Subject: Re: [bitcoin-dev] OP_CAT Makes Bitcoin Quantum Secure [was CheckSigFromStack for Arithmetic Values]

From: ZmnSCPxj <[email-masked]>
Date: Thu, 08 Jul 2021 08:12:01 +0000

Good morning Jeremy,

Yes, quite neat indeed, too bad Lamport signatures are so huge (a couple kilobytes)... blocksize increase *cough*

Since a quantum computer can derive the EC privkey from the EC pubkey and this scheme is resistant to that, I think you can use a single well-known EC privkey, you just need a unique Lamport keypair for each UTXO (uniqueness being mandatory due to Lamport requiring preimage revelation).

Regards,
ZmnSCPxj

> Dear Bitcoin Devs,
>
> As mentioned previously, OP_CAT (or similar operation) can be used to make Bitcoin "quantum safe" by signing an EC signature. This should work in both Segwit V0 and Tapscript, although you have to use HASH160 for it to fit in Segwit V0.
>
> See [my blog](https://rubin.io/blog/2021/07/06/quantum-bitcoin/) for the specific construction, reproduced below.
>
> Yet another entry to the "OP_CAT can do that too" list.
>
> Best,
>
> Jeremy
> -----
>
> I recently published [a blog
> post](https://rubin.io/blog/2021/07/02/signing-5-bytes/) about signing up to a
> 5 byte value using Bitcoin script arithmetic and Lamport signatures.
>
> By itself, this is neat, but a little limited. What if we could sign longer
> messages? If we can sign up to 20 bytes, we could sign a HASH160 digest which
> is most likely quantum safe...
>
> What would it mean if we signed the HASH160 digest of a signature? What the
> what? Why would we do that?
>
> Well, as it turns out, even if a quantum computer were able to crack ECDSA, it
> would yield revealing the private key but not the ability to malleate the
> content of what was actually signed.  I asked my good friend and cryptographer
> [Madars Virza](https://madars.org/) if my intuition was correct, and he
> confirmed that it should be sufficient, but it's definitely worth closer
> analysis before relying on this. While the ECDSA signature can be malleated to a
> different, negative form, if the signature is otherwise made immalleable there
> should only be one value the commitment can be opened to.
>
> If we required the ECDSA signature be signed with a quantum proof signature
> algorithm, then we'd have a quantum proof Bitcoin! And the 5 byte signing scheme
> we discussed previously is a Lamport signature, which is quantum secure.
> Unfortunately, we need at least 20 contiguous bytes... so we need some sort of
> OP\_CAT like operation.
>
> OP\_CAT can't be directly soft forked to Segwit v0 because it modifies the
> stack, so instead we'll (for simplicity) also show how to use a new opcode that
> uses verify semantics, OP\_SUBSTRINGEQUALVERIFY that checks a splice of a string
> for equality.
>
> ```
> ... FOR j in 0..=5
>     <0>
>     ... FOR i in 0..=31
>         SWAP hash160 DUP <H(K_j_i_1)> EQUAL IF DROP <2**i> ADD ELSE <H(K_j_i_0)> EQUALVERIFY ENDIF
>     ... END FOR
>     TOALTSTACK
> ... END FOR
>
> DUP HASH160
>
> ... IF CAT AVAILABLE
>     FROMALTSTACK
>     ... FOR j in 0..=5
>         FROMALTSTACK
>         CAT
>     ... END FOR
>     EQUALVERIFY
> ... ELSE SUBSTRINGEQUALVERIFY AVAILABLE
>     ... FOR j in 0..=5
>         FROMALTSTACK <0+j*4> <4+j*4> SUBSTRINGEQUALVERIFY DROP DROP DROP
>     ...  END FOR
>     DROP
> ... END IF
>
> <pk> CHECKSIG
> ```
>
> That's a long script... but will it fit? We need to verify 20 bytes of message
> each bit takes around 10 bytes script, an average of 3.375 bytes per number
> (counting pushes), and two 21 bytes keys = 55.375 bytes of program space and 21
> bytes of witness element per bit.
>
> It fits! `20*8*55.375 = 8860`, which leaves 1140 bytes less than the limit for
> the rest of the logic, which is plenty (around 15-40 bytes required for the rest
> of the logic, leaving 1100 free for custom signature checking). The stack size
> is 160 elements for the hash gadget, 3360 bytes.
>
> This can probably be made a bit more efficient by expanding to a ternary
> representation.
>
> ```
>         SWAP hash160 DUP <H(K_j_i_0)> EQUAL  IF DROP  ELSE <3**i> SWAP DUP <H(K_j_i_T)> EQUAL IF DROP SUB ELSE <H(K_j_i_1)> EQUALVERIFY ADD  ENDIF ENDIF
> ```
>
> This should bring it up to roughly 85 bytes per trit, and there should be 101
> trits (`log(2**160)/log(3) == 100.94`), so about 8560 bytes... a bit cheaper!
> But the witness stack is "only" `2121` bytes...
>
> As a homework exercise, maybe someone can prove the optimal choice of radix for
> this protocol... My guess is that base 4 is optimal!
>
> ## Taproot?
>
> What about Taproot? As far as I'm aware the commitment scheme (`Q = pG + hash(pG
> || m)G`) can be securely opened to m even with a quantum computer (finding `q`
> such that `qG = Q` might be trivial, but suppose key path was disabled, then
> finding m and p such that the taproot equation holds should be difficult because
> of the hash, but I'd need to certify that claim better).  Therefore this
> script can nest inside of a Tapscript path -- Tapscript also does not impose a
> length limit, 32 byte hashes could be used as well.
>
> Further, to make keys reusable, there could be many Lamport keys comitted inside
> a taproot tree so that an address could be used for thousands of times before
> expiring. This could be used as a measure to protect accidental use rather than
> to support it.
>
> Lastly, Schnorr actually has a stronger non-malleability property than ECDSA,
> the signatures will be binding to the approved transaction and once Lamport
> signed, even a quantum computer could not steal the funds.
>
> --
> @JeremyRubin