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Subject: [bitcoin-dev] `OP_EVICT`: An Alternative to `OP_TAPLEAFUPDATEVERIFY`
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`OP_EVICT`: An Alternative to `OP_TAPLEAFUPDATEVERIFY`
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=3D=
=3D=3D=3D=3D

In late 2021, `aj` proposed `OP_TAPLEAFUPDATEVERIFY` in order to
implement CoinPools and similar constructions.

`Jeremy` observed that due to the use of Merkle tree paths, an
`OP_TLUV` would require O(log N) hash revelations in order to
reach a particular tapleaf, which, in the case of a CoinPool,
would then delete itself after spending only a particular amount
of funds.
He then observed that `OP_CTV` trees also require a similar
revelation of O(log N) transactions, but with the advantage that
once revealed, the transactions can then be reused, thus overall
the expectation is that the number of total bytes onchain is
lesser compared to `OP_TLUV`.

After some thinking, I realized that it was the use of the
Merkle tree to represent the promised-but-offchain outputs of
the CoinPool that lead to the O(log N) space usage.
I then started thinking of alternative representations of
sets of promised outputs, which would not require O(log N)
revelations by avoiding the tree structure.

Promised Outputs
----------------

Fundamentally, we can consider that a solution for scaling
Bitcoin would be to *promise* that some output *can* appear
onchain at some point in the future, without requiring that the
output be shown onchain *right now*.
Then, we can perform transactional cut-through on spends of the
promised outputs, without requiring onchain activity ("offchain").
Only if something Really Bad (TM) happens do we need to actually
drop the latest set of promised outputs onchain, where it has to
be verified globally by all fullnodes (and would thus incur scaling
and privacy costs).

As an example of the above paradigm, consider the Lightning
Network.
Outputs representing the money of each party in a channel are
promised, and *can* appear onchain (via the unilateral close
mechanism).
In the meantime, there is a mechanism for performing cut-through,
allowing transfers between channel participants; any number of
transactions can be performed that are only "solidified" later,
without expensive onchain activity.

Thus:

* A CoinPool is really a way to commit to promised outputs.
  To change the distribution of those promised outputs, the
  CoinPool operators need to post an onchain transaction, but
  that is only a 1-input-1-output transaction, and with Schnorr
  signatures the single input requires only a single signature.
  But in case something Really Bad (TM) happens, any participant
  can unilaterally close the CoinPool, instantiating the promised
  outputs.
* A statechain is really just a CoinPool hosted inside a
  Decker-Wattenhofer or Decker-Russell-Osuntokun construction.
  This allows changing the distribution of those promised outputs
  without using an onchain transaction --- instead, a new state
  in the Decker-Wattenhofer/Decker-Russell-Osuntokun construction
  is created containing the new state, which invalidates all older
  states.
  Again, any participant can unilaterally shut it down, exposing
  the state of the inner CoinPool.
* A channel factory is really just a statechain where the
  promised outputs are not simple 1-of-1 single-owner outputs,
  but are rather 2-of-2 channels.
  This allows graceful degradation, where even if the statechain
  ("factory") layer has missing participants, individual 2-of-2
  channels can still continue operating as long as they do not
  involve missing participants, without requiring all participants
  to be online for large numbers of transactions.

We can then consider that the base CoinPool usage should be enough,
as other mechanisms (`OP_CTV`+`OP_CSFS`, `SIGHASH_NOINPUT`) can be
used to implement statechains and channels and channel factories.

I therefore conclude that what we really need is "just" a way to
commit ourselves to exposing a set of promised outputs, with the
proviso that if we all agree, we can change that set (without
requiring that the current or next set be exposed, for both
scaling and privacy).

(To Bitcoin Cashers: this is not an IOU, this is *committed* and
can be enforced onchain, that is enough to threaten your offchain
counterparties into behaving correctly.
They cannot gain anything by denying the outputs they promised,
you can always drop it onchain and have it enforced, thus it is
not just merely an IOU, as IOUs are not necessarily enforceable,
but this mechanism *would* be.
Blockchain as judge+jury+executioner, not noisy marketplace.)

