:: ZIP: 312 Title: FROST for Spend Authorization Signatures Owners: Conrado Gouvea Chelsea Komlo Deirdre Connolly Status: Draft Category: Wallet Created: 2022-08-dd License: MIT Discussions-To: Pull-Request: Terminology =========== {Edit this to reflect the key words that are actually used.} The key words "MUST", "MUST NOT", "SHOULD", and "MAY" in this document are to be interpreted as described in RFC 2119. [#RFC2119]_ The terms below are to be interpreted as follows: Unlinkability The property of statistical independence of signatures from the signers' long-term keys, ensuring that (for perfectly uniform generation of Randomizers and no leakage of metadata) it is impossible to determine whether two transactions were generated by the same party. Abstract ======== This proposal adapts FROST [#FROST]_, a threshold signature scheme, to make it unlinkable, which is a requirement for its use in the Zcash protocol. The adapted scheme generates signatures compatible with spend authorization signatures in the Zcash protocol, for the Sapling and Orchard network upgrades. Motivation ========== In the Zcash protocol, Spend Authorization Signatures are employed to authorize a transaction. The ability to generate these signatures with the user's private key is what effectively allows the user to spend funds. This is a security-critical step, since anyone who obtains access to the private key will be able to spend the user's funds. For this reason, one interesting possibility is to require multiple parties to allow the transaction to go through. This can be accomplished with threshold signatures, where the private key is split between parties in a way that a threshold (e.g. 2 out of 3) of them must sign the transaction in order to create the final signature. This enables scenarios such as users and exchanges sharing custody of a wallet, for example. FROST is one of such threshold signature protocols. However, it can't be used as-is since the Zcash protocol also requires re-randomizing public and private keys to ensure unlinkability between transactions. This ZIP specifies a variant of FROST with re-randomization support. Requirements ============ - All signatures generated by following this ZIP must be verified successfully as Sapling or Orchard spend authorization signatures using the appropriate validating key. - The signatures generated by following this ZIP should meet the security criteria for Signature with Re-Randomizable Keys as specified in the Zcash protocol [#protocol-concretereddsa]_. - The threat model described below must be taken into account. Threat Model ------------ In normal usage, a Zcash user follows multiple steps in order to generate a shielded transaction: - The transaction is created. - The transaction is signed with a re-randomized version of the user's spend authorization private key. - The zero-knowledge proof for the transaction is created with the randomizer as an auxiliary (secret) input, among others. When employing re-randomizable FROST as specified in this ZIP, the goal is to split the spend authorization private key :math:`\mathsf{ask}` among multiple possible signers. This means that the proof generation will still be performed by a single participant, likely the one that created the transaction in the first place. Note that this user already controls the privacy of the transaction since they are responsible for creating the proof. This fits well into the "Coordinator" role from the FROST specification [#frost-protocol]_. The Coordinator is responsible for sending the message to be signed to all participants, and to aggregate the signature shares. With those considerations in mind, the threat model considered in this ZIP is: - The Coordinator is trusted with the privacy of the transaction (which includes the unlinkability property). A rogue Coordinator will be able to break unlinkability and privacy, but should not be able to create signed transactions without the approval of `MIN_SIGNERS` participants, as specified in FROST. - All key share holders are also trusted with the privacy of the transaction, thus a rogue key share holder will be able to break its privacy and unlinkability. Non-requirements ================ - This ZIP does not support removing the Coordinator role, as described in #[frost-removingcoordinator]_. - This ZIP does not prevent key share holders from linking the signing operation to a transaction in the blockchain. - Like the FROST specification [#FROST], this ZIP does not specify a key generation procedure; but refer to that specification for guidelines. - Network privacy is not in scope for this ZIP, and must be obtained with other tools if desired. Specification ============= Algorithms in this section are specified using Python pseudo-code, in the same fashion as the FROST specification [#FROST]_. The types Scalar, Element, and G are defined in #[frost-primeordergroup]_, as well as the notation for elliptic-curve arithmetic, which uses the additive notation. Note that this notation differs from that used in the Zcash Protocol Specification. For example, `G.ScalarMult(P, k)` is used for scalar multiplication, where the protocol spec would use :math:`[k] P` with the group implied by :math:`P`. Re-randomizable FROST --------------------- To add re-randomization to FROST, follow the specification [#FROST]_ with the following modifications. Key Generation '''''''''''''' While key generation is out of scope for this ZIP and the FROST spec [#FROST]_, it needs to be consistent with FROST, see [#frost-tdkg]_ for guidance. The spend authorization private key :math:`\mathsf{ask}` [#protocol-spendauthsig]_ is the particular key that must be used in the context of this ZIP. Note that the :math:`\mathsf{ask}` is usually derived from the spending key :math:`\mathsf{sk}`, though that is not required. Doing so might require a trusted dealer key generation process as detailed in [#frost-tdkg]_ (as opposed to distributed key generation). Randomizer Generation ''''''''''''''''''''' A new helper function is defined, which computes :math:`\mathsf{RedDSA.GenRandom}`: :: randomizer_generate(): Inputs: - None Outputs: randomizer, a Scalar def randomizer_generate(): randomizer_input = random_bytes(64) return H3(randomizer_input) Binding Factor Computation '''''''''''''''''''''''''' The `compute_binding_factors` function is changed to receive the `randomizer_point` as follows: :: Inputs: - commitment_list = [(i, hiding_nonce_commitment_i, binding_nonce_commitment_i), ...], a list of commitments issued by each participant, where each element in the list indicates a NonZeroScalar identifier i and two commitment Element values (hiding_nonce_commitment_i, binding_nonce_commitment_i). This list MUST be sorted in ascending order by identifier. - msg, the message to be signed. - randomizer_point, an element in G. Outputs: - binding_factor_list, a list of (NonZeroScalar, Scalar) tuples representing the binding factors. def compute_binding_factors(commitment_list, msg, randomizer_point): msg_hash = H4(msg) encoded_commitment_hash = H5(encode_group_commitment_list(commitment_list)) rho_input_prefix = msg_hash || encoded_commitment_hash || G.SerializeElement(randomizer_point) binding_factor_list = [] for (identifier, hiding_nonce_commitment, binding_nonce_commitment) in commitment_list: rho_input = rho_input_prefix || G.SerializeScalar(identifier) binding_factor = H1(rho_input) binding_factor_list.append((identifier, binding_factor)) return binding_factor_list Round One - Commitment '''''''''''''''''''''' Roune One is exactly the same as specified #[FROST]_. But for context, it involves these steps: - Each signer generates nonces and their corresponding public commitments. A nonce is a pair of Scalar values, and a commitment is a pair of Element values. - The nonces are stored locally by the signer and kept private for use in the second round. - The commitments are sent to the Coordinator. Round Two - Signature Share Generation '''''''''''''''''''''''''''''''''''''' In Round Two, the Coordinator generates a random scalar `randomizer` by calling `randomizer_generate`. Then it computes `randomizer_point = G.ScalarBaseMult(randomizer)` and sends it to each signer, over a confidential and authenticated channel, along with the message and the set of signing commitments. (Note that this differs from regular FROST which just requires an authenticated channel.) In Zcash, the message that needs to be signed is actually the SIGHASH transaction hash, which is does not convey enough information for the signers to decide if they want to authorize the transaction or not. Therefore, in practice, more data is needed to be sent from the Coordinator to the signers, possibly the transaction itself, openings of value commitments, decryption of note ciphertexts, etc.; and the signers must check that the given SIGHASH matches the data sent from the Coordinator, or compute the SIGHASH themselves from that data. However, the specific mechanism for that process is outside the scope of this ZIP. The `sign` function is changed to receive `randomizer_point` and incorporate it into the computation of the binding factor. It is specified as the following: :: Inputs: - identifier, identifier i of the participant, a NonZeroScalar. - sk_i, Signer secret key share, a Scalar. - group_public_key, public key corresponding to the group signing key, an Element. - nonce_i, pair of Scalar values (hiding_nonce, binding_nonce) generated in round one. - msg, the message to be signed, a byte string. - commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a list of commitments issued in Round 1 by each participant and sent by the Coordinator. Each element in the list indicates a NonZeroScalar identifier j and two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j). This list MUST be sorted in ascending order by identifier. - randomizer_point, an element in G (sent by the Coordinator). Outputs: - sig_share, a signature share, a Scalar. def sign(identifier, sk_i, group_public_key, nonce_i, msg, commitment_list): # Compute the randomized group public key randomized_group_public_key = group_public_key + randomizer_point # Compute the binding factor(s) binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_point) binding_factor = binding_factor_for_participant(binding_factor_list, identifier) # Compute the group commitment group_commitment = compute_group_commitment(commitment_list, binding_factor_list) # Compute the interpolating value participant_list = participants_from_commitment_list(commitment_list) lambda_i = derive_interpolating_value(identifier, participant_list) # Compute the per-message challenge challenge = compute_challenge(group_commitment, randomized_group_public_key, msg) # Compute the signature share (hiding_nonce, binding_nonce) = nonce_i sig_share = hiding_nonce + (binding_nonce * binding_factor) + (lambda_i * sk_i * challenge) return sig_share Signature Share Verification and Aggregation '''''''''''''''''''''''''''''''''''''''''''' The `aggregate` function is changed to incorporate the randomizer as follows: :: Inputs: - commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a list of commitments issued in Round 1 by each participant, where each element in the list indicates a NonZeroScalar identifier j and two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j). This list MUST be sorted in ascending order by identifier. - msg, the message to be signed, a byte string. - sig_shares, a set of signature shares z_i, Scalar values, for each participant, of length NUM_PARTICIPANTS, where MIN_PARTICIPANTS <= NUM_PARTICIPANTS <= MAX_PARTICIPANTS. - group_public_key, public key corresponding to the group signing key, - randomizer, the randomizer Scalar. Outputs: - (R, z), a Schnorr signature consisting of an Element R and Scalar z. - randomized_group_public_key, the randomized group public key def aggregate(commitment_list, msg, sig_shares, group_public_key, randomizer): # Compute the randomized group public key randomizer_point = G.ScalarBaseMult(randomizer) randomized_group_public_key = group_public_key + randomizer_point # Compute the binding factors binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_point) # Compute the group commitment group_commitment = compute_group_commitment(commitment_list, binding_factor_list) # Compute the challenge challenge = compute_challenge(group_commitment, randomized_group_public_key, msg) # Compute aggregated signature z = Scalar(0) for z_i in sig_shares: z = z + z_i return (group_commitment, z + randomizer * challenge), randomized_group_public_key The `verify_signature_share` is changed to incorporate the randomizer point, as follows: :: Inputs: - identifier, identifier i of the participant, a NonZeroScalar. - PK_i, the public key for the i-th participant, where PK_i = G.ScalarBaseMult(sk_i), an Element. - comm_i, pair of Element values in G (hiding_nonce_commitment, binding_nonce_commitment) generated in round one from the i-th participant. - sig_share_i, a Scalar value indicating the signature share as produced in round two from the i-th participant. - commitment_list = [(j, hiding_nonce_commitment_j, binding_nonce_commitment_j), ...], a list of commitments issued in Round 1 by each participant, where each element in the list indicates a NonZeroScalar identifier j and two commitment Element values (hiding_nonce_commitment_j, binding_nonce_commitment_j). This list MUST be sorted in ascending order by identifier. - group_public_key, public key corresponding to the group signing key, an Element. - msg, the message to be signed, a byte string. - randomizer_point, an element in G. Outputs: - True if the signature share is valid, and False otherwise. def verify_signature_share(identifier, PK_i, comm_i, sig_share_i, commitment_list, group_public_key, msg, randomizer_point): # Compute the randomized group public key randomized_group_public_key = group_public_key + randomizer_point # Compute the binding factors binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_point) binding_factor = binding_factor_for_participant(binding_factor_list, identifier) # Compute the group commitment group_commitment = compute_group_commitment(commitment_list, binding_factor_list) # Compute the commitment share (hiding_nonce_commitment, binding_nonce_commitment) = comm_i comm_share = hiding_nonce_commitment + G.ScalarMult(binding_nonce_commitment, binding_factor) # Compute the challenge challenge = compute_challenge(group_commitment, randomized_group_public_key, msg) # Compute the interpolating value participant_list = participants_from_commitment_list(commitment_list) lambda_i = derive_interpolating_value(identifier, participant_list) # Compute relation values l = G.ScalarBaseMult(sig_share_i) r = comm_share + G.ScalarMult(PK_i, challenge * lambda_i) return l == r Ciphersuites ------------ FROST(Jubjub, BLAKE2b-512) '''''''''''''''''''''''''' This ciphersuite uses Jubjub for the Group and BLAKE2b-512 for the Hash function `H` meant to produce signatures indistinguishable from RedJubjub Sapling Spend Authorization Signatures as specified in [#protocol-concretespendauthsig]_. - Group: Jubjub [#protocol-jubjub]_ with base point :math:``\mathcal{G}^{\mathsf{Sapling}}` as defined in [#protocol-concretespendauthsig]_. - Order: :math:`r_\mathbb{J}` as defined in [#protocol-jubjub]_. - Identity: as defined in [#protocol-jubjub]_. - RandomScalar(): Implemented by returning a uniformly random Scalar in the range \[0, `G.Order()` - 1\]. Refer to {{frost-randomscalar}} for implementation guidance. - SerializeElement(P): Implemented as :math:`\mathsf{repr}_\mathbb{J}(P)` as defined in [#protocol-jubjub]_ - DeserializeElement(P): Implemented as :math:`\mathsf{abst}_\mathbb{J}(P)` as defined in [#protocol-jubjub]_, returning an error if :math:`\bot` is returned. Additionally, this function validates that the resulting element is not the group identity element, returning an error if the check fails. - SerializeScalar: Implemented by outputting the little-endian 32-byte encoding of the Scalar value. - DeserializeScalar: Implemented by attempting to deserialize a Scalar from a little-endian 32-byte string. This function can fail if the input does not represent a Scalar in the range \[0, `G.Order()` - 1\]. - Hash (`H`): BLAKE2b-512 [#BLAKE]_ (BLAKE2b with 512-bit output and 16-byte personalization string), and Nh = 64. - H1(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubR", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. - H2(m): Implemented by computing BLAKE2b-512("Zcash_RedJubjubH", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. (This is equivalent to :math:`\mathsf{H}^\circledast(m)`, as defined by the :math:`\mathsf{RedJubjub}` scheme instantiated in [#protocol-concretereddsa]_.) - H3(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubN", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. - H4(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubM", m). - H5(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubC", m). FROST(Pallas, BLAKE2b-512) '''''''''''''''''''''''''' This ciphersuite uses Pallas for the Group and BLAKE2b-512 for the Hash function `H` meant to produce signatures indistinguishable from RedPallas Orchard Spend Authorization Signatures as specified in [#protocol-concretespendauthsig]_. - Group: Pallas [#protocol-pallasandvesta]_ with base point :math:``\mathcal{G}^{\mathsf{Orchard}}` as defined in [#protocol-concretespendauthsig]_. - Order: :math:`r_\mathbb{P}` as defined in [#protocol-pallasandvesta]_. - Identity: as defined in [#protocol-pallasandvesta]_. - RandomScalar(): Implemented by returning a uniformly random Scalar in the range \[0, `G.Order()` - 1\]. Refer to {{frost-randomscalar}} for implementation guidance. - SerializeElement(P): Implemented as :math:`\mathsf{repr}_\mathbb{P}(P)` as defined in [#protocol-pallasandvesta]_. - DeserializeElement(P): Implemented as :math:`\mathsf{abst}_\mathbb{P}(P)` as defined in [#protocol-pallasandvesta]_, failing if :math:`\bot` is returned. Additionally, this function validates that the resulting element is not the group identity element, returning an error if the check fails. - SerializeScalar: Implemented by outputting the little-endian 32-byte encoding of the Scalar value. - DeserializeScalar: Implemented by attempting to deserialize a Scalar from a little-endian 32-byte string. This function can fail if the input does not represent a Scalar in the range \[0, `G.Order()` - 1\]. - Hash (`H`): BLAKE2b-512 [#BLAKE]_ (BLAKE2b with 512-bit output and 16-byte personalization string), and Nh = 64. - H1(m): Implemented by computing BLAKE2b-512("FROST_RedPallasR", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. - H2(m): Implemented by computing BLAKE2b-512("Zcash_RedPallasH", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. (This is equivalent to :math:`\mathsf{H}^\circledast(m)`, as defined by the :math:`\mathsf{RedPallas}` scheme instantiated in [#protocol-concretereddsa]_.) - H3(m): Implemented by computing BLAKE2b-512("FROST_RedPallasN", m), interpreting the 64 bytes as a little-endian integer, and reducing the resulting integer modulo `G.Order()`. - H4(m): Implemented by computing BLAKE2b-512("FROST_RedPallasM", m). - H5(m): Implemented by computing BLAKE2b-512("FROST_RedPallasC", m). Reference implementation ======================== TODO: add links to implementation References ========== .. [#BLAKE] `BLAKE2: simpler, smaller, fast as MD5 `_ .. [#RFC2119] `RFC 2119: Key words for use in RFCs to Indicate Requirement Levels `_ .. [#FROST] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST `_ .. [#frost-protocol] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 5: Two-Round FROST Signing Protocol `_ .. [#frost-removingcoordinator] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 7.3: Removing the Coordinator Role `_ .. [#frost-primeordergroup] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 3.1: Prime-Order Group `_ .. [#frost-tdkg] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix B: Trusted Dealer Key Generation `_ .. [#frost-randomscalar] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix C: Random Scalar Generation `_ .. [#protocol-concretereddsa] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.7: RedDSA, RedJubjub, and RedPallas `_ .. [#protocol-concretespendauthsig] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.7.1: Spend Authorization Signature (Sapling and Orchard) `_ .. [#protocol-spendauthsig] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 4.15: Spend Authorization Signature (Sapling and Orchard) `_ .. [#protocol-jubjub] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.9.3: Jubjub `_ .. [#protocol-pallasandvesta] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.9.6: Pallas and Vesta `_