Hi Jeremy,

Thanks for bringing back to awareness covenant-based drivechain designs
again!

I'm not super familiar with the thousands shades of sidechains, and
especially how the variants of pegging mechanisms influence the soundness
of the game-theory backing up the functionaries execution. However it could
be interesting to give security bounds to the defect of any trusted
component, such as the one-time trusted setup, and the impacts on funds. If
it's a full-blown loss, a timevalue loss, a privacy leak, etc...

Started at least an entry for the ZmnSCPxj design:
https://github.com/ariard/bitcoin-contracting-primitives-wg/pull/9

One interesting point from the OG post:
> The recursive covenant could, with the help of `OP_CAT` and
> `OP_CTV`, check that every transaction spending the UTXO has a
> second output that is an `OP_RETURN` with a commitment to the
> sidechain block.
> We can ensure that only one such transaction exists in each
> mainchain block by adding a `<1> OP_CSV`, ensuring that only one
> sidechain-commitment transaction can occur on each mainchain
> block.

Such recursive-covenant "embedded" sidechains could be used as solution to
the double-spend of payment pools and channel factories partitions, as an
instantiation of a "on-chain authoritative board" for partitions statement,
as described earlier this year, in a quest to solve the high interactivity
issue affecting those constructions [0].

Best,
Antoine

[0]
https://lists.linuxfoundation.org/pipermail/bitcoin-dev/2022-April/020370.html

Le mer. 14 sept. 2022 à 14:32, Jeremy Rubin via bitcoin-dev <
bitcoin-dev@lists.linuxfoundation.org> a écrit :

