Introduction

Redundancy can be a useful tool for speeding up payment channel networks and improving throughput. We presented last week at Financial Crypto. You just received a nice introduction to payment channels and payment channel networks.

Payment channels

Alice and Bob are connected through a payment channel. This is a channel and in that channel there are some coins that are escrowed. Some coins belong to Alice some belong to Bob and they want to transact them back and forth. The problem is that one side of the channel might run out of liquidity which makes routing in payment channel networks to be particularly difficult.

Payment channel networks

Alice and Bob are connected through payment channel networks. In that network there's some intermedaries that we visualize like this. Each payment channel has collateral locked up in the payment channels.

Atomic multipath routing (AMP)

We're looking at a scheme here called atomic multipath routing (AMP) which was mentioned in the previous talk. The payments get split up into three second paths through the network, for example, instead of a single path. A conditional payment using HTLC conditional on the revelation of a certain preimage... for some time, these funds are locked up along the paths. Only after revealing the preimage or the secret are the intermedaries able to claim their funds.

Another crucial detail in the lightning network is how exactly these routing decisions are being made. Namely the channel balances are undisclosed to the network participants so as a network participant I only know the topology of the network but I don't know how many coins are allocated to what side of each channel. Alice tries to figure out a route to send her funds to Bob but she doesn't really know if there's liquidity available in the network. Once Alice tries one path, she needs to roll back a payment and attempt a second path and hopefully the payment goes through.

Problems

There are undisclosed balances, which makes routing difficult. "Everyone-waits-for-the-last" behavior in AMP: a payment can only be completed once the last path makes it to the other side, and that last path is probably one that several previous attempts failed to deliver to, and it has to be repeated over and over again and it takes time to complete transfers. This leads to high latency due to high time to complete transfers. It consequently leads to low throughput: while transfers are pending, you can't use the liquidity to route other transfers. The funds get locked up, and they aren't available for other transfers, which reduces throughput.

Redundancy

Another problem with having n paths and having to wait for the slowest of them is called a straggler problem in distributed computing. One answer that people propose in these contexts is to introduce redundancy. Here, introducing redundancy means attempting more units of work or send more paths upfront and you complete the transfer not when the last one makes it but when a quorum of the paths have made it.

Simulation results

We did some prelimnary simulation results to show that there is an effect and we could hope to get something with those intuitons. There's a retry scheme and a redundancy scheme. If we don't allow for no retry and no redundancy then of course the throughput is the same for both schemes. But as we increase the number of additional paths, the redundancy scheme outperforms the retry scheme. There's a peak at some point and it plates and then comes back down. The other metric we're interested in is latency, which is a function of the number of additional paths. Again if we allow for no retries and no redundancy then there's no difference; but the more additional paths we allow in the retry scheme, the longer it takes because we keep trying whereas in the redundancy scheme we're sending all these redundant paths upfront and we can complete faster. This makes sense. Note that this is a side remark more than anything else: we're not playing with the success probability here. The success probability of a payment under this redundancy scheme is still strictly higher than under the retry scheme for an average payment.

Implementing redundancy

Hopefully I have made the point that redundancy is an interesting technique to look at. How can we implement it? If we just send more paths upfront, then what prevents the intermediary from taking more money than he is supposed to? That's a counterparty risk that we don't want in our Boomerang protocol.

Conditional payments

We saw something about conditional payments in the previous presentation using HTLCs. Alice makes a payment that is conditional upon revelation of that preimage that Bob created. Ingrid, the intermediary, can only claim the coin or fee if she has the secret from Bob and she knows where to go to get the secret from Bob by paying Bob. If she has the secret, then she will get her money. She is willing to extend the same offer to Bob.

Boomerang contract

Alice makes a conditional payment via Ingrid to Bob of 1 BTC. Now we add a backwards transaction that allows Alice to take her money back if she receives a proof that Bob withdrew too many funds from her. This condition on this backwards path is a strengthening of the forward path condition. If you're able to activate the backwards path, then you also know everything that is necessary to activate the forward path.

