Subject: [bitcoin-dev] Merkleize All The Things

From: Salvatore Ingala <[email-masked]>
Date: Tue, 8 Nov 2022 10:17:42 +0100

Hi list,

I have been working on some notes to describe an approach that uses
covenants in order to enable general smart contracts in bitcoin. You can
find them here:

    https://merkle.fun

The approach has a number of desirable features:

- small impact to layer 1;
- not application-specific, very general;
- it fits well into P2TR;
- it does not require new cryptographic assumptions, nor any construction
that has not withstood the test of time.

This content was presented at the BTCAzores unconference, where it received
the name of MATT − short for Merkleize All The Things.
In fact, no other cryptographic primitive is required, other than Merkle
trees.

I believe this construction gets close to answering the question of how
small a change on bitcoin's layer 1 would suffice to enable arbitrary smart
contracts.

It is not yet at the stage where a formal proposal can be made, therefore
the proposed specs are only for illustrative purposes.

The same content is reformatted below for the mailing list.

Looking forward to hearing about your comments and improvements.
Salvatore Ingala

==========================================

# General smart contracts in bitcoin via covenants

Covenants are UTXOs that are encumbered with restrictions on the outputs of
the transaction spending the UTXO. More formally, we can define a covenant
any UTXO such that at least one of its spending conditions is valid only if
one or more of the outputs’ scriptPubKey satisfies certain restrictions.

Generally, covenant proposals also add some form of introspection (that is,
the ability for Script to access parts of the inputs/outputs, or the
blockchain history).

In this note, we want to explore the possibilities unleashed by the
addition of a covenant with the following properties:

- introspection limited to a single hash attached to the UTXO (the
“covenant data”), and input/output amounts;
- pre-commitment to every possible future script (but not their data);
- few simple opcodes operating with the covenant data.

We argue that such a simple covenant construction is enough to extend the
power of bitcoin’s layer 1 to become a universal settlement layer for
arbitrary computation.

Moreover, the covenant can elegantly fit within P2TR transactions, without
any substantial increase for the workload of bitcoin nodes.

A preliminary version of these notes was presented and discussed at the
BTCAzores Unconference [1], on 23rd September 2022.

# Preliminaries

We can think of a smart contract as a “program” that updates a certain
state according to predetermined rules (which typically include access
control by authorizing only certain public keys to perform certain
actions), and that can possibly lock/unlock some coins of the underlying
blockchain according to the same rules.

The exact definition will be highly dependent on the properties of the
underlying blockchain.

In bitcoin, the only state upon which all the nodes reach consensus is the
UTXO set; other blockchains might have other data structures as part of the
consensus, like a key-value store that can be updated as a side effect of
transaction execution.

In this section we explore the following concepts in order to set the
framework for a definition of smart contracts that fits the structure of
bitcoin:

- the contract’s state: the “memory” the smart contract operates on;
- state transitions: the rules to update the contract’s state;
- covenants: the technical means that can allow contracts to function in
the context of a bitcoin UTXO.

In the following, an on-chain smart contract is always represented as a
single UTXO that implicitly embeds the contract’s state and possibly
controls some coins that are “locked” in it. More generally, one could
think of smart contracts that are represented in a set of multiple UTXOs;
we leave the exploration of generalizations of the framework to future
research.

## State

Any interesting “state” of a smart contract can ultimately be encoded as a
list, where each element is either a bit, a fixed-size integers, or an
arbitrary byte string.

Whichever the choice, it does not really affect what kinds of computations
are expressible, as long as one is able to perform some basic computations
on those elements.

In the following, we will assume without loss of generality that
computations happen on a state which is a list of fixed length S = [s_1,
s_2, …, s_n], where each s_i is a byte string.

