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From: Antoine Riard <antoine.riard@gmail.com>
Date: Fri, 18 Feb 2022 13:09:07 -0500
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To: ZmnSCPxj <ZmnSCPxj@protonmail.com>, 
 Bitcoin Protocol Discussion <bitcoin-dev@lists.linuxfoundation.org>
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Cc: Anthony Towns <aj@erisian.com.au>
Subject: Re: [bitcoin-dev] `OP_EVICT`: An Alternative to
	`OP_TAPLEAFUPDATEVERIFY`
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Hi Zeeman,

> 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.

In the context of payment pools, I think the O(log N) revelations can be
avoided already today by pre-signing all the combinations of
promised-but-offchain outputs publications order. However, this approach
presents a factorial complexity and appears as an intractable problem for
high-number of pool users.

I think this factorial complexity issue is the primary problem to enable
scalable payment pools. This issue appears to be solvable by introducing an
accumulator at the script interpreter level. IMO, the efficiency of the
accumulated set representations comes as a second-order issue.

In the comparison of different covenant primitives, I believe we should ask
first if the flexibility offered is enough to solve the factorial
complexity. I would say performance trade-offs analysis can only be
conducted in logically equivalent primitives.

> A statechain is really just a CoinPool hosted inside a
>  Decker-Wattenhofer or Decker-Russell-Osuntokun construction.

Note, to the best of my knowledge, how to use LN-Penalty in the context of
multi-party construction is still an unsolved issue. If an invalidated
state is published on-chain, how do you guarantee that the punished output
value is distributed "fairly" among the "honest" set of users ? At least
where fairness is defined as a reasonable proportion of the balances they
owned in the latest state.

> (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.)

To be fair towards the Bitcoin Cashers, I think there are still limitations
of LN, we have not solved yet. Especially, w.r.t to mass exits from the
off-chain layers to the chain, where the blocks would stay fulfilled longer
than the standard HTLC timelocks, at  a fee price point that the average
user can't buy... I'm not sure if we have outlawed the "bank runs" scenario
yet of LN.

I would say yes the Blockchain is a juge authority, but in the worst-case
we might be all in market competition to get enforcement.

> 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.

I think we should dissociate a) *outputs publication ordering* from the b)
*spends paths ordering* itself. Even if to each spend path a output
publication is attached, the ordering constraint might not present the same
complexity.

Under this distinction, are you sure that TLUV imposes an ordering on the
output publication ?

> 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.

I think we should dissociate two types of pool spends : a) eviction by the
pool unanimity in case of irresponsive participants and b) unilateral
withdrawal by a participant because of the liquidity allocation policy. I
think the distinction is worthy, as the pool participant should be stable
and the eviction not abused.

I'm not sure if TLUV enables b), at least without transforming the
unilateral withdrawal into an eviction. To ensure the TLUV operation is
correct  (spent leaf is removed, withdrawing participant point removed,
etc), the script content must be inspected by *all* the participant.
However, I believe
knowledge of this content effectively allows you to play it out against the
pool at any time ? It's likely solvable at the price of a CHECKSIG.

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

>  * If it is `1` that simply means "use the Taproot internal
>    pubkey", as is usual for `OP_CHECKSIG`.

IIUC, this assumes the deployment of BIP118, where if the  public key is a
single byte 0x01, the internal pubkey is used
for verification.

>  * Output indices must not be duplicated, and indicated
>    outputs must be SegWit v1 ("Taproot") outputs.

I think public key duplication must not be verified. If a duplicated public
key is present, the point is subtracted twice from the internal pubkey and
therefore the aggregated
key remains unknown ? So it sounds to me safe against replay attacks.

>  * The public key is the input point (i.e. stack top)
>    **MINUS** all the public keys of the indicated outputs.

Can you prevent eviction abuse where one counterparty threatens to evict
everyone as all the output signatures are known among participants and free
to sum ? (at least not considering fees)

> 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.

I think in the context of (off-chain) payment pool, OP_EVICT requires
participant cooperation *after* the state update to allow a single
participant to withdraw her funds.

I believe this is unsafe if we retain as an off-chain construction security
requirement that a participant should have the unilateral means to enforce
the latest agreed upon state at any time during the construction lifetime.

I would say an OP_EVICT construction could solve the issue where the pool
participants exchange pre-signatures of the internal pubkey with the
withdrawing participant point removed. However, I believe such fix would a)
block promised outputs batching (or at least in a pre-committed way like
radix pools) and b) be grieved by the factorial complexity described above.

> The combined fund cannot be spent except if all participants
> agree.

If all participants agree minus the evicted ones, correct ? The output
promises signatures are shared at state setup, therefore no additional
contribution from the evicted participant (I think).

