Strong XOR Lemma for Information Complexity

TL;DR

This paper establishes a strong XOR lemma for information complexity, showing that computing the XOR of n instances of a function requires linearly more information than computing a single instance, confirming the optimality of naive methods.

Contribution

It proves a tight lower bound on the information complexity of the XOR of multiple instances, extending the XOR lemma to the realm of information complexity in communication models.

Findings

Naive protocol is information-theoretically optimal.

Computing f^{}n requires n times the information of a single computation.

Lower bounds are tight in error trade-offs.

Abstract

For any $\{0,1\}$-valued function $f$, its \emph{$n$-folded XOR} is the function $f^{\oplus n}$ where $f^{\oplus n}(X_1, \ldots, X_n) = f(X_1) \oplus \cdots \oplus f(X_n)$. Given a procedure for computing the function $f$, one can apply a ``naive" approach to compute $f^{\oplus n}$ by computing each $f(X_i)$ independently, followed by XORing the outputs. This approach uses $n$ times the resources required for computing $f$. In this paper, we prove a strong XOR lemma for \emph{information complexity} in the two-player randomized communication model: if computing $f$ with an error probability of $O(n^{-1})$ requires revealing $I$ bits of information about the players' inputs, then computing $f^{\oplus n}$ with a constant error requires revealing $\Omega(n) \cdot (I - 1 - o_n(1))$ bits of information about the players' inputs. Our result demonstrates that the naive protocol for computing $f^{\oplus n}$ is both information-theoretically optimal and asymptotically tight in error trade-offs.