BIP 0327

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  BIP: 327
  Title: MuSig2 for BIP340-compatible Multi-Signatures
  Author: Jonas Nick <jonasd.nick@gmail.com>
          Tim Ruffing <crypto@timruffing.de>
          Elliott Jin <elliott.jin@gmail.com>
  Status: Draft
  License: BSD-3-Clause
  Type: Informational
  Created: 2022-03-22
  Post-History: 2022-04-05: https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2022-April/020198.html [bitcoin-dev] MuSig2 BIP
                2022-10-11: https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2022-October/021000.html [bitcoin-dev] MuSig2 BIP
  Comments-URI: https://github.com/bitcoin/bips/wiki/Comments:BIP-0327

Introduction

Abstract

This document proposes a standard for the MuSig2 multi-signature scheme. The standard is compatible with BIP340 public keys and signatures. It supports tweaking, which allows deriving BIP32 child keys from aggregate public keys and creating BIP341 Taproot outputs with key and script paths.

Copyright

This document is licensed under the 3-clause BSD license.

Motivation

MuSig2 is a multi-signature scheme that allows multiple signers to create a single aggregate public key and cooperatively create ordinary Schnorr signatures valid under the aggregate public key. Signing requires interaction between all signers involved in key aggregation. (MuSig2 is a n-of-n multi-signature scheme and not a t-of-n threshold-signature scheme.)

The primary motivation is to create a standard that allows users of different software projects to jointly control Taproot outputs (BIP341). Such an output contains a public key which, in this case, would be the aggregate of all users' individual public keys. It can be spent using MuSig2 to produce a signature for the key-based spending path.

The on-chain footprint of a MuSig2 Taproot output is essentially a single BIP340 public key, and a transaction spending the output only requires a single signature cooperatively produced by all signers. This is more compact and has lower verification cost than each signer providing an individual public key and signature, as would be required by an n-of-n policy implemented using OP_CHECKSIGADD as introduced in (BIP342). As a side effect, the number n of signers is not limited by any consensus rules when using MuSig2.

Moreover, MuSig2 offers a higher level of privacy than OP_CHECKSIGADD: MuSig2 Taproot outputs are indistinguishable for a blockchain observer from regular, single-signer Taproot outputs even though they are actually controlled by multiple signers. By tweaking an aggregate public key, the shared Taproot output can have script spending paths that are hidden unless used.

There are multi-signature schemes other than MuSig2 that are fully compatible with Schnorr signatures. The MuSig2 variant proposed below stands out by combining all the following features:

• Simple Key Setup: Key aggregation is non-interactive and fully compatible with BIP340 public keys.
• Two Communication Rounds: MuSig2 is faster in practice than previous three-round multi-signature schemes such as MuSig1, particularly when signers are connected through high-latency anonymous links. Moreover, the need for fewer communication rounds simplifies the algorithms and reduces the probability that implementations and users make security-relevant mistakes.
• Provable security: MuSig2 has been proven existentially unforgeable under the algebraic one-more discrete logarithm (AOMDL) assumption (instead of the discrete logarithm assumption required for single-signer Schnorr signatures). AOMDL is a falsifiable and weaker variant of the well-studied OMDL problem.
• Low complexity: MuSig2 has a substantially lower computational and implementation complexity than alternative schemes like MuSig-DN. However, this comes at the cost of having no ability to generate nonces deterministically and the requirement to securely handle signing state.

Design

• Compatibility with BIP340: In this proposal, the aggregate public key is a BIP340 X-only public key, and the signature output at the end of the signing protocol is a BIP340 signature that passes BIP340 verification for the aggregate public key and a message. The individual public keys that are input to the key aggregation algorithm are plain public keys in compressed format.
• Tweaking for BIP32 derivations and Taproot: This proposal supports tweaking aggregate public keys and signing for tweaked aggregate public keys. We distinguish two modes of tweaking: Plain tweaking can be used to derive child aggregate public keys per BIP32. X-only tweaking, on the other hand, allows creating a BIP341 tweak to add script paths to a Taproot output. See below for details.
• Non-interactive signing with preprocessing: The first communication round, exchanging the nonces, can happen before the message or the exact set of signers is determined. Once the parameters of the signing session are finalized, the signers can send partial signatures without additional interaction.
• Key aggregation optionally independent of order: The output of the key aggregation algorithm depends on the order in which the individual public keys are provided as input. Key aggregation does not sort the individual public keys by default because applications often already have a canonical order of signers. Nonetheless, applications can mandate sorting before aggregation, and this proposal specifies a canonical order to sort the individual public keys before key aggregation. Sorting will ensure the same output, independent of the initial order.
• Third-party nonce and partial signature aggregation: Instead of every signer sending their nonce and partial signature to every other signer, it is possible to use an untrusted third-party aggregator in order to reduce the communication complexity from quadratic to linear in the number of signers. In each of the two rounds, the aggregator collects all signers' contributions (nonces or partial signatures), aggregates them, and broadcasts the aggregate back to the signers. A malicious aggregator can force the signing session to fail to produce a valid Schnorr signature but cannot negatively affect the unforgeability of the scheme.
• Partial signature verification: If any signer sends a partial signature contribution that was not created by honestly following the signing protocol, the signing session will fail to produce a valid Schnorr signature. This proposal specifies a partial signature verification algorithm to identify disruptive signers. It is incompatible with third-party nonce aggregation because the individual nonce is required for partial verification.
• MuSig2* optimization: This proposal uses an optimized scheme MuSig2*, which allows saving a point multiplication in key aggregation as compared to MuSig2. MuSig2* is proven secure in the appendix of the MuSig2 paper. The optimization consists of assigning the constant key aggregation coefficient 1 to the second distinct key in the list of individual public keys to be aggregated (as well as to any key identical to this key).
• Size of the nonce and security: In this proposal, each signer's nonce consists of two elliptic curve points. The MuSig2 paper gives distinct security proofs depending on the number of points that constitute a nonce. See section Choosing the Size of the Nonce for a discussion.

Overview

Implementers must make sure to understand this section thoroughly to avoid subtle mistakes that may lead to catastrophic failure.

Optionality of Features

The goal of this proposal is to support a wide range of possible application scenarios. Given a specific application scenario, some features may be unnecessary or not desirable, and implementers can choose not to support them. Such optional features include:

• Applying plain tweaks after x-only tweaks.
• Applying tweaks at all.
• Dealing with messages that are not exactly 32 bytes.
• Identifying a disruptive signer after aborting (aborting itself remains mandatory).
• Dealing with duplicate individual public keys in key aggregation.

If applicable, the corresponding algorithms should simply fail when encountering inputs unsupported by a particular implementation. (For example, the signing algorithm may fail when given a message which is not 32 bytes.) Similarly, the test vectors that exercise the unimplemented features should be re-interpreted to expect an error, or be skipped if appropriate.

General Signing Flow

The signers start by exchanging their individual public keys and computing an aggregate public key using the KeyAgg algorithm. Whenever they want to sign a message, the basic order of operations to create a multi-signature is as follows:

First broadcast round: The signers start the signing session by running NonceGen to compute secnonce and pubnonce. Then, the signers broadcast their pubnonce to each other and run NonceAgg to compute an aggregate nonce.

Second broadcast round: At this point, every signer has the required data to sign, which, in the algorithms specified below, is stored in a data structure called Session Context. Every signer computes a partial signature by running Sign with the secret signing key, the secnonce and the session context. Then, the signers broadcast their partial signatures to each other and run PartialSigAgg to obtain the final signature. If all signers behaved honestly, the result passes BIP340 verification.

Both broadcast rounds can be optimized by using an aggregator who collects all signers' nonces or partial signatures, aggregates them using NonceAgg or PartialSigAgg, respectively, and broadcasts the aggregate result back to the signers. A malicious aggregator can force the signing session to fail to produce a valid Schnorr signature but cannot negatively affect the unforgeability of the scheme, i.e., even a malicious aggregator colluding with all but one signer cannot forge a signature.

IMPORTANT: The Sign algorithm must not be executed twice with the same secnonce. Otherwise, it is possible to extract the secret signing key from the two partial signatures output by the two executions of Sign. To avoid accidental reuse of secnonce, an implementation may securely erase the secnonce argument by overwriting it with 64 zero bytes after it has been read by Sign. A secnonce consisting of only zero bytes is invalid for Sign and will cause it to fail.

To simplify the specification of the algorithms, some intermediary values are unnecessarily recomputed from scratch, e.g., when executing GetSessionValues multiple times. Actual implementations can cache these values. As a result, the Session Context may look very different in implementations or may not exist at all. However, computation of GetSessionValues and storage of the result must be protected against modification from an untrusted third party. This party would have complete control over the aggregate public key and message to be signed.

Public Key Aggregation

We distinguish between two public key types, namely plain public keys, the key type traditionally used in Bitcoin, and X-only public keys. Plain public keys are byte strings of length 33 (often called compressed format). In contrast, X-only public keys are 32-byte strings defined in BIP340.

The individual public keys of signers as input to the key aggregation algorithm KeyAgg (and to GetSessionValues and PartialSigVerify) are plain public keys. The output of KeyAgg is a KeyAgg Context which stores information required for tweaking the aggregate public key (see below), and it can be used to produce an X-only aggregate public key, or a plain aggregate public key. In order to obtain an X-only public key compatible with BIP340 verification, implementations call the GetXonlyPubkey function with the KeyAgg Context. To get the plain aggregate public key, which is required for some applications of tweaking, implementations call GetPlainPubkey instead.

The aggregate public key produced by KeyAgg (regardless of the type) depends on the order of the individual public keys. If the application does not have a canonical order of the signers, the individual public keys can be sorted with the KeySort algorithm to ensure that the aggregate public key is independent of the order of signers.

The same individual public key is allowed to occur more than once in the input of KeyAgg and KeySort. This is by design: All algorithms in this proposal handle multiple signers who (claim to) have identical individual public keys properly, and applications are not required to check for duplicate individual public keys. In fact, applications are recommended to omit checks for duplicate individual public keys in order to simplify error handling. Moreover, it is often impossible to tell at key aggregation which signer is to blame for the duplicate, i.e., which signer came up with an individual public key honestly and which disruptive signer copied it. In contrast, MuSig2 is designed to identify disruptive signers at signing time (see Identifying Disruptive Signers).

While the algorithms in this proposal are able to handle duplicate individual public keys, there are scenarios where applications may choose to abort when encountering duplicates. For example, we can imagine a scenario where a single entity creates a MuSig2 setup with multiple signing devices. In that case, duplicates may not result from a malicious signing device copying an individual public key of another signing device but from accidental initialization of two devices with the same seed. Since MuSig2 key aggregation would accept the duplicate keys and not error out, which would in turn reduce the security compared to the intended key setup, applications may reject duplicate individual public keys before passing them to MuSig2 key aggregation and ask the user to investigate.

Nonce Generation

IMPORTANT: NonceGen must have access to a high-quality random generator to draw an unbiased, uniformly random value rand' . In contrast to BIP340 signing, the values k₁ and k₂ must not be derived deterministically from the session parameters because otherwise active adversaries can trick the victim into reusing a nonce.

The optional arguments to NonceGen enable a defense-in-depth mechanism that may prevent secret key exposure if rand' is accidentally not drawn uniformly at random. If the value rand' was identical in two NonceGen invocations, but any other argument was different, the secnonce would still be guaranteed to be different as well (with overwhelming probability), and thus accidentally using the same secnonce for Sign in both sessions would be avoided. Therefore, it is recommended to provide the optional arguments sk, aggpk, and m if these session parameters are already determined during nonce generation. The auxiliary input extra_in can contain additional contextual data that has a chance of changing between NonceGen runs, e.g., a supposedly unique session id (taken from the application), a session counter wide enough not to repeat in practice, any nonces by other signers (if already known), or the serialization of a data structure containing multiple of the above. However, the protection provided by the optional arguments should only be viewed as a last resort. In most conceivable scenarios, the assumption that the arguments are different between two executions of NonceGen is relatively strong, particularly when facing an active adversary.

In some applications, it is beneficial to generate and send a pubnonce before the other signers, their individual public keys, or the message to sign is known. In this case, only the available arguments are provided to the NonceGen algorithm. After this preprocessing phase, the Sign algorithm can be run immediately when the message and set of signers is determined. This way, the final signature is created quicker and with fewer round trips. However, applications that use this method presumably store the nonces for a longer time and must therefore be even more careful not to reuse them. Moreover, this method is not compatible with the defense-in-depth mechanism described in the previous paragraph.

Instead of every signer broadcasting their pubnonce to every other signer, the signers can send their pubnonce to a single aggregator node that runs NonceAgg and sends the aggnonce back to the signers. This technique reduces the overall communication. A malicious aggregator can force the signing session to fail to produce a valid Schnorr signature but cannot negatively affect the unforgeability of the scheme.

In general, MuSig2 signers are stateful in the sense that they first generate secnonce and then need to store it until they receive the other signers' pubnonces or the aggnonce. However, it is possible for one of the signers to be stateless. This signer waits until it receives the pubnonce of all the other signers and until session parameters such as a message to sign, individual public keys, and tweaks are determined. Then, the signer can run NonceGen, NonceAgg and Sign in sequence and send out its pubnonce along with its partial signature. Stateless signers may want to consider signing deterministically (see Modifications to Nonce Generation) to remove the reliance on the random number generator in the NonceGen algorithm.

Identifying Disruptive Signers

The signing protocol makes it possible to identify malicious signers who send invalid contributions to a signing session in order to make the signing session abort and prevent the honest signers from obtaining a valid signature. This property is called "identifiable aborts" and ensures that honest parties can assign blame to malicious signers who cause an abort in the signing protocol.

Aborts are identifiable for an honest party if the following conditions hold in a signing session:

• The contributions received from all signers have not been tampered with (e.g., because they were sent over authenticated connections).
• Nonce aggregation is performed honestly (e.g., because the honest signer performs nonce aggregation on its own or because the aggregator is trusted).
• The partial signatures received from all signers are verified using the algorithm PartialSigVerify.

If these conditions hold and an honest party (signer or aggregator) runs an algorithm that fails due to invalid protocol contributions from malicious signers, then the algorithm run by the honest party will output the index of exactly one malicious signer. Additionally, if the honest parties agree on the contributions sent by all signers in the signing session, all the honest parties who run the aborting algorithm will identify the same malicious signer.

Further Remarks

Some of the algorithms specified below may also assign blame to a malicious aggregator. While this is possible for some particular misbehavior of the aggregator, it is not guaranteed that a malicious aggregator can be identified. More specifically, a malicious aggregator (whose existence violates the second condition above) can always make signing abort and wrongly hold honest signers accountable for the abort (e.g., by claiming to have received an invalid contribution from a particular honest signer).

The only purpose of the algorithm PartialSigVerify is to ensure identifiable aborts, and it is not necessary to use it when identifiable aborts are not desired. In particular, partial signatures are not signatures. An adversary can forge a partial signature, i.e., create a partial signature without knowing the secret key for the claimed individual public key. However, if PartialSigVerify succeeds for all partial signatures then PartialSigAgg will return a valid Schnorr signature.

Tweaking the Aggregate Public Key

The aggregate public key can be tweaked, which modifies the key as defined in the Tweaking Definition subsection. In order to apply a tweak, the KeyAgg Context output by KeyAgg is provided to the ApplyTweak algorithm with the is_xonly_t argument set to false for plain tweaking and true for X-only tweaking. The resulting KeyAgg Context can be used to apply another tweak with ApplyTweak or obtain the aggregate public key with GetXonlyPubkey or GetPlainPubkey.

In addition to individual public keys, the KeyAgg algorithm accepts tweaks, which modify the aggregate public key as defined in the Tweaking Definition subsection. For example, if KeyAgg is run with v = 2, is_xonly_t₁ = false, is_xonly_t₂ = true, then the aggregate key is first plain tweaked with tweak₁ and then X-only tweaked with tweak₂.

The purpose of supporting tweaking is to ensure compatibility with existing uses of tweaking, i.e., that the result of signing is a valid signature for the tweaked public key. The MuSig2 algorithms take arbitrary tweaks as input but accepting arbitrary tweaks may negatively affect the security of the scheme. Instead, signers should obtain the tweaks according to other specifications. This typically involves deriving the tweaks from a hash of the aggregate public key and some other information. Depending on the specific scheme that is used for tweaking, either the plain or the X-only aggregate public key is required. For example, to do BIP32 derivation, you call GetPlainPubkey to be able to compute the tweak, whereas BIP341 TapTweaks require X-only public keys that are obtained with GetXonlyPubkey.

The tweak mode provided to ApplyTweak depends on the application: Plain tweaking can be used to derive child public keys from an aggregate public key using BIP32. On the other hand, X-only tweaking is required for Taproot tweaking per BIP341. A Taproot-tweaked public key commits to a script path, allowing users to create transaction outputs that are spendable either with a MuSig2 multi-signature or by providing inputs that satisfy the script path. Script path spends require a control block that contains a parity bit for the tweaked X-only public key. The bit can be obtained with GetPlainPubkey(keyagg_ctx)[0] & 1.

Algorithms

The following specification of the algorithms has been written with a focus on clarity. As a result, the specified algorithms are not always optimal in terms of computation and space. In particular, some values are recomputed but can be cached in actual implementations (see General Signing Flow).

Notation

The following conventions are used, with constants as defined for secp256k1. We note that adapting this proposal to other elliptic curves is not straightforward and can result in an insecure scheme.

• Lowercase variables represent integers or byte arrays.
  • The constant p refers to the field size, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F.
  • The constant n refers to the curve order, 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141.
• Uppercase variables refer to points on the curve with equation y² = x³ + 7 over the integers modulo p.
  • is_infinite(P) returns whether P is the point at infinity.
  • x(P) and y(P) are integers in the range 0..p-1 and refer to the X and Y coordinates of a point P (assuming it is not infinity).
  • The constant G refers to the base point, for which x(G) = 0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798 and y(G) = 0x483ADA7726A3C4655DA4FBFC0E1108A8FD17B448A68554199C47D08FFB10D4B8.
  • Addition of points refers to the usual elliptic curve group operation.
  • Multiplication (⋅) of an integer and a point refers to the repeated application of the group operation.
• Functions and operations:
  • || refers to byte array concatenation.
  • The function x[i:j], where x is a byte array and i, j ≥ 0, returns a (j - i)-byte array with a copy of the i-th byte (inclusive) to the j-th byte (exclusive) of x.
  • The function bytes(n, x), where x is an integer, returns the n-byte encoding of x, most significant byte first.
  • The constant empty_bytestring refers to the empty byte array. It holds that len(empty_bytestring) = 0.
  • The function xbytes(P), where P is a point for which not is_infinite(P), returns bytes(32, x(P)).
  • The function len(x) where x is a byte array returns the length of the array.
  • The function has_even_y(P), where P is a point for which not is_infinite(P), returns y(P) mod 2 == 0.
  • The function with_even_y(P), where P is a point, returns P if is_infinite(P) or has_even_y(P). Otherwise, with_even_y(P) returns -P.
  • The function cbytes(P), where P is a point for which not is_infinite(P), returns a || xbytes(P) where a is a byte that is 2 if has_even_y(P) and 3 otherwise.
  • The function cbytes_ext(P), where P is a point, returns bytes(33, 0) if is_infinite(P). Otherwise, it returns cbytes(P).
  • The function int(x), where x is a 32-byte array, returns the 256-bit unsigned integer whose most significant byte first encoding is x.
  • The function lift_x(x), where x is an integer in range 0..2²⁵⁶-1, returns the point P for which x(P) = x and has_even_y(P), or fails if x is greater than p-1 or no such point exists. The function lift_x(x) is equivalent to the following pseudocode:
    • Fail if x > p-1.
    • Let c = x³ + 7 mod p.
    • Let y' = c^(p+1)/4 mod p.
    • Fail if c ≠ y'² mod p.
    • Let y = y' if y' mod 2 = 0, otherwise let y = p - y' .
    • Return the unique point P such that x(P) = x and y(P) = y.
  • The function cpoint(x), where x is a 33-byte array (compressed serialization), sets P = lift_x(int(x[1:33])) and fails if that fails. If x[0] = 2 it returns P and if x[0] = 3 it returns -P. Otherwise, it fails.
  • The function cpoint_ext(x), where x is a 33-byte array (compressed serialization), returns the point at infinity if x = bytes(33, 0). Otherwise, it returns cpoint(x) and fails if that fails.
  • The function hash_tag(x) where tag is a UTF-8 encoded tag name and x is a byte array returns the 32-byte hash SHA256(SHA256(tag) || SHA256(tag) || x).
• Other:
  • Tuples are written by listing the elements within parentheses and separated by commas. For example, (2, 3, 1) is a tuple.

Key Generation and Aggregation

Key Generation of an Individual Signer

Algorithm IndividualPubkey(sk):

• Inputs:
  • The secret key sk: a 32-byte array, freshly generated uniformly at random
• Let d' = int(sk).
• Fail if d' = 0 or d' ≥ n.
• Return cbytes(d'⋅G).

KeyAgg Context

The KeyAgg Context is a data structure consisting of the following elements:

• The point Q representing the potentially tweaked aggregate public key: an elliptic curve point
• The accumulated tweak tacc: an integer with 0 ≤ tacc < n
• The value gacc : 1 or -1 mod n

We write "Let (Q, gacc, tacc) = keyagg_ctx" to assign names to the elements of a KeyAgg Context.

Algorithm GetXonlyPubkey(keyagg_ctx):

• Let (Q, _, _) = keyagg_ctx
• Return xbytes(Q)

Algorithm GetPlainPubkey(keyagg_ctx):

• Let (Q, _, _) = keyagg_ctx
• Return cbytes(Q)

Key Sorting

Algorithm KeySort(pk_1..u):

• Inputs:
  • The number u of individual public keys with 0 < u < 2^32
  • The individual public keys pk_1..u: u 33-byte arrays
• Return pk_1..u sorted in lexicographical order.

Key Aggregation

Algorithm KeyAgg(pk_1..u):

• Inputs:
  • The number u of individual public keys with 0 < u < 2^32
  • The individual public keys pk_1..u: u 33-byte arrays
• Let pk2 = GetSecondKey(pk_1..u)
• For i = 1 .. u:
  • Let P_i = cpoint(pk_i); fail if that fails and blame signer i for invalid individual public key.
  • Let a_i = KeyAggCoeffInternal(pk_1..u, pk_i, pk2).
• Let Q = a₁⋅P₁ + a₂⋅P₂ + ... + a_u⋅P_u
• Fail if is_infinite(Q).
• Let gacc = 1
• Let tacc = 0
• Return keyagg_ctx = (Q, gacc, tacc).

Internal Algorithm HashKeys(pk_1..u):

• Return hash_{KeyAgg list}(pk₁ || pk₂ || ... || pk_u)

Internal Algorithm GetSecondKey(pk_1..u):

• For j = 1 .. u:
  • If pk_j ≠ pk₁:
    • Return pk_j
• Return bytes(33, 0)

Internal Algorithm KeyAggCoeff(pk_1..u, pk'):

• Let pk2 = GetSecondKey(pk_1..u):
• Return KeyAggCoeffInternal(pk_1..u, pk', pk2)

Internal Algorithm KeyAggCoeffInternal(pk_1..u, pk', pk2):

• Let L = HashKeys(pk_1..u)
• If pk' = pk2:
  • Return 1
• Return int(hash_{KeyAgg coefficient}(L || pk')) mod n

Applying Tweaks

Algorithm ApplyTweak(keyagg_ctx, tweak, is_xonly_t):

• Inputs:
  • The keyagg_ctx: a KeyAgg Context data structure
  • The tweak: a 32-byte array
  • The tweak mode is_xonly_t: a boolean
• Let (Q, gacc, tacc) = keyagg_ctx
• If is_xonly_t and not has_even_y(Q):
  • Let g = -1 mod n
• Else:
  • Let g = 1
• Let t = int(tweak); fail if t ≥ n
• Let Q' = g⋅Q + t⋅G
  • Fail if is_infinite(Q')
• Let gacc' = g⋅gacc mod n
• Let tacc' = t + g⋅tacc mod n
• Return keyagg_ctx' = (Q', gacc', tacc')

Nonce Generation

Algorithm NonceGen(sk, pk, aggpk, m, extra_in):

• Inputs:
  • The secret signing key sk: a 32-byte array (optional argument)
  • The individual public key pk: a 33-byte array (see Signing with Tweaked Individual Keys for the reason that this argument is mandatory)
  • The x-only aggregate public key aggpk: a 32-byte array (optional argument)
  • The message m: a byte array (optional argument)
  • The auxiliary input extra_in: a byte array with 0 ≤ len(extra_in) ≤ 2³²-1 (optional argument)
• Let rand' be a 32-byte array freshly drawn uniformly at random
• If the optional argument sk is present:
  • Let rand be the byte-wise xor of sk and hash_MuSig/aux(rand')
• Else:
  • Let rand = rand'
• If the optional argument aggpk is not present:
  • Let aggpk = empty_bytestring
• If the optional argument m is not present:
  • Let m_prefixed = bytes(1, 0)
• Else:
  • Let m_prefixed = bytes(1, 1) || bytes(8, len(m)) || m
• If the optional argument extra_in is not present:
  • Let extra_in = empty_bytestring
• Let k_i = int(hash_MuSig/nonce(rand || bytes(1, len(pk)) || pk || bytes(1, len(aggpk)) || aggpk || m_prefixed || bytes(4, len(extra_in)) || extra_in || bytes(1, i - 1))) mod n for i = 1,2
• Fail if k₁ = 0 or k₂ = 0
• Let R_⁎,1 = k₁⋅G, R_⁎,2 = k₂⋅G
• Let pubnonce = cbytes(R_⁎,1) || cbytes(R_⁎,2)
• Let secnonce = bytes(32, k₁) || bytes(32, k₂) || pk
• Return (secnonce, pubnonce)

Nonce Aggregation

Algorithm NonceAgg(pubnonce_1..u):

• Inputs:
  • The number u of pubnonces with 0 < u < 2^32
  • The public nonces pubnonce_1..u: u 66-byte arrays
• For j = 1 .. 2:
  • For i = 1 .. u:
    • Let R_i,j = cpoint(pubnonce_i[(j-1)*33:j*33]); fail if that fails and blame signer i for invalid pubnonce.
  • Let R_j = R_1,j + R_2,j + ... + R_u,j
• Return aggnonce = cbytes_ext(R₁) || cbytes_ext(R₂)

Session Context

The Session Context is a data structure consisting of the following elements:

• The aggregate public nonce aggnonce: a 66-byte array
• The number u of individual public keys with 0 < u < 2^32
• The individual public keys pk_1..u: u 33-byte arrays
• The number v of tweaks with 0 ≤ v < 2^32
• The tweaks tweak_1..v: v 32-byte arrays
• The tweak modes is_xonly_t_1..v : v booleans
• The message m: a byte array

We write "Let (aggnonce, u, pk_1..u, v, tweak_1..v, is_xonly_t_1..v, m) = session_ctx" to assign names to the elements of a Session Context.

Algorithm GetSessionValues(session_ctx):

• Let (aggnonce, u, pk_1..u, v, tweak_1..v, is_xonly_t_1..v, m) = session_ctx
• Let keyagg_ctx₀ = KeyAgg(pk_1..u); fail if that fails
• For i = 1 .. v:
  • Let keyagg_ctx_i = ApplyTweak(keyagg_ctx_i-1, tweak_i, is_xonly_t_i); fail if that fails
• Let (Q, gacc, tacc) = keyagg_ctx_v
• Let b = int(hash_{MuSig/noncecoef}(aggnonce || xbytes(Q) || m)) mod n
• Let R₁ = cpoint_ext(aggnonce[0:33]), R₂ = cpoint_ext(aggnonce[33:66]); fail if that fails and blame nonce aggregator for invalid aggnonce.
• Let R' = R₁ + b⋅R₂
• If is_infinite(R'):
  • Let final nonce R = G (see Dealing with Infinity in Nonce Aggregation)
• Else:
  • Let final nonce R = R'
• Let e = int(hash_{BIP0340/challenge}(xbytes(R) || xbytes(Q) || m)) mod n
• Return (Q, gacc, tacc, b, R, e)

Algorithm GetSessionKeyAggCoeff(session_ctx, P):

• Let (_, u, pk_1..u, _, _, _, _) = session_ctx
• Let pk = cbytes(P)
• Fail if pk not in pk_1..u
• Return KeyAggCoeff(pk_1..u, pk)

Signing

Algorithm Sign(secnonce, sk, session_ctx):

• Inputs:
  • The secret nonce secnonce that has never been used as input to Sign before: a 97-byte array
  • The secret key sk: a 32-byte array
  • The session_ctx: a Session Context data structure
• Let (Q, gacc, _, b, R, e) = GetSessionValues(session_ctx); fail if that fails
• Let k₁' = int(secnonce[0:32]), k₂' = int(secnonce[32:64])
• Fail if k_i' = 0 or k_i' ≥ n for i = 1..2
• Let k₁ = k₁', k₂ = k₂' if has_even_y(R), otherwise let k₁ = n - k₁', k₂ = n - k₂'
• Let d' = int(sk)
• Fail if d' = 0 or d' ≥ n
• Let P = d'⋅G
• Let pk = cbytes(P)
• Fail if pk ≠ secnonce[64:97]
• Let a = GetSessionKeyAggCoeff(session_ctx, P); fail if that fails
• Let g = 1 if has_even_y(Q), otherwise let g = -1 mod n
• Let d = g⋅gacc⋅d' mod n (See Negation Of The Secret Key When Signing)
• Let s = (k₁ + b⋅k₂ + e⋅a⋅d) mod n
• Let psig = bytes(32, s)
• Let pubnonce = cbytes(k₁'⋅G) || cbytes(k₂'⋅G)
• If PartialSigVerifyInternal(psig, pubnonce, pk, session_ctx) (see below) returns failure, fail
• Return partial signature psig

Partial Signature Verification

Algorithm PartialSigVerify(psig, pubnonce_1..u, pk_1..u, tweak_1..v, is_xonly_t_1..v, m, i):

• Inputs:
  • The partial signature psig: a 32-byte array
  • The number u of public nonces and individual public keys with 0 < u < 2^32
  • The public nonces pubnonce_1..u: u 66-byte arrays
  • The individual public keys pk_1..u: u 33-byte arrays
  • The number v of tweaks with 0 ≤ v < 2^32
  • The tweaks tweak_1..v: v 32-byte arrays
  • The tweak modes is_xonly_t_1..v : v booleans
  • The message m: a byte array
  • The index of the signer i in the of public nonces and individual public keys with 0 < i ≤ u
• Let aggnonce = NonceAgg(pubnonce_1..u); fail if that fails
• Let session_ctx = (aggnonce, u, pk_1..u, v, tweak_1..v, is_xonly_t_1..v, m)
• Run PartialSigVerifyInternal(psig, pubnonce_i, pk_i, session_ctx)
• Return success iff no failure occurred before reaching this point.

Internal Algorithm PartialSigVerifyInternal(psig, pubnonce, pk, session_ctx):

• Let (Q, gacc, _, b, R, e) = GetSessionValues(session_ctx); fail if that fails
• Let s = int(psig); fail if s ≥ n
• Let R_⁎,1 = cpoint(pubnonce[0:33]), R_⁎,2 = cpoint(pubnonce[33:66])
• Let Re_⁎' = R_⁎,1 + b⋅R_⁎,2
• Let effective nonce Re_⁎ = Re_⁎' if has_even_y(R), otherwise let Re_⁎ = -Re_⁎'
• Let P = cpoint(pk); fail if that fails
• Let a = GetSessionKeyAggCoeff(session_ctx, P)
• Let g = 1 if has_even_y(Q), otherwise let g = -1 mod n
• Let g' = g⋅gacc mod n (See Negation Of The Individual Public Key When Partially Verifying)
• Fail if s⋅G ≠ Re_⁎ + e⋅a⋅g'⋅P
• Return success iff no failure occurred before reaching this point.

Partial Signature Aggregation

Algorithm PartialSigAgg(psig_1..u, session_ctx):

• Inputs:
  • The number u of signatures with 0 < u < 2^32
  • The partial signatures psig_1..u: u 32-byte arrays
  • The session_ctx: a Session Context data structure
• Let (Q, _, tacc, _, _, R, e) = GetSessionValues(session_ctx); fail if that fails
• For i = 1 .. u:
  • Let s_i = int(psig_i); fail if s_i ≥ n and blame signer i for invalid partial signature.
• Let g = 1 if has_even_y(Q), otherwise let g = -1 mod n
• Let s = s₁ + ... + s_u + e⋅g⋅tacc mod n
• Return sig = xbytes(R) || bytes(32, s)

Test Vectors and Reference Code

We provide a naive, highly inefficient, and non-constant time pure Python 3 reference implementation of the key aggregation, partial signing, and partial signature verification algorithms.

Standalone JSON test vectors are also available in the same directory, to facilitate porting the test vectors into other implementations.

The reference implementation is for demonstration purposes only and not to be used in production environments.

Remarks on Security and Correctness

Signing with Tweaked Individual Keys

The scheme in this proposal has been designed to be secure even if signers tweak their individual secret keys with tweaks known to the adversary (e.g., as in BIP32 unhardened derivation) before providing the corresponding individual public keys as input to key aggregation. In particular, the scheme as specified above requires each signer to provide a final individual public key pk already to NonceGen, which writes it into the secnonce array so that it can be checked against IndividualPubkey(sk) in the Sign algorithm. The purpose of this check in Sign is to ensure that pk, and thus the secret key sk that will be provided to Sign, is determined before the signer sends out the pubnonce.

If the check in Sign was omitted, and a signer supported signing with at least two different secret keys sk₁ and sk₂ which have been obtained via tweaking another secret key with tweaks known to the adversary, then the adversary could, after having seen the pubnonce, influence whether sk₁ or sk₂ is provided to Sign. This degree of freedom may allow the adversary to perform a generalized birthday attack and thereby forge a signature (see bitcoin-dev mailing list post and writeup for details).

Checking pk against IndividualPubkey(sk) is a simple way to ensure that the secret key provided to Sign is fully determined already when NonceGen is invoked. This removes the adversary's ability to influence the secret key after having seen the pubnonce and thus rules out the attack. Note that the scheme as given in the MuSig2 paper does not perform the check in Sign. However, the security model in the paper does not cover tweaking at all and assumes a single fixed secret key.

Modifications to Nonce Generation

Implementers must avoid modifying the NonceGen algorithm without being fully aware of the implications. We provide two modifications to NonceGen that are secure when applied correctly and may be useful in special circumstances, summarized in the following table.

	needs secure randomness	needs secure counter	needs to keep state securely	needs aggregate nonce of all other signers (only possible for one signer)
NonceGen	✓		✓
CounterNonceGen		✓	✓
DeterministicSign				✓

First, on systems where obtaining uniformly random values is much harder than maintaining a global atomic counter, it can be beneficial to modify NonceGen. The resulting algorithm CounterNonceGen does not draw rand' uniformly at random but instead sets rand' to the value of an atomic counter that is incremented whenever it is read. With this modification, the secret signing key sk of the signer generating the nonce is not an optional argument and must be provided to NonceGen. The security of the resulting scheme then depends on the requirement that reading the counter must never yield the same counter value in two NonceGen invocations with the same sk.

Second, if there is a unique signer who is supposed to send the pubnonce last, it is possible to modify nonce generation for this single signer to not require high-quality randomness. Such a nonce generation algorithm DeterministicSign is specified below. Note that the only optional argument is rand, which can be omitted if randomness is entirely unavailable. DeterministicSign requires the argument aggothernonce which should be set to the output of NonceAgg run on the pubnonce value of all other signers (but can be provided by an untrusted party). Hence, using DeterministicSign is only possible for the last signer to generate a nonce and makes the signer stateless, similar to the stateless signer described in the Nonce Generation section.

Deterministic and Stateless Signing for a Single Signer

Algorithm DeterministicSign(sk, aggothernonce, pk_1..u, tweak_1..v, is_xonly_t_1..v, m, rand):

• Inputs:
  • The secret signing key sk: a 32-byte array
  • The aggregate public nonce aggothernonce (see above): a 66-byte array
  • The number u of individual public keys with 0 < u < 2^32
  • The individual public keys pk_1..u: u 32-byte arrays
  • The number v of tweaks with 0 ≤ v < 2^32
  • The tweaks tweak_1..v: v 32-byte arrays
  • The tweak methods is_xonly_t_1..v: v booleans
  • The message m: a byte array
  • The auxiliary randomness rand: a 32-byte array (optional argument)
• If the optional argument rand is present:
  • Let sk' be the byte-wise xor of sk and hash_MuSig/aux(rand)
• Else:
  • Let sk' = sk
• Let keyagg_ctx₀ = KeyAgg(pk_1..u); fail if that fails
• For i = 1 .. v:
  • Let keyagg_ctx_i = ApplyTweak(keyagg_ctx_i-1, tweak_i, is_xonly_t_i); fail if that fails
• Let aggpk = GetPubkey(keyagg_ctx_v)
• Let k_i = int(hash_{MuSig/deterministic/nonce}(sk' || aggothernonce || aggpk || bytes(8, len(m)) || m || bytes(1, i - 1))) mod n for i = 1,2
• Fail if k₁ = 0 or k₂ = 0
• Let R_⁎,1 = k₁⋅G, R_⁎,2 = k₂⋅G
• Let pubnonce = cbytes(R_⁎,2) || cbytes(R_⁎,2)
• Let d = int(sk)
• Fail if d = 0 or d ≥ n
• Let pk = cbytes(d⋅G)
• Let secnonce = bytes(32, k₁) || bytes(32, k₂) || pk
• Let aggnonce = NonceAgg((pubnonce, aggothernonce)); fail if that fails and blame nonce aggregator for invalid aggothernonce.
• Let session_ctx = (aggnonce, u, pk_1..u, v, tweak_1..v, is_xonly_t_1..v, m)
• Return (pubnonce, Sign(secnonce, sk, session_ctx))

Tweaking Definition

Two modes of tweaking the aggregate public key are supported. They correspond to the following algorithms:

Algorithm ApplyPlainTweak(P, t):

• Inputs:
  • P: a point
  • The tweak t: an integer with 0 ≤ t < n
• Return P + t⋅G

Algorithm ApplyXonlyTweak(P, t):

• Return with_even_y(P) + t⋅G

Negation Of The Secret Key When Signing

In order to produce a partial signature for an X-only aggregate public key that is an aggregate of u individual public keys and tweaked v times (X-only or plain), the Sign algorithm may need to negate the secret key during the signing process.

