The communication complexity of addition

Suppose each of k ≤ no(1) players holds an n-bit number xi in its hand. The players wish to determine if ∑ i≤k xi = s. We give a public-coin protocol with error 1% and communication O(k lg k). The communication bound is independent of n, and for k ≥ 3 improves on the O(k lg n) 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 lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Ω(k lg lg k) for protocols that are one-way, like ours. We give a protocol to determine if ∑ xi > s with error 1% and communication O(k lg k) lg n. For k ≥ 3 this improves on Nisan’s O(k lg n) 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(lg n) lower bound to determine if x1 > x2. This improves on the Ω( √ lg n) bound by Smirnov (1988). 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). ∗Supported by NSF grant CCF-0845003. Email: [email protected]