Subject: [bitcoindev] Re: jpeg resistance of various post-quantum signature schemes

From: "'Bas Westerbaan' via Bitcoin Development Mailing List" <[email-masked]>
Date: Thu, 22 May 2025 05:57:33 -0700 (PDT)

On Wednesday, May 21, 2025 at 10:58:00 PM UTC+2 Hunter Beast wrote:

Thank you for this! It's definitely informing how we approach development 
of BIP-360. SLH-DSA is concering, in that 7/8 arbitrary data would make it 
about on par with the de facto witness discount. I don't want to sacrifice 
SLH-DSA because it's favored due to hash-based signatures having more 
confidence due to not introducing as many novel security assumptions as are 
introduced with lattice cryptography.

At present, lattices are the only viable approach to post-quantum key 
agreement in TLS. If come Q-day they're broken, then it's not just Bitcoin 
that's in big trouble. If you do want the certainty of hashes, you might 
want to consider XMSS: that's JPEG resistant. With parameters n=16, h=20, 
d=1, w=16 it has 32 byte public key and 880 byte signature can sign a 
million messages, and only requires 3,000 hashes for verification [1] 
(which can actually be reduced threefold.) The big downside is that if you 
use the same OTS leaf twice, probably anyone can forge another signature on 
that leaf. In this case you might make this mistake harder by keeping track 
of the last leaf that was used for each public key. If you see a public key 
sign using the same leaf a second time, you simply ignore the second 
signature. This helps against an oopsie that's at least a few hours apart, 
but not if you're using the same leaf twice in short succession.

Another concern regarding SLH-DSA might be its performance, it's an order 
of magnitude more costly to run than FALCON, which itself is an order of 
magnitude more costly to run than secp256k1 Schnorr...

I assume you're talking about signature size? Falcon-512 requires fewer 
cycles to verify than secp256k1. SLH-DSA's verification is a bit slower. 
There is some flexibility: SLH-DSA today assumes that a signer will make 
2^64 signatures. If you drop that to say one million, then you can get 
smaller parameters. You can also vary parameters to smoothly vary signature 
size, verification time, and signing time. There is some momentum between 
standardising new variants of SLH-DSA. See also this paper [2]. If XMSS is 
too scary, you might want to consider a Bitcoin tailored variant of SLH-DSA.

We'll also be deprecating ML-DSA because it's too similar to FALCON in 
terms of performance and size.

Falcon has great signature size and verification performance. Its 
verification routine is also simple to implement. I do have to warn about 
it's signing routine: it's quite complicated and tricky to implement 
securily, especially if you want it to be fast. I don't think speed is 
critical here, so I would stay away from implementations that use 
floating-point accelerators. Another thing to note is that if lattice 
cryptanalysis improves, the first step above Falcon-512 is Falcon-1024. A 
Falcon-768 is possible (and used to be specified), but it's quite a bit 
more complex.

Best,

 Bas

JPEG resistance and scaling will need to be solved through separate means, 
perhaps with BitZip, which is what I'm calling Ethan's proposal a couple 
weeks back for block-wide transaction compression scaling PQC signatures 
through STARK proofs.

Will be making those changes to the BIP soon. Feedback is always welcome!

On Wednesday, May 21, 2025 at 5:20:02 AM UTC-6 Bas Westerbaan wrote:

Hi all,

My colleague Ethan asked me the fun question which post-quantum signature 
schemes have the following security property, which he called jpeg 
resistance.

Attacker wins if for a (partially specified) signature and full message, 
they can find a completed signature and public key, such that the completed 
signature verifies under the public key.

A naive hash-based signature is not jpeg resistant. Schoolbook Winternitz 
one-time signatures, forest-of-trees few-time signatures, and Merkle trees 
all validate signatures (/authentication paths) by recomputing the public 
key (/Merkle tree root) from the signature and the message, and checking 
whether the recomputed public key matches the actual public key. That means 
we can pick anything for the signature, and just set the public key to the 
recomputed public key.

