On that last point about "proof of knowledge of R", I suddenly realised it's not a viable suggestion: of course it defends against key subtraction attacks, but does not defend at all against the ability to grind nonces adversarially in a Wagner type attack, so that you could get a forgery on the victim's single key from a bunch of parallel signing sessions. So, question retracted, there. On Saturday, April 26, 2025 at 10:14:36 AM UTC-6 waxwing/ AdamISZ wrote: > Some comments/questions on the general structure of the scheme: > > When I started thinking about ways to change the algorithm, I started to > appreciate it more :) Although this algo is not specific to Bitcoin I'm > viewing it 100% through that lens here. Some thoughts: > > We want this CISA algorithm to have the property that it doesn't require > the blockchain (and its verifiers) to incur linear cost in the number of > signers/signatures. For a 100 input transaction, we get big gains from the > owner or owners of the inputs choosing to use this algo, but that would > mostly be lost if either the verifying was linear in the number, or if the > size of the signature was linear in the number. So to avoid that we want a > (R, s) structure to be actually published, not an (R1..Rn, s) or a (R, > s1..sn). That pretty much forces us to make a sum R for all the individual > component's R-values, and the same for s. > > However it doesn't quite force us to add literally everything. The pubkeys > *can* be kept separate, because they are retrieved implicitly from the > existing blockchain record, they are not published with the signature > (taproot). (Technically the same comment applies to the message being > signed). This allows us to use the more "pedestrian", "safe" idea; we are > not aggregating keys (as in MuSig) so we can actually add each with its own > challenge hash: sum( challenge_hash_i * X_i). This may worry you that there > is a performance issue because the verifier has to iterate through that > whole list ( the verification equation being: sG =?= R + c_1 X_1 + c_2X_2 + > .. ), but the paper specifically claims that comparing this with just batch > verifying the individual signatures (i.e. without CISA), this is twice as > fast. > > So one could simplistically say "OK that's the pubkey side, they're > treated individually so we don't have to worry about that, but what about > adding the R values?" ("worry" here means: trivial key subtraction attacks > or sophisticated Wagner/ROS grinding). And here what is done is basically > the same as in MuSig2, which is to say, by breaking the nonce into two > components and including an additional challenge hash, you prevent the > counterparty/adversary from grinding R values successfully. Note that the > "b" coefficient used here is more explicit about hashing the full context, > than it was in MuSig2: it's hashing each individual pubkey and message as > well as the R2 subcomponents for each party. This is vaguely similar to > "client side validation" ideas: it's not really "validation" as in state > updates, but it's having the more complex/expensive part of the calculation > being done in the coordination before anything goes on-chain, and allowing > us to just use a single "R" value onchain that we know is safe. > > (Side note: it's worth remembering that a lot (maybe a huge majority?) of > the usage of CISA will be a single signer of multiple inputs; for these > cases there is not the same security arguments required, only that the > final signature is not leaking the private key!). > > That side note reminds me of my first question: would it not be > appropriate to include a proof of the zero knowledgeness property of the > scheme, and not only the soundness? I can kind of accept the answer "it's > trivial" based on the structure of the partial sig components (s_k = r_k1 + > br_k2 + c_k x_k) being "identical" to baseline Schnorr? > > The side note also raises this point: would it be a good idea to > explicitly write down ways in which the usage of the scheme/structure can, > and cannot, be optimised for the single-party case? Intuitively it's > "obvious" that you may be able to streamline it for the case where all > operations happen on the same device, with a single owner of all the > private keys. I realize that this is a thorny point, because we explicitly > want to account for the corruption of parties that are "supposed" to be the > same as the honest signer, but aren't. > > And my last question is about this multi-component-nonce technique: > > Did you consider the idea of e.g. sending proofs of knowledge of R along > with R in the coordination step? This would keep the same number of rounds, > and I'm assuming (though not sure exactly) that it makes the security proof > significantly simpler, but my guess is you mostly dismiss such approaches > as being too expensive for, say, constrained devices? (I imagine something > like: 2 parties say, X1 sends (R1, pi_R1) and same for X2, to coordinator, > then sum directly for overall R; here pi_R1 is ofc just a schnorr sig on > r). If we're talking about bandwidth the current "ctx" object is already > pretty large, right, because it contains all the pubkeys and all the > messages (though in bitcoin they could be implicit perhaps). > > (I won't mention the other idea, which is going back to MuSig1 style and > just committing to R, because that's what both MuSig2 and FROST went away > from, preferring fewer rounds.) > > By the way after writing this overly long post I realised I didn't even > get in to the really tricky part of the algorithm, the "check our key and > message appears once" part because of the multisig-to-aggregated-sig > transformation and the hole previously identified in it, which to be fair > is the most interesting bit. Oh well, another time! > > Cheers, > AdamISZ/waxwing > On Thursday, April 17, 2025 at 10:38:46 AM UTC-6 Jonas Nick wrote: > >> Hi list, >> >> Cross-Input Signature Aggregation (CISA) has been a recurring topic here, >> aiming >> to reduce transaction sizes and verification cost [0]. Tim Ruffing, >> Yannick >> Seurin and I recently published DahLIAS, the first interactive aggregate >> signature scheme with constant-size signatures (64 bytes) compatible with >> secp256k1. >> >> https://eprint.iacr.org/2025/692.pdf >> >> Recall that in an aggregate signature scheme, each signer contributes >> their own >> message, which distinguishes it from multi- and threshold signatures, >> where all >> signers sign the same message. This makes aggregate signature schemes the >> natural cryptographic primitive for cross-input signature aggregation >> because >> each transaction input typically requires signing a different message. >> >> Previous candidates for constant-size aggregate signatures either: >> - Required cryptographic assumptions quite different from the discrete >> logarithm >> problem on secp256k1 currently used in Bitcoin signatures (e.g., groups >> with >> efficient pairings). >> - Were "folklore" constructions, lacking detailed descriptions and >> security >> proofs. >> >> Besides presenting DahLIAS, the paper provides a proof that a class of >> these >> folklore constructions are indeed secure if the signer does _not_ use key >> tweaking (e.g., no Taproot commitments or BIP 32 derivation). Moreover, >> we show >> that there exists a concrete attack against a folklore aggregate >> signature >> scheme derived from MuSig2 when key tweaking is used. >> >> In contrast, DahLIAS is proven to be compatible with key tweaking. >> Moreover, it >> requires two rounds of communication for signing, where the first round >> can be >> run before the messages to be signed are known. Verification of DahLIAS >> signatures is asymptotically twice as fast as half-aggregate Schnorr >> signatures >> and as batch verification of individual Schnorr signatures. >> >> We believe DahLIAS offers an attractive building block for a potential >> CISA >> proposal and welcome any feedback or discussion. >> >> Jonas Nick, Tim Ruffing, Yannick Seurin >> >> >> [0] See, e.g., https://cisaresearch.org/ for a summary of various CISA >> discussions. >> > -- You received this message because you are subscribed to the Google Groups "Bitcoin Development Mailing List" group. 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