Importantly: both `OP_CTV` and `OP_TLUV` force the user to
decide on a particular, but ultimately arbitrary, ordering for
promised outputs.
In principle, a set of promised outputs, if the owners of those
outputs are peers, does not have *any* inherent order.
Thus, I started to think about a commitment scheme that does not
impose any ordering during commitment.

Digression: N-of-N With Eviction
--------------------------------

An issue with using an N-of-N construction is that if any single
participant is offline, the construction cannot advance its state.

This has lead to some peopple proposing to instead use K-of-N
once N reaches much larger than 2 participants for CoinPools/statechains/
channel factories.

However, even so, K-of-N still requires that K participants remain
online, and the level K is a security parameter.
If less than K participants are online, then the construction
*still* cannot advance its state.

Worse, because K < N, a single participant can have its funds
outright stolen by a quorum of K participants.
There is no way to prove that the other participants in the same
construction are not really sockpuppets of the same real-world
entity, thus it is entirely possible that the K quorum is actually
just a single participant that is now capable of stealing the
funds of all the other participants.
The only way to avoid this is to use N-oF-N: N-of-N requires
*your* keys, thus the coins are *your* coins.
In short: K-of-N, as it allows the state to be updated without your
keys (on the excuse that "if you are offline, we need to be able to
update state"), is *not your keys not your coins*.

K-of-N should really only be used if all N are your sockpuppets,
and you want to HODL your funds.
This is the difference between consensus "everyone must agree" and
voting "enough sockpuppets can be used to overpower you".

With `OP_TLUV`, however, it is possible to create an "N-of-N With
Eviction" construction.
When a participant in the N-of-N is offline, but the remaining
participants want to advance the state of the construction, they
instead evict the offline participant, creating a smaller N-of-N
where *all* participants are online, and continue operating.

This avoids the *not your keys not your coins* problem of K-of-N
constructions, while simultaneously providing a way to advance
the state without the full participant set being online.

The only real problem with `OP_TLUV` is that it takes O(log N)
hash revelations to evict one participant, and each evicted
participant requires one separate transaction.

K-of-N has the "advantage" that even if you are offline, the state
can be advanced without evicting you.
However, as noted, as the coins can be spent without your keys,
the coins are not your coins, thus this advantage may be considered
dubious --- whether you are online or offline, a quorum of K can
outright steal your coins.
Eviction here requires that your coins be returned to your control.

Committing To An Unordered Set
------------------------------

In an N-of-N CoinPool/statechain/channel factory, the ownership
of a single onchain UTXO is shared among N participants.
That is, there are a number of promised outputs, not exposed
onchain, which the N participants agree on as the "real" current
state of the construction,
However, the N participants can also agree to change the current
state of the construction, if all of them sign off on the change.

Each of the promised outputs has a value, and the sum of all
promised values is the value of the onchain UTXO.
Interestingly, each of the promised outputs also has an SECP256K1
point that can be used as a public key, and the sum of all
promised points is the point of the onchain UTXO.

Thus, the onchain UTXO can serve as a commitment to the sum of
the promised outputs.
The problem is committing to each of the individual promised
outputs.

We can observe that a digital signature not only proves knowledge
of a private key, it also commits to a particular message.
Thus, we can make each participant sign their own expected
promised output, and share the signature for their promised
output.

When a participant is to be evicted, the other participants
take the signature for the promised output of the to-be-evicted
participant, and show it onchain, to attest to the output.
Then, the onchain mechanism should then allow the rest of the
funds to be controlled by the N-of-N set minus the evicted
participant.

`OP_EVICT`
----------

With all that, let me now propose the `OP_EVICT` opcode.

`OP_EVICT` accepts a variable number of arguments.

