On the very important argument laid out in Appendix B:
(for readers, this part of the paper deals with this problem: if the scheme can allow a scenario where the output "effective nonce" of an honest signer is the same for two different contexts and thus have two difference signature hashes (in DahLIAS the sighash is H(L, R, X_i, m_i)), then with some deft mathematical footwork, we can extract a forgery on some other message by that signer, as long as we can do a bunch of parallel signing sessions; you can think of this as a sophisticated extension of the fundamental idea of "nonce reuse is insecure").
... I find myself stepping back through the arguments for this structure R1 + bR2 in MuSig2. In that case, it is vital that we end with a verification that looks like : sG =?= R + H(R,P,m)P, precisely, because we need MuSig2 verification to be indistinguishable from single-key Schnorr verification.
In DahLIAS we have (obviously, reasonably) relaxed that restriction. The verification now allows for multiple pubkeys and messages to be inputs; that's the whole point, we're publically verifying authorization of a bunch of (pubkey, message) pairs, so it would be incoherent or at least unnecessary to *require* some kind of aggregate pubkey as input. (though as the paper discussed, you can *try* to build the IAS from an IMS, which would mean actually doing that, but as demonstrated, it doesn't really seem to work, anyway).
So when we look at the R component, which, distinct from the pubkeys, *must actually be published*, we do need to have an aggregated R value (to get a short signature), so at the end, we publish (R, s) and can do a verification with the implied pubkey message pairs ((P1, m1), (P2, m2),..) like: sG =?= R + sigma (sighash(Pn,mn) * Pn).
So here's my question: why does the signing context, represented by "b", in the aggregate R-value, need to be a fixed value across signing indices? Clearly if we have one b-value, H-non(ctx), where ctx is ((P1, m1), (P2, m2),..) [1], then it is easy to sum all the R1,i = R1 and then sum all the R2,i values = R2 and then R = R1 + bR2, exploiting the linearity. But why do we have to? If coefficient b were different per participant, i.e. b_i = H(ctx, m_i, P_i) then it makes that sum "harder" but still trivial for all participants to create/calculate. All participants can still agree on the correct aggregate "R" before making their second stage output s_i.
If I am right that that is possible, then the gain is clear (I claim!): the attacks previously described, involving "attacker uses same key with different message" fail. The first thing I'd note is that the basic thwarting of ROS/Wagner style attacks still exists, because the b_i values still include the whole context, meaning grinding your nonce doesn't allow you to control the victim's effective nonce. But because in this case, you cannot create scenarios like in Appendix B, i.e. in the notation there:
F(X1, m1(0), out1, ctx) = F(X1, m1(1), out1, ctx) is no longer true because b no longer only depends on global ctx, but also on m1 (b_1 = Hnon(ctx, m1, P1) is my proposal),
then the "Property 3" does not apply and so (maybe? I haven't thought it through properly) the duplication checks as currently described, would not be needed.
I feel like this alternate version is more intuitive: I see it as analogous to (though not the same) as Fiat-Shamir hashing, where the main idea is to fix the actual context of the proof; but the context of *my* partial signature for this aggregate, is not only ((P1, m1), (P2, m2),..) but also my particular entry in that list.
[1] Here I am ignoring that actual DahLIAS uses ctx including R2 values because I understand that that is part of the checking procedure that I am (somewhat vaguely) here trying to argue might be omitted.
Cheers,
AdamISZ/waxwing
On Thursday, May 1, 2025 at 12:14:30 AM UTC-3 waxwing/ AdamISZ wrote:
> 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.
On Wednesday, April 30, 2025 at 9:03:34 AM UTC-6 Jonas Nick wrote:
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'.
You received this message because you are subscribed to the Google Groups "Bitcoin Development Mailing List" group.
To unsubscribe from this group and stop receiving emails from it, send an email to bitcoindev+unsubscribe@googlegroups.com.
To view this discussion visit https://groups.google.com/d/msgid/bitcoindev/3f23ebaa-02c7-45d1-bf57-9baf48c133a3n%40googlegroups.com.