> 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.

I'm not even sure if it's required with OP_EVICT, as the publication of the
promised output are ultimately restrained by a signature of the updated
internal pubkey, this set of signers verify that promised output N does
bind to the published state N ?

> 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.

I believe you can slightly modify TLUV to make it functional for CoinPool
revival, where you want to prevent equivocation among the remaining set of
signers. Though, I'm leaning to agree that you may require at least one
signature validation  (first to restrain spend authorization inside the
pool participants, second to attach fees at broadcast-time).

> 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.

I agree that in the context of payment pools, aggregation of
non-cooperative unilateral spends is a scalability bottleneck, especially
in the face of mempools congestion. If we rely on merkle
trees as the accumulator primitive, there is still the path to aggregate
many branches in-flight.

Any misunderstandings of this proposal are my own.
,
Antoine

Le jeu. 17 f=C3=A9vr. 2022 =C3=A0 21:45, ZmnSCPxj via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> a =C3=A9crit :

> `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.
> _______________________________________________
> bitcoin-dev mailing list
> bitcoin-dev@lists.linuxfoundation.org
> https://lists.linuxfoundation.org/mailman/listinfo/bitcoin-dev
>

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<div dir=3D"ltr">Hi Zeeman,<br><br>&gt; After some thinking, I realized tha=
t it was the use of the<br>&gt; Merkle tree to represent the promised-but-o=
ffchain outputs of<br>&gt; the CoinPool that lead to the O(log N) space usa=
ge.<br>&gt; I then started thinking of alternative representations of<br>&g=
t; sets of promised outputs, which would not require O(log N)<br>&gt; revel=
ations by avoiding the tree structure.<br><br>In the context of payment poo=
ls, I think the O(log N) revelations can be avoided already today by pre-si=
gning all the combinations of promised-but-offchain outputs publications or=
der. However, this approach presents a factorial complexity and appears as =
an intractable problem for high-number of pool users.<br><br>I think this f=
actorial complexity issue is the primary problem to enable scalable payment=
 pools. This issue appears to be solvable by introducing an accumulator at =
the script interpreter level. IMO, the efficiency of the accumulated set re=
presentations comes as a second-order issue.<br><br>In the comparison of di=
fferent covenant primitives, I believe we should ask first if the flexibili=
ty offered is enough to solve the factorial complexity. I would say perform=
ance trade-offs analysis can only be conducted in logically equivalent prim=
itives.<br><br>&gt; A statechain is really just a CoinPool hosted inside a<=
br>&gt; =C2=A0Decker-Wattenhofer or Decker-Russell-Osuntokun construction.<=
br><br>Note, to the best of my knowledge, how to use LN-Penalty in the cont=
ext of multi-party construction is still an unsolved issue. If an invalidat=
ed state is published on-chain, how do you guarantee that the punished outp=
ut value is distributed &quot;fairly&quot; among the &quot;honest&quot; set=
 of users ? At least<br>where fairness is defined as a reasonable proportio=
n of the balances they owned in the latest state.<br><br>&gt; (To Bitcoin C=
ashers: this is not an IOU, this is *committed* and<br>&gt; can be enforced=
 onchain, that is enough to threaten your offchain<br>&gt; counterparties i=
nto behaving correctly.<br>&gt; They cannot gain anything by denying the ou=
tputs they promised,<br>&gt; you can always drop it onchain and have it enf=
orced, thus it is<br>&gt; not just merely an IOU, as IOUs are not necessari=
ly enforceable,<br>&gt; but this mechanism *would* be.<br>&gt; Blockchain a=
s judge+jury+executioner, not noisy marketplace.)<br><br>To be fair towards=
 the Bitcoin Cashers, I think there are still limitations of LN, we have no=
t solved yet. Especially, w.r.t to mass exits from the off-chain layers to =
the chain, where the blocks would stay fulfilled longer than the standard H=
TLC timelocks, at=C2=A0 a fee price point that the average user can&#39;t b=
uy... I&#39;m not sure if we have outlawed the &quot;bank runs&quot; scenar=
io yet of LN.<br><br>I would say yes the Blockchain is a juge authority, bu=
t in the worst-case we might be all in market competition to get enforcemen=
t.<br><br>&gt; In principle, a set of promised outputs, if the owners of th=