> *also available here on my blog with nicer
> formatting: https://rubin.io/bitcoin/2022/09/14/drivechain-apo/
> <https://rubin.io/bitcoin/2022/09/14/drivechain-apo/>*
>
> This post draws heavily from Zmnscpxj's fantastic post showing how to
> make drivechains with recursive covenants. In this post, I will show
> similar tricks that can accomplish something similar using ANYPREVOUT
> with a one time trusted setup ceremony.
>
> This post presents general techniques that could be applied to many
> different types of covenant.
>
> # Peano Counters
>
> The first component we need to build is a Peano counter graph. Instead
> of using sha-256, like in Zmnscpxj's scheme, we will use a key and
> build a simple 1 to 5 counter that has inc / dec.
>
> Assume a key K1...K5, and a point NUMS which is e.g.
> HashToCurve("Spookchains").
>
> Generate scripts as follows:
>
> ```
> <1 || K1> CHECKSIG
> ...
> <1 || K5> CHECKSIG
> ```
>
> Now generate 2 signatures under Ki with flags `SIGHASH_SINGLE |
> SIGHASH_ANYONECANPAY | SIGHASH_ANYPREVOUT`.
>
>
> ## Rule Increment
> For each Ki, when `i < 5`, create a signature that covers a
> transaction described as:
>
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K{i+1}> CHECKSIG})
> ```
>
> ## Rule Decrement
> For each Ki, when `i > 1` The second signature should cover:
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K{i-1}> CHECKSIG})
> ```
>
>
>
> _Are these really Peano?_ Sort of. While a traditional Peano numeral
> is defined as a structural type, e.g. `Succ(Succ(Zero))`, here we
> define them via a Inc / Dec transaction operator, and we have to
> explicitly bound these Peano numbers since we need a unique key per
> element. They're at least spiritually similar.
>
> ## Instantiation
> Publish a booklet of all the signatures for the Increment and
> Decrement rules.
>
> Honest parties should destroy the secret key sets `k`.
>
>
> To create a counter, simply spend to output C:
>
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K1> CHECKSIG})
> ```
>
>
> The signature from K1 can be bound to C to 'transition' it to (+1):
>
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K2> CHECKSIG})
> ```
>
> Which can then transition to (+1):
>
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K3> CHECKSIG})
> ```
>
> Which can then transition (-1) to:
>
> ```
> Amount: 1 satoshi
> Key: Tr(NUMS, {<1 || K2> CHECKSIG})
> ```
>
> This can repeat indefinitely.
>
>
> We can generalize this technique from `1...5` to `1...N`.
>
>
>
> # Handling Arbitrary Deposits / Withdrawals
>
>
> One issue with the design presented previously is that it does not
> handle arbitrary deposits well.
>
> One simple way to handle this is to instantiate the protocol for every
> amount you'd like to support.
>
> This is not particularly efficient and requires a lot of storage
> space.
>
> Alternatively, divide (using base 2 or another base) the deposit
> amount into a counter utxo per bit.
>
> For each bit, instead of creating outputs with 1 satoshi, create
> outputs with 2^i satoshis.
>
> Instead of using keys `K1...KN`, create keys `K^i_j`, where i
> represents the number of sats, and j represents the counter. Multiple
> keys are required per amount otherwise the signatures would be valid
> for burning funds.
>
> ## Splitting and Joining
>
> For each `K^i_j`, it may also be desirable to allow splitting or
> joining.
>
> Splitting can be accomplished by pre-signing, for every `K^i_j`, where
> `i!=0`, with `SIGHASH_ALL | SIGHASH_ANYPREVOUT`:
>
> ```
> Input: 2^i sats with key K^i_j
> Outputs:
>     - 2^i-1 sats to key K^{i-1}_j
>     - 2^i-1 sats to key K^{i-1}_j
> ```
>
> Joining can be accomplished by pre-signing, for every `K^i_j`, where
> `i!=MAX`, with `SIGHASH_ALL | SIGHASH_ANYPREVOUT`:
>
> ```
> Inputs:
>     - 2^i sats with key K^i_j
>     - 2^i sats with key K^i_j
> Outputs:
>     - 2^i+1 sats to key K^{i+1}_j
> ```
>
> N.B.: Joining allows for third parties to deposit money in externally,
> that is not a part of the covenant.
>
>
> The splitting and joining behavior means that spookchain operators
> would be empowered to consolidate UTXOs to a smaller number, while
> allowing arbitrary deposits.
>
>
> # One Vote Per Block
>
> To enforce that only one vote per block mined is allowed, ensure that
> all signatures set the input sequence to 1 block. No CSV is required
> because nSequence is in the signatures already.
>
> # Terminal States / Thresholds
>
> When a counter reaches the Nth state, it represents a certain amount
> of accumulated work over a period where progress was agreed on for
> some outcome.
>
> There should be some viable state transition at this point.
>
> One solution would be to have the money at this point sent to an
> `OP_TRUE` output, which the miner incrementing that state is
> responsible for following the rules of the spookchain. Or, it could be
> specified to be some administrator key / federation for convenience,
> with a N block timeout that degrades it to fewer signers (eventually
> 0) if the federation is dead to allow recovery.
>
> This would look like, from any `K^i_j`, a signature for a transaction
> putting it into an `OP_TRUE` and immediately spending it. Other
> spookchain miners would be expected to orphan that miner otherwise.
>
>
> # Open States / Proposals
>
> From a state `K^i_1`, the transaction transitioning to `K^i_2` can be
> treated as 'special' and the `OP_RETURN` output type can be used to
> commit to, e.g., the outputs that must be created in when the Terminal
> State is reached. This clarifies the issue of "what is being voted
> on".
>
> This method does not *lock in* at a consensus layer what Terminal
> State is being voted on.
>
> In certain circumstances, without violating the one-time-setup
> constraint, if a fixed list of withdrawer's addresses is known in
> advance, the Open States could cover withdrawals to specific
> participants, which then must collect a certain number of votes from
> miners.  However, it seems impossible, without new primitives, for an
> arbitrary transaction proposal to be voted on.
>
> # Setup Variants
>
> ## xpubs
>
> Instead of using randomly generated keys for each state, define each
> to be an xpub and derive a path where it is k/i/j for each
> state/satoshi amount. This saves some data, and also requires less
> entropy.
>
> ### Trustless Data Commit:
>
> commit to the hash of the entire program spec as a tweak to the xpub,
> so that someone can quickly verify if they have all the signatures you
> are expected to generate if honest.
>
> One way to do this is to convert a hash to a list of HD Child Numbers
> (9 of them) deterministically, and tweak the xpub by that. This is a
> convenient, yet inefficient, way to tweak an xpub because the child
> has a normal derivation path for signing devices.
