Strong XOR Lemma for Communication with Bounded Rounds
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In this paper, we prove a strong XOR lemma for bounded-round two-player randomized communication. For a function f:๐ณร๐ดโ{0,1}, the n-fold XOR function f^โ n:๐ณ^nร๐ด^nโ{0,1} maps n input pairs (X_1,โฆ,X_n,Y_1,โฆ,Y_n) to the XOR of the n output bits f(X_1,Y_1)โโฏโ f(X_n, Y_n). 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^โ n with probability 1/2+exp(-O(n)) must use nยท(r^-O(r)ยท C-1) bits. When r is a constant and C is sufficiently large, this is ฮฉ(nยท C) bits. It matches the communication cost and the success probability of the trivial protocol that computes the n bits f(X_i,Y_i) 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 [Shaltiel'03]. By the equivalence between the discrepancy and the correlation with 2-bit communication protocols [Viola-Wigderson'08], our new XOR lemma implies the previous result.
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