The situation is more subtle for actual standardized hash-based signatures. 
RFC 8391 XMSS doesn’t sign the message itself, but first hashes in (among 
others) the public key. Basically the best we can do for XMSS (except for 
setting the signature randomizer) is to guess the public key. Thus it’s 
pretty much jpeg resistant.

The situation is different again for RFC 8391 XMSSMT. XMSSMT is basically a 
certificate chain of XMSS signatures. An XMSSMT public key is an XMSS 
public key. An XMSSMT signature is a chain of XMSS signatures: the XMSSMT 
public key signs another XMSS public key; which signs another public XMSS 
public key; …; which signs the message. Again the top XMSSMT public key is 
hashed into the message signed, but that only binds the first XMSS 
signature. We can’t mess with the first signature, but the other signatures 
we can choose freely, as those roots are not bound. Thus XMSSMT with two 
subtrees is only half jpeg resistant and it gets worse with more subtrees.

Similarly SLH-DSA (FIPS 205, née SPHINCS+) is a certificate chain of (a 
variant of) XMSS signing another XMSS public key, which signs another XMSS 
public key, etc, which signs a FORS public key, which signs the final 
message. The SLH-DSA public key is the first XMSS public key. From the 
message and the public key it derives the FORS key pair (leaf) in the hyper 
tree to use to sign, and the message to actually sign. This means we can’t 
mess with the first XMSS keypair. Thus to attack SLH-DSA we honestly 
generate the first XMSS keypair. Then given a message, we just pick the 
signature arbitrarily for all but the first XMSS signature. We run the 
verification routine to recompute the root to sign by the first XMSS 
keypair. Then we sign it honestly. It depends a bit on the parameters, but 
basically we get to pick roughly ⅞ of the signature for free.

ML-DSA (FIPS 204, née Dilithium) is a Fiat–Shamir transform of a 
(module-)lattice identification scheme. In the identification scheme the 
prover picks a nonce y, and sends the commitment w1 = HighBits(A y) to the 
verifier, where A is a matrix that’s part of the public key and HighBits 
drops the lower bits (of the coefficients of the polynomials in the 
vector). The verifier responds with a challenge c, to which the prover 
returns the response z = y + c s1, where s1 is part of the private key. The 
verifier checks, among other things, whether HighBits(Az-ct) = w1, where t 
= As1+s2 is part of the public key. As usual with Fiat–Shamir, in ML-DSA 
the challenge c is the hash of the commitment, message, and public key. The 
scheme has commitment recovery, so the signature itself consists of the 
response z and the challenge c. (There is also a hint h, but that’s small 
and we can ignore it.) If we set s1 to zero, then z=y, which is free to 
choose. So we can freely choose z, which is by far the largest part of the 
signature. Such a public key t is easy to detect, as it has small 
coefficients. Instead we can set s1 to zero on only a few components. That 
allows us to choose z arbitrarily for those components, still breaking jpeg 
resistance, while being hard to detect. There could well be other 
approaches here.

Falcon. A Falcon private key are small polynomials f,g. Its public key is h 
= g f-1. With the private key, for any polynomial c, we can compute small s1 
and s2 with s1 + s2h = c. A Falcon signature is a pair r, s2 where s1 = 
H(r, m) - s2 h is small. s2 is Guassian distributed, and is encoded using 
an Elias–Fano approach. It’s then padded to make signatures fixed-length. 
Clearly the randomizer r can be set arbitrarily, but it’s only 40 bytes. 
Putting arbitrary bytes in most of the encoding of s2 will likely yield a 
sufficiently small s2. Now, I thought about using this s2 as a new g and 
construct a signature that way by finding s’1 and s’2 with s’1 + s’2s1f-1 = 
H(r,m), but my brother suggested a simpler approach. s2 is likely 
invertible and we can set h = H(r, m)/s2. Both approaches would be thwarted 
by using H(H(h), r, m) instead of H(r, m). I do not know if there is still 
another attack.

Best,

 Bas

[1] https://westerbaan.name/~bas/hashcalc/ 
[2] https://eprint.iacr.org/2024/018.pdf

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