* The stack top is either the constant `1`, or an SECP256K1
  point.
  * If it is `1` that simply means "use the Taproot internal
    pubkey", as is usual for `OP_CHECKSIG`.
* The next stack item is a number, equal to the number of
  outputs that were promised, and which will now be evicted.
* The next stack items will alternate:
  * A number indicating an output index.
  * A signature for that output.
  * Output indices must not be duplicated, and indicated
    outputs must be SegWit v1 ("Taproot") outputs.
    The public key of the output will be taken as the public
    key for the corresponding signature, and the signature
    only covers the output itself (i.e. value and
    `scriptPubKey`).
    This means the signature has no `SIGHASH`.
  * As the signature covers the public key, this prevents
    malleation of a signature using one public key to a
    signature for another public key.
* After that is another signature.
  * This signature is checked using `OP_CHECKSIG` semantics
    (including `SIGHASH` support).
  * The public key is the input point (i.e. stack top)
    **MINUS** all the public keys of the indicated outputs.

As a concrete example, suppose A, B, C, and D want to make a
CoinPool (or offchain variant of such) with the following
initial state:

* A :=3D 10
* B :=3D 6
* C :=3D 4
* D :=3D 22

Let us assume that A, B, C, and D have generated public
keys in such a way to avoid key cancellation (e.g.
precommitment, or the MuSig scheme).

The participants then generate promised outputs for the
above, and each of them shares signatures for the promised
outputs:

* sign(a, "A :=3D 10")
* sign(b, "B :=3D 6")
* sign(c, "C :=3D 4")
* sign(d, "D :=3D 22")

Once that is done, they generate:

* Q =3D A + B + C + D
* P =3D h(Q|`<1> OP_EVICT`) * Q

Then they spend their funds, creating a Taproot output:

* P :=3D 42

If all participants are online, they can move funds between
each other (or to other addresses) by cooperatively signing
using the point P, and the magic of Taproot means that use
of `OP_EVICT` is not visible.

Suppose however that B is offline.
Then A, C, and D then decide to evict B.
To do so, they create a transaction that has an output
with "B :=3D 6", and they reveal the `OP_EVICT` Tapscript
as well as sign(b, "B :=3D 6").
This lets them change state and spend their funds without
B being online.
And B remains secure, as they cannot evict B except using
the pre-signed output, which B certifies as their expected
promised output.

Note that the opcode as described above allows for multiple
evictions in the same transaction.
If B and C are offline, then the remaining participants
simply need to expose multiple outputs in the same
transaction.

Security
--------

I am not a cryptographer.
Thus, the security of this scheme is a conjecture.

As long as key cancellation is protected against, it should
be secure.
The combined fund cannot be spent except if all participants
agree.
A smaller online participant set can be created only if a
participant is evicted, and eviction will force the owned
funds of the evicted participant to be instantiated.
The other participants cannot synthesize an alternate
signature signing a different value without knowledge of the
privkey of the evicted participant.

To prevent signature replay, each update of an updateable
scheme like CoinPool et al should use a different pubkey
for each participant for each state.
As the signature covers the pubkey, it should be safe to
use a non-hardened derivation scheme so that only a single
root privkey is needed.

Additional Discussion
---------------------

### Eviction Scheme

We can consider that the eviction scheme proposed here is the
following contract:

* Either all of us agree on some transfer, OR,
* Give me my funds and the rest of you can all go play with
  your funds however you want.

The signature that commits to a promised output is then the
agreement that the particular participant believes they are
entitled to a particular amount.

We can consider that a participant can re-sign their output
with a different amount, but that is why `OP_EVICT` requires
the *other* participants to cooperatively sign as well.
If the other participants cooperatively sign, they effectively
agree to the participant re-signing for a different amount,
and thus actually covered by "all of us agree".

### Pure SCRIPT Contracts

A "pure SCRIPT contract" is a Taproot contract where the
keyspend path is not desired, and the contract is composed of
Tapscript branches.

In such a case, the expected technique would be for the
contract participants to agree on a NUMS point where none
of the participants can know the scalar (private key) behind
the point, and to use that as the internal Taproot pubkey
`Q`.
For complete protocols, the NUMS point can be a protocol-defined
constant.

As the `OP_EVICT` opcode requires that each promised output
be signed, on the face of it, this technique cannot be used
for `OP_EVICT`-promised outputs, as it is impossible to sign
using the NUMS point.