>
> ## Single Party
>
> A single party pre-signs all the transactions for the spookchain, and
> then deletes their xpriv.
>
> You trust them to have deleted the key, and signed properly, but you
> do not trust whoever served you the spookchain blob to have given you
> all the state transitions because of the trustless data commitment.
>
> ## MuSig Multi-Party
>
> Define a MuSig among all participants in the setup ceremony, N-of-N.
>
> Now you simply trust that any one person in the one-time-setup was
> honest! Very good.
>
> ## Unaggregated Multi-Party
>
>
> Allow for unaggregated multi-sig keys in the spec. This grows with
> O(signers), however, it means that a-la-carte you can aggregate setups
> from random participants who never interacted / performed setup
> ceremonies independently if they signed the same specs.
>
> Can also combine multiple MuSig Multi-Parties in this way.
>
> This is nice because MuSig inherently implies the parties colluded at
> one point to do a MuSig setup, whereas unaggregated multi-sig could be
> performed with no connectivity between parties.
>
> ## Soft Forking Away Trust
>
> Suppose a spookchain becomes popular. You could configure your client
> to reject invalid state transitions, or restrict the spookchain keys
> to only sign with the known signatures. This soft fork would smoothly
> upgrade the trust assumption.
>
> ## Symmetry of State Transition Rules & DAG Covenants
>
> We could have our increment state transitions be done via a trustless
> covenant, and our backwards state transitions be done via the setup.
>
> This would look something like the following for state i:
>
> ```
> Tr(NUMS, {
>     `<sig for state K_{i+1}> <1 || PK_nonsecret> CHECKSIG`,
>     `<1 || Ki> CHECKSIG`
> })
> ```
>
> The advantage of such an optimization is theoretically nice because it
> means that *only* the non-destructuring recursive part of the
> computation is subject to the one-time-setup trust assumption, which
> might be of use in various other protocols, where recursivity might
> only be unlocked e.g. after a timeout (but for spookchains it is used
> at each step).
>
> A compiler writer might perform this task by starting with an
> arbitrary abstract graph, and then removing edges selectively (a
> number of heuristics may make sense, e.g., to minimize reliance on
> one-time-setup or minimize costs) until the graph is a Directed
> Acyclic Graph, consisting of one or more components, compiling those
> with committed covenants, and then adding the removed edges back using
> the one-time-setup key materials.
>
>
> # Commentary on Trust and Covenantiness
>
> Is this a covenant? I would say "yes". When I defined covenants in my
> _Calculus of Covenants_ post, it was with a particular set of
> assumptions per covenant.
>
> Under that model, you could, e.g., call a 7-10 multi-sig with specific
> committed instructions as 4-10 honest (requires 4 signatories to be
> honest to do invalid state transition) and 4-10 killable (requires 4
> signatories to die to have no way of recovering).
>
> For emulations that are pre-signed, like the varieties used to emulate
> CTV, it is a different model because if your program is correct and
> you've pre-gotten the signatures for N-N it is 1-N honest (only 1
> party must be honest to prevent an invalid state transition) and
> unkillable (all parties can safely delete keys).
>
> I model these types of assumptions around liveness and honesty as
> different 'complexity classes' than one another.
>
> What I would point out is that with the counter model presented above,
> this is entirely a pre-signed 1-N honest and unkillable covenant that
> requires no liveness from signers. Further, with APO, new instances of
> the covenant do not require a new set of signers, the setup is truly
> one-time. Therefore this type of covenant exists in an even lower
> trust-complexity class than CTV emulation via presigneds, which
> requires a new federation to sign off on each contract instance.
>
>
> With that preface, let us analyze this covenant:
>
>
> 1) A set of sets of transaction intents (a family), potentially
> recursive or co-recursive (e.g., the types of state transitions that
> can be generated).  These intents can also be represented by a
> language that generates the transactions, rather than the literal
> transactions themselves. We do the family rather than just sets at
> this level because to instantiate a covenant we must pick a member of
> the family to use.
>
>
> The set of sets of transaction intents is to increment / decrement to
> a successor or predecessor, or to halve into two instances or double
> value by adding funds. Each successor or predecessor is the same type
> of covenant, with the excetion of the first and last, which have some
> special rules.
>
>
> 2) A verifier generator function that generates a function that
> accepts an intent that is any element of one member of the family of
> intents and a proof for it and rejects others.
>
> The verifier generator is the simple APO CHECKSIG script.
>
> 3) A prover generator function that generates a function that takes an
> intent that is any element of one member of the family and some extra
> data and returns either a new prover function, a finished proof, or a
> rejection (if not a valid intent).
>
> The prover generator is the selection of the correct signature from a
> table for a given script.
>
> Run the prover generator with the private keys present *once* to
> initialize over all reachable states, and cache the signatures, then
> the keys may be deleted for future runs.
>
> 4) A set of proofs that the Prover, Verifier, and a set of intents are
> "impedance matched", that is, all statements the prover can prove and
> all statements the verifier can verify are one-to-one and onto (or
> something similar), and that this also is one-to-one and onto with one
> element of the intents (a set of transactions) and no other.
>
> At a given key state the only things that may happen are signed
> transactions, no other data is interpreted off of the stack. Therefore
> there is perfect impedance match.
>
>
> 5) A set of assumptions under which the covenant is verified (e.g., a
> multi-sig covenant with at least 1-n honesty, a multisig covenant with
> any 3-n honesty required, Sha256 collision resistance, Discrete Log
> Hardness, a SGX module being correct).
>
> Uniquely, that during the setup phase at least one of the keys
> were faithfully deleted.
>
> The usual suspects for any bitcoin transaction are also assumed for
> security.
>
>
> 6) Composability:
>
> The Terminal State can pay out into a pre-specified covenant if
> desired from any other family of covenants.
> --
> @JeremyRubin <https://twitter.com/JeremyRubin>
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