Homomorphic secret splitting

This is an old tool from cryptography. We have an elliptic curve d here. The homomorphic property is that H(ca + dB) = H(a)^c * H(B)^d. If I give you a bunch of evaluations of H on alpha and beta, then you're able to compute this function on some linear combinations of alpha and beta. Equipped with this instantiation, here's how the protocol works.

Bob comes up with a bunch of random numbers. These are the coefficients of a low-degree polynomial P(x). He computes the hashes of them using this hash function that we specified up here. Then he sends this over to Alice and Alice can use them to compute using the homomorphic property hashes of evaluations of that polynomial. So you can convince yourself using this homomorphic property allows Alice to compute the hash of polynomial evaluations given only the hashes of the coefficients of the polynomial.

Then she uses h_i as payment challenges on the ith path. If Bob reveals too many of these polynomial evaluations which he has to in order to claim the funds on the respective paths, then because this is a low degree polynomial, Alice is able to reconstruct the coefficients of the polynomial, and ALice is able to get a proof that Bob overdrew.

Implementation

On the payment channel, there's a settling transaction and we're implementing the contract as an output on that settlement transaction. The contract says that there's one bitcoin that we're deciding over here. If nothing happens and there's just a simple timeout, then this coin remains with the first party. If this condition is met and the preimage is released, then this path gets activated and the funds go into a retaliation transaction and this retaliation transaction has an output that is either if p1 reveals the proof of Bob overdrawing then the money goes to p1, or if there is a timeout and this backwards path is not activated then the funds stay with p2.

Homomorphic hash function

There's one catch: we needed this hash function with this special homomorphic property. One straightforward way to do this is to add a bitcoin script opocde like ECEXP, but this could also be implemented using adaptor signatures which was originally popularized by Andrew Poelstra in 2018 for Schnorr signatures. Adaptor signatures are now available for ECDSA as well.

Adaptor signatures

Adaptor signatures are an interesting cryptographic object. You have two parties p1 and p2 and both of them have a key and they have a hash challenge and they have some randomness. They bring this together and they do some crypto dance in order to produce an adaptor signature. This adaptor signature has the following properties: (1) if you ever learn the preimage that corresponds to the hash challeng,e then using the adaptor signature you can produce a proper signature, and (2) if you have the adaptor signature and you see a proper signature, then you can derive the preimage. The first property is necessary for p2 and the second property is necessary for p1. We can replace the preimage revelation step for the homomorphic hash function with just adaptor signatures.

Formal guarantees

Previously we proved some non-surprising statements about the properties you would want from a scheme like this... For Alice we prove that if Bob steals money from her and does more than he should, we prove that she can get her money back. If Bob behaves honestly then we prove there's no way for Alice to get her money back and all of this follows from standard cryptographic assumptions.

Food for thought

There seems to be a tradeoff between privacy and performance. We don't see this in other communication networks really. It's the fact that channel balances are not being revealed, makes routing hard, and this reduces the performance of payment channel networks. It would be interesting to retain privacy and increase performance.

All of these routing protocols, it's nice to describe them in a setting assuming honest nodes but what about rational nodes? What if the players in the network can gain from not following the protocol as you specified it? How do you incentivize rational pyament channel network participants to actually follow your protocol?

https://arxiv.org/abs/1910..... went by too fast.

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Disclaimer: These are unpaid transcriptions, performed in real-time and in-person during the actual source presentation. Due to personal time constraints they are usually not reviewed against the source material once published. Errors are possible. If the original author/speaker or anyone else finds errors of substance, please email me at kanzure@gmail.com for corrections or contribute online via github/git. I sometimes add annotations to the transcription text. These will always be denoted by a standard editor's note in parenthesis brackets ((like this)), or in a numbered footnote. I welcome feedback and discussion of these as well.

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