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To: Russell O'Connor <roconnor@blockstream.com>
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Cc: Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
Subject: Re: [bitcoin-dev] Speedy covenants (OP_CAT2)
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Good morning Russell,

> On Wed, May 11, 2022 at 7:42 AM ZmnSCPxj via bitcoin-dev <bitcoin-dev@lis=
ts.linuxfoundation.org> wrote:
>
> > REMEMBER: `OP_CAT` BY ITSELF DOES NOT ENABLE COVENANTS, WHETHER RECURSI=
VE OR NOT.
>
>
> I think the state of the art has advanced to the point where we can say "=
OP_CAT in tapscript enables non recursive covenants and it is unknown wheth=
er OP_CAT can enable recursive covenants or not".
>
> A. Poelstra in https://www.wpsoftware.net/andrew/blog/cat-and-schnorr-tri=
cks-i.html show how to use CAT to use the schnorr verification opcode to ge=
t the sighash value + 1 onto the stack, and then through some grinding and =
some more CAT, get the actual sighash value on the stack. From there we can=
 use SHA256 to get the signed transaction data onto the stack and apply int=
rospect (using CAT) to build functionality similar to OP_CTV.
>
> The missing bits for enabling recursive covenants comes down to needing t=
o transform a scriptpubkey into an taproot address, which involves some twe=
aking. Poelstra has suggested that it might be possible to hijack the ECDSA=
 checksig operation from a parallel, legacy input, in order to perform the =
calculations for this tweaking. But as far as I know no one has yet been ab=
le to achieve this feat.

Hmm, I do not suppose it would have worked in ECDSA?
Seems like this exploits linearity in the Schnorr.
For the ECDSA case it seems that the trick in that link leads to `s =3D e +=
 G[x]` where `G[x]` is the x-coordinate of `G`.
(I am not a mathist, so I probably am not making sense; in particular, ther=
e may be an operation to add two SECP256K1 scalars that I am not aware of)

In that case, since Schnorr was added later, I get away by a technicality, =
since it is not *just* `OP_CAT` which enabled this style of covenant, it wa=
s `OP_CAT` + BIP340 v(^^);;;;;

Also holy shit math is scary.

Seems this also works with `OP_SUBSTR`, simply by inverting it into "valida=
te that the concatenation is correct" rather than "concatenate it ourselves=
".




So really: are recursive covenants good or...?
Because if recursive covenants are good, what we should really work on is m=
aking them cheap (in CPU load/bandwidth load terms) and private, to avoid c=
entralization and censoring.

Regards,
ZmnSCPxj