The communication complexity of addition

Suppose each of k <= n(o(1)) players holds an n-bit number xi in its hand. The players wish to determine if Sigma(i <= k) x(i) = 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) O(k lg k) lower bound for various m. We also obtain some lower bounds over the integers, including Omega(k lg lg k) for protocols that are one-way, like ours. We give a protocol to determine if P xi > s with error 1% and communication O(k lg k) lg n. For k 3 this improves on Nisan's O(k lg(2) n) bound. A similar improvement holds for computing degree-(k polynomialthreshold functions in the number-on-forehead model. We give a (public-coin, 2-player, tight) Omega(lg n) lower bound to determine if x(1) > x(2). This improves on the (p lg n) bound by Smirnov (1988). As an application, we show that polynomial-size AC 0 circuits augmented with O(1) threshold (or symmetric) gates cannot compute cryptographic pseudorandom functions, extