### Interactive Coding Resilient to an Unknown Number of Erasures

Ran Gelles, Siddharth Iyer, Michael Wagner
2020 International Conference on Principles of Distributed Systems
We consider distributed computations between two parties carried out over a noisy channel that may erase messages. Following a noise model proposed by , the noise level observed by the parties during the computation in our setting is arbitrary and a priori unknown to the parties. We develop interactive coding schemes that adapt to the actual level of noise and correctly execute any two-party computation. Namely, in case the channel erases T transmissions, the coding scheme will take N + 2T
more » ... missions using an alphabet of size 4 (alternatively, using 2N + 4T transmissions over a binary channel) to correctly simulate any binary protocol that takes N transmissions assuming a noiseless channel. We can further reduce the communication to N + T by relaxing the communication model and allowing parties to remain silent rather than forcing them to communicate in every round of the coding scheme. Our coding schemes are efficient, deterministic, have linear overhead both in their communication and round complexity, and succeed (with probability 1) regardless of the number of erasures T . Alternatively, there exists an efficient, deterministic, binary noise-resilient protocol Π 2 of length 2N + 4T that simulates π assuming an arbitrary and a priori unknown number T of erasures. It holds that Since T , the amount of noise, is unknown to begin with, the length of the coding scheme must adapt to the actual noise that the parties observe. Such coding schemes are called adaptive [3] . Adaptivity raises several issues that must be dealt with appropriately. The main issue is termination. Since the coding scheme adapts its length to the observed noise, and since the different parties observe different noise patterns, their termination is not necessarily synchronized. As a matter of fact, obtaining synchronized termination is impossible. Lemma 2. Let Π be a protocol for exchanging messages between Alice and Bob, that is resilient to an unbounded amount T of erasures. Then, there always exists a noise pattern for which Alice and Bob terminate in different rounds. This can be seen as a variant of the famous "coordinated attack problem" [24] , where reaching full synchronization between two parties is known to be impossible. See [18] for a proof and an elaborated discussion about unsynchronized termination. Unsynchronized termination means that one party may terminate while the other party continues to send (and receive) transmissions as dictated by its protocol. In this case, the communication model should specify what happens in those rounds where only one party is active and the other has terminated. Specifically, it should specify what messages the active party receives in this case. In our setting we define a special symbol we call silence (cf. [3]). We assume that silence is (implicitly) communicated by a terminated party. That is, the still-active party hears silence in every round it is set to listen and the other party has already terminated. We note that silence is corruptible -the channel may erase silent transmissions, and the active party will see an erasure mark instead. On the other hand, these implicit silent transmissions are not considered part of the communication of the protocol (i.e., we do not count them towards the communication complexity). Hearing a silence is a univocal indication that the other side has terminated, allowing the other party to terminate as well and bypassing the impossibility of synchronized termination. O P O D I S 2 0 1 9