Efficient Multiparty Interactive Coding for Insertions, Deletions, and Substitutions

Abstract

In the field of interactive coding, two or more parties wish to carry out a distributed computation over a communication network that may be noisy. The ultimate goal is to develop efficient coding schemes that can tolerate a high level of noise while increasing the communication by only a constant factor (i.e., constant rate).In this work we consider synchronous communication networks over an arbitrary topology, in the powerful adversarial insertion-deletion noise model. Namely, the noisy channel may adversarially alter the content of any transmitted symbol, as well as completely remove a transmitted symbol or inject a new symbol into the channel.We provide efficient, constant rate schemes that successfully conduct any computation with high probability as long as the adversary corrupts at most ε m fraction of the total communication, where m is the number of links in the network and ε is a small constant. This scheme assumes the parties share a random string to which the adversarial noise is oblivious. We can remove this assumption at the price of being resilient to ε m log m adversarial error. While previous work considered the insertion-deletion noise model in the two-party setting, to the best of our knowledge, our scheme is the first multiparty scheme that is resilient to insertions and deletions. Furthermore, our scheme is the first computationally efficient scheme in the multiparty setting that is resilient to adversarial noise.1 By "arbitrary" we mean that the topology of the network can be an arbitrary graph G = (V, E) where each node is a party and each edge is a communication channel connecting the parties associated with these nodes.2 That is, a channel that flips every bit with some constant probability ε ∈ (0, 1/2). 3 [JKL15] obtained this result for the specific star network, whereas [HS16] generalized this result to a network with arbitrary topology. 4 We assume that each insertion, deletion, or substitution counts as a single corruption.Theorem 1.2 (no common randomness, non-oblivious noise, informal). Let G = (V, E) be an arbitrary synchronous network with n = |V | nodes and m = |E| links. For any noiseless protocol Π over G with a predetermined order of speaking, and for any sufficiently small constant ε, there exists an efficient coding scheme that simulates Π over a noisy network G. The simulated protocol is robust to adversarial insertion, deletion, and substituition noise, assuming at most (ε/m log m)-fraction of the communication is corrupted. The simulated protocol communicates O(CC(Π)) bits, and succeeds with probability at least 1 − exp(−CC(Π)/m).In Appendix B we consider the case where the adversarial channel is non-oblivious, however, the parties pre-share randomness. In this case we show a coding scheme (Algorithm C) that is resilient to a somewhat higher noise level of ε/m log log m-fraction of insertion and deletion noise, while still incurring a constant blowup in the communication. See Appendix B for the complete details.3 Lemma 6.3. Let |Π| = CC(Π) 5m log(m) and let ε > 0 be a su...

Small Memory Robust Simulation of Client-Server Interactive Protocols over Oblivious Noisy Channels

“…• Silent Rounds. As mentioned in [12], Hoza observed that the ability to have silent rounds can encode information. Essentially, speaking in an odd-numbered round means 1, and speaking in an even-numbered round means 0.…”

“…As mentioned in [15], a subsequent work [11] has achieved a better rate of 1 − O( log 1 ) for channels with feedback or erasure channels.Multi-party Case. The multi-party case has also been studied for random [5,25] and adversarial [12,17] corruptions. However, these constructions use large memory and guarantees only O(1) communication overhead for some restricted error rates.…”

“…RobustReceive( , t) //Bob is the receiver 10 core * ← {0} l ; Counter CNT ← 0 t/l + 1 iterations do //Less than half of the codewords can be corrupted12 Receive W from Alice //Receive (corrupted) codeword from Alice Received message core * The total number of bits sent in Algorithm 3 is Θ( + t). The memory usage is O( + log t ).…”