CFRG C. Komlo Internet-Draft University of Waterloo, Zcash Foundation Intended status: Informational I. Goldberg Expires: 13 February 2022 University of Waterloo T. Wilson-Brown Zcash Foundation 12 August 2021 Two-Round Threshold Signatures with FROST draft-irtf-cfrg-frost-01 Abstract 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 EdDSA verification of signatures. However, this draft does not generate deterministic nonces as defined by EdDSA, 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 mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/browse/cfrg/. Source for this draft and an issue tracker can be found at https://github.com/chelseakomlo/frost-spec. 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/. 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 13 February 2022. Copyright Notice Copyright (c) 2021 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 Simplified BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Simplified BSD License. Table of Contents 1. Introduction 2. Change Log 3. Terminology 4. Overview 5. Conventions and Terminology 6. Cryptographic Dependencies 6.1. Prime-Order Group 6.1.1. Input Validation 6.2. Cryptographic Hash Function 7. Helper functions 7.1. Schnorr Signature Operations 7.2. Polynomial Operations 7.2.1. Evaluation of a polynomial 7.2.2. Lagrange coefficients 7.2.3. Deriving the constant term of a polynomial 7.3. Shamir Secret Sharing 7.4. Verifiable Secret Sharing 8. Two-Round FROST 8.1. Round One 8.1.1. Round Two 8.1.2. Aggregate 9. Ciphersuites 9.1. FROST(ristretto255, SHA-512) 10. Security Considerations 10.1. Nonce Reuse Attacks 10.2. Protocol Failures 10.3. External Requirements / Non-Goals 11. Contributors 12. References 12.1. Normative References 12.2. Informative References Appendix A. Acknowledgments Appendix B. Trusted Dealer Key Generation Authors' Addresses 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/chelseakomlo/frost-spec. 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. This draft specifies only two-round signing operations. This draft specifies signatures that are compatible with EdDSA verification of signatures, but not EdDSA nonce generation. EdDSA-style nonce- generation, where the nonce is derived deterministically, is insecure in a multi-party signature setting. [OPEN ISSUE: EdDSA compatibility is still an open issue, see: https://github.com/chelseakomlo/frost-spec/issues/5] 2. Change Log draft-01 * Submitted a draft that specifies operations, notation and cryptographic dependencies. draft-00 * Submitted a basic draft after adoption of draft-komlo-frost as a working group item. 3. Terminology 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. 4. Overview FROST is a threshold signature protocol involving the following parties: * Signers: Entities with signing key shares that participate in the threshold signing protocol * Coordinator: An entity responsible for performing coordination among signers and for aggregating signature shares at the end of the protocol, resulting in the final signature. This party may be a signer themselves or an external party. FROST assumes the selection of all participants, including the dealer, signer, and Coordinator are all chosen external to the protocol. In FROST, Signers participate in two rounds to sign an input message and produce a single, aggregate signature. All signers are assumed to have the group state and their corresponding signing keys; see Section 8 for details about how this state is generated. At the end of the second round, the Coordinator performs an aggregation function to produce the final signature This is sketched below. (group info) (group info, (group info, | signing key) signing key) | | | v v v Coordinator Signer-1 ... Signer-n ------------------------------------------------------------ message ------------> | == Round 1 (Commitment) == | SigningComitment | | |<-----------------------+ | | ... | | SigningComitment | |<-----------------------------------------+ == Round 2 (Signing) == | | SigningPackage | | +------------------------> | | SignatureShare | | |<-----------------------+ | | ... | | SigningPackage | +------------------------------------------> | SignatureShare | <------------------------------------------+ == Aggregation == | signature | <-----------+ Details about each of these rounds and the corresponding protocol messages is in Section 8. 