DahLIAS: Discrete Logarithm-Based Interactive Aggregate Signatures

[Jonas Nick]

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.

[waxwing/ AdamISZ]

Hi Jonas and list.

So I'm reading the paper and it's very interesting. I have other questions but this one seems more important so I'll just stick with this one:

Appendix A2 explains an attack on Musig2-IAS, in which you can forge a partial signature on a tweaked key of the honest signer. I don't understand why this same attack cannot be applied to MuSig2 itself?

the multisig-to-IAS "translation" makes sense, given the caveat of the weakness identified in the 2018 paper and explained here in detail, other than that it's basically about the message being a concat of the individual messages (and keys). But surely that doesn't change the structure of the attack? (i.e. multiply your R-vals by a2/a1, then take partial sig and multiply by a2/a1 and add the tweak). I note that 3 round musig is not vulnerable to it, nor would some PoK of R be.

Obviously I missed something.

Cheers,

AdamISZ/waxwing

[Jonas Nick]

Thanks for bringing this up. It's an interesting question and it made us realize
that we should clarify this section of the paper, as there are indeed some
subtleties here that are currently unmentioned.

> I don't understand why this same attack cannot be applied to MuSig2 itself?

There are nuances, but I think it's fair to say that the same attack cannot be
applied to MuSig2 itself. During the attack, the adversary requests a partial
signature for public key X and message m from the honest signer. Using this, the
adversary is able to create a partial signature for public key X' = TweakPK(X,
t), where t is some tweak chosen by the adversary, and message m'. When applying
the attack to MuSig2, we have that m' = m, and when applying it to MuSig2-IAS,
we may have m != m'.

So, using the attack, the adversary is able to produce a signature sigma_1 for
MuSig2 and sigma_2 for MuSig2-IAS such that

- MuSig2.Verify(KeyAgg(X, X'), m, sigma_1) = 1, and
- MuSig2-IAS.Verify((X, m), (X', m'), sigma_2) = 1.

sigma_2 is clearly a forgery under the EUF-CMA-TK security model defined in the
DahLIAS paper because it is a signature for a message m' that the honest signer
hasn't signed. In contrast, sigma_1 only covers the message that the honest
signer actually signed. Whether sigma_1 counts as a forgery depends on the
abstract security notion that you consider for multisignature tweaking. We
didn't provide such a model in the MuSig2 paper and I am not aware of a standard
one. It would be easy to design a security model where sigma_1 constitutes a
forgery and one where it doesn't.

More importantly, could this be a problem for MuSig2 in practice? I can only
come up with contrived scenarios, but it may still be worth mentioning in the
BIP, for example.

[waxwing/ AdamISZ]

I'm struggling to understand one detail in DahLIA's algorithm: the use of R2 as a check and not R1 (or both). Is it just that only one is needed? Is it just an optimization?

Thanks,

AdamISZ/waxwing

On Thursday, April 17, 2025 at 10:38:46 AM UTC-6 Jonas Nick wrote:

[Jonas Nick]

Yes, just one is needed and this is an optimization. The signer does not require
every individual R1 value as input, only the R2 values. This is different to
MuSig2, where both R1 and R2 can be aggregated by the coordinator.

[waxwing/ AdamISZ]

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:

[waxwing/ AdamISZ]

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.

[Jonas Nick]

Thanks for your comments.

> 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?

That partial signatures do not leak information about the secret key x_k is
implied by the security theorem for DahLIAS: If information would leak, the
adversary could use that to win the unforgeability game. However, the adversary
doesn't win the game unless the adversary solves the DL problem or finds a
collision in hash function Hnon.

> 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?

This is a very interesting point, probably out of scope for the paper. A
single-party signer, given secret keys xi, ..., xn for public keys X1, ..., Xn
can draw r at random, compute R := r*G and then set s := r + c1*x1 + ... +
cn*xn. So this would only require a single group multiplication.

> 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

We believe Appendix B provides a helpful characterization of "Wagner-style"
vulnerabilities. Roughly speaking, it shows that schemes where the adversary can
ask the signer to produce a partial signature s = r + c*x or s' = r + c'*x such
that c != c' then the scheme is vulnerable. In your "proof of knowledge of R
idea", the adversary can choose to provide either R2 or R2' in a signing
request, which would result in the same "effective nonce" r being used be the
signer but different challenges c and c'.

[waxwing/ AdamISZ]

> That partial signatures do not leak information about the secret key x_k is
implied by the security theorem for DahLIAS: If information would leak, the
adversary could use that to win the unforgeability game. However, the adversary
doesn't win the game unless the adversary solves the DL problem or finds a

collision in hash function Hnon.

OK, so that's maybe a theoretical confusion on my part, I'm thinking of the HVZK property of the Schnorr ID scheme, which "kinda" carries over into the FS transformed version with a simulator (maybe? kinda?). Anyway this is a sidetrack and not relevant to the paper, so I'll stop on that.

> This is a very interesting point, probably out of scope for the paper. A
single-party signer, given secret keys xi, ..., xn for public keys X1, ..., Xn
can draw r at random, compute R := r*G and then set s := r + c1*x1 + ... +
cn*xn. So this would only require a single group multiplication.

I feel bad for saying so, but I absolutely do believe it's in scope of the paper :) If there is a concrete, meaningful optimisation that's both possible and sensible (and as you say, there is such an ultra-simple optimisation ... I guess that's entirely correct!), then it should be included there and not elsewhere. Why? Because it's exactly the kind of thing an engineer might want to do, but it's definitely not their place to make a judgement as to whether it's safe or not, given that these protocols are such a minefield. I'd say even if there is *no* such optimisation possible it's worth saying so.

I guess the counterargument is that it's suitable for a BIP not the paper? But I'd disagree, this isn't purely a bitcoin thing.

On the third paragraph, yeah, as per earlier email, I realised that that just doesn't work.