Stellar work Antoine and Gleb! Really excited to see designs come out on payment pools.

I've also been designing some payment pools (I have some not ready code I can share with you guys off list), and I wanted to share what I learned here in case it's useful.

In my design of payment pools, I don't think the following requirement: "A CoinPool must satisfy the following *non-interactive any-order withdrawal* property: at any point in time and any possible sequence of previous CoinPool events, a participant should be able to move their funds from the CoinPool to any address the participant wants without cooperation with other CoinPool members." is desirable in O(1) space. I think it's much better to set the requirement to O(log(n)), and this isn't just because of wanting to use CTV, although it does help.

Let me describe a quick CTV based payment pool:

Build a payment pool for N users as N/2 channels between participants created in a payment tree with a radix of R, where every node has a multisig path for being used as a multi-party channel and the CTV branch has a preset timeout. E.g., with radix 2:

                                      Channel(a,b,c,d,e,f,g,h)

                                     /                                   \

               Channel(a,b,c,d)                                Channel(e,f,g,h)

                    /    \                                                    /                 \

Channel(a,b)    Channel(c,d)                          Channel(e,f)    Channel(g,h)

All of these channels can be constructed and set up non-interatively using CTV, and updated interactively. By default payments can happen with minimal coordination of parties by standard lightning channel updates at the leaf nodes, and channels can be rebalanced at higher layers with more participation.

Now let's compare the first-person exit non cooperative scenario across pools:

CTV-Pool:

Wait time: Log(N). At each branch, you must wait for a timeout, and you have to go through log N to make sure there are no updated states. You can trade off wait time/fees by picking different radixes.

TXN Size: Log(N) 1000 people with radix 4 --> 5 wait periods. 5*4 txn size. Radix 20 --> 2 wait periods. 2*20 txn size.

Accumulator-Pool:

Wait Time: O(1)

TXN Size: Depending on accumulator: O(1), O(log N), O(N) bits. Let's be favorable to Accumulators and assumer O(1), but keep in mind constant may be somewhat large/operations might be expensive in validation for updates.

This *seems* like a clear win for Accumulators. But not so fast. Let's look at the case where *everyone* exits non cooperatively from a payment pool. What is the total work and time?

CTV Pool:

Wait time: Log(N)

Txn Size: O(N) (no worse than 2x factor overhead with radix 2, higher radixes dramatically less overhead)

Accumulator Pool:

Wait time: O(N)

Txn Size: O(N) (bear in mind *maybe* O(N^2) or O(N log N) if we use an sub-optimal accumulator, or validation work may be expensive depending on the new primitive)

So in this context, CTV Pool has a clear benefit. The last recipient can always clear in Log(N) time whereas in the accumulator pool, the last recipient has to wait much much longer. There's no asymptotic difference in Tx Size, but I suspect that CTV is at least as good or cheaper since it's just one tx hash and doesn't depend on implementation.

Another property that is nice about the CTV pool style is the bisecting property. Every time you have to do an uncooperative withdrawal, you split the group into R groups. If your group is not cooperating because one person is permanently offline, then Accumulator pools *guarantee* you need to go through a full on-chain redemption. Not so with a CTV-style pool, as if you have a single failure among [1,2,3,4,5,6,7,8,9,10] channels (let's say channel 8 fails), then with a radix 4 setup your next steps are:

[1,2,3,4,5,6,7,8,9,10]

[1,2,3,4,5,6,7,X,9,10]

[1,2,3,4] [5,6,7,X] [9,10]

[1,2,3,4] 5 6 7 X [9,10]

So you only need to do Log(N) chain work to exit the bad actor, but then it amortizes! A future failure (let's say of 5) only causes 5 to have to close their channel, and does not affect anyone else.

With an accumulator based pool, if you re-pool after one failure, a second failure causes another O(N) work. So then total work in that case is O(N^2). You can improve the design by making the evict in any order option such that you can *kick out* a member in any order, that helps solve some of this nastiness (rather than them opting to leave). But I'm unclear how to make this safe w.r.t. updated states. You could also allow, perhaps, any number of operators to simultaneously leave in a tx. Also not sure how to do that.

Availability:

With CTV Pools, you can make a payment if just your immediate conterparty is online in your channel. Opportunistically, if people above you are online, you can make channel updates higher up in the tree which have better timeout properties. You can also create new channels, binding yourself to different parties if there is a planned exit.

With Accumulator pools, you need all parties online to make payments.

Cooperation Case:

CTV Pools and Accumulator pools, in a cooperative case, both just act like a N of N multisig.

Privacy:

Because Accumulator pools always require N signers, it's possible to build a better privacy model where N parties are essentially managing a chaumian ecash like server for updates, giving good privacy of who initiated payments. You *could* do this with CTV pools as well, but I expect users to prefer making updates at the 2 party channel layer for low latency, and to get privacy benefits out of the routability of the channels and ability to connect to the broader lightning network.

Technical Complexity:

Both protocols require new features in Bitcoin. CTV is relatively simple, I would posit that accumulators + sighashnoinput are relatively not simple.

The actual protocol design for CTV pools is pretty simple and can be compatible with LN, I already have a rudimentary implementation of the required transactions (but not servers).

Interactivity:

In both designs, the payment pool can be created non-interactively. This is *super* important as it means third parties (e.g., an exchange) can withdraw users funds *into* a payment pool.

Thanks for reading!

Jeremy