5. Conventions and Terminology The following notation and terminology are used throughout this document. * "s" denotes a secret that is Shamir secret shared among the participants. * "s[i]" denotes the i-th share of the secret "s". * A participant is an entity that is trusted to hold a secret share. * "n" denotes the number of participants, and the number of shares that "s" is split into. * "t" denotes the threshold number of participants required to issue a signature. More specifically, at least "t" shares must be combined to issue a valid signature. * "L_i" represents the ith Lagrange coefficient. * "sig = (R, z)" denotes a Schnorr signature with public commitment "R" and response "z". * "PK" is the group public key. * "sk_i" is each ith individual's private key, consisting of the tuple "sk_i = (i, s[i])". * "len(x)" is the length of integer input "x" as an 8-byte, big- endian integer. * || denotes contatenation, i.e., x || y = xy. This specification makes use of the following utility functions: * SUM(START, END){TERMS}: this function denotes the summation from START to END (inclusive) of TERMS. For example, SUM(N=0, 3){2N} is equal to 2*(1+2+3)=12. * PROD(START, END){TERMS}: this function denotes the product from START to END of TERMS in similar manner. 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. Let "B" be a generator, or distiguished element, of "G", a finite group of with order "l", a large prime. Throughout this document, and in practice, we assume this group to be instantiated as an arbitrary abstraction of an elliptic curve subgroup, defined over a finite field; however, that does not restrict an implementation from instantiating FROST signatures over other groups, provided their order be prime. We denote group elements with capital Roman letters, and scalars with lower-cased Roman letters. We use "+" to denote the group operation, and "-" to denote inversion. We use "*" to denote multiplication of a scalar by a group element, that is, the group element added to itself in succession a number of times equal to the value of the scalar. Testing equality between two group elements is denoted as "?=", where it is assumed that the elements are in some canonical, serialised form. 6. Cryptographic Dependencies FROST depends on the following cryptographic constructs: * Prime-order Group, Section 6.1; * Cryptographic hash function, Section 6.2; These are described in the following sections. 6.1. 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"). * HashToScalar(x): A member function of "G" that deterministically maps an array of bytes "x" to an element in GF(p). This function is optionally parameterized by a DST. * 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 6.1.1. * SerializeScalar(s): A member function of "G" that maps a scalar element "s" to a unique byte array "buf" of fixed length "Ns". The output type of this function is "SerializedScalar". * DeserializeScalar(buf): A member function of "G" that maps a byte array "buf" to a scalar "s", or fails if the input is not a valid byte representation of a scalar. This function can raise a DeserializeError if deserialization fails; see Section 6.1.1. 6.1.1. 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()". 6.2. 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 9. Using H, we introduce two separate domain-separated hashes, H1 and H2, where H1(m) = H("rho" || len(m) || m) and H2(m) = H("chal" || len(m) || m). 7. Helper functions Beyond the core dependencies, the protocol in this document depends on the following helper operations: * Schnorr signatures, Section 7.1; * Polynomial operations, {dep-polynomial}; * Shamir Secret Sharing, {dep-shamir}; and * Verifiable Secret Sharing committments, Section 7.4. This sections describes these operations in more detail. 7.1. 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, sig, SK): r = RandomScalar() R = ScalarBaseMult(r) c = Hash(m, R) z = (r + c) * SK return (c), z) The corresponding verification operation is as follows. schnorr_signature_verify(msg, sig, PK): Inputs: - msg, signed message, an octet string - sig, a tuple (c, z) output from schnorr_signature_generate - PK, public key, a group element Outputs: 1 if signature is valid, and 0 otherwise def schnorr_signature_verify(msg, sig, PK): (c, z) = sig c_inv = c^-1 R' = ScalarBaseMult(z) + (PK * c_inv) c' = Hash(m, R') if c == c': return 1 return 0 7.2. Polynomial Operations This section describes operations on and associated with polynomials that are used in the main signing protocol. 7.2.1. Evaluation of a polynomial This section describes a method for evaluating a polynomial "f" at a particular input "x", i.e., "f(x)" using Horner's method. polynomial_evaluate(x, coeffs): Inputs: - x, input at which to evaluate the polynomial, a scalar - coeffs, the polynomial coefficients Outputs: Scalar result of the polynomial evaluated at input x def polynomial_evaluate(xcoord, coeffs): value = 0 for (counter, coeff) in coeffs.reverse(): if counter = coeffs.len(): value += coeff // add the constant term else: value += coeff value *= x return value 7.2.2. 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(i, L): Inputs: - i, an index, contained in L - L, the set of x-coordinates Outputs: L_i, the i-th Lagrange coefficient def derive_lagrange_coefficient(i, L): numerator = 0 denominator = 0 for j in L: if j == i: continue numerator *= j denominator *= j - i L_i = numerator / denominator return L_i 7.2.3. 