ose<br>&gt; outputs are peers, does not have *any* inherent order.<br>&gt; =
Thus, I started to think about a commitment scheme that does not<br>&gt; im=
pose any ordering during commitment.<br><br>I think we should dissociate a)=
 *outputs publication ordering* from the b) *spends paths ordering* itself.=
 Even if to each spend path a output publication is attached, the ordering =
constraint might not present the same complexity.<br><br>Under this distinc=
tion, are you sure that TLUV imposes an ordering on the output publication =
?<br><br>&gt; With `OP_TLUV`, however, it is possible to create an &quot;N-=
of-N With<br>&gt; Eviction&quot; construction.<br>&gt; When a participant i=
n the N-of-N is offline, but the remaining<br>&gt; participants want to adv=
ance the state of the construction, they<br>&gt; instead evict the offline =
participant, creating a smaller N-of-N<br>&gt; where *all* participants are=
 online, and continue operating.<br><br>I think we should dissociate two ty=
pes of pool spends : a) eviction by the pool unanimity in case of irrespons=
ive participants and b) unilateral withdrawal by a participant because of t=
he liquidity allocation policy. I think the distinction is worthy, as the p=
ool participant should be stable and the eviction not abused.<br><br>I&#39;=
m not sure if TLUV enables b), at least without transforming the unilateral=
 withdrawal into an eviction. To ensure the TLUV operation is correct=C2=A0=
 (spent leaf is removed, withdrawing participant point removed, etc), the s=
cript content must be inspected by *all* the participant. However, I believ=
e <br>knowledge of this content effectively allows you to play it out again=
st the pool at any time ? It&#39;s likely solvable at the price of a CHECKS=
IG.<br><br>`OP_EVICT`<br>----------<br><br>&gt; =C2=A0* If it is `1` that s=
imply means &quot;use the Taproot internal<br>&gt; =C2=A0 =C2=A0pubkey&quot=
;, as is usual for `OP_CHECKSIG`.<br><br>IIUC, this assumes the deployment =
of BIP118, where if the=C2=A0 public key is a single byte 0x01, the interna=
l pubkey is used<br>for verification.<br><br>&gt; =C2=A0* Output indices mu=
st not be duplicated, and indicated<br>&gt; =C2=A0 =C2=A0outputs must be Se=
gWit v1 (&quot;Taproot&quot;) outputs.<br><br>I think public key duplicatio=
n must not be verified. If a duplicated public key is present, the point is=
 subtracted twice from the internal pubkey and therefore the aggregated<br>=
key remains unknown ? So it sounds to me safe against replay attacks.<br><b=
r>&gt; =C2=A0* The public key is the input point (i.e. stack top)<br>&gt; =
=C2=A0 =C2=A0**MINUS** all the public keys of the indicated outputs.<br><br=
>Can you prevent eviction abuse where one counterparty threatens to evict e=
veryone as all the output signatures are known among participants and free =
to sum ? (at least not considering fees)<br><br>&gt; Suppose however that B=
 is offline.<br>&gt; Then A, C, and D then decide to evict B.<br>&gt; To do=
 so, they create a transaction that has an output<br>&gt; with &quot;B :=3D=
 6&quot;, and they reveal the `OP_EVICT` Tapscript<br>&gt; as well as sign(=
b, &quot;B :=3D 6&quot;).<br>&gt; This lets them change state and spend the=
ir funds without<br>&gt; B being online.<br>&gt; And B remains secure, as t=
hey cannot evict B except using<br>&gt; the pre-signed output, which B cert=
ifies as their expected<br>&gt; promised output.<br><br>I think in the cont=
ext of (off-chain) payment pool, OP_EVICT requires participant cooperation =
*after* the state update to allow a single participant to withdraw her fund=
s.<br><br>I believe this is unsafe if we retain as an off-chain constructio=
n security requirement that a participant should have the unilateral means =
to enforce the latest agreed upon state at any time during the construction=
 lifetime.<br><br>I would say an OP_EVICT construction could solve the issu=
e where the pool participants exchange pre-signatures of the internal pubke=
y with the withdrawing participant point removed. However, I believe such f=
ix would a) block promised outputs batching (or at least in a pre-committed=
 way like radix pools) and b) be grieved by the factorial complexity descri=
bed above.<br><br>&gt; The combined fund cannot be spent except if all part=
icipants<br>&gt; agree.<br><br>If all participants agree minus the evicted =
ones, correct ? The output promises signatures are shared at state setup, t=
herefore no additional contribution from the evicted participant (I think).=
<br><br>&gt; To prevent signature replay, each update of an updateable<br>&=