### Merkleized state

By constructing a Merkle tree that has the (hashes of) the elements of S in
the leaves, we can produce a short commitment h_S to the entire list S with
the following properties (that hold for a verifier that only knows h_S):

- a (log n)-sized proof can prove the value of an element s_i;
- a (log n + |x|)-sized proof can prove the new commitment h_S’, where S’
is a new list obtained by replacing the value of a certain leaf with x.

This allows to compactly commit to a RAM, and to prove correctness of RAM
updates.

In other words, a stateful smart contract can represent an arbitrary state
in just a single hash, for example a 32-byte SHA256 output.

### State transitions and UTXOs

We can conveniently represent a smart contract as a finite state machine
(FSM), where exactly one node can be active at a given time. Each node has
an associated state as defined above, and a set of transition rules that
define:

- who can use the rule;
- what is the next active node in the FSM;
- what is the state of the next active node.

It is then easy to understand how covenants can conveniently represent and
enforce the smart contracts in this framework:

- The smart contract is instantiated by creating a UTXO encumbered with a
covenant; the smart contract is in the initial node of the FSM.
- The UTXO’s scriptPubKey specifies the current state and the valid
transitions.
- The UTXO(s) produced after a valid transition might or might not be
further encumbered, according to the rules.

Therefore, what is necessary in order to enable this framework in bitcoin
Script is a covenant that allows the enforcement of such state transitions,
by only allowing outputs that commit to a valid next node (and
corresponding state) in the FSM.

It is not difficult to show that arbitrary computation is possible over the
committed state, as long as relatively simple arithmetic or logical
operations are available over the state.

Remark: using an acyclic FSM does not reduce the expressivity of the smart
contracts, as any terminating computation on bounded-size inputs which
requires cycles can be unrolled into an acyclic one.

### Merkleized state transitions

Similarly to how using Merkle trees allows to succinctly represent
arbitrary data with a short, 32-byte long summary, the same trick allows to
succinctly represent arbitrary state transitions (the smart contract’s
code) with a single 32-byte hash. Each of the possible state transitions is
encoded as a Script which is put in a leaf of a Merkle tree; the Merkle
root of this tree is a commitment to all the possible state transitions.
This is exactly what the taptree achieves in Taproot (see BIP-0341 [2]).

Later sections in this document will suggest a possible way of how both the
contract’s state and valid transition rules could be represented in UTXOs.

## On-chain computation?!

Should the chain actually do computation?

If naively designed, the execution of a contract might require a large
number of transactions, which is not feasible.

While the covenant approach does indeed enable a chain of transactions to
perform arbitrary computation, simple economic considerations will push
protocol designers to perform any non-trivial computation off-chain, and
instead use the blockchain consensus only to verify the computation; or, if
possible, skip the verification altogether.

The fundamental fact that a blockchain’s layer 1 never actually needs to
run complex programs in order to enable arbitrary complex smart contracting
was observed in the past, for example in a 2016 post by Greg Maxwell [3].

Vitalik Buterin popularized the concept of "functionality escape velocity"
[4] to signify the minimum amount of functionality required on layer 1 in
order to enable anything else to be built on top (that is, on layer 2 and
beyond).

In the following section, we will argue that a simple covenant construction
suffices to achieve the functionality escape velocity in the UTXO model.

# Commitments to computation and fraud challenges

In this section, we explore how a smart contract that requires any
non-trivial computation f : X --> Y (that is too expensive or not feasible
with on-chain Script state transitions) can be implemented with the simple
covenants described in the previous section.

The ideas in this section appeared in literature; the reader is referred to
the references for a more comprehensive discussion.

We want to be able to build contracts that allow conditions of the type
"f(x) = y"; yet, we do not want layer 1 to be forced to perform any
expensive computation.

In the following, we assume for simplicity that Alice and Bob are the only
participants of the covenant, and they both locked some funds bond_A and
bond_B (respectively) inside the covenant’s UTXO.

1. Alice posts the statement “f(x) = y”.
2. After a challenge period, if no challenge occurs, Alice is free to
continue and unlock the funds; the statement is true.
3. At any time before the challenge period expires, Bob can start a
challenge: “actually, f(x) = z”.