<poem> The following elliptic curve points arise as intermediate steps when creating a signature: • P_i as computed in KeyAgg is the point corresponding to the i-th signer's individual public key. Defining d_i' to be the i-th signer's secret key as an integer, i.e., the d' value as computed in the Sign algorithm of the i-th signer, we have

   Pi = di'⋅G .

• Q₀ is the aggregate of the individual public keys. It is identical to value Q computed in KeyAgg and therefore defined as

   Q0 = a1⋅P1 + a2⋅P2 + ... + au⋅Pu.

• Q_i is the tweaked aggregate public key after the i-th execution of ApplyTweak for 1 ≤ i ≤ v. It holds that

   Qi = f(i-1) + ti⋅G for i = 1, ..., v where
       f(i-1) := with_even_y(Qi-1) if is_xonly_ti and
       f(i-1) := Qi-1 otherwise.

• with_even_y(Q_v) is the final result of the key aggregation and tweaking operations. It corresponds to the output of GetXonlyPubkey applied on the final KeyAgg Context. </poem>

The signer's goal is to produce a partial signature corresponding to the final result of key aggregation and tweaking, i.e., the X-only public key with_even_y(Q_v).

<poem> For 1 ≤ i ≤ v, we denote the value g computed in the i-th execution of ApplyTweak by g_i-1. Therefore, g_i-1 is -1 mod n if and only if is_xonly_t_i is true and Q_i-1 has an odd Y coordinate. In other words, g_i-1 indicates whether Q_i-1 needed to be negated to apply an X-only tweak:

   f(i-1) = gi-1⋅Qi-1 for 1 ≤ i ≤ v.

Furthermore, the Sign and PartialSigVerify algorithms set value g depending on whether Q_v needed to be negated to produce the (X-only) final output. For consistency, this value g is referred to as g_v in this section.

   with_even_y(Qv) = gv⋅Qv.

</poem>

<poem> So, the (X-only) final public key is

   with_even_y(Qv)
       = gv⋅Qv
       = gv⋅(f(v-1) + tv⋅G)
       = gv⋅(gv-1⋅(f(v-2) + tv-1⋅G) + tv⋅G)
       = gv⋅gv-1⋅f(v-2) + gv⋅(tv + gv-1⋅tv-1)⋅G
       = gv⋅gv-1⋅f(v-2) + (sumi=v-1..v ti⋅prodj=i..v gj)⋅G
       = gv⋅gv-1⋅...⋅g1⋅f(0) + (sumi=1..v ti⋅prodj=i..v gj)⋅G
       = gv⋅...⋅g0⋅Q0 + gv⋅taccv⋅G
   where tacci is computed by KeyAgg and ApplyTweak as follows:
     tacc0 = 0
     tacci = ti + gi-1⋅tacci-1 for i=1..v mod n
   for which it holds that gv⋅taccv = sumi=1..v ti⋅prodj=i..v gj.

</poem>

<poem> KeyAgg and ApplyTweak compute

   gacc0 = 1
   gacci = gi-1⋅gacci-1 for i=1..v mod n

So we can rewrite above equation for the final public key as

 with_even_y(Qv) = gv⋅gaccv⋅Q0 + gv⋅taccv⋅G.

</poem>

<poem> Then we have

   with_even_y(Qv) - gv⋅taccv⋅G
       = gv⋅gaccv⋅Q0
       = gv⋅gaccv⋅(a1⋅P1 + ... + au⋅Pu)
       = gv⋅gaccv⋅(a1⋅d1'⋅G + ... + au⋅du'⋅G)
       = sumi=1..u(gv⋅gaccv⋅ai⋅di')*G.

</poem>

Intuitively, gacc_i tracks accumulated sign flipping and tacc_i tracks the accumulated tweak value after applying the first i individual tweaks. Additionally, g_v indicates whether Q_v needed to be negated to produce the final X-only result. Thus, signer i multiplies its secret key d_i' with g_v⋅gacc_v in the Sign algorithm.

Negation Of The Individual Public Key When Partially Verifying

<poem> As explained in Negation Of The Secret Key When Signing the signer uses a possibly negated secret key

   d = gv⋅gaccv⋅d' mod n

when producing a partial signature to ensure that the aggregate signature will correspond to an aggregate public key with even Y coordinate. </poem>

<poem> The PartialSigVerifyInternal algorithm is supposed to check

 s⋅G = Re⁎ + e⋅a⋅d⋅G.

</poem>

<poem> The verifier doesn't have access to d⋅G but can construct it using the individual public key pk as follows: d⋅G

   = gv⋅gaccv⋅d'⋅G
   = gv⋅gaccv⋅cpoint(pk)

Note that the aggregate public key and list of tweaks are inputs to partial signature verification, so the verifier can also construct g_v and gacc_v. </poem>

Dealing with Infinity in Nonce Aggregation

If the nonce aggregator provides aggnonce = bytes(33,0) || bytes(33,0), either the nonce aggregator is dishonest or there is at least one dishonest signer (except with negligible probability). If signing aborted in this case, it would be impossible to determine who is dishonest. Therefore, signing continues so that the culprit is revealed when collecting and verifying partial signatures.

However, the final nonce R of a BIP340 Schnorr signature cannot be the point at infinity. If we would nonetheless allow the final nonce to be the point at infinity, then the scheme would lose the following property: if PartialSigVerify succeeds for all partial signatures, then PartialSigAgg will return a valid Schnorr signature. Since this is a valuable feature, we modify MuSig2* (which is defined in the appendix of the MuSig2 paper) to avoid producing an invalid Schnorr signature while still allowing detection of the dishonest signer: In GetSessionValues, if the final nonce R would be the point at infinity, set it to the generator instead (an arbitrary choice).

