Randomness Requirements for Security Security systems are built on strong cryptographic algorithms that foil pattern analysis attempts. However, the security of these systems is dependent on generating secret quantities for passwords, cryptographic keys, and similar quantities. The use of pseudo-random processes to generate secret quantities can result in pseudo-security. A sophisticated attacker may find it easier to reproduce the environment that produced the secret quantities and to search the resulting small set of possibilities than to locate the quantities in the whole of the potential number space. Choosing random quantities to foil a resourceful and motivated adversary is surprisingly difficult. This document points out many pitfalls in using poor entropy sources or traditional pseudo-random number generation techniques for generating such quantities. It recommends the use of truly random hardware techniques and shows that the existing hardware on many systems can be used for this purpose. It provides suggestions to ameliorate the problem when a hardware solution is not available, and it gives examples of how large such quantities need to be for some applications. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. Elliptic Curves for Security This memo specifies two elliptic curves over prime fields that offer a high level of practical security in cryptographic applications, including Transport Layer Security (TLS). These curves are intended to operate at the ~128-bit and ~224-bit security level, respectively, and are generated deterministically based on a list of required properties. The Transport Layer Security (TLS) Protocol Version 1.3 This document specifies version 1.3 of the Transport Layer Security (TLS) protocol. TLS allows client/server applications to communicate over the Internet in a way that is designed to prevent eavesdropping, tampering, and message forgery. This document updates RFCs 5705 and 6066, and obsoletes RFCs 5077, 5246, and 6961. This document also specifies new requirements for TLS 1.2 implementations.
Acknowledgments This document was improved based on input and contributions by the Zcash Foundation engineering team.
Trusted Dealer Key Generation One possible key generation mechanism is to depend on a trusted dealer, wherein the dealer generates a group secret s uniformly at random and uses Shamir and Verifiable Secret Sharing as described in and to create secret shares of s to be sent to all other participants. We highlight at a high level how this operation can be performed. The dealer is trusted to 1) generate good randomness, and 2) delete secret values after distributing shares to each participant, and 3) keep secret values confidential. It is assumed the dealer then sends one secret key share to each of the NUM_SIGNERS participants, along with C. After receiving their secret key share and C, participants MUST abort if they do not have the same view of C. Otheriwise, each participant MUST perform vss_verify(secret_key_share_i, C), and abort if the check fails. The trusted dealer MUST delete the secret_key and secret_key_shares upon completion. Use of this method for key generation requires a mutually authenticated secure channel between the dealer and participants to send secret key shares, wherein the channel provides confidentiality and integrity. Mutually authenticated TLS is one possible deployment option.
Shamir Secret Sharing In Shamir secret sharing, a dealer distributes a secret Scalar s to n participants in such a way that any cooperating subset of MIN_SIGNERS participants can recover the secret. There are two basic steps in this scheme: (1) splitting a secret into multiple shares, and (2) combining shares to reveal the resulting secret. This secret sharing scheme works over any field F. In this specification, F is the scalar field of the prime-order group G. The procedure for splitting a secret into shares is as follows. MAX_SIGNERS or if MIN_SIGNERS is less than 2 def secret_share_shard(s, MAX_SIGNERS, MIN_SIGNERS): if MIN_SIGNERS > MAX_SIGNERS: raise "invalid parameters" if MIN_SIGNERS < 2: raise "invalid parameters" # Generate random coefficients for the polynomial, yielding # a polynomial of degree (MIN_SIGNERS - 1) coefficients = [s] for i in range(1, MIN_SIGNERS): coefficients.append(G.RandomScalar()) # Evaluate the polynomial for each point x=1,...,n secret_key_shares = [] for x_i in range(1, MAX_SIGNERS + 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 MIN_SIGNERS to recover the secret s is as follows.
Verifiable Secret Sharing Feldman's Verifiable Secret Sharing (VSS) builds upon Shamir secret sharing, adding a verification step to demonstrate the consistency of a participant's share with a public commitment to the polynomial f for which the secret s is the constant term. This check 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 MIN_SIGNERS-1 is as follows. The procedure for verification of a participant's share is as follows. If vss_verify fails, the participant MUST abort the protocol, and failure should be investigated out of band. We now define how the Coordinator and signing participants can derive group info, which is an input into the FROST signing protocol.
Test Vectors This section contains test vectors for all ciphersuites listed in . All Element and Scalar values are represented in serialized form and encoded in hexadecimal strings. Signatures are represented as the concatenation of their constituent parts. The input message to be signed is also encoded as a hexadecimal string. Each test vector consists of the following information.
  • Configuration: This lists the fixed parameters for the particular instantiation of FROST, including MAX_SIGNERS, MIN_SIGNERS, and NUM_SIGNERS.
  • Group input parameters: This lists the group secret key and shared public key, generated by a trusted dealer as described in , as well as the input message to be signed. 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 identifier, the hiding and binding commitment values produced in , as well as the resulting group binding factor input, computed in part from the group commitment list encoded as described in , and group binding factor as computed in ).
  • Round two parameters and outputs: This lists the NUM_SIGNERS participants engaged in the protocol, identified by their integer identifier, along with their corresponding output signature share as produced in .
  • Final output: This lists the aggregate signature as produced in .
FROST(Ed25519, SHA-512)
FROST(Ed448, SHAKE256)
FROST(ristretto255, SHA-512)
FROST(P-256, SHA-256)