In case of a challenge, Alice and Bob enter a challenge resolution
protocol, arbitrated by layer 1; the winner takes the other party’s bond
(details and the exact game theory vary based on the type of protocol the
challenge is part of; choosing the right amount of bonds is crucial for
protocol design).

The remainder of this section sketches an instantiation of the challenge
protocol.

## The bisection protocol for arbitrary computation

In this section, we sketch the challenge protocol for an arbitrary
computation f : X --> Y.

### Computation trace

Given the function f, it is possible to decompose the entire computation in
simple elementary steps, each performing a simple, atomic operation. For
example, if the domain of x and y is that of binary strings of a fixed
length, it is possible to create a boolean circuit that takes x and
produces y; in practice, some form of assembly-like language operating on a
RAM might be more efficient and fitting for bitcoin Script.

In the following, we assume each elementary operation is operating on a
RAM, encoded in the state via Merkle trees as sketched above. Therefore,
one can represent all the steps of the computation as triples tri = (st_i,
op_i, st_{i + 1}), where st_i is the state (e.g. a canonical Merkle tree of
the RAM) before the i-th operation, st_{i + 1} is the state after, and op_i
is the description of the operation (implementation-specific; it could be
something like “add a to b and save the result in c).

Finally, a Merkle tree M_T is constructed that has as leaves the values of
the individual computation steps T = {tr_0, tr_1, …, tr_{N - 1}} if the
computation requires N steps, producing the Merkle root h_T. The height of
the Merkle tree is log N. Observe that each internal node commits to the
portion of the computation trace corresponding to its own subtree.

Let’s assume that the Merkle tree commitments for internal nodes are
further augmented with the states st_{start} and st_{end}, respectively the
state before the operation of in the leftmost leaf of the subtree, and
after the rightmost leaf of the subtree.

### Bisection protocol

The challenge protocol begins with Alice posting what she claims is the
computation trace h_A, while Bob disagrees with the trace h_B != h_A;
therefore, the challenge starts at the root of M_T, and proceeds in steps
in order to find a leaf where Alice and Bob disagree (which is guaranteed
to exist, hence the disagreement). Note that the arbitration mechanism
knows f, x and y, but not the correct computation trace hash h_T.

(Bisection phase): While the challenge is at a non-leaf node of M_T, Alice
and Bob take turns to post the two hashes corresponding to the left and
right child of their claimed computation trace hash; moreover, they post
the start/end state for each child node. The protocol enforces that Alice’s
transaction is only valid if the posted hashes h_{l; A} and h_{r; A}, and
the declared start/end state for each child are consistent with the
commitment in the current node.

(Arbitration phase): If the protocol has reached the i-th leaf node, then
each party reveals (st_i, op_i, st_{i + 1}); in fact, only the honest party
will be able to reveal correct values, therefore the protocol can
adjudicate the winner.

Remark: there is definitely a lot of room for optimizations; it is left for
future work to find the optimal variation of the approach; moreover,
different challenge mechanisms could be more appropriate for different
functions f.

### Game theory (or why the chain will not see any of this)

With the right economic incentives, protocol designers can guarantee that
playing a losing game always loses money compared to cooperating.
Therefore, the challenge game is never expected to be played on-chain. The
size of the bonds need to be appropriate to disincentivize griefing attacks.

### Implementing the bisection protocol's state transitions

It is not difficult to see that the entire challenge-response protocol
above can be implemented using the simple state transitions described above.

Before a challenge begins, the state of the covenant contains the value of
x, y and the computation trace computed by Alice. When starting the
challenge, Bob also adds its claim for the correct computation trace, and
the covenant enters the bisection phase.

