Aggregate delegated exit for L2 pools

It’s often been claimed that “covenants don’t solve the L2 exit problem” because it’s always not economical to exit a layer 2 system for a user whose balance is too low to cover for the fees.

In this post, I will argue that fraud proofs might provide an interesting solution to this problem.

TL;DR: a withdrawal protocol based on fraud proofs can allow many parties that are individually too poor to withdraw their balance from a pooled UTXO to cooperate off-chain and perform a single, larger withdrawal.

Intro: fraud proofs

Readers who are familiar with the concept of fraud proofs can safely skip this section.

The idea of fraud proofs is to replace a (possibly expensive) computation in a smart contract with just a claim that a computation would give a certain result.

As long as the interested parties in a smart contracts have a way of disproving fraudulent claims - and a protocol that punishes any fraud attempt, then frauds (and fraud proofs) are never expected to end up on chain.

The saving generally comes not only from avoiding an expensive operation in Script, but also by not putting the entire witness data that the computation would need: a commitment to it is generally sufficient, as long as the counterparties are able to independently compute the correct data themselves.

The trade-off of using protocols based on fraud proofs (optimistic protocols) in comparison to a full on-chain validation of the contract transition are:

  • fraud proofs require a challenge period, in comparison to direct computations that are confirmed in a single transaction; moreover, some capital lockup is generally needed during the challenge period;
  • fraud proof protocols rely on the assumption that miners are not actively censoring the transactions of the challenger.

The most commonly used optimistic protocol in bitcoin today is in fact a Lightning channel: the “justice transaction” is a fraud proof.

Remark. While MATT admits a generic fraud proof protocol for arbitrary computations (that scales logarithmically on its computational cost), there are many special cases for which the fraud proof protocol is a lot simpler. Fraud proofs on statements about the leaves of a Merkle tree (for example “if I change the i-th leaf of the Merkle tree with root R, the root of the new Merkle tree is R'” have fairly straightforward fraud proof protocols).

The optimistic pool setting

Imagine a layer 2 system where a single UTXO stores the state as a list of user:balance for n users (where user is a public key, and balance is an amount in satoshis), and the operator (or a Musig2 of all the participants) publishes new states according to some rules.

An important assumption in what follows is that all the users completely know the state of the contract - that is, each user’s balance.

If the operator messes up (for example, fraud is proven, or the operator becomes unresponsive and does not publish the next state), then the UTXO can be spent towards an unwinding state, where the only action allowed is getting your money out of the pool.

There are multiple constructions that might fit this model (rollups, coinpools, etc.). In the following, we only focus on the unwinding stage of such constructions: something went wrong, and all the users have to withdraw their coins.

The “Unwind” contract

Once the panic button is pressed, the pool is in the Unwind contract (a UTXO with a covenant): the only allowed action is “TTMAR” (Take The Money And Run). This UTXO contains a vector commitment of all the user:balance pairs; that is, the root of the Merkle tree of the vector.

High value users can exit directly and instantly by proving their balances (with a Merkle proof) – but low value users can’t, as even publishing the Merkle proof is more expensive than their balance!

Optimistic withdrawal

If the Merkle proof is large, one can think of a cheaper way: withdraw with a fraud proof.

User Will who is withdrawing claims “my balance is X, and the remaining pool is Unwind(all-users-minus-Will)”; he additionally attaches 1 ₿ as a bond, and the money is locked for the challenge period. If he lied, whoever proves the fraud can take (1+X)/2 ₿ (half of Will’s money), while (1+X)/2 ₿ is burned. Otherwise, at the end of the challenge period, Will can take out his money, and his bond, and produce the output Unwind(all-users-minus-Will)

This optimistic withdrawal might be cheaper as it does not require a Merkle proof, lowering the balance threshold for which withdrawals are economically feasible.

However, users who want to withdraw their 10000 sats are still out of luck.

Optimistic aggregated withdrawal

The idea is to generalize the optimistic withdrawal to an arbitrary subset of pool users, and add an intermediary Ingrid.

A set S of users of users wants to delegate to Ingrid the right to withdraw their “aggregate balance”. Each of them signs a delegation message with their pubkey in the pool saying “I authorize Ingrid to withdraw my balance”.

Ingrid then spends the Unwind UTXO, claiming “I’m authorized to withdraw a total of X on behalf of the users in S = {s_0, s_1, …, s_k}; the new state will be Unwind(all users, except the ones in S)” (note: only the index of each user in S is posted); the output also contains the Merkle root of all the delegations, and is timelocked to allow fraud proofs, and Ingrid also posts a juicy 10 ₿ bond.

If Ingrid is lying about the total aggregate balance of S, or about the new state of the contract (with the updated Merkle root), any user in the system can challenge Ingrid: since they know the full state of the system, a fraud proof protocol on the Merkle tree of the UTXO state will allow to expose the lie. upon winning the challenge, they will 5 ₿ from Ingrid’s bond and burn the other 5 ₿; the withdrawal attempt is reversed.

Any user in S can challenge Ingrid if they didn’t sign a delegation (but they also have to put a bond, and if Ingrid proves they did sign a delegation, Ingrid pockets half of their bond).

If there is no challenge, Ingrid takes the money of all the users in S, and gets back her bond.

Ingrid then can (custodially) return the funds in some other way - on lightning, on Liquid, in fiat, etc.


