Falcon Post-Quantum Signature Scheme Proposal

[Giulio Golinelli]

Hi everyone,

I am to share a technical demonstration and benchmarking project that integrates the Falcon post-quantum signature scheme (Falcon-512) into Bitcoin Core, implemented as a soft-fork within the classic P2WPKH mode. This work aims to provide a practical reference for possible future Falcon adoption, especially as it approaches FIPS standardization.
You can find details at this fork.

Why Falcon?

Falcon is a lattice-based, post-quantum digital signature scheme designed to be secure against quantum attacks. Unlike other PQC candidates such as SPHINCS+ and ML-DSA, Falcon offers significantly smaller signature and public key sizes, as well as efficient signing and verification times. It is implemented in pure C and does not require external dependencies.

Benchmarking & Results
Aspect                           Falcon    ECDSA
Public Key Size (B) 897         33
Signature Size (B) 655         71
Verification Time (μs) 57         120

Verification time is more critical than signature creation time in Bitcoin, since signature creation is performed by clients (wallets), while nodes focus on verification.

Integration

• Falcon was included into the codebase from the original GitHub repository.
• The build system (CMakeLists.txt) was updated to support Falcon.
• Falcon verification has been soft-fork enabled via a new script verification flag.

Next Steps & Reference
This project serves as a practical demonstration of Falcon’s promising performance, highlighting its advantages over currently selected post-quantum signature algorithms such as SPHINCS+ and ML-DSA, which face significant time and space limitations. As Falcon approaches FIPS standardization, this work aims to provide a reference for future adoption and integration in Bitcoin.

Let me know what you think and if this could be of interest for which case I can complement the project by integrating Falcon into all the other spending paths. I also look forward to development/integration corrections.

Best regards,
Giulio

[waxwing/ AdamISZ]

Thanks for the report!

Forgive the rather ignorant question here, but:

Given the obvious that we have a problem with size on-chain (and I'm aware you've focused here specifically on the most plausible scheme that has the least ridiculously large size, and yet it's still 20x larger), has there been comparison of the possibility of batched signing (not batched *verification*, but signing) in different PQ schemes, with a view to a CISA like approach to transactions in a future with much larger keys and sigs? A nice side effect might be a pure economic motivation for much better fungibility (coinjoin becoming much more desirable for the base layer, albeit I think it's in higher layers where we are/will be get(ting) most privacy).

A cursory search tells me that Falcon specifically can't support any kind of batched signing, but I have no idea whether that's correct.

Cheers,

AdamISZ/waxwing

[conduition]

Falcon (FN-DSA) relies on discrete gaussian sampling using constant-time floating point arithmetic for signers, which is very hard to implement quickly and in constant time (securely). Despite being significantly harder to implement than ML-DSA, it only provides a mild (factor of two or so) improvement in signature + pubkey size. This is why we're probably not including FN-DSA in our PQ signature opcode BIP following BIP360.

https://blog.cloudflare.com/nist-post-quantum-surprise/#floating-points-falcons-achilles

While I wouldn't rule out Falcon permanently, I personally feel more research is needed to explore Falcon, its weaknesses, and how flexibly it can be adapted to schemes like CISA, BIP32, and multisignatures. Let it bake a little longer.

If small signatures are your goal, then I'd look into SQIsign, which uses isogeny-based cryptography to produce very small sigs (148b) and pubkeys (65b) using some convoluted mathematical tricks. However, much like Falcon, it is still immature and needs more researchers to optimize its verification, explore its strengths, and attack its weaknesses. 

If you want a PQC scheme that's ready today and also provides small signatures, I'll point you to XMSS, and Jonas Nick's SHRINCS proposal. You can configure an unbalanced XMSS tree to get 272 byte signatures, potentially smaller if you crank up the parameters. The catch is a dependence on statefulness. 

regards,

conduition

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[Giulio Golinelli]

Hi Adam,

thanks a lot for the thoughtful question — it’s definitely not ignorant at all, and it touches one of the core reasons Schnorr ended up being such a good fit for Bitcoin.