gt; scheme like CoinPool et al should use a different pubkey<br>&gt; for ea=
ch participant for each state.<br><br>I&#39;m not even sure if it&#39;s req=
uired with OP_EVICT, as the publication of the promised output are ultimate=
ly restrained by a signature of the updated internal pubkey, this set of si=
gners verify that promised output N does bind to the published state N ?<br=
><br>&gt; Its advantage is reduced number of eviction transactions,<br>&gt;=
 as multiple evictions, plus the revival of the CoinPool,<br>&gt; can be pu=
t in a single transaction.<br>&gt; It has the disadvantage relative to `OP_=
TLUV` of requiring<br>&gt; point operations.<br>&gt; I have not explored co=
mpletely, but my instinct suggests<br>&gt; that `OP_TLUV` use may require a=
t least one signature<br>&gt; validation anyway.<br><br>I believe you can s=
lightly modify TLUV to make it functional for CoinPool revival, where you w=
ant to prevent equivocation among the remaining set of signers. Though, I&#=
39;m leaning to agree that you may require at least one signature validatio=
n=C2=A0 (first to restrain spend authorization inside the pool participants=
, second to attach fees at broadcast-time).<br><br>&gt; It may be possible =
to design an `OP_CSFS` variant that<br>&gt; allows batch validation, such a=
s by extending the virtual<br>&gt; machine with an accumulator for pending =
signature<br>&gt; validations.<br><br>I agree that in the context of paymen=
t pools, aggregation of non-cooperative unilateral spends is a scalability =
bottleneck, especially in the face of mempools congestion. If we rely on me=
rkle<br>trees as the accumulator primitive, there is still the path to aggr=
egate many branches in-flight.<br><br>Any misunderstandings of this proposa=
l are my own.<br>,<br>Antoine<br></div><br><div class=3D"gmail_quote"><div =
dir=3D"ltr" class=3D"gmail_attr">Le=C2=A0jeu. 17 f=C3=A9vr. 2022 =C3=A0=C2=
=A021:45, ZmnSCPxj via bitcoin-dev &lt;<a href=3D"mailto:bitcoin-dev@lists.=
linuxfoundation.org">bitcoin-dev@lists.linuxfoundation.org</a>&gt; a =C3=A9=
crit=C2=A0:<br></div><blockquote class=3D"gmail_quote" style=3D"margin:0px =
0px 0px 0.8ex;border-left:1px solid rgb(204,204,204);padding-left:1ex">`OP_=
EVICT`: An Alternative to `OP_TAPLEAFUPDATEVERIFY`<br>
=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<br>
<br>
In late 2021, `aj` proposed `OP_TAPLEAFUPDATEVERIFY` in order to<br>
implement CoinPools and similar constructions.<br>
<br>
`Jeremy` observed that due to the use of Merkle tree paths, an<br>
`OP_TLUV` would require O(log N) hash revelations in order to<br>
reach a particular tapleaf, which, in the case of a CoinPool,<br>
would then delete itself after spending only a particular amount<br>
of funds.<br>
He then observed that `OP_CTV` trees also require a similar<br>
revelation of O(log N) transactions, but with the advantage that<br>
once revealed, the transactions can then be reused, thus overall<br>
the expectation is that the number of total bytes onchain is<br>
lesser compared to `OP_TLUV`.<br>
<br>
After some thinking, I realized that it was the use of the<br>
Merkle tree to represent the promised-but-offchain outputs of<br>
the CoinPool that lead to the O(log N) space usage.<br>
I then started thinking of alternative representations of<br>
sets of promised outputs, which would not require O(log N)<br>
revelations by avoiding the tree structure.<br>
<br>
Promised Outputs<br>
----------------<br>
<br>
Fundamentally, we can consider that a solution for scaling<br>
Bitcoin would be to *promise* that some output *can* appear<br>
onchain at some point in the future, without requiring that the<br>
output be shown onchain *right now*.<br>
Then, we can perform transactional cut-through on spends of the<br>
promised outputs, without requiring onchain activity (&quot;offchain&quot;)=
.<br>
Only if something Really Bad (TM) happens do we need to actually<br>
drop the latest set of promised outputs onchain, where it has to<br>
be verified globally by all fullnodes (and would thus incur scaling<br>
and privacy costs).<br>
<br>
As an example of the above paradigm, consider the Lightning<br>
Network.<br>
Outputs representing the money of each party in a channel are<br>
promised, and *can* appear onchain (via the unilateral close<br>
mechanism).<br>
In the meantime, there is a mechanism for performing cut-through,<br>
allowing transfers between channel participants; any number of<br>
transactions can be performed that are only &quot;solidified&quot; later,<b=
r>
without expensive onchain activity.<br>
<br>
Thus:<br>
<br>
* A CoinPool is really a way to commit to promised outputs.<br>