Summing up, the properties of the system are such that:

  • the intermediary can be chosen at any time after the Unwind utxo is created; users can cooperate at any time with other users to find an intermediary who still didn’t withdraw (and they have an incentive to do so);
  • the cost for Ingrid to claim the money is (almost) constant (except for the list of indices that grows with the user indices, that is pretty small anyway);
  • any unsuccessful fraud loses money;
  • any set of users with sufficient aggregate balance can withdraw using the optimistic approach.


Non-custodial Ingrid

For simplicity, we said that Ingrid “custodially” helps S above, but one can build on top of this scheme: for example, the delegation could already specify some other UTXO(s) where the money has to go to once the challenge period is over – maybe a new pool with a different operators agreed upon by the users in S).

OP_CHECKTEMPLATEVERIFY or OP_TXHASH would provide an efficient to pre-program the future of the coins after the aggregated withdrawal.

Constant withdrawal cost? (open problem)

In the above, the index of each user who is withdrawing is part of the initial aggregate withdrawal message. While that doesn’t require too many bytes, it still make the transaction size increase linearly with the number of users withdrawing.

For a pool with n users, one could use 1 bit per user (listing for each of the n users whether they are in S or not), so the total cost of withdrawing is fixed at n + O(1) bits; this might be good enough in practice.

It would be great to find a protocol with constant cost independent on n or |S|.

Parallel optimistic withdrawals? (open problem)

With an opcode like OP_CHECKCONTRACTVERIFY that allows cross-input introspection, the optimistic withdrawal protocol (and possible challenge protocol) could be performed on a separate UTXO instead of spending the pool UTXO; only at maturity (after the challenge period is over), both UTXOs could be spent together. However, it’s not clear if there is a clean/easy way to make one withdrawal not interfere with other ones happening in parallel.

Script requirements

  • Amount introspection (equality checks should be enough)


  • OP_CHECKSIGFROMSTACK to sign delegations (there are workarounds to do it without, but it’s just awkward)
  • 64 bit arithmetic


There are many details to figure out, but the approach seems sound to me. I look forward to your comments.


You mention (1+X)/2 BTC as half of Bill’s money. Shouldn’t you use 1/2 instead.

We already know there is fraud. So the X probably doesn’t belong to bill.


You are correct, X should just stay in the pool, 1/2 goes to the challenger, 1/2 is burned.

As additional context: the reason for burning some of the bond is to make sure that prevent sybil attacks where the claimant and the challenger are the same entity, which otherwise would only incur the fee cost. That’s standard in fraud proofs.


A little postscript to detail how the fraud proofs needed in the protocol might look like, as I had time to brainstorm it a bit more.

The only fraud-sensitive statement that is more complicated is Ingrid’s statement:

“I’m withdrawing for users S = \left\{s_0, s_1, \dots, s_{k - 1}\right\}, whose total balance is T”, where each s_i is the index of a user within the vector commitment of all the user:balance pairs. That’s because this is a statement about multiple leaves of the Merkle tree simultaneously, as it is a statement about their sum.

Script itself can compute root_S, the root of the Merkle tree of the vector S, so that this root_S can be passed around in the next UTXOs.

Using MATT generic fraud proof protocols as a blackbox, we could use a fraud proof on the following computation:

\sum_{i=1}^k \operatorname{balance}(s_i)

This is a computation with k steps, so it would require about 2\log k transactions to resolve.

However, we can make an ad-hoc fraud proof protocol that requires less rounds:

  1. Ingrid claims “I’m withdrawing for users S = \left\{s_0, s_1, \dots, s_{k - 1}\right\}, whose total balance is T”.
  2. Anyone who spots that T is wrong challenges Ingrid. “No, $T is wrong”.
  3. Ingrid now has to reveal on-chain the individual balances of S: “Here is the list of user:balance in S: [(s_0, b_0), \dots, (s_k, b_k)]”. The Script is only valid if the provided indices are the same (recomputing root_S, which must match), and the sum of the balances is T. The new root_S' of the user:balance pairs is computed, and it is verified if indeed T = \sum_{i=0}^k b_i).
  4. If Ingrid lied, at least 1 of the balances (say b_t was wrong, and it can be exposed with two Merkle proofs (one to show the balance Ingrid claimed for s_t, and the other to show the real balance of user s_t).

Therefore, in case Ingrid lies, just 3 transactions are enough to expose the fraud.

As usual, only (1) is expected to happen in practice, and (2-4) only happens in case of fraud (or claims of fraud). That’s why (1) and (3) are separated; publishing the list of users and balances is significantly more expensive than only the user indices (a few bytes for each user).

In this version, step (3) would benefit from 64-bit arithmetic (although it is not too hard to simulate it with OP_CAT).

Note that if the size of S is large (which might make sense in practice, since all the users want to get out anyway), then representing S as an n-bit bitmap is way more efficient for very large S, and the O(\log n)-size fraud proof protocol is probably much more efficient in terms of bytes (as it does not require to post on-chain all the user balances).

Of course, multiple versions of the fraud proof protocol might co-exist in different tapleaves.

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I really like this line of thinking, and I have a question.

Do you think it’s possible to get rid of the script requirements in the interactive case just like Ark is trying to do? That is, build a v1 without the covenant requirements and instead require interactivity.

No, I don’t think that’s possible.

Interactivity “before the fact” can only replace covenants when the possible futures can be enumerated in advance. That’s not the case for fraud proofs, except for very simple ones.

Note that Ark can’t avoid the unilateral exit costs, with or without covenants. As far as I understand, the protocols sketched in this post could also be used in Ark for the case where the operator stops responding.

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