musig-spec: More minor cleanup

Author: robot-dreams

doc/musig-spec.mediawiki

@@ -24,14 +24,14 @@ 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 key. 
+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 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.) 
+(MuSig2 is a ''n-of-n'' multi-signature scheme and not a ''t-of-n' threshold-signature scheme.)
 
 The primary motivation for MuSig2 is the activation of Taproot ([https://github.com/bitcoin/bips/blob/master/bip-0341.mediawiki BIP341]) on the Bitcoin network, which introduced the ability to authorize transactions with Schnorr signatures.
 This standard allows the creation of aggregate public keys that can be used in Taproot outputs.
 
-The on-chain footprint of a MuSig2 Taproot output is 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 <code>OP_CHECKSIGADD</code> as introduced in ([https://github.com/bitcoin/bips/blob/master/bip-0342.mediawiki BIP342]). 
+The on-chain footprint of a MuSig2 Taproot output is 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 <code>OP_CHECKSIGADD</code> as introduced in ([https://github.com/bitcoin/bips/blob/master/bip-0342.mediawiki 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 <code>OP_CHECKSIGADD</code>: 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 key, the shared Taproot output can have script spending paths that are hidden unless used.
@@ -48,10 +48,10 @@ The MuSig2 variant in this specification stands out by combining all of the foll
 * '''Compatibility with BIP340''': The aggregate public key created as part of this MuSig2 specification is a BIP340 X-only public key, and the signature output at the end of the protocol is a BIP340 signature that passes BIP340 verification for the aggregate key and a message. The public keys that are input to the key aggregation algorithm are also X-only public keys. Compared to compressed serialization, this adds complexity to the specification, but as X-only keys are becoming more common, the full key may not be available.
 * '''Tweaking for BIP32 derivations and Taproot''': The specification supports tweaking aggregate public keys and signing for tweaked aggregate public keys. We distinguish two modes of tweaking: ''Ordinary'' tweaking can be used to derive child aggregate public keys per [https://github.com/bitcoin/bips/blob/master/bip-0032.mediawiki BIP32]. ''X-only'' tweaking, on the other hand, allows creating a [https://github.com/bitcoin/bips/blob/master/bip-0341.mediawiki BIP341] tweak to add script paths to a Taproot output. See section [[#tweaking|Tweaking]] below for details.
 * '''Non-interactive signing with preprocessing''': The first communication round, exchanging the nonces, can happen before the message or even the exact set of signers is determined. Therefore, the signers can view it as a preprocessing step. Later, when the parameters of the signing session are chosen, they can send partial signatures without additional interaction.
-* '''Key aggregation optionally independent of order''': The output of the key aggregation algorithm depends on the order of the input public keys. The specification defines an algorithm to sort the public keys before key aggregation. This will ensure the same output, independent of the initial order. Key aggregation does not sort the public keys by default because applications often already have a canonical order of signers. Then, sorting is unnecessary and very slow for a large set of signers compared to the rest of the MuSig2 protocol. In the worst case, sorting algorithms in standard libraries can have quadratic run time, which is undesirable in adversarial settings. Nonetheless, applications using this specification can mandate sorting before aggregation.
+* '''Key aggregation optionally independent of order''': The output of the key aggregation algorithm depends on the order of the input public keys. The specification defines a function to sort the public keys before key aggregation. This will ensure the same output, independent of the initial order. Key aggregation does not sort the public keys by default because applications often already have a canonical order of signers. Nonetheless, applications using this specification can mandate sorting before aggregation<ref>Applications that sort input public keys before aggregation should ensure that the sort implementation is reasonably efficient, and in particular does not degenerate to quadratic runtime on pathological inputs.</ref>.
 * '''Third party nonce aggregation''': Instead of every signer sending their nonce to every other signer, it is possible to use an untrusted third party that collects all signers' nonces, computes an aggregate nonce, and broadcasts it to the signers. This reduces the communication complexity from quadratic to linear in the number of signers. If the aggregator sends an incorrect aggregate nonce, the signing session will fail to produce a valid Schnorr signature. However, the aggregator 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 protocol, the signing session will fail to produce a valid Schnorr signature. This standard 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''': The specification uses an optimization that allows saving a point multiplication in key aggregation. The MuSig2 scheme with this optimization is called MuSig2* and proven secure in the appendix of the [https://eprint.iacr.org/2020/1261 MuSig2 paper]. The optimization is that the second key in the list of public keys given to the key aggregation algorithm (as well as any keys identical to this key) gets the constant key aggregation coefficient ''1''.
+* '''MuSig2* optimization''': The specification uses an optimization that allows saving a point multiplication in key aggregation. The MuSig2 scheme with this optimization is called MuSig2* and proven secure in the appendix of the [https://eprint.iacr.org/2020/1261 MuSig2 paper]. The optimization is that the second distinct key in the list of public keys given to the key aggregation algorithm (as well as any keys identical to this key) gets the constant key aggregation coefficient ''1''.
 * '''Parameterization of MuSig2 and security''': In this specification, each signer's nonce consists of two elliptic curve points. The [https://eprint.iacr.org/2020/1261 MuSig2 paper] gives distinct security proofs depending on the number of points that constitute a nonce. See section [[#choosing-the-size-of-the-nonce|Choosing the Size of the Nonce]] for a discussion.
 
 This specification is written with a focus on clarity.