=C2=A0 To change the distribution of those promised outputs, the<br>
=C2=A0 CoinPool operators need to post an onchain transaction, but<br>
=C2=A0 that is only a 1-input-1-output transaction, and with Schnorr<br>
=C2=A0 signatures the single input requires only a single signature.<br>
=C2=A0 But in case something Really Bad (TM) happens, any participant<br>
=C2=A0 can unilaterally close the CoinPool, instantiating the promised<br>
=C2=A0 outputs.<br>
* A statechain is really just a CoinPool hosted inside a<br>
=C2=A0 Decker-Wattenhofer or Decker-Russell-Osuntokun construction.<br>
=C2=A0 This allows changing the distribution of those promised outputs<br>
=C2=A0 without using an onchain transaction --- instead, a new state<br>
=C2=A0 in the Decker-Wattenhofer/Decker-Russell-Osuntokun construction<br>
=C2=A0 is created containing the new state, which invalidates all older<br>
=C2=A0 states.<br>
=C2=A0 Again, any participant can unilaterally shut it down, exposing<br>
=C2=A0 the state of the inner CoinPool.<br>
* A channel factory is really just a statechain where the<br>
=C2=A0 promised outputs are not simple 1-of-1 single-owner outputs,<br>
=C2=A0 but are rather 2-of-2 channels.<br>
=C2=A0 This allows graceful degradation, where even if the statechain<br>
=C2=A0 (&quot;factory&quot;) layer has missing participants, individual 2-o=
f-2<br>
=C2=A0 channels can still continue operating as long as they do not<br>
=C2=A0 involve missing participants, without requiring all participants<br>
=C2=A0 to be online for large numbers of transactions.<br>
<br>
We can then consider that the base CoinPool usage should be enough,<br>
as other mechanisms (`OP_CTV`+`OP_CSFS`, `SIGHASH_NOINPUT`) can be<br>
used to implement statechains and channels and channel factories.<br>
<br>
I therefore conclude that what we really need is &quot;just&quot; a way to<=
br>
commit ourselves to exposing a set of promised outputs, with the<br>
proviso that if we all agree, we can change that set (without<br>
requiring that the current or next set be exposed, for both<br>
scaling and privacy).<br>
<br>
(To Bitcoin Cashers: this is not an IOU, this is *committed* and<br>
can be enforced onchain, that is enough to threaten your offchain<br>
counterparties into behaving correctly.<br>
They cannot gain anything by denying the outputs they promised,<br>
you can always drop it onchain and have it enforced, thus it is<br>
not just merely an IOU, as IOUs are not necessarily enforceable,<br>
but this mechanism *would* be.<br>
Blockchain as judge+jury+executioner, not noisy marketplace.)<br>
<br>
Importantly: both `OP_CTV` and `OP_TLUV` force the user to<br>
decide on a particular, but ultimately arbitrary, ordering for<br>
promised outputs.<br>
In principle, a set of promised outputs, if the owners of those<br>
outputs are peers, does not have *any* inherent order.<br>
Thus, I started to think about a commitment scheme that does not<br>
impose any ordering during commitment.<br>
<br>
Digression: N-of-N With Eviction<br>
--------------------------------<br>
<br>
An issue with using an N-of-N construction is that if any single<br>
participant is offline, the construction cannot advance its state.<br>
<br>
This has lead to some peopple proposing to instead use K-of-N<br>
once N reaches much larger than 2 participants for CoinPools/statechains/<b=
r>
channel factories.<br>
<br>
However, even so, K-of-N still requires that K participants remain<br>
online, and the level K is a security parameter.<br>
If less than K participants are online, then the construction<br>
*still* cannot advance its state.<br>
<br>
Worse, because K &lt; N, a single participant can have its funds<br>
outright stolen by a quorum of K participants.<br>
There is no way to prove that the other participants in the same<br>
construction are not really sockpuppets of the same real-world<br>
entity, thus it is entirely possible that the K quorum is actually<br>
just a single participant that is now capable of stealing the<br>
funds of all the other participants.<br>
The only way to avoid this is to use N-oF-N: N-of-N requires<br>
*your* keys, thus the coins are *your* coins.<br>
In short: K-of-N, as it allows the state to be updated without your<br>
keys (on the excuse that &quot;if you are offline, we need to be able to<br=
>
update state&quot;), is *not your keys not your coins*.<br>
<br>
K-of-N should really only be used if all N are your sockpuppets,<br>
and you want to HODL your funds.<br>
This is the difference between consensus &quot;everyone must agree&quot; an=
d<br>
