diff --git a/zip-0312.html b/zip-0312.html index c5d670b1..85f1793e 100644 --- a/zip-0312.html +++ b/zip-0312.html @@ -1,16 +1,389 @@ - ZIP 312: Shielded Multisignatures using FROST + ZIP 312: FROST for Spend Authorization Signatures + 
ZIP: 312
-Title: Shielded Multisignatures using FROST
-Status: Reserved
-Category: Standards / RPC / Wallet
-Discussions-To: <https://github.com/zcash/zips/issues/382>
+Title: FROST for Spend Authorization Signatures +Owners: Conrado Gouvea <conrado@zfnd.org> + Chelsea Komlo <ckomlo@uwaterloo.ca> + Deirdre Connolly <deirdre@zfnd.org> +Status: Draft +Category: Wallet +Created: 2022-08-dd +License: MIT +Discussions-To: <https://github.com/zcash/zips/issues/382> +Pull-Request: <https://github.com/zcash/zips/pull/662> +

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. 2

+

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 3, 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

+ +

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 + \(\mathsf{ask}\) + 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 4. 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_PARTICIPANTS 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

+ +
+

Specification

+

Algorithms in this section are specified using Python pseudo-code, in the same fashion as the FROST specification 3.

+

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 + \([k] P\) + with the group implied by + \(P\) + .

+

Re-randomizable FROST

+

To add re-randomization to FROST, follow the specification 3 with the following modifications.

+

Key Generation

+

While key generation is out of scope for this ZIP and the FROST spec 3, it needs to be consistent with FROST, see 8 for guidance. The spend authorization private key + \(\mathsf{ask}\) + 12 is the particular key that must be used in the context of this ZIP. Note that the + \(\mathsf{ask}\) + is usually derived from the spending key + \(\mathsf{sk}\) + , though that is not required. This allows using distributed key generation, since the key it generates is unpredictable. Note however that note deriving + \(\mathsf{ask}\) + from + \(\mathsf{sk}\) + prevents using seed phrases to recover the original secret (which may be something desirable in the context of FROST).

+
+

Randomizer Generation

+

A new helper function is defined, which computes + \(\mathsf{RedDSA.GenRandom}\) + :

+ 
randomizer_generate():
+
+Inputs:
+- None
+
+Outputs: randomizer, a Scalar
+
+def randomizer_generate():
+  randomizer_input = random_bytes(64)
+  return H3(randomizer_input)
+
+

Round One - Commitment

+

Roune One is exactly the same as specified 3. 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 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 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 (over the same encrypted, authenticated channel) 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 remains unchanged, but its inputs must be modified relative to the randomizer as following:

+
    +
  • sk_i = sk_i + randomizer
    • +
  • group_public_key = group_public_key + G.ScalarBaseMult(randomizer)
    • +
+
+

Signature Share Verification and Aggregation

+

The aggregate function remains unchanged, but its inputs must be modified relative to the randomizer as following:

+
    +
  • group_public_key = group_public_key + G.ScalarBaseMult(randomizer)
    • +
+

The verify_signature_share function remains unchanged, but its inputs must be modified relative to the randomizer as following:

+
    +
  • PK_i = PK_i + G.ScalarBaseMult(randomizer)
    • +
  • group_public_key = group_public_key + G.ScalarBaseMult(randomizer)
    • +
+
+
+

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 11.

+
    +
  • Group: Jubjub 13 with base point + \(\mathcal{G}^{\mathsf{Sapling}}\) + as defined in 11. +
      +
    • Order: + \(r_\mathbb{J}\) + as defined in 13.
      • +
    • Identity: as defined in 13.
      • +
    • 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 + \(\mathsf{repr}_\mathbb{J}(P)\) + as defined in 13
      • +
    • DeserializeElement(P): Implemented as + \(\mathsf{abst}_\mathbb{J}(P)\) + as defined in 13, returning an error if + \(\bot\) + 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 1 (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 + \(\mathsf{H}^\circledast(m)\) + , as defined by the + \(\mathsf{RedJubjub}\) + scheme instantiated in 10.)
      • +
    • 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).
      • +
    +
    • +
+

Signature verification is as specified in 11 for RedJubjub.

+
+

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 11.

+
    +
  • Group: Pallas 14 with base point + \(\mathcal{G}^{\mathsf{Orchard}}\) + as defined in 11. +
      +
    • Order: + \(r_\mathbb{P}\) + as defined in 14.
      • +
    • Identity: as defined in 14.
      • +
    • 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 + \(\mathsf{repr}_\mathbb{P}(P)\) + as defined in 14.
      • +
    • DeserializeElement(P): Implemented as + \(\mathsf{abst}_\mathbb{P}(P)\) + as defined in 14, failing if + \(\bot\) + 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 1 (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 + \(\mathsf{H}^\circledast(m)\) + , as defined by the + \(\mathsf{RedPallas}\) + scheme instantiated in 10.)
      • +
    • 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).
      • +
    +
    • +
+

Signature verification is as specified in 11 for RedPallas.

+
+
+
+

Rationale

+

FROST is a threshold Schnorr signature scheme, and Zcash Spend Authorization are also Schnorr signatures, which allows the usage of FROST with Zcash. However, since there is no widespread standard for Schnorr signatures, it must be ensured that the signatures generated by the FROST variant specified in this ZIP can be verified successfully by a Zcash implementation following its specification. In practice this entails making sure that the generated signature can be verified by the + \(\mathsf{RedDSA.Validate}\) + function specified in 10:

+ +

The second step is adding the re-randomization functionality so that each FROST signing generates a re-randomized signature:

+ +
+

Reference implementation

+

TODO: add links to implementation

+
+

References

+ + + + + + + +
1BLAKE2: simpler, smaller, fast as MD5
+ + + + + + + +
2RFC 2119: Key words for use in RFCs to Indicate Requirement Levels
+ + + + + + + +
3Draft RFC: Two-Round Threshold Schnorr Signatures with FROST
+ + + + + + + +
4Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 5: Two-Round FROST Signing Protocol
+ + + + + + + +
5Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 7.3: Removing the Coordinator Role
+ + + + + + + +
6Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Section 3.1: Prime-Order Group
+ + + + + + + +
7Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix B: Schnorr Signature Generation and Verification for Prime-Order Groups
+ + + + + + + +
8Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix B: Trusted Dealer Key Generation
+ + + + + + + +
9Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix C: Random Scalar Generation
+ + + + + + + +
10Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.7: RedDSA, RedJubjub, and RedPallas
+ + + + + + + +
11Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.7.1: Spend Authorization Signature (Sapling and Orchard)
+ + + + + + + +
12Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 4.15: Spend Authorization Signature (Sapling and Orchard)
+ + + + + + + +
13Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.9.3: Jubjub
+ + + + + + + +
14Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.9.6: Pallas and Vesta
+ + + + + + + +
15Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1
+
\ No newline at end of file diff --git a/zip-0312.rst b/zip-0312.rst index 76cf1527..89168bc1 100644 --- a/zip-0312.rst +++ b/zip-0312.rst @@ -98,7 +98,7 @@ 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. + without the approval of ``MIN_PARTICIPANTS`` 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. @@ -110,7 +110,7 @@ Non-requirements #[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 +- 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. @@ -125,7 +125,7 @@ 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 +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`. @@ -145,8 +145,10 @@ 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). +though that is not required. This allows using distributed key generation, since +the key it generates is unpredictable. Note however that note deriving :math:`\mathsf{ask}` +from :math:`\mathsf{sk}` prevents using seed phrases to recover the original +secret (which may be something desirable in the context of FROST). Randomizer Generation @@ -168,41 +170,10 @@ A new helper function is defined, which computes :math:`\mathsf{RedDSA.GenRandom return H3(randomizer_input) -Binding Factor Computation -'''''''''''''''''''''''''' - -The `compute_binding_factors` function is changed to receive the `randomizer_commitment` -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_commitment, 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_commitment): - 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_commitment) - - 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 +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. @@ -214,11 +185,11 @@ involves these steps: Round Two - Signature Share Generation '''''''''''''''''''''''''''''''''''''' -In Round Two, the Coordinator generates a random scalar `randomizer` by calling -`randomizer_generate`. Then it computes `randomizer_commitment = 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 Round Two, the Coordinator generates a random scalar ``randomizer`` by calling +``randomizer_generate`` 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 does not convey enough information for the signers to @@ -230,149 +201,26 @@ that the given SIGHASH matches the data sent from the Coordinator, or compute th 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_commitment` and incorporate it -into the computation of the binding factor. It is specified as the following: :: +The ``sign`` function remains unchanged, but its inputs must be modified relative +to the ``randomizer`` as 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_commitment, 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_commitment - - # Compute the binding factor(s) - binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_commitment) - 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 +- ``sk_i = sk_i + randomizer`` +- ``group_public_key = group_public_key + G.ScalarBaseMult(randomizer)`` Signature Share Verification and Aggregation '''''''''''''''''''''''''''''''''''''''''''' -The `aggregate` function is changed to incorporate the randomizer as follows: :: +The ``aggregate`` function remains unchanged, but its inputs must be modified +relative to the ``randomizer`` as following: - 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. +- ``group_public_key = group_public_key + G.ScalarBaseMult(randomizer)`` - Outputs: - - (R, z), a Schnorr signature consisting of an Element R and Scalar z. - - randomized_group_public_key, the randomized group public key +The ``verify_signature_share`` function remains unchanged, but its inputs must be modified +relative to the ``randomizer`` as following: - def aggregate(commitment_list, msg, sig_shares, group_public_key, randomizer): - # Compute the randomized group public key - randomizer_commitment = G.ScalarBaseMult(randomizer) - randomized_group_public_key = group_public_key + randomizer_commitment - - # Compute the binding factors - binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_commitment) - - # 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_commitment, 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_commitment): - # Compute the randomized group public key - randomized_group_public_key = group_public_key + randomizer_commitment - - # Compute the binding factors - binding_factor_list = compute_binding_factors(commitment_list, msg, randomizer_commitment) - 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 +- ``PK_i = PK_i + G.ScalarBaseMult(randomizer)`` +- ``group_public_key = group_public_key + G.ScalarBaseMult(randomizer)`` @@ -384,17 +232,17 @@ Ciphersuites FROST(Jubjub, BLAKE2b-512) '''''''''''''''''''''''''' -This ciphersuite uses Jubjub for the Group and BLAKE2b-512 for the Hash function `H` +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}}` +- 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. + \[0, ``G.Order()`` - 1\]. 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 @@ -404,22 +252,22 @@ Authorization Signatures as specified in [#protocol-concretespendauthsig]_. 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 \[0, `G.Order()` - 1\]. + 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), +- 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()`. + 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()`. + 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()`. + modulo ``G.Order()``. - H4(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubM", m). - H5(m): Implemented by computing BLAKE2b-512("FROST_RedJubjubC", m). @@ -430,17 +278,17 @@ for RedJubjub. FROST(Pallas, BLAKE2b-512) '''''''''''''''''''''''''' -This ciphersuite uses Pallas for the Group and BLAKE2b-512 for the Hash function `H` +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}}` +- 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. + \[0, ``G.Order()`` - 1\]. 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 @@ -449,22 +297,22 @@ Authorization Signatures as specified in [#protocol-concretespendauthsig]_. 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 \[0, `G.Order()` - 1\]. + 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), +- 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()`. + 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()`. + 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()`. + modulo ``G.Order()``. - H4(m): Implemented by computing BLAKE2b-512("FROST_RedPallasM", m). - H5(m): Implemented by computing BLAKE2b-512("FROST_RedPallasC", m). @@ -487,6 +335,7 @@ by the :math:`\mathsf{RedDSA.Validate}` function specified in :math:`\mathsf{RedDSA.Validate}`, must yield the values expected by the function. This is ensured by defining SerializeElement and SerializeScalar in each ciphersuite to yield those values. + - The challenge c used during FROST signing must be equal to the challenge c computed during :math:`\mathsf{RedDSA.Validate}`. This requires defining the ciphersuite H2 function as the :math:`\mathsf{H}^\circledast(m)` Zcash @@ -494,13 +343,14 @@ by the :math:`\mathsf{RedDSA.Validate}` function specified in Fortunately FROST and Zcash use the same input order (R, public key, message) so we just need to make sure that SerializeElement (used to compute the encoded public key before passing to the hash function) matches what - :math:`\mathsf{RedDSA.Validate}` expects; which is possible since both - :underline:`R` and :underline:`vk` (the public key) are encoded in the same - way in Zcash. -- Note that `r` (and thus `R`) will not be generated as specified in RedDSA.Sign. + :math:`\mathsf{RedDSA.Validate}` expects; which is possible since both `R` and + `vk` (the public key) are encoded in the same way in Zcash. + +- Note that ``r`` (and thus ``R``) will not be generated as specified in RedDSA.Sign. This is not an issue however, since with Schnorr signatures it does not matter - for the verifier how the `r` value was chosen, it just needs to be generated + for the verifier how the ``r`` value was chosen, it just needs to be generated uniformly at random, which is true for FROST. + - The above will ensure that the verification equation in :math:`\mathsf{RedDSA.Validate}` will pass, since FROST ensures the exact same equation will be valid as described in [#frost-primeorderverify]_. @@ -512,19 +362,25 @@ signing generates a re-randomized signature: This is exactly what is done in the functions defined above. - The re-randomization must be done in each signature share generation, such that the aggregated signature must be valid under verification with the - randomized public key. The `R` value from the signature is not influenced by - the randomizer so we just need to focus on the `z` value (using FROST - notation). Recall that `z` must equal to `r + (c * sk)`. FROST generates - signature shares so that when they are all add up to this value. Under - re-randomization it must be equal to `r + (c * (sk + randomizer))` (see + randomized public key. The ``R`` value from the signature is not influenced by + the randomizer so we just need to focus on the ``z`` value (using FROST + notation). Recall that ``z`` must equal to ``r + (c * sk)``, and that each + signature share is ``z_i = (hiding_nonce + (binding_nonce * binding_factor)) + + (lambda_i * c * sk_i)``. The first terms are not influenced by the randomizer + so we can only look into the second term of each top-level addition, i.e. ``c + * sk`` must be equal to ``sum(lambda_i * c * sk_i)`` for each participant + ``i``. Under re-randomization these become ``c * (sk + randomizer)`` (see :math:`\mathsf{RedDSA.RandomizedPrivate}`, which refers to the randomizer as - :math:`\alpha`). This can be rewritten as `r + (c * sk) + (c * randomizer)`. - In other words, we can simply generate the signature shares using the original - FROST procedure, and then add `(c * randomizer)` to `z` in the aggregate step. + :math:`\alpha`) and ``sum(lambda_i * c * (sk_i + randomizer))``. The latter + can be rewritten as ``c * (sum(lambda_i * sk_i) + randomizer * + sum(lambda_i)``. Since ``sum(lambda_i * sk_i) == sk`` per the Shamir secret + sharing mechanism used by FROST, and since ``sum(lambda_i) == 1`` + [#sum-lambda-proof]_, we arrive at ``c * (sk + randomizer)`` as required. + - The re-randomization procedure must be exactly the same as in [#protocol-concretereddsa]_ to ensure that re-randomized keys are uniformly distributed and signatures are unlinkable. This is also true; observe that - `randomizer_generate` is exactly the same as + ``randomizer_generate`` is exactly the same as :math:`\mathsf{RedDSA.GenRandom}`; and signature generation is compatible with :math:`\mathsf{RedDSA.RandomizedPrivate}`, :math:`\mathsf{RedDSA.RandomizedPublic}`, :math:`\mathsf{RedDSA.Sign}` and @@ -542,15 +398,16 @@ 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-primeorderverify] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix B: Schnorr Signature Generation and Verification for Prime-Order Groups `_ -.. [#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 `_ +.. [#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-primeorderverify] `Draft RFC: Two-Round Threshold Schnorr Signatures with FROST. Appendix B: Schnorr Signature Generation and Verification for Prime-Order Groups `_ +.. [#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 `_ +.. [#protocol-pallasandvesta] `Zcash Protocol Specification, Version 2022.3.4 [NU5]. Section 5.4.9.6: Pallas and Vesta `_ +.. [#sum-lambda-proof] `Prove that the sum of the Lagrange (interpolation) coefficients is equal to 1 `_