However, we should note that the requirement of a "pure SCRIPT"
contract is that none of the participants can unilaterally
sign an alternate spend.
Using an N-of-N of the participants as the Taproot internal
pubkey is sufficient to ensure this.

As a concrete example: suppose we want an HTLC, which has a
hashlock branch requiring participant A, and a timelock branch
requiring participant B.
Such a simple scheme would not require that both A and B be
able to cooperatively spend the output, thus we might have
preferred the technique of using a NUMS point as Taproot
internal pubkey.
But using a NUMS point would not allow any signature, even the
`OP_EVICT`-required signatures-of-promised-outputs.

Instead of using a NUMS point for the Taproot internal pubkey,
we can use the sum of `A[tmp] + B[tmp]` (suitably protected
against key cancellation).
Then both A and B can cooperatively sign the promised output,
and keep the promised output in an `OP_EVICT`-enforced UTXO.
After creating the signature for the promised output, A and B
can ensure that the keypath branch cannot be used by securely
deleting the private keys for `A[tmp]` and `B[tmp]`
respectively.

### Signature Half-Aggregation

It is possible to batch-validate, and as `OP_EVICT` must
validate at least two signatures (an eviction and the
signature of the remaining) it makes sense to use batch
validation for `OP_EVICT`.

Of note is that Schnorr signatures allow for third-party
half-aggregation, where the `s` components of multiple
signatures are summed together, but the `R` components
are not.

(Warning: I am not aware of any security proofs that
half-aggregation is actually **safe**!
In particular, BIP-340 does not define half-aggregation,
and its batch validation algorithm is not, to my naivete,
extensible to half-aggregation.)

Basically, if we are batch validating two signatures
`(R[0], s[0])`, `(R[1], s[1])` of two messages `m[0]`
and `m[1]` signed by two keys `A[0]` and `A[1]`, we
would do:

* For `i =3D 0, 1`: `e[i] =3D h(R[i]|m[i])`
* Check: `(s[0] + s[1]) * G` is equal to `R[0] + e[0] * A[0] + R[1] + e[1] =
* A[1]`.

As we can see, the `s` can be summed before being
posted on the blockchain, as validators do not need
individual `s[i]`.
However, `R` cannot be summed as each one needs to be
hashed.

This half-aggregation is third-party, i.e. someone
without any knowledge of any private keys can simply
sum the `s` components of multiple signatures.

As `OP_EVICT` always validates at least two signatures,
using half-aggregation can remove at least 32 weight
units, and each additional promised output being evicted
is another signature whose `s` can be added to the sum.
Of course, **that depends on half-aggregation being
secure**.

### Relationship to Other Opcodes

`OP_CTV` does other things than this opcode, and cannot
be used as a direct alternative.
In particular while `OP_CTV` *can* commit to a set of
promised outputs, if a promised output needs to be
published, the remaining funds are now distributed over a
set of UTXOs.
Thus, "reviving" the CoinPool (or offchain variant thereof)
requires consuming multiple UTXOs, and the consumption of
multiple UTXOs is risky unless specifically designd for it.
(In particular, if the UTXOs have different signer sets,
one signer set can initially cooperate to revive the
CoinPool, then spend their UTXO to a different transaction,
which if confirmed will invalidate the revival transaction.)

This opcode seems largely in direct competitiong with
`OP_TLUV`, with largely the same design goal.
Its advantage is reduced number of eviction transactions,
as multiple evictions, plus the revival of the CoinPool,
can be put in a single transaction.
It has the disadvantage relative to `OP_TLUV` of requiring
point operations.
I have not explored completely, but my instinct suggests
that `OP_TLUV` use may require at least one signature
validation anyway.

It may be possible to implement `OP_EVICT` in terms of
`OP_TX`/`OP_TXHASH`, `OP_CSFS`, and a point-subtraction
operation.
However, `OP_EVICT` allows for the trivial implementation
of batch validation (and, if half-aggregation is safe, to
use half-aggregation instead), whereas we expect multiple
`OP_CSFS` to be needed to implement this, without any
possibility of batch validation.
It may be possible to design an `OP_CSFS` variant that
allows batch validation, such as by extending the virtual
machine with an accumulator for pending signature
validations.