The communication complexity of addition

Abstract

Suppose each of k≤no(1) players holds an n-bit number xi in its hand. The players wish to determine if ?i≤kxi=s. We give a public-coin protocol with error 1% and communication O(k logk). The communication bound is independent of n, and for k?3 improves on the O(k logn) bound by Nisan (Bolyai Soc. Math. Studies; 1993). Our protocol also applies to addition modulo m. In this case we give a matching (public-coin) Ω(k logk) lower bound for various m. We also obtain some lower bounds over the integers, including Ω (k log logk) for protocols that are one-way, like ours. We give a protocol to determine if ?xi >s with error 1% and communication O(k logk) log n. For k?3 this improves on Nisan's O(k log2n) bound. A similar improvement holds for computing degree-(k?1) polynomial-threshold functions in the number-on-forehead model. We give a (public-coin, 2-player, tight) Ω(logn) lower bound to determine if x1 >x2. This improves on the Ω(?logn) bound by Smirnov (1988). Troy Lee informed us in January 2013 that an Ω(logn) lower bound may also be obtained by combining a result in learning theory by Forster et al. (2003) with a result by Linial and Shraibman (2009). As an application, we show that polynomial-size AC0 circuits augmented with O(1) threshold (or symmetric) gates cannot compute cryptographic pseudorandom functions, extending the result about AC0 by Linial, Mansour, and Nisan (J. ACM; 1993).