An XOR Lemma for Deterministic Communication Complexity

TL;DR

This paper establishes a new lower bound on the deterministic communication complexity of the n-fold XOR of a function, linking it to the function's complexity and rank, using innovative information-theoretic methods.

Contribution

It introduces a novel XOR lemma for deterministic communication complexity that relates complexity bounds to function rank and employs new information-theoretic techniques.

Findings

Proves a lower bound on $D(f^{igoplus n})$ in terms of $D(f)$ and $ ext{rank}(f)$

Develops a new approach using information theory in communication complexity

Provides insights into the complexity of XOR functions in communication models

Abstract

We prove a lower bound on the communication complexity of computing the $n$-fold xor of an arbitrary function $f$, in terms of the communication complexity and rank of $f$. We prove that $D(f^{\oplus n}) \geq n \cdot \Big(\frac{\Omega(D(f))}{\log \mathsf{rk}(f)} -\log \mathsf{rk}(f)\Big )$, where here $D(f), D(f^{\oplus n})$ represent the deterministic communication complexity, and $\mathsf{rk}(f)$ is the rank of $f$. Our methods involve a new way to use information theory to reason about deterministic communication complexity.