What you’re referring to is technically called signature aggregation, which is distinct from batch verification. Signature aggregation relies on a linear homomorphic algebraic structure that is inherent to Schnorr signatures, and it’s precisely this property that enables constructions like MuSig. Unfortunately, this kind of structure is not something we get across the post-quantum signature landscape (whether standardized or exotic schemes).

L2 ZK rollups can mitigate this by verifying many user signatures off-chain and proving the resulting state transition with a single succinct proof, effectively collapsing many verifications into one on L1. In a post-quantum setting, LaBRADOR ([https://eprint.iacr.org/2022/1341.pdf](https://eprint.iacr.org/2022/1341.pdf)), a lattice-based SNARK could be adopted. Bitcoin-oriented ZK rollups along these lines, such as Citrea, already explore this approach.

My current view is that L2 ZK constructions may be a key part of the toolbox to mitigate the absence of aggregation and efficient batching — at least until we discover a PQ signature scheme with Schnorr-like advantages. Considering it took roughly 30 years from RSA to reach Schnorr, we should expect that fully realizing such a post-quantum analogue will likely take several years of research and development.

Very happy to hear your thoughts on this, and thanks again for raising the point. I’m still relatively new to Bitcoin, so if I’ve made any incorrect assumptions here, please feel free to correct me.

Best regards,

Giulio

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[Giulio Golinelli]

Falcon (FN-DSA) relies on discrete gaussian sampling using constant-time floating point arithmetic for signers, which is very hard to implement quickly and in constant time (securely).

This is true for the first Falcon version published (randomized mode of operation). This implementation uses the author-recommended deterministic Falcon mode (see author’s notes) which uses software floating-point emulation . This eliminates side-channel risks associated with non-constant-time hardware FPUs. It is also SNARK-friendly and overcomes portability limitation. While this sacrifices the performance optimizations of true FPUs, signing speed is not critical in Bitcoin, where verification dominates node activity.

If small signatures are your goal, then I'd look into SQIsign

This would be good like many other PQ exotic schemes but all of these are far from being standardized soon.

If you want a PQC scheme that's ready today and also provides small signatures, I'll point you to XMSS, and Jonas Nick's SHRINCS proposal. You can configure an unbalanced XMSS tree to get 272 byte signatures, potentially smaller if you crank up the parameters. The catch is a dependence on statefulness. 

SPHINCS+ cannot be considered a valid fallback as it introduces large signature overhead (it's not meant for bitcoin-like use-cases). Any TPM-based state management would reduce performance and compatibility across architectures. The hash-based nature of SHRINCS is highly SNARK-unfriendly, making them incompatible with emerging L2 ZK rollup constructions. Moreover in high-throughput L2 environments, state management, limits on the number of signatures and performance degradation proportional to published signatures are critical bottlenecks.

[Mikhail Kudinov]

Hi everyone,
I am happy that the discussion on the PQ topic is active. I wanted to add my view on the raised issues. 

For the fallback in SHRINCS, one option is to use SPHINCS+ as a fallback with a limited number of signatures. By setting an upper bound as large as 2^30 or 2^40, the signature size can be significantly reduced, and the scheme would only be invoked in exceptional circumstances. In most realistic scenarios, the fallback would consist of generating a single signature to move assets to a new address. As for the statefulness problems, I agree that this is an important drawback that we should keep in mind.

The SHA-based SPHINCS+ is indeed not particularly efficient in SNARK settings. But one could replace the hash functions with SNARK-friendly alternatives (for example, Poseidon) in the future, which will make it much, much more efficient.

It is also a question: how much weight should we put on adopting an explicitly SNARK-friendly signature scheme? While such compatibility is clearly advantageous, it does not seem to me to be a decisive point on its own. What would you say?
I am also unsure to what extent Falcon can be considered SNARK-friendly. Has there been any research in this direction, or are there benchmarks evaluating its performance in SNARK environments?

Finally, regarding SQIsign: although the signature sizes are remarkable, I agree that we need more time for the scheme to mature before considering adoption.

Best,
Mike

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[waxwing/ AdamISZ]

Hi Mike,

> It is also a question: how much weight should we put on adopting an explicitly SNARK-friendly signature scheme? While such compatibility is clearly advantageous, it does not seem to me to be a decisive point on its own. What would you say?

