On Thu, Sep 09, 2021 at 04:41:38PM +1000, Anthony Towns wrote:
> I'll split this into two emails, this one's the handwavy overview,
> the followup will go into some of the implementation complexities.

(This is informed by discussions with Greg, Matt Corallo, David Harding
and Jeremy Rubin; opinions and mistakes my own, of course)

First, let's talk quickly about IN_OUT_AMOUNT. I think the easiest way to
deal with it is just a single opcode that pushes two values to the stack;
however it could be two opcodes, or it could even accept a parameter
letting you specify which input (and hence which corresponding output)
you're talking about (-1 meaning the current input perhaps). 

Anyway, a big complication here is that amounts in satoshis require up
to 51 bits to represent them, but script only allows you to do 32 bit
maths. However introducing IN_OUT_AMOUNT already means using an OP_SUCCESS
opcode, which in turn allows us to arbitrarily redefine the behaviour
of other opcodes -- so we can use the presence of IN_OUT_AMOUNT in the
script to upgrade ADD, SUB, and the comparison operators to support 64
bit values. Enabling MUL, DIV and MOD might also be worthwhile.

Moving onto TLUV. My theory is that it pops three items off the stack. The
top of the stack is "C" the control integer; next is "H" the
additional path step; and finally "X" the tweak for the internal
pubkey. If "H" is the empty vector, no additional path step is
added; otherwise it must be 32 bytes. If "X" is the empty vector,
the internal pubkey is not tweaked; otherwise it must be a 32 byte
x-only pubkey.

The low bit of C indicates the parity of X; if it's 0, X has even y,
if it's 1, X has odd y.

The next bit of C indicates whether the current script is dropped from
the merkle path, if it's 0, the current script is kept, if it's 1 the
current script is dropped.

The remaining bits of C (ie C >> 2) are the number of steps in the merkle
path that are dropped. (If C is negative, behaviour is to be determined
-- either always fail, or always succeed and left for definition via
future soft-fork)

For example, suppose we have a taproot utxo that had 5 scripts
(A,B,C,D,E), calculated as per the example in BIP 341 as:

    AB = H_TapBranch(A, B)
    CD = H_TapBranch(C, D)
    CDE = H_TapBranch(CD, E)
    ABCDE = H_TapBranch(AB, CDE)

And we're spending using script E, in that case the control block includes
the script E, and the merkle path to it, namely (AB, CD).

So here's some examples of what you could do with TLUV to control how
the spending scripts can change, between the input sPK and the output sPK.

At it's simplest, if we used the script "0 0 0 TLUV", then that says we
keep the current script, keep all steps in the merkle path, don't add
any new ones, and don't change the internal public key -- that is that
we want to resulting sPK to be exactly the same as the one we're spending.

If we used the script "0 F 0 TLUV" (H=F, C=0) then we keep the current
script, keep all the steps in the merkle path (AB and CD), and add
a new step to the merkle path (F), giving us:

    EF = H_TapBranch(E, F)
    CDEF =H_TapBranch(CD, EF)
    ABCDEF = H_TapBranch(AB, CDEF)

If we used the script "0 F 2 TLUV" (H=F, C=2) then we drop the current
script, but keep all the other steps, and add a new step (effectively
replacing the current script with a new one):

    CDF = H_TapBranch(CD, F)
    ABCDF = H_TapBranch(AB, CDF)

If we used the script "0 F 4 TLUV" (H=F, C=4) then we keep the current
script, but drop the last step in the merkle path, and add a new step
(effectively replacing the *sibling* of the current script):

    EF = H_TapBranch(E, F)
    ABEF = H_TapBranch(AB, EF)

If we used the script "0 0 4 TLUV" (H=empty, C=4) then we keep the current
script, drop the last step in the merkle path, and don't add anything new
(effectively dropping the sibling), giving just:

    ABE = H_TapBranch(AB, E)

Implementing the release/locked/available vault construct would then
look something like this:

Locked script = "OP_RETURN"
<D> CSV"
HOT = 32B hot wallet pubkey
X = maximum amount spendable via hot wallet at any time
D = compulsory delay between releasing funds and being able to spend them
H_LOCKED = H_TapLeaf(locked script)
H_AVAILABLE= H_TapLeaf(available script)
Internal public key = 32B cold wallet pubkey

Moving on to the pooled scheme and actually updating the internal pubkey
is, unfortunately, where things start to come apart. In particular,
since taproot uses 32-byte x-only pubkeys (with implicit even-y) for the
scriptPubKey and the internal public key, we have to worry about what
happens if, eg, A,B,C and A+B+C all have even-y, but (A+B)=(A+B+C)-C does
not have even-y. In that case allowing C to remove herself from the pool,
might result in switching from the scriptPubKey Qabc to the scriptPubKey
Qab as follows:

     Qabc = (A+B+C) + H(A+B+C, (Sa, (Sb, Sc)))*G
     Qab = -(A+B) + H( -(A+B), (Sa, Sb)*G

That's fine so far, but what happens if B then removes himself from the
pool? You take the internal public key, which turns out to be -(A+B)
since (A+B) did not have even y, and then subtract B, but that gives you
-A-2B instead of just A. So B obtains his funds, but B's signature hasn't
been cancelled out from the internal public key, so is still required
in order to do key path spends, which is definitely not what we want.