voting &quot;enough sockpuppets can be used to overpower you&quot;.<br>
<br>
With `OP_TLUV`, however, it is possible to create an &quot;N-of-N With<br>
Eviction&quot; construction.<br>
When a participant in the N-of-N is offline, but the remaining<br>
participants want to advance the state of the construction, they<br>
instead evict the offline participant, creating a smaller N-of-N<br>
where *all* participants are online, and continue operating.<br>
<br>
This avoids the *not your keys not your coins* problem of K-of-N<br>
constructions, while simultaneously providing a way to advance<br>
the state without the full participant set being online.<br>
<br>
The only real problem with `OP_TLUV` is that it takes O(log N)<br>
hash revelations to evict one participant, and each evicted<br>
participant requires one separate transaction.<br>
<br>
K-of-N has the &quot;advantage&quot; that even if you are offline, the stat=
e<br>
can be advanced without evicting you.<br>
However, as noted, as the coins can be spent without your keys,<br>
the coins are not your coins, thus this advantage may be considered<br>
dubious --- whether you are online or offline, a quorum of K can<br>
outright steal your coins.<br>
Eviction here requires that your coins be returned to your control.<br>
<br>
Committing To An Unordered Set<br>
------------------------------<br>
<br>
In an N-of-N CoinPool/statechain/channel factory, the ownership<br>
of a single onchain UTXO is shared among N participants.<br>
That is, there are a number of promised outputs, not exposed<br>
onchain, which the N participants agree on as the &quot;real&quot; current<=
br>
state of the construction,<br>
However, the N participants can also agree to change the current<br>
state of the construction, if all of them sign off on the change.<br>
<br>
Each of the promised outputs has a value, and the sum of all<br>
promised values is the value of the onchain UTXO.<br>
Interestingly, each of the promised outputs also has an SECP256K1<br>
point that can be used as a public key, and the sum of all<br>
promised points is the point of the onchain UTXO.<br>
<br>
Thus, the onchain UTXO can serve as a commitment to the sum of<br>
the promised outputs.<br>
The problem is committing to each of the individual promised<br>
outputs.<br>
<br>
We can observe that a digital signature not only proves knowledge<br>
of a private key, it also commits to a particular message.<br>
Thus, we can make each participant sign their own expected<br>
promised output, and share the signature for their promised<br>
output.<br>
<br>
When a participant is to be evicted, the other participants<br>
take the signature for the promised output of the to-be-evicted<br>
participant, and show it onchain, to attest to the output.<br>
Then, the onchain mechanism should then allow the rest of the<br>
funds to be controlled by the N-of-N set minus the evicted<br>
participant.<br>
<br>
`OP_EVICT`<br>
----------<br>
<br>
With all that, let me now propose the `OP_EVICT` opcode.<br>
<br>
`OP_EVICT` accepts a variable number of arguments.<br>
<br>
* The stack top is either the constant `1`, or an SECP256K1<br>
=C2=A0 point.<br>
=C2=A0 * If it is `1` that simply means &quot;use the Taproot internal<br>
=C2=A0 =C2=A0 pubkey&quot;, as is usual for `OP_CHECKSIG`.<br>
* The next stack item is a number, equal to the number of<br>
=C2=A0 outputs that were promised, and which will now be evicted.<br>
* The next stack items will alternate:<br>
=C2=A0 * A number indicating an output index.<br>
=C2=A0 * A signature for that output.<br>
=C2=A0 * Output indices must not be duplicated, and indicated<br>
=C2=A0 =C2=A0 outputs must be SegWit v1 (&quot;Taproot&quot;) outputs.<br>
=C2=A0 =C2=A0 The public key of the output will be taken as the public<br>
=C2=A0 =C2=A0 key for the corresponding signature, and the signature<br>
=C2=A0 =C2=A0 only covers the output itself (i.e. value and<br>
=C2=A0 =C2=A0 `scriptPubKey`).<br>
=C2=A0 =C2=A0 This means the signature has no `SIGHASH`.<br>
=C2=A0 * As the signature covers the public key, this prevents<br>
=C2=A0 =C2=A0 malleation of a signature using one public key to a<br>
=C2=A0 =C2=A0 signature for another public key.<br>
* After that is another signature.<br>
=C2=A0 * This signature is checked using `OP_CHECKSIG` semantics<br>
=C2=A0 =C2=A0 (including `SIGHASH` support).<br>
=C2=A0 * The public key is the input point (i.e. stack top)<br>
=C2=A0 =C2=A0 **MINUS** all the public keys of the indicated outputs.<br>
<br>
As a concrete example, suppose A, B, C, and D want to make a<br>
CoinPool (or offchain variant of such) with the following<br>
initial state:<br>
<br>