During the bisaction phase, the covenant contains the claimed computation
trace for that node of the computation protocol, according to each party.
In turns, each party has to reveal the corresponding computation trace for
both the children of the current node; the transaction is only valid if the
hash of the current node can be computed correctly from the information
provided by each party about the child nodes. The protocol repeats on one
of the two child nodes on whose computation trace the two parties disagree
(which is guaranteed to exist). If a leaf of M_T is reached, the covenant
enters the final arbitration phase.

During the arbitration phase (say at the i-th leaf node of M_T), any party
can win the challenge by providing correct values for tr_i = (st_i, op_i,
st_{i + 1}). Crucially, only one party is able to provide correct values,
and Script can verify that indeed the state moves from st_i to st_{i + 1}
by executing op_i. The challenge is over.

At any time, the covenant allows one player to automatically win the
challenge after a certain timeout if the other party (who is expected to
“make his move”) does not spend the covenant. This guarantees that the
protocol can always find a resolution.

### Security model

As for other protocols (like the lightning network), a majority of miners
can allow a player to win a challenge by censoring the other player’s
transactions. Therefore, the bisection protocol operates under the honest
miner majority assumption. This is acceptable for many protocols, but it
should certainly be taken into account during protocol design.

# MATT covenants

We argued that the key to arbitrary, fully general smart contracts in the
UTXO model is to use Merkle trees, at different levels:

1. succinctly represent arbitrary state with a single hash. Merkleize the
state!
2. succinctly represent the possible state transitions with a single hash.
Merkleize the Script!
3. succinctly represent arbitrary computations with a single hash.
Merkleize the execution!

(1) and (2) alone allow contracts with arbitrary computations; (3) makes
them scale.

   Merkleize All The Things!

In this section we sketch a design of covenant opcodes that are
taproot-friendly and could easily be added in a soft fork to the existing
SegWitv1 Script.

## Embedding covenant data in P2TR outputs

We can take advantage of the double-commitment structure of taproot outputs
(that is, committing to both a public key and a Merkle tree of scripts) to
compactly encode both the covenant and the state transition rules inside
taproot outputs.

The idea is to replace the internal pubkey Q with a key Q’ obtained by
tweaking Q with the covenant data (the same process that is used to commit
to the root of the taptree). More precisely, if d is the data committed to
the covenant, the covenant-data-augmented internal key Q’ is defined as:

    Q’ = Q + int(hashTapCovenantData(Q || h_{data}))G

where h_{data} is the sha256-hash of the covenant data. It is then easy to
prove that the point is constructed in this way, by repeating the
calculation.

If there is no useful key path spend, similarly to what is suggested in
BIP-0341 [5] for the case of scripts with no key path spends, we can use
the NUMS point:
    H =
lift_x(0x0250929b74c1a04954b78b4b6035e97a5e078a5a0f28ec96d547bfee9ace803ac0).

TODO: please double check if the math above is sound.

## Changes to Script

The following might be some minimal new opcodes to add for taproot
transactions in order to enable the construction above. This is a very
preliminary proposal, and not yet complete nor correct.

- OP_SHA256CAT: returns the SHA256 hash of the concatenation of the second
and the first (top) element of the stack. (redundant if OP_CAT is enabled,
even just on operands with total length up to 64 bytes)
- OP_CHECKINPUTCOVENANTVERIFY: let x, d be the two top elements of the
stack; behave like OP_SUCCESS if any of x and d is not exactly 32 bytes;
otherwise, check that the x is a valid x-only pubkey, and the internal
pubkey P is indeed obtained by tweaking lift_x(x) with d.
- OP_INSPECTNUMINPUTS, OP_INSPECTNUMOUTPUTS, OP_INSPECTINPUTVALUE and
OP_INSPECTOUTPUTVALUE - opcodes to push number on the stack of
inputs/outputs and their amounts.
- OP_CHECKOUTPUTCOVENANTVERIFY: given a number out_i and three 32-byte hash
elements x, d and taptree on top of the stack, verifies that the out_i-th
output is a P2TR output with internal key computed as above, and tweaked
with taptree. This is the actual covenant opcode.

