XOR Lemmas for Communication via Marginal Information

Abstract

We define the $\text{marginal information}$ of a communication protocol, and use it to prove XOR lemmas for communication complexity. We show that if every $C$-bit protocol has bounded advantage for computing a Boolean function $f$, then every $\tilde{\Omega }(C\sqrt{n})$-bit protocol has advantage $\exp (-\Omega (n))$ for computing the $n$-fold xor $f^{\oplus n}$. We prove exponentially small bounds in the average case setting, and near optimal bounds for product distributions and for bounded-round protocols.

Fixed typos