This modification to GetSessionValues does not affect the unforgeability of the scheme. Given a successful adversary against the unforgeability game (EUF-CMA) for the modified scheme, a reduction can win the unforgeability game for the original scheme by simulating the modification towards the adversary: When the adversary provides aggnonce' = bytes(33, 0) || bytes(33, 0), the reduction sets aggnonce = cbytes_ext(G) || bytes(33, 0). For any other aggnonce' , the reduction sets aggnonce = aggnonce' . (The case that the adversary provides an aggnonce' ≠ bytes(33, 0) || bytes(33, 0) but nevertheless R' in GetSessionValues is the point at infinity happens only with negligible probability.)

Choosing the Size of the Nonce

The MuSig2 paper contains two security proofs that apply to different variants of the scheme. The first proof relies on the random oracle model (ROM) and applies to a scheme variant where each signer's nonce consists of four elliptic curve points. The second proof requires a stronger model, namely the combination of the ROM and the algebraic group model (AGM), and applies to an optimized scheme variant where the signers' nonces consist of only two points. This proposal uses the latter, optimized scheme variant. Relying on the stronger model is a legitimate choice for the following reasons:

First, an approach widely taken is interpreting a Forking Lemma proof in the ROM merely as design justification and ignoring the loss of security due to the Forking Lemma. If one believes in this approach, then the ROM may not be the optimal model in the first place because some parts of the concrete security bound are arbitrarily ignored. One may just as well move to the ROM+AGM model, which produces bounds close to the best-known attacks, e.g., for Schnorr signatures.

Second, as of this writing, there is no instance of a serious cryptographic scheme with a security proof in the AGM that is not secure in practice. There are, however, insecure toy schemes with AGM security proofs, but those explicitly violate the requirements of the AGM. Broken AGM proofs of toy schemes provide group elements to the adversary without declaring them as group element inputs. In contrast, in MuSig2, all group elements that arise in the scheme are known to the adversary and declared as group element inputs. A scheme very similar to MuSig2 and with two-point nonces was independently proven secure in the ROM and AGM by Alper and Burdges.

Backwards Compatibility

This document proposes a standard for the MuSig2 multi-signature scheme that is compatible with BIP340. MuSig2 is not compatible with ECDSA signatures traditionally used in Bitcoin.

Change Log

To help implementers understand updates to this document, we attach a version number that resembles semantic versioning (MAJOR.MINOR.PATCH). The MAJOR version is incremented if changes to the BIP are introduced that are incompatible with prior versions. An exception to this rule is MAJOR version zero (0.y.z) which is for development and does not need to be incremented if backwards incompatible changes are introduced. The MINOR version is incremented whenever the inputs or the output of an algorithm changes in a backward-compatible way or new backward-compatible functionality is added. The PATCH version is incremented for other changes that are noteworthy (bug fixes, test vectors, important clarifications, etc.).

• 1.0.1 (2024-05-14):
  • Fix minor issue in PartialSigVerify vectors.
• 1.0.0 (2023-03-26):
  • Number 327 was assigned to this BIP.
• 1.0.0-rc.4 (2023-03-02):
  • Add expected value of pubnonce to NonceGen test vectors.
• 1.0.0-rc.3 (2023-02-28):
  • Improve NonceGen test vectors by not using an all-zero hex string as rand_ values. This change addresses potential issues in some implementations that interpret this as a special value indicating uninitialized memory or a broken random number generator and therefore return an error.
  • Fix invalid length of a pubnonce in the PartialSigVerify test vectors.
  • Improve KeySort test vector.
  • Add explicit IndividualPubkey algorithm.
  • Rename KeyGen Context to KeyAgg Context.
• 1.0.0-rc.2 (2022-10-28):
  • Fix vulnerability that can occur in certain unusual scenarios (see bitcoin-dev mailing list: Add mandatory pk argument to NonceGen, append pk to secnonce and check in Sign that the pk in secnonce matches. Update test vectors.
  • Make sure that signer's key is in list of individual public keys by adding failure case to GetSessionKeyAggCoeff and add test vectors.
• 1.0.0-rc.1 (2022-10-03): Submit draft BIP to the BIPs repository
• 0.8.6 (2022-09-15): Clarify that implementations do not need to support every feature and add a test vector for signing with a tweaked key
• 0.8.5 (2022-09-05): Rename some functions to improve clarity.
• 0.8.4 (2022-09-02): Make naming of nonce variants R in specifications of the algorithms and reference code easier to read and more consistent.
• 0.8.3 (2022-09-01): Overwrite secnonce in sign reference implementation to help prevent accidental reuse and add test vector for invalid secnonce.
• 0.8.2 (2022-08-30): Fix KeySort input length and add test vectors
• 0.8.1 (2022-08-26): Add DeterministicSign algorithm
• 0.8.0 (2022-08-26): Switch from X-only to plain public key for individual public keys. This requires updating a large portion of the test vectors.
• 0.7.2 (2022-08-17): Add NonceGen and Sign/PartialSigVerify test vectors for messages longer than 32 bytes.
• 0.7.1 (2022-08-10): Extract test vectors into separate JSON file.
• 0.7.0 (2022-07-31): Change NonceGen such that output when message is not present is different from when message is present but has length 0.
• 0.6.0 (2022-07-31): Allow variable length messages, change serialization of the message in the NonceGen hash function, and add test vectors
• 0.5.2 (2022-06-26): Fix aggpk in NonceGen test vectors.
• 0.5.1 (2022-06-22): Rename "ordinary" tweaking to "plain" tweaking.
• 0.5.0 (2022-06-21): Separate ApplyTweak from KeyAgg and introduce KeyGen Context.
• 0.4.0 (2022-06-20): Allow the output of NonceAgg to be infinity and add test vectors
• 0.3.2 (2022-06-02): Add a lot of test vectors and improve handling of invalid contributions in reference code.
• 0.3.1 (2022-05-24): Add NonceGen test vectors
• 0.3.0 (2022-05-24): Hash i - 1 instead of i in NonceGen
• 0.2.0 (2022-05-19): Change order of arguments in NonceGen hash function
• 0.1.0 (2022-05-19): Publication of draft BIP on the bitcoin-dev mailing list

Acknowledgements

We thank Brandon Black, Riccardo Casatta, Lloyd Fournier, Russell O'Connor, and Pieter Wuille for their contributions to this document.