The optimal error resilience of interactive communication over binary channels

Abstract

In interactive coding, Alice and Bob wish to compute some function f of their individual private inputs x and y. They do this by engaging in a non-adaptive (fixed order, fixed length) interactive protocol to jointly compute f(x,y). The goal is to do this in an error-resilient way, such that even given some fraction of adversarial corruptions to the protocol, both parties still learn f(x,y). In this work, we study the optimal error resilience of such a protocol in the face of adversarial bit flip or erasures. While the optimal error resilience of such a protocol over a large alphabet is well understood, the situation over the binary alphabet has remained open. Firstly, we determine the optimal error resilience of an interactive coding scheme over the binary erasure channel to be 1/2, by constructing a protocol that achieves this previously known upper bound. The communication complexity of our binary erasure protocol is linear in the size of the minimal noiseless protocol computing f. Secondly, we determine the optimal error resilience over the binary bit flip channel for the message exchange problem (where f(x,y)=(x,y)) to be 1/6. The communication complexity of our protocol is polynomial in the size of the parties' inputs. Note that this implies an interactive coding scheme for any f resilient to 1/6 errors with an exponential blowup in communication complexity.