Subject: Re: [bitcoindev] Signing a Bitcoin Transaction with Lamport Signatures (no changes needed)

From: Ethan Heilman <[email-masked]>
Date: Wed, 1 May 2024 16:02:30 -0400

Hi Antoine,

> On the usage of such emulated Lamport signature scheme in a public transaction-relay network, there is one fundamental security property of Lamport signature to be aware off, mainly the one-time usage. So this is very unclear if as soon you're broadcasting the transaction, miners coalition could withhold the transaction inclusion to trigger the initial signer to reveal more a lot of pre-committed fixed-nonce ECDSA signatures.

I'm not sure I understand what you are saying here. I agree that once
Alice broadcasts a transaction spending such an output, she can not
double spend that output without losing security. You'd want to define
the unforgeability property to be EU-CMA with only a single signature.
Why would Alice simply just not rebroadcast her original transaction?
If she wants the capability to bump the fees without changing the sig
hash, she can use the SIGHASH flag anyone can pay or CPFP.

> using the point at infinity as your point R, and if from them you could play tricks with coordinates.

What do you mean by this? k=0? I do agree that this scheme is making
some very odd assumptions about ecdsa signatures and they may not
hold. Certainly no one should expect this scheme to be secure without
a proof of security or at the least some sort of justification for why
anyone expects these assumptions to hold.

> Beyond, 2^64 bit of security doesn't seem a lot in considerations of state-of-the-art collision attacks on hash functions from recent academic literature.

I agree that 64 bits of security is nowhere near safe. I used 80 bits
of security in the example because that is the collision resistance of
P2SH. 80-bits is still probably too low.

> And one have to consider how you can take the short-cut by pre-computing rainbow tables for ECDSA r-value of a given signature size.

It's worse than that! Storage and lookup would not require anything so
fancy as rainbow tables. Once you have precomputed a 20 byte r-value
you can use it everywhere. Grinding such an r-value would cost 2^96
queries for 50% success rate, but would let you trivially break any of
these signatures which used a 21 byte r-value with a pen and some
paper. Still, if you could do 2^96 ecdsa queries, it would be
computationally expensive as mining 1,125,899,906,842,620 bitcoin
blocks. You'd probably be better off mining those blocks than grinding
the r-value.

