On (in)ability to embed data into Schnorr

[waxwing/ AdamISZ]

Hi all,

https://github.com/AdamISZ/schnorr-unembeddability/

Here I'm analyzing whether the following statement is true: "if you can embed data into a (P, R, s) tuple (Schnorr pubkey and signature, BIP340 style), without grinding or using a sidechannel to "inform" the reader, you must be leaking your private key".

See the abstract for a slightly more fleshed out context.

I'm curious about the case of P, R, s published in utxos to prevent usage of utxos as data. I think this answers in the half-affirmative: you can only embed data by leaking the privkey so that it (can) immediately fall out of the utxo set.

(To emphasize, this is different to the earlier observations (including by me!) that just say it is *possible* to leak data by leaking the private key; here I'm trying to prove that there is *no other way*).

However I still am probably in the large majority that thinks it's appalling to imagine a sig attached to every pubkey onchain.

Either way, I found it very interesting! Perhaps others will find the analysis valuable.

Feedback (especially of the "that's wrong/that's not meaningful" variety) appreciated.

Regards,

AdamISZ/waxwing

[Greg Maxwell]

Intuitively it sounds likely, -- just in that the available values are a image on the curve and a value summed with a hash dependent on everything else.  I think it would be hard to prove.

But is it even really worth the analysis when grinding gets you a 12% embedding rate in that signature at not that significant cost? (because you can independently grind the nonce and signature itself, or nonce and pubkey) -- and when beyond the cost of the additional signature (making the output 3x its cost) requiring signing when forming the address completely kills public derivation, multisig with cold keys. etc?  ... and then any of whatever spam concerns people have would likely be exacerbated by the spammers using more resources due to the embedding rate?

Also re private key leaking an utxo set, well not so if it's part of an explicit multisig. E.g. 2 of 2 with leaked key and a secure one.

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[Andrew Poelstra]

On Wed, Oct 01, 2025 at 10:10:16PM +0000, Greg Maxwell wrote:
> Intuitively it sounds likely, -- just in that the available values are a
> image on the curve and a value summed with a hash dependent on everything
> else. I think it would be hard to prove.
>
> But is it even really worth the analysis when grinding gets you a 12%
> embedding rate in that signature at not that significant cost? (because you
> can independently grind the nonce and signature itself, or nonce and
> pubkey) -- and when beyond the cost of the additional signature (making the
> output 3x its cost) requiring signing when forming the address completely
> kills public derivation, multisig with cold keys. etc? ... and then any of
> whatever spam concerns people have would likely be exacerbated by the
> spammers using more resources due to the embedding rate?
>

Some time ago, I talked to Ethan Heilman about this in the context of PQ
signatures, and he made the interesting point that you can think of
12% embedding rate as representing an 8x discount for real signatures vs
embedded data. And that maybe that's okay, incentive-wise.

Needing to grind out portions of 32-byte blocks probably also reduces
the risk from people trying to embed virus signatures or other malicious
data.

As for waxwing's original question -- I also intuitively believe that
the only way to embed data in a Schnorr signature is by grinding or
revealing your key ... and I'm not convinced you can do it even by
revealing your key. (R is an EC point that you can't force to be any
particular value except by making a NUMS point, which you then can't use
to sign; and s = k + ex where e is a hash of kG (among other things)
so I don't think you can force that value at all.)

--
Andrew Poelstra
Director, Blockstream Research
Email: apoelstra at wpsoftware.net
Web: https://www.wpsoftware.net/andrew

The sun is always shining in space
-Justin Lewis-Webster

[waxwing/ AdamISZ]

Hi Greg, Andrew, list,

Answers to Greg then Andrew:

> E.g. 2 of 2 with leaked key and a secure one.

That's a very good point! I was narrowly focused on the signature scheme, but Bitcoin is more than a signature scheme!

>   But is it even really worth the analysis when grinding gets you a 12% embedding rate in that signature at not that significant cost? (because you can independently grind the nonce and signature itself, or nonce and pubkey) -- and when beyond the cost of the additional signature (making the output 3x its cost) requiring signing when forming the address completely kills public derivation, multisig with cold keys. etc?  ... and then any of whatever spam concerns people have would likely be exacerbated by the spammers using more resources due to the embedding rate?

I certainly don't think it's worth *doing* (hence my use of the term "appalling idea" :) ), as per the things you mention there.

I wrote the document as a mostly academic investigation. It would be nice to be surer what the limits are, although I suspect we're all reasonably confident of what is/isn't possible.

>  12% embedding rate

Where do you get that number from? 33% for embedding 256 bits in (P, R, s) (but as per this discussion, according to me, at the cost of key leakage). If we include the other bytes in a (taproot anyway) utxo that's not much less, I guess 30% ish. I could try to guess but it'd be easier if you told me :)

to Andrew:

> As for waxwing's original question -- I also intuitively believe that

the only way to embed data in a Schnorr signature is by grinding or
revealing your key ... and I'm not convinced you can do it even by
revealing your key. (R is an EC point that you can't force to be any
particular value except by making a NUMS point, which you then can't use
to sign; and s = k + ex where e is a hash of kG (among other things)
so I don't think you can force that value at all.)

Ah, I see what you're saying, it's a subtly different target. ECDSA allows that s be controlled, Schnorr doesn't, but I set up the game as "adversary must be able to publish a function f such that f(any published R, s, (e)) = data", i.e. not just f = identity function. That was why I wrote in the introduction (copied here for convenience:)

"Data can effectively be embedded in signatures by using a publically-inferrable nonce, as was noted \href{https://groups.google.com/g/bitcoindev/c/d6ZO7gXGYbQ/m/Y8BfxMVxAAAJ}{here} and was later fleshed out in detail \href{https://blog.bitmex.com/the-unstoppable-jpg-in-private-keys/}{here} (\textbf{note}: both these sources discuss nonce-reuse but it's worse than that: any \emph{publically inferrable} nonce can achieve the same thing, such as, the block hash of the parent block; this will have the same embedding rate and cannot be disallowed)."

It may be a different target "politically" :) but I was only thinking technically, in terms of how people might end up using outputs. From a technical point of view it makes no difference if f is the identity or something more complex (as long as it's efficiently computable).

Cheers,

AdamISZ/waxwing