TODO:

- Many contracts need parties to provide additional data; simply passing it
via the witness faces the problem that it could be malleated. Therefore, a
way of passing signed data is necessary. One way to address this problem
could be to add a commitment to the data in the annex, and add an opcode to
verify such commitment. Since the annex is covered by the signature, this
removes any malleability. Another option is an OP_CHECKSIGFROMSTACK opcode,
but that would cost an additional signature check.
- Bitcoin numbers in current Script are not large enough for amounts.

Other observations:

- OP_CHECKINPUTCOVENANTVERIFY and OP_CHECKOUTPUTCOVENANTVERIFY could have a
mode where x is replaced with a NUMS pubkey, for example if the first
operand is an empty array of bytes instead of a 32 byte pubkey; this saves
about 31 bytes when no internal pubkey is needed (so about 62 bytes for a
typical contract transition using both opcodes)
- Is it worth adding other introspection opcodes, for example
OP_INSPECTVERSION, OP_INSPECTLOCKTIME? See Liquid's Tapscript Opcodes [6].
- Is there any malleability issue? Can covenants “run” without signatures,
or is a signature always to be expected when using spending conditions with
the covenant encumbrance? That might be useful in contracts where no
signature is required to proceed with the protocol (for example, any party
could feed valid data to the bisection protocol above).
- Adding some additional opcodes to manipulate stack elements might also
bring performance improvements in applications (but not strictly necessary
for feasibility).

Remark: the additional introspection opcodes available in Blockstream
Liquid [6] do indeed seem to allow MATT covenants; in fact, the opcodes
OP_CHECKINPUTCOVENANTVERIFY and OP_CHECKOUTPUTCOVENANTVERIFY could be
replaced by more general opcodes like the group {OP_TWEAKVERIFY,
OP_INSPECTINPUTSCRIPTPUBKEY, OP_PUSHCURRENTINPUTINDEX,
OP_INSPECTOUTPUTSCRIPTPUBKEY }.

### Variant: bounded recursivity

In the form described above, the covenant essentially allows fully
recursive constructions (an arbitrary depth of the covenant execution tree
is in practice equivalent to full recursion).

If recursivity is not desired, one could modify the covenants in a way that
only allows a limited depth: a counter could be attached to the covenant,
with the constraint that the counter must be decreased for
OP_CHECKOUTPUTCOVENANTVERIFY. That would still allow arbitrary fraud proofs
as long as the maximum depth is sufficient.

However, that would likely reduce its utility and prevent certain
applications where recursivity seems to be a requirement.

The full exploration of the design space is left for future research.

# Applications

This section explores some of the potential use cases of the techniques
presented above. The list is not exhaustive.

Given the generality of fraud proofs, some variant of every kind of smart
contracts or layer two construction should be possible with MATT covenants,
although the additional requirements (for example the capital lockup and
the challenge period delays) needs to be accurately considered; further
research is necessary to assess for what applications the tradeoffs are
acceptable.

## State channels

A state channel is a generalization of a payment channel where,
additionally to the balance at the end of each channel, some additional
state is stored. The state channel also specifies what are the rules on how
to update the channel’s state.

For example, two people might play a chess game, where the state encodes
the current configuration of the board. The valid state transitions
correspond to the valid moves; and, once the game is over, the winner takes
a specified amount of the channel’s money.

With eltoo-style updates, such a game could be played entirely off-chain,
as long as both parties are cooperating (by signing the opponent’s state
update).

The role of the blockchain is to guarantee that the game can be moved
forward and eventually terminated in case the other party does not
cooperate.

In stateful blockchain, this is simply achieved by publishing the latest
state (Merkleized or not) and then continuing the entire game on-chain.
This is expensive, especially if the state transitions require some complex
computation.

