Strong XOR Lemma for Information Complexity

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.