The Optimal Error Resilience of Interactive Communication Over Binary Channels

TL;DR

This paper determines the maximum error resilience achievable in binary interactive communication protocols, achieving a sixth for bit flips and a half for erasures, matching known upper bounds.

Contribution

It constructs protocols that attain the optimal error resilience over binary channels, resolving an open problem in the field.

Findings

Achieves 1/6 error resilience for binary bit flip channels.

Achieves 1/2 error resilience for binary erasure channels.

Protocols' communication complexity is polynomial and linear in input size, respectively.

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) 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. In this work, we resolve this problem of determining the optimal error resilience over binary channels. In particular, we construct protocols achieving $\frac16$ error resilience over the binary bit flip channel and $\frac12$ error resilience over the binary erasure channel, for both of which matching upper bounds are known. We remark that the communication complexity of our binary bit flip protocol is polynomial in the size of the inputs, and the communication complexity of our binary erasure protocol is linear in the size of the minimal noiseless protocol computing $f$.