Strong XOR Lemma for Communication with Bounded Rounds: (extended abstract)

In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function f:X×Y?{0,1}, the n-fold XOR function f?n: Xn× Yn ? {0,1} maps n input pairs (X1,. . . , Xn, Y1,. . ., Yn) to the XOR of the n output bits f(X1, Y1)?. . .? f(Xn, Yn). We prove that if every r-round communication protocols that computes f with probability 2/3 uses at least C bits of communication, then any r-round protocol that computes f? n with probability 1/2+exp(-O(n)) must use n (r-O(r) C-1) bits. When r is a constant and C is sufficiently large, this is O(n C) bits. It matches the communication cost and the success probability of the trivial protocol that computes the n bits f(Xi, Yi) independently and outputs their XOR, up to a constant factor in n. A similar XOR lemma has been proved for f whose communication lower bound can be obtained via bounding the discrepancy [17]. By the equivalence between the discrepancy and the correlation with 2-bit communication protocols [19], our new XOR lemma implies the previous result.