Subject: Re: [bitcoindev] Re: DahLIAS: Discrete Logarithm-Based Interactive Aggregate Signatures

From: Jonas Nick <[email-masked]>
Date: Thu, 17 Jul 2025 13:15:40 +0000

Hi waxwing,

Thanks again for your comments.

 > My initial reaction would be, since it's not worsening the scaling of the
 > verifier, does it matter?

I think saving time in signing does matter (3 group exponentiations requiring
O(1) group operations in total vs. O(n/log n) group operations); for example, in
constrained signing devices as you mention. In particular, the "single-b"
variant with the larger signing cost doesn't appear to have advantages (see
below) compared to "multi-b" which has lower signing cost.

 > The scheme is explicitly not limited to Bitcoin, nor blockchains, though,
 > so there's that; is that relevant here?

The scheme is not limited to Bitcoin, but the main application we designed for
is Bitcoin. I agree that verification performance is of primary importance. We
would choose a scheme with lower signing performance, if it gives us a better
verification performance in return (if the trade-off is reasonable).

 > Yes, those are some interesting points to consider. On one detail: "In any
 > case, identifying disruptive participants will work reliably only if the
 > coordinator is honest, so let's assume this." -- this could also be addressed
 > with proofs of knowledge, no?

Maybe I misunderstand what you're getting at, but I don't understand how proofs
of knowledge would get rid of the honest coordinator requirement for identifying
disruptive signers. Moreover, both R_{2,i} and R_{2,j} could have a valid proof
of knowledge attached (for example, if parties i and j share the dlog of R_{2,i}
= R_{2,j}).

 > Anyway, for me it was more a sort of preference for purely algebraic
 > algorithms. It's a little fanciful, but algebraic algorithms are easier to
 > encode in circuits in zero knowledge (though things like equality checks are
 > entirely doable ofc!) and maybe easier to "encode" into modular schemes that
 > use them as a building block. Maybe. Less conditional branches / loops to
 > traverse in the code?

Why exactly would it be easier to encode the multi-b variant in a circuit? The
single-b variant requires checking whether there exists i such that R_{2,i}
matches a fixed R_{2,j}. In the multi-b variant we'd need to compute the product
of all R_{2,i}^{b_i}, which, even with a multiexp implementation, requires at
least visiting all elements plus the actual multiexponentiation involving
O(n/log n) group operations. So encoding the single-b variant appears to be
strictly easier.

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