* A :=3D 10<br>
* B :=3D 6<br>
* C :=3D 4<br>
* D :=3D 22<br>
<br>
Let us assume that A, B, C, and D have generated public<br>
keys in such a way to avoid key cancellation (e.g.<br>
precommitment, or the MuSig scheme).<br>
<br>
The participants then generate promised outputs for the<br>
above, and each of them shares signatures for the promised<br>
outputs:<br>
<br>
* sign(a, &quot;A :=3D 10&quot;)<br>
* sign(b, &quot;B :=3D 6&quot;)<br>
* sign(c, &quot;C :=3D 4&quot;)<br>
* sign(d, &quot;D :=3D 22&quot;)<br>
<br>
Once that is done, they generate:<br>
<br>
* Q =3D A + B + C + D<br>
* P =3D h(Q|`&lt;1&gt; OP_EVICT`) * Q<br>
<br>
Then they spend their funds, creating a Taproot output:<br>
<br>
* P :=3D 42<br>
<br>
If all participants are online, they can move funds between<br>
each other (or to other addresses) by cooperatively signing<br>
using the point P, and the magic of Taproot means that use<br>
of `OP_EVICT` is not visible.<br>
<br>
Suppose however that B is offline.<br>
Then A, C, and D then decide to evict B.<br>
To do so, they create a transaction that has an output<br>
with &quot;B :=3D 6&quot;, and they reveal the `OP_EVICT` Tapscript<br>
as well as sign(b, &quot;B :=3D 6&quot;).<br>
This lets them change state and spend their funds without<br>
B being online.<br>
And B remains secure, as they cannot evict B except using<br>
the pre-signed output, which B certifies as their expected<br>
promised output.<br>
<br>
Note that the opcode as described above allows for multiple<br>
evictions in the same transaction.<br>
If B and C are offline, then the remaining participants<br>
simply need to expose multiple outputs in the same<br>
transaction.<br>
<br>
Security<br>
--------<br>
<br>
I am not a cryptographer.<br>
Thus, the security of this scheme is a conjecture.<br>
<br>
As long as key cancellation is protected against, it should<br>
be secure.<br>
The combined fund cannot be spent except if all participants<br>
agree.<br>
A smaller online participant set can be created only if a<br>
participant is evicted, and eviction will force the owned<br>
funds of the evicted participant to be instantiated.<br>
The other participants cannot synthesize an alternate<br>
signature signing a different value without knowledge of the<br>
privkey of the evicted participant.<br>
<br>
To prevent signature replay, each update of an updateable<br>
scheme like CoinPool et al should use a different pubkey<br>
for each participant for each state.<br>
As the signature covers the pubkey, it should be safe to<br>
use a non-hardened derivation scheme so that only a single<br>
root privkey is needed.<br>
<br>
Additional Discussion<br>
---------------------<br>
<br>
### Eviction Scheme<br>
<br>
We can consider that the eviction scheme proposed here is the<br>
following contract:<br>
<br>
* Either all of us agree on some transfer, OR,<br>
* Give me my funds and the rest of you can all go play with<br>
=C2=A0 your funds however you want.<br>
<br>
The signature that commits to a promised output is then the<br>
agreement that the particular participant believes they are<br>
entitled to a particular amount.<br>
<br>
We can consider that a participant can re-sign their output<br>
with a different amount, but that is why `OP_EVICT` requires<br>
the *other* participants to cooperatively sign as well.<br>
If the other participants cooperatively sign, they effectively<br>
agree to the participant re-signing for a different amount,<br>
and thus actually covered by &quot;all of us agree&quot;.<br>
<br>
### Pure SCRIPT Contracts<br>
<br>
A &quot;pure SCRIPT contract&quot; is a Taproot contract where the<br>
keyspend path is not desired, and the contract is composed of<br>
Tapscript branches.<br>
<br>
In such a case, the expected technique would be for the<br>
contract participants to agree on a NUMS point where none<br>
of the participants can know the scalar (private key) behind<br>
the point, and to use that as the internal Taproot pubkey<br>
`Q`.<br>
For complete protocols, the NUMS point can be a protocol-defined<br>
constant.<br>
<br>
As the `OP_EVICT` opcode requires that each promised output<br>
be signed, on the face of it, this technique cannot be used<br>
for `OP_EVICT`-promised outputs, as it is impossible to sign<br>
using the NUMS point.<br>
<br>
However, we should note that the requirement of a &quot;pure SCRIPT&quot;<b=
r>
contract is that none of the participants can unilaterally<br>
sign an alternate spend.<br>
Using an N-of-N of the participants as the Taproot internal<br>
pubkey is sufficient to ensure this.<br>
<br>
As a concrete example: suppose we want an HTLC, which has a<br>