On Wed, May 1, 2024 at 4:57 AM Antoine Riard <[email-masked]> wrote:
>
> Hi Ethan,
>
> From my understanding, you're proposing to emulate Lamport signature verification / generation
> scheme by leveraging the implicit signature digest of the OP_CHECKSIG operation, which has been
> a valid Bitcoin Script opcode since genesis block. Signature digests is a commitment to a bitcoin
> transaction fields, and this is verified by the interpreter both for ECDSA and Schnorr schemes.
>
> Here you're proposing to use the ECDSA's `k` nonce as a fixed public value by committing the
> ECDSA-signature length as a parameter of an OP_SIZE and the cleartext `r,s` signature itself as
> the verification parameter of a OP_SHA256, emulating the h(x) = y for Lamport signature range of
> bits, all in a redeem script (i.e P2SH).
>
> I don't know if your proposed scheme is secure against message forgery attacks or invalid curve
> domain parameters, e.g using the point at infinity as your point R, and if from them you could play
> tricks with coordinates.
>
> On the usage of such emulated Lamport signature scheme in a public transaction-relay network,
> there is one fundamental security property of Lamport signature to be aware off, mainly the one-time
> usage. So this is very unclear if as soon you're broadcasting the transaction, miners coalition could
> withhold the transaction inclusion to trigger the initial signer to reveal more a lot of pre-committed
> fixed-nonce ECDSA signatures.
>
> Apart of concerns of this scheme in a blockchain and assuming it's not utterly broken due to
> message forgery attacks, I'm skeptical on the robustness of the scheme as the number of on-chain
> pre-committed signatures is a parameter of the preimage-resistance of the Lamport signature scheme
> itself. Sounds a classic time-space tradeoff, by increasing the lot of fixed-nonce signatures we're making
> it harder to break, even for observers with significant computational ressources.
>
> Beyond, 2^64 bit of security doesn't seem a lot in considerations of state-of-the-art collision attacks
> on hash functions from recent academic literature. And one have to consider how you can take the
> short-cut by pre-computing rainbow tables for ECDSA r-value of a given signature size.
>
> I think a more interesting open question of this post is if we have already hash-chain-based covenant
> "today" in Bitcoin. If by fixing the integer `z` of the s-value of ECDSA signature in redeem script, and
> computing backward the chain of chids redeem scripts to avoid hash-chain dependencies. This is unclear
> what would be the security guarantees and average btc units cost for scriptSig / witness under current block
> size limit of 4MWU.
>
> Best,
> Antoine
> Le mardi 30 avril 2024 à 22:18:36 UTC+1, Ethan Heilman a écrit :
>>
>> One could redesign this scheme to minimize the number of opcodes.
>>
>> Back of the napkin scheme follows:
>>
>> Alice, rather than Lamport signing the length of each ECDSA signature, instead Lamport (or WOTS) signs a vector of the positions of the low-length ECDSA signatures (low length here means length=58 rather than length=59). Then the redeem script would only check the length of those particular signatures and could drop the other other public keys. This saves significantly on the number of opcodes because we only need to check the Lamport signature on the vector, no one each signature length and we can drop unused checked signatures without evaluating them.
>>
>> Alice's advantage over the attacker is that she gets to fix the positions of the low length ECDSA signatures after she generates them. This means if the total number of signatures is N and the number of low length signatures is M, her advantage over the attacker is (N choose M). For instance if N=M=10, to generate 10 59-length signatures, Alice needs to perform 2^(8*10) work. This is because we assume the probability of generating a 58-byte ECDSA signature is 1/256 (1/2^8). However if N=40, M=10 she only needs to perform 2^(8*10)/(40 choose 10) work.
>>
>> If we assume that we want 80 bits of security, this means we need M=10 low length ECDCSA signatures (2^(8*10)). The next parameter is how cheap we want this to be for Alice to compute, we can boost Alice's signing time by increasing N and remove log2(N choose 10) from the work she needs to compute the signature. If we want to ensure Alice has to do no more than 2^32 work to sign, then we need N=46 or 46 ecdsa signatures.
>>
>> On Tue, Apr 30, 2024 at 10:21 AM Andrew Poelstra <[email-masked]> wrote:
>>>
>>> On Tue, Apr 30, 2024 at 08:32:42AM -0400, Matthew Zipkin wrote:
>>> > > if an attacker managed to grind a 23-byte r-value at a cost of 2^72
>>> > computations, it would provide the attacker some advantage.
>>> >
>>> > If we are assuming discrete log is still hard, why do we need Lamport
>>> > signatures at all? In a post-quantum world, finding k such that r is 21
>>> > bytes or less is efficient for the attacker.
>>> >
>>>
>>> Aside from Ethan's point that a variant of this technique is still
>>> secure in the case that discrete log is totally broken (or even
>>> partially broken...all we need is that _somebody_ is able to find the
>>> discrete log of the x=1 point and for them to publish this).
>>>
>>> Another reason this is useful is that if you have a Lamport signature on
>>> the stack which is composed of SIZE values, all of which are small
>>> enough to be manipulated with the numeric script opcodes, then you can
>>> do covenants in Script.
>>>
>>> (Sadly(?), I think none of this works in the context of the 201-opcode
>>> limit...and absent BitVM challenge-response tricks it's unlikely you can
>>> do much in the context of the 4MWu block size limit..), but IMO it's a
>>> pretty big deal that size limits are now the only reason that Bitcoin
>>> doesn't have covenants.)
>>>
>>> --
>>> 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
>>>
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