An alternative that avoids moving computations on-chain is the use of a
challenge-response protocol, as sketched above.

Similarly to the security model of lightning channels, an honest party can
always win a challenge under the honest-majority of miners. Therefore, it
is game-theoretically losing to attempt cheating in a channel.

## CoinPool

Multiparty state channels are possible as well; therefore, constructions
like CoinPool [7] should be possible, enabling multiple parties to share a
single UTXO.

## Zero knowledge proofs in L2 protocols

Protocols based on ZK-proofs require the blockchain to be the verifier; the
verifier is a function that takes a zero-knowledge proof and returns
true/false based on its correctness.

Instead of an OP_STARK operator in L1, one could think of compiling the
OP_STARK as the function f in the protocol above.

Note that covenants with a bounded “recursion depth” are sufficient to
express OP_STARK, which in turns imply the ability to express arbitrary
functions within contracts using the challenge protocol.

One advantage of this approach is that no new cryptographic assumptions are
added to bitcoin’s layer 1 even if OP_STARK does require it; moreover, if a
different or better OP_STARK2 is discovered, the innovation can reach layer
2 contracts without any change needed in layer 1.

## Optimistic rollups

John Light recently posted a research report on how Validity Rollups could
be added to bitcoin’s layer 1 [8]. While no exact proposal is pushed
forward, the suggested changes required might include a combination of
recursive covenants, and specific opcodes for validity proof verification.

Fraud proofs are the core for optimistic rollups; exploring the possibility
of implementing optimistic rollups with MATT covenants seems a promising
direction. Because of the simplicity of the required changes to Script,
this might answer some of the costs and risks analyzed in the report, while
providing many of the same benefits. Notably, no novel cryptography needs
to become part of bitcoin’s layer 1.

Optimistic Rollups would probably require a fully recursive version of the
covenant (while fraud proofs alone are possible with a limited recursion
depth).

# Acknowledgments

Antoine Poinsot suggested an improvement to the original proposed covenant
opcodes, which were limited to taproot outputs without a valid key-path
spend.

The author would also like to thank catenocrypt, Antoine Riard, Ruben
Somsen and the participants of the BTCAzores unconference for many useful
discussions and comments on early versions of this proposal.

# References

The core idea of the bisection protocol appears to have been independently
rediscovered multiple times. In blockchain research, it is at the core of
fraud proof constructions with similar purposes, although not focusing on
bitcoin or covenants; see for example:

- Harry Kalodner et al. “Arbitrum: Scalable, private smart contracts.” −
27th USENIX Security Symposium. 2018.
https://www.usenix.org/system/files/conference/usenixsecurity18/sec18-kalodner.pdf
- Jason Teutsch and Christian Reitwiessner. “A scalable verification
solution for blockchains” − TrueBit protocol. 2017.
https://people.cs.uchicago.edu/~teutsch/papers/truebit.pdf

The same basic idea was already published prior to blockchain use cases;
see for example:

Ran Canetti, Ben Riva, and Guy N. Rothblum. “Practical delegation of
computation using multiple servers.” − Proceedings of the 18th ACM
conference on Computer and communications security. 2011.
http://diyhpl.us/~bryan/papers2/bitcoin/Practical%20delegation%20of%20computation%20using%20multiple%20servers.pdf

# Footnotes

[1] - https://btcazores.com
[2] - https://github.com/bitcoin/bips/blob/master/bip-0341.mediawiki
[3] -
https://bitcointalk.org/index.php?topic=1427885.msg14601127#msg14601127
[4] - https://vitalik.ca/general/2019/12/26/mvb.html
[5] -
https://github.com/bitcoin/bips/blob/master/bip-0341.mediawiki#constructing-and-spending-taproot-outputs
[6] -
https://github.com/ElementsProject/elements/blob/master/doc/tapscript_opcodes.md
[7] - https://coinpool.dev/v0.1.pdf
[8] - https://bitcoinrollups.org