hashlock branch requiring participant A, and a timelock branch<br>
requiring participant B.<br>
Such a simple scheme would not require that both A and B be<br>
able to cooperatively spend the output, thus we might have<br>
preferred the technique of using a NUMS point as Taproot<br>
internal pubkey.<br>
But using a NUMS point would not allow any signature, even the<br>
`OP_EVICT`-required signatures-of-promised-outputs.<br>
<br>
Instead of using a NUMS point for the Taproot internal pubkey,<br>
we can use the sum of `A[tmp] + B[tmp]` (suitably protected<br>
against key cancellation).<br>
Then both A and B can cooperatively sign the promised output,<br>
and keep the promised output in an `OP_EVICT`-enforced UTXO.<br>
After creating the signature for the promised output, A and B<br>
can ensure that the keypath branch cannot be used by securely<br>
deleting the private keys for `A[tmp]` and `B[tmp]`<br>
respectively.<br>
<br>
### Signature Half-Aggregation<br>
<br>
It is possible to batch-validate, and as `OP_EVICT` must<br>
validate at least two signatures (an eviction and the<br>
signature of the remaining) it makes sense to use batch<br>
validation for `OP_EVICT`.<br>
<br>
Of note is that Schnorr signatures allow for third-party<br>
half-aggregation, where the `s` components of multiple<br>
signatures are summed together, but the `R` components<br>
are not.<br>
<br>
(Warning: I am not aware of any security proofs that<br>
half-aggregation is actually **safe**!<br>
In particular, BIP-340 does not define half-aggregation,<br>
and its batch validation algorithm is not, to my naivete,<br>
extensible to half-aggregation.)<br>
<br>
Basically, if we are batch validating two signatures<br>
`(R[0], s[0])`, `(R[1], s[1])` of two messages `m[0]`<br>
and `m[1]` signed by two keys `A[0]` and `A[1]`, we<br>
would do:<br>
<br>
* For `i =3D 0, 1`: `e[i] =3D h(R[i]|m[i])`<br>
* Check: `(s[0] + s[1]) * G` is equal to `R[0] + e[0] * A[0] + R[1] + e[1] =
* A[1]`.<br>
<br>
As we can see, the `s` can be summed before being<br>
posted on the blockchain, as validators do not need<br>
individual `s[i]`.<br>
However, `R` cannot be summed as each one needs to be<br>
hashed.<br>
<br>
This half-aggregation is third-party, i.e. someone<br>
without any knowledge of any private keys can simply<br>
sum the `s` components of multiple signatures.<br>
<br>
As `OP_EVICT` always validates at least two signatures,<br>
using half-aggregation can remove at least 32 weight<br>
units, and each additional promised output being evicted<br>
is another signature whose `s` can be added to the sum.<br>
Of course, **that depends on half-aggregation being<br>
secure**.<br>
<br>
### Relationship to Other Opcodes<br>
<br>
`OP_CTV` does other things than this opcode, and cannot<br>
be used as a direct alternative.<br>
In particular while `OP_CTV` *can* commit to a set of<br>
promised outputs, if a promised output needs to be<br>
published, the remaining funds are now distributed over a<br>
set of UTXOs.<br>
Thus, &quot;reviving&quot; the CoinPool (or offchain variant thereof)<br>
requires consuming multiple UTXOs, and the consumption of<br>
multiple UTXOs is risky unless specifically designd for it.<br>
(In particular, if the UTXOs have different signer sets,<br>
one signer set can initially cooperate to revive the<br>
CoinPool, then spend their UTXO to a different transaction,<br>
which if confirmed will invalidate the revival transaction.)<br>
<br>
This opcode seems largely in direct competitiong with<br>
`OP_TLUV`, with largely the same design goal.<br>
Its advantage is reduced number of eviction transactions,<br>
as multiple evictions, plus the revival of the CoinPool,<br>
can be put in a single transaction.<br>
It has the disadvantage relative to `OP_TLUV` of requiring<br>
point operations.<br>
I have not explored completely, but my instinct suggests<br>
that `OP_TLUV` use may require at least one signature<br>
validation anyway.<br>
<br>
It may be possible to implement `OP_EVICT` in terms of<br>
`OP_TX`/`OP_TXHASH`, `OP_CSFS`, and a point-subtraction<br>
operation.<br>
However, `OP_EVICT` allows for the trivial implementation<br>
of batch validation (and, if half-aggregation is safe, to<br>
use half-aggregation instead), whereas we expect multiple<br>
`OP_CSFS` to be needed to implement this, without any<br>
possibility of batch validation.<br>
It may be possible to design an `OP_CSFS` variant that<br>
allows batch validation, such as by extending the virtual<br>
machine with an accumulator for pending signature<br>
validations.<br>
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