Interactive communication with unknown noise rate

Abstract

Alice and Bob want to run a protocol over an noisy channel, where a certain number of bits are flipped adversarially. Several results take a protocol requiring L bits of noise-free communication and make it robust over such a channel. In a recent breakthrough result, Haeupler described an algorithm that sends a number of bits that is conjectured to be near optimal in such a model. However, his algorithm critically requires a priori knowledge of the number of bits that will be flipped by the adversary. We describe an algorithm requiring no such knowledge. If an adversary flips T bits, our algorithm sends L+O ((T + √LT + L) log(LT + L)) bits in expectation and succeeds with high probability in L. It does so without any a priori knowledge of T. Assuming a conjectured lower bound by Haeupler, ourresult is optimal up to logarithmic factors. Our algorithm critically relies on the assumption of a private channel. We show that privacy is necessary when the amount of noise is unknown.