Fast Algorithms for Interactive Coding

Abstract

Consider two parties who wish to communicate in order to execute some interactive protocol π. However, the communication channel between them is noisy: An adversary sees everything that is transmitted over the channel and can change a constant fraction of the bits as he pleases, thus interrupting the execution of π (which was designed for an errorless channel). If π only contained one message, then a good error correcting code would have overcame the noise with only a constant overhead in communication, but this solution is not applicable to interactive protocols with many short messages. Schulman (FOCS 92, STOC 93) presented the notion of interactive coding: A simulator that, given any protocol π, is able to simulate it (i.e. produce its intended transcript) even with constant rate adversarial channel errors, and with only constant (multiplicative) communication overhead. Until recently, however, the running time of all known simulators was exponential (or sub-exponential) in the communication complexity of π (denoted N in this work). Brakerski and Kalai (FOCS 12) recently presented a simulator that runs in time poly(N). Their simulator is randomized (each party flips private coins) and has failure probability roughly 2−-N. In this work, we improve the computational complexity of interactive coding. While at least N computational steps are required (even just to output the transcript of π), the BK simulator runs in time . We present two efficient algorithms for interactive coding: The first with computational complexity O(N log N) and exponentially small failure probability; and the second with computational complexity O(N), but failure probability 1/poly(N). (Computational complexity is measured in the RAM model.)