Does it depend on what we really mean by SNARK, apart from the literal definition?

What SNARK schemes are we thinking of that are post quantum? I presume I can say "all the pairings based SNARKs (and the DLOG based ones) are not quantum resistant". Are STARKs the only realistic option in this scenario? I am enormously ignorant about STARKs, but I do seem to recall that they have large proofs, so at least in theory that's a problem for such planning?

Or maybe there's another PQ SNARK scheme that I'm just unaware of.

I guess this is orthogonal to the point you're raising about arithmetic-circuit friendly hash functions like Poseidon, though. That part I find a bit brain-melting because: what mechanism is assumed to exist to translate a STARK or SNARK proof into an onchain effect? If there was say a STARK op-code then, job done. You'd just somehow have to have sufficiently small STARK proofs which is AIUI not trivial, even with nice hash functions. If not, we're back to the current hyper-sophisticated scenarios of things like Glock and BABE where you use garbled circuits, witness encryption, and also need something like the adaptor primitive of "swap signature for secret" which we currently get kind of "for free" in Schnorr with the linearity. I suppose there might be alternatives (e.g. HTLC instead of PTLC) that might be more clunky, but viable. And all of that is of course "fraud proof" style, with "slashing", and not anywhere near as clean a design as "just verify the STARK".

How does one take the 10000ft view on this needed to make a design decision? Obviously I don't know, at all, just raising points here. I guess the most interesting one is: "is STARK realistically the only game in town here?".

Cheers,

AdamISZ/waxwing

[cassio gusson]

hey guys

I don't know if this is important for the implementation being discussed, but I think the study that identified a flaw in quantum key distribution (QKD) is worthwhile.

https://ieeexplore.ieee.org/document/11223709

Best regards
Cássio Gusson

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Abraços
Cássio Gusson

[Mikhail Kudinov]

The investigation of Lattice-Based approaches is on the table, so I think we will be able to make a thoughtful comparison at some point. I support the idea that SNARK/STARK-friendliness can not be a crucial point, as it is not currently supported, but can be an argument for tie-breaking cases.  

All pairing-based schemes are broken by a quantum computer, yes. STARKs are plausibly post-quantum. There are some schemes based on Lattices as well. 

The QKD problems are not important for our discussion; they do not affect PQ transition.

Best,
Mike 

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[waxwing/ AdamISZ]

> I support the idea that SNARK/STARK-friendliness can not be a crucial point, as it is not currently supported, but can be an argument for tie-breaking cases.

It's kind of contextual, right: it's a naturally correct default position that "one shouldn't impose arbitrary requirements on new PQ schemes that are not in place for even the existing schnorr/ecdsa schemes, all the more so with respect to speculative concepts around offchain transacting systems that aren't proven or don't even exist in prototype form". But the "contextual" here is: does the proposed QC scheme have an enormous onchain footprint (or verification cost) without any offsetting quality that could ameliorate what that implies about transaction throughputs (or centralization pressures)? That's why I mentioned batched (I should have said aggregated) signing and CISA, even though it might seem like a bit of a side issue. In case we really are looking at 20x multipliers and there is *no* ameliorating factor then ... it feels hard to support a move in that direction, except of course, in extremis.

A natural counterpoint might be: "we don't have some set-in-stone plan to develop and support only one PQ scheme, so if one is proposed and prototyped which has no scaling amelioration, it's OK, we're not outlawing a future better version".

Another natural counterpoint might be "doing anything remotely viable is hard enough, so *demanding* some compatibility with STARK or some PQ SNARK is kind of ridiculous". Fair :)

May I also ask whether support for adaptors is considered in scope (or indeed, do any of the plausible candidates have this property?)? I guess you'd consider it out of scope, though it'd be an interesting detail.

AdamISZ/waxwing

[conduition]

Hi all,

I don't think arithmetic circuit optimization for SNARKs should be a main course for the PQ transition. Maybe it could be a dessert if we have appetite for it later.

I am confident that no ZK-SNARK will ever make it into consensus on Bitcoin unless it has transparent setup, and I doubt a ZK-SNARK will survive long term if it is not quantum-secure. AFAIK that limits our options to ZK-STARKs (correct me if you know of any other transparent PQ-secure ZK-SNARK).