If we ignore that caveat (eg, having TLUV consider it to be an error if
you end up an internal public key that has odd-y) then the scripts for
exiting the pool are straightforward (if your balance is BAL and your
key is KEY):

    <KEY> DUP "" 1 TLUV

It seems like "just ignore it" might be feasible for modest sized pools --
just choose A, B, C, D.. so that every combination of them (A+B+C, A+D,
etc) sums to a point that happens to have even-y and have each participant
in the pool verify that prior to using the pool. If I got my maths right,
you'll need to do about (2**n) trials to find a set of lucky points,
but each unlucky set will tend to fail quickly, leading to amortized
constant time for each test, so something like 3*(2**n) work overall. So
as long as n is no more than 20 or 30, that should be reasonably feasible.

To deal with it properly, you need to have the utxo commit to the parity
of the internal public key and have some way to find out that value when
using TLUV. There are probably three plausible ways of doing this.

The straightforward way is just to commit to it in the scriptPubKey --
that is, rather than taproot's approach of setting Q = P + H(P, S)*G where
P is a 32 byte x-only pubkey, also commit to the parity of P in the H(P,
S) step, and reveal the parity of the internal public key as part of the
control block when spending via the script path, in addition to revealing
the parity of the scriptPubKey point as we do already. Since taproot is
already locked in for activation, it's too late to change this behaviour
for taproot addresses, but we could include this in a future soft-fork
that enabled entroot or similar, or we could make this the behaviour of
(eg) 33B segwit v1 addresses that begin with 0x00, or similar.

If we don't commit to the parity in the scriptPubKey, there are two other
ways to commit to it in the utxo: either by having script ensure it is
committed to it in the value, or by extending the data that's saved in
the utxo database.

To commit to it in the value, you might do something like:


and change TLUV's control parameter to be: C&1 = add/subtract the point,
C&2 = require the result to be even/odd y (with C&4 and C>>3 controlling
whether the current script and how many merkle paths are dropped). The
idea being to require that, if the utxo's value in satoshis is 0 mod
2, you subtract the point, and if it's 1 mod 2, you add the point,
and that the *output* amount's value in satoshis is different (mod 2)
from the input amount's value (mod 2), exactly when the resulting point
ends up with odd y.  Combined with a rule to ensure the output amount
doesn't decrease by more than your balance, this would effectively mean
that if half the time when you withdraw your balance you'll have to pay
a 1 satoshi fee to the remaining pool members so the the parity of the
remaining value is correct, which is inelegant, but seems like workable.

The other approach sits somewhere between those two, and would involve
adding a flag to each entry in the utxo database to say whether the
internal public key had been inverted. This would only be set if the
utxo had been created via a spending script that invoked TLUV, and TLUV
would use the flag to determine whether to add/subtract the provided
point. That seems quite complicated to implement to me, particularly if
you want to allow the flag to be able to be set by future opcodes that
we haven't thought of yet.

All of this so far assumed that the hashes for any new merkle steps are
fixed when the contract is created. If "OP_CAT" or similar were enabled,
however, you could construct those hashes programmatically in script,
which might lead to some interesting behaviour. For example, you could
construct a script that says "allow anyone to add themselves to the
buy-a-citadel pool, as long as they're contributing at least 10 BTC",
which would then verify they have control of the pubkey they're adding,
and allow them to add a script that lets them pull their 10 BTC back
out via that pubkey, and participate in key path spends in the same
way as everyone else. Of course, that sort of feature probably also
naturally extends to many of the "covenants considered harmful" cases,
eg a dollar-auction-like-contract: "Alice can spend this utxo after 1000
confirmations" or "anyone who increases the balance by 0.1 BTC can swap
Alice's pubkey for their own in the sibling script to this".

An interesting thing to note is that constructing the script can sometimes
be more efficient than hardcoding it, eg, I think

    "TapLeaf" SHA256 DUP CAT [0xc0016a] CAT SHA256

is correct for calculating the hash for the "OP_RETURN" script, and at
~17 bytes should be cheaper than the ~33 bytes it would take to hardcode
the hash.

To construct a new script programmatically you almost certainly need to
use templates, eg

    SIZE 32 EQUALVERIFY [0xc02220] SWAP CAT [0xac] CAT
    "TapLeaf" SHA256 DUP CAT SWAP CAT SHA256

might take a public key off the stack and turn it into the hash for a
script that expects a signature from that pubkey. I believe you could
construct multiple scripts and combine them via

    CAT "TapBranch" SHA256 DUP CAT SWAP CAT SHA256

or similar as well.

There's a serious caveat with doing that in practice though: if you allow
people to add in arbitrary opcodes when constructing the new script,
they could choose to have that opcode be one of the "OP_SUCCESS" opcodes,
and, if they're a miner, use that to bypass the covenant constraints
entirely. So if you want to think about this, the template being filled
in probably has to be very strict, eg including the specific PUSH opcode
for the data being provided in the witness, and checking that the length
of the witness data exactly matches the PUSH opcode being used.

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