Date: Mon, 17 Aug 2020 05:04:42 +0000
To: Tier Nolan <tier.nolan@gmail.com>,
 Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
From: ZmnSCPxj <ZmnSCPxj@protonmail.com>
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Subject: Re: [bitcoin-dev] reviving op_difficulty
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Good morning Tier, Thomas, and aj,

> On Sun, Aug 16, 2020 at 4:50 PM Thomas Hartman via bitcoin-dev <bitcoin-d=
ev@lists.linuxfoundation.org> wrote:
>
> > My understanding is that adding a single op_difficulty operation as
> > proposed would enable not true difficulty futures but binary options
> > on difficulty.
> >
> > https://en.wikipedia.org/wiki/Binary_option
>
> Any kind of opcode is a binary option.=C2=A0 Either the output can be spe=
nt or it can't.
>
> You could get a pseudo-continuous future by having lots of outputs with d=
ifferent thresholds.
>
> Alice and Bob create a transaction with 100 outputs and each having 1% of=
 the future's value.
>
> Output 0:=C2=A0 Pay Alice if diff < 1.00 trillion else Bob
> Output 1:=C2=A0 Pay Alice if diff < 1.01 trillion else Bob
> ....
> Output 98:=C2=A0 Pay Alice if diff < 1.98 trillion else Bob
> Output 99:=C2=A0 Pay Alice if diff < 1.99 trillion else Bob
>
> If the difficulty is 1.25 trillion, then Alice gets outputs 0-24 and Bob =
gets outputs 25-99.=C2=A0 The future has a tick size of 1%.=C2=A0 It isn't =
very efficient though

Taproot MAST to the rescue.

* Alice and Bob agree on the number of ticks N and payout schedule.
* Alice and Bob generate N fresh keypairs and share them.
* Alice and Bob generate tapleaf scripts of the form:
  * script[i] =3D Alice[i] && Bob[i] && diff < 1.00 trillion + i * tick_siz=
e && CLTV(deadline)
* Alice and Bob generate the taproot MAST for the above scripts.
* Alice and Bob generate, but do ***NOT*** sign, a funding transaction payi=
ng out to the generated taproot MAST.
* Bob generates partial signatures for N payout transactions, with lower-di=
fficulty-targets paying out less to Alice and more to Bob, and higher-diffi=
culty-targets paying out more to Alice and less to Bob.
  * This requires spending the [i]th tapleaf script with the appropriate di=
fficulty target.
* Alice saves all the Bob signatures.
* At deadline, Alice rationally selects the highest-paying version that is =
still acceptable, based on the actual difficulty target at the time.

This requires publishing only O(log N) data (the merkle path to the selecte=
d tapleaf).
This translates to the 100-tick example requiring only one TXO, 1 scripthas=
h, and 7 or so Merkle-tree-path hashes, compared to the above example which=
 requires 100 TXOs and 100 script hashes.

The same scheme can be used with `OP_CTV` and without keypairs being involv=
ed, but basically anything `OP_CTV` can do, signing keypairs with pre-gener=
ated signatures from all participants can do just as well, with higher stor=
age and setup costs.

Regards,
ZmnSCPxj