In my admittedly shallow foray into ZK-STARKs, the biggest problem I found is the scaling floor of STARKs. Simple proofs over small circuits have very large communication complexity (hundreds of kilobytes) compared to their classical witnesses (a few bytes). However, as the circuit gate depth grows, communication complexity grows as O(n log n) (see page 12 of this paper for a pretty graph), which is really awesome. Even the biggest STARKs don't seem to exceed 1MB. This makes STARKS most suitable for very large computations which would either be impractical to recompute naively, or impractical to store classical witness data for. Common examples are zk-rollups, as Giulio pointed out, or chain validity proofs like ZeroSync. 

The main point I want to stress is that for STARKs to be useful for PQ signature aggregation, we need to aggregate a lot of signatures at once. Ethan Heilman played with this idea.

Or you could avoid uploading the STARK proof to the chain at all. See this recent paper by Nick, Eagen, and Linus called "Shielded CSV", which describes a (non-quantum secure) way to use recursive ZK proofs to make payments on Bitcoin with perfect privacy, without needing to post the full ZK proofs to the network. Only a payment's recipient needs to see the proof. Lots of good ideas to explore there for any who want to make a quantum-secure CSV protocol.

As for uploading STARKs directly to the chain, to be worth our while in terms of blockspace savings we'd need to aggregate at least a few dozen PQ signatures per proof... hundreds of sigs if the PQ scheme is more succinct like Falcon. Because STARKs scale so well for the verifier, it doesn't really matter to the verifier how big the computation is - Proof sizes stay capped below the megabyte range, and verifier runtime remains very fast. The choice of computation matters far more to the prover, who will have to spend minutes or hours of heavy compute time proving it, depending on circuit size. Unless there exists some obvious demand to make these hypothetical PQ signature provers very fas,. I don't see any reason to use arithmetic circuit size as a high weight metric for our PQ signature scheme choices.

Also, remember that optimizing for arithmetic circuit size and ZK prover efficiency sometimes means hobbling classical computational efficiency. The Poseidon hash function is about an order of magnitude slower than SHA2, for example.

Mind you, STARKs is still a young field with active research. For example there's Circle STARKs which improve STARK proving time. I recently spoke with a grad student who is considering a research project on the application of ZK technologies to post-quantum signature schemes like SPHINCS. I know he lurks on this list, so maybe he'd have some more nuanced opinions to share.

PS. I'm very curious to know how the arithmetized circuit sizes of different signing schemes compares, e.g. SPHINCS vs XMSS vs Dilithium vs EC Schnorr. If anyone has data on that please pipe in!

regards,

conduition

[conduition]

I would be happy to be proven wrong, and to learn that Falcon is actually very easy to implement. Of all the NIST PQC schemes, Falcon is the one I understand least, so I may be mistaken. But software emulation of floating point arithmetic doesn't sound easy, especially if getting it wrong means a potential forgery attack. Also, there is still discrete Gaussian sampling to contend with.

> SPHINCS+ cannot be considered a valid fallback as it introduces large signature overhead (it's not meant for bitcoin-like use-cases).

I disagree. I think it's crucial for us to have a conservative stateless signing scheme ready as a fallback to authenticate the UTXO set if a CRQC appears. Though the signatures are indeed large, that can be mitigated by smaller parameter sets as Mikhail mentioned, or if you're OK with losing NIST compatibility, using the SPHINCS+C variant and friends. 

> Any TPM-based state management would reduce performance and compatibility across architectures

It doesn't have to be a TPM. My point is that if your wallet can manage state in some way, XMSS makes for really tiny, fast, and easy-to-implement signatures. Even if some platforms can't hack it, others would happily make the trade-off. It doesn't hurt that unbalanced XMSS disincentivizes address reuse as well.

regards,

conduition

[Giulio Golinelli]

Hi Mike,

how much weight should we put on adopting an explicitly SNARK-friendly signature scheme? While such compatibility is clearly advantageous, it does not seem to me to be a decisive point on its own. What would you say?

From a security perspective it's irrelevant.

Best,

Giulio