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. polynomial_interpolation(points): Inputs: - points, a set of `t` points on a polynomial f Outputs: the constant term of f def polynomial_interpolation(points): let L = [] for point in points: L.append(point[0]) // add the x-coordinate f_zero = SUM(points[0], points[t]){point}: point[1] * derive_lagrange_coefficient(point[0], L) return f_zero 7.3. 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): 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: a list of n secret shares, each of which is an element of F Errors: - "invalid parameters", if t > n def secret_share(s, n, t): if t > n: raise "invalid parameters" # Generate random coefficients for the polynomial coefficients = [s] for i in range(t - 1): coefficients.append(RandomScalar()) # Evaluate the polynomial for each participant, identified by their index i points = [] for i in range(n): point_i = polynomial_evaluate(1, coefficients) points.append(point_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 funtionally the randomly generated polynomial evaluated at "i". We denote a secret share as the tuple "(i, points[i])", and the list of these shares as "shares". The procedure for combining a "shares" list of length "t" to recover the secret "s" is as follows. secret_share_combine(shares): Inputs: - shares, a list of t secret shares, each a tuple (i, f(i)) - 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 def secret_share_combine(shares): s = polynomial_interpolation(shares) return s 7.4. 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. vss_commit(coeffs): Inputs: - 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) C.append(A_i) return C The procedure for verification of a participant's share is as follows. vss_verify(sk_i, C): Inputs: - 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 8. Two-Round FROST The FROST protocol produces a standard Schnorr signature over an input message of at most 2^16-1 bytes long. The protocol assumes that each participant "P_i" knows the following: * Group public key, denoted "PK = s * B", corresponding to the group secret key "s" * Participant signing key, which is the tuple "sk = (i, s[i])", where "s[i]" 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. FROST assumes the existence of a _Coordinator_, which is a Signer responsible for the following: 1. Determining which signers will participate (at least "t" 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. We describe the protocol in two rounds: commitment and signing. 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. The Coordinator then performs an aggregation round at the end and outputs the final signature. This protocol 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 10. 8.1. Round One Each signer in round one generates a nonce "nonce = (d, e)" and commitment "comm = (D, E)" for each signer. frost_commit(): Inputs: None Outputs: (nonce, comm), a tuple of nonce and nonce commitment pairs def frost_commit(): d = RandomScalar() e = RandomScalar() D = ScalarBaseMult(d) E = ScalarBaseMult(e) nonce = (d, e) comm = (D, E) return nonce, comm The output "nonce" from Participant "P_i" is stored locally and kept private for use in the second round. The output "comm" from Participant "P_i" is sent to the Coordinator. Both group elements in this tuple are serialized and encoded in a "SigningComitment", along with the participant ID, as follows. 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 blinding factor encoded as a serialized group element. 8.1.1. Round Two 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 "t"). 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 a "SigningPackage", composed as follows. struct { SigningComitment signing_commitments<1..2^16-1>; opaque msg<0..2^16-1>; } SigningPackage; signing_commitments An list of w SigningComitment values, where t <= w <= n, ordered in ascending order by SigningComitment.id. This list MUST NOT contain more than one SigningComitment value corresponding to each signer. Signers MUST ignore SigningPackage values with duplicate SignerIDs. msg The message to be signed. Each signer then runs the following procedure. frost_sign(sk_i, (d_i, e_i), m, B, L): Inputs: - sk_i: secret key that is the tuple sk_i= (i, s[i]) - nonce (d_i, e_i) generated in round one - m: the message to be signed (sent by the Coordinator). - B={(D_j, E_j), ...}: a set of commitments issued by each signer in round one, of length w, where t <= w <= n (sent by the Coordinator). - L: a set containing identifiers for each signer, similarly of length w (sent by the Coordinator). Outputs: a signature share z_i, to be sent to the Coordinator. frost_sign(sk_i, (d_i, e_i), m, B, L): binding_factor = H1(B) R = SUM(B[1], B[l]){(j, D_j, E_j)}: D_j + (E_j * binding_factor ) L_i = derive_lagrange_coefficient(i, L) c = H2(R, m) z_i = d_i + (e_i * binding_factor) + L_i + s[i] + c return z_i The output of this procedure is a signature share. Each signer then sends this share back to the collector in a "SignatureShare", which is constructed as follows. struct { SignerID id; opaque package_id[Nh]; opaque signature_share[Ns]; } SignatureShare; id The SignerID. package_id The cryptographic hash of the corresponding SigningPackage, i.e., package_id = H(SigningPackage). signature_share The signature share from this signer encoded as a serialized scalar. The coordinator uses SignatureShare.package_id to group signature shares for the same SigningPackage. Given a set of SignatureShare values, the Coordinator MAY elect to verify these using the following procedure. frost_verify_signature_share(PK_i, z_i, R, R_i, L_i, m): Inputs: - PK_i, the public key for the ith signer, where PK_i = ScalarBaseMult(s[i]). - z_i, the signature share for the ith signer - R_i, the commitment for the ith signer, where R_i = F_i + E_i * rho - R, the group commitment - L_i, the ith Lagrange coefficient for the signing set. - m, the message to be signed Outputs: 1 if the signature share is valid, and 0 otherwise frost_verify_signature_share(PK_i, z_i, R_i, L_i, m) c' = H2(R, m) Z_i = HashToGroup(z_i) R_i' = Z_i + (PK_i * -c') if R_i == R_i': return 1 return 0 8.1.2. Aggregate 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 8.1.1 frost_aggregate(R, Z): Inputs: - R: the group commitment. - Z: a set of signature shares z_i for each signer, of length w, where t <= w <= n. Outputs: sig=(R, z), a Schnorr signature. frost_aggregate(R, Z): z = SUM(Z[1], Z[w]){z_i}: z_i return sig=(R, z) 9. 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 9.1. 9.1. FROST(ristretto255, SHA-512) * Group: ristretto255 [RISTRETTO] - HashToGroup(): Use hash_to_ristretto255 [I-D.irtf-cfrg-hash-to-curve] with DST = "HashToGroup-" || contextString, and "expand_message" = "expand_message_xmd" using SHA-512. - HashToScalar(): Compute "uniform_bytes" using "expand_message" = "expand_message_xmd", DST = "HashToScalar-" || contextString, and output length 64, interpret "uniform_bytes" as a 512-bit integer in little-endian order, and reduce the integer modulo "Order()". - Serialization: Both group elements and scalars are encoded in Ne = Ns = 32 bytes. For group elements, use the 'Encode' and 'Decode' functions from [RISTRETTO]. For scalars, ensure they are fully reduced modulo "Order()" and in little-endian order. * Hash: SHA-512, and Nh = 64. 10. Security Considerations A security analysis of FROST exists in [FROST21]. 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. * 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 rest of this section documents issues particular to implementations or deployments. 10.1. 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. 10.2. 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. 10.3. External Requirements / Non-Goals FROST does not target the following goals. * Post quantum security. FROST 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. 11. Contributors * Chris Wood * Isis Lovecruft 12. References 12.1. Normative References [I-D.irtf-cfrg-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- 11, 13 April 2021, . [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, . [RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, May 2017, . [RISTRETTO] 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, 4 August 2021, . 12.2. Informative References [FROST21] Komlo, C., Goldberg, I., and T. Wilson-Brown, "Two-Round Threshold Signatures with FROST", 1 June 2021, . [RFC4086] Eastlake 3rd, D., Schiller, J., and S. Crocker, "Randomness Requirements for Security", BCP 106, RFC 4086, DOI 10.17487/RFC4086, June 2005, . 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 Section 7.3 and Section 7.4 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. trusted_dealer_keygen(n, t): Inputs: - n, the number of shares to generate, an integer - t, the threshold of the secret sharing scheme, an integer Outputs: a list of secret keys, each which is an element of F, and a public key which is an element of G. It is assumed the dealer then sends one secret key to each of the n participants, and afterwards deletes the secrets from their local device. def trusted_dealer_keygen(n, t): s = RandomScalar() points = secret_share_split(s, n, t) secret_keys = [] sk_i = (i, points[i]) secret_keys.append(sk_i) public_key = ScalarBaseMult(s) return secret_keys, public_key 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. Authors' Addresses Chelsea Komlo University of Waterloo, Zcash Foundation Email: ckomlo@uwaterloo.ca Ian Goldberg University of Waterloo Email: iang@uwaterloo.ca T Wilson-Brown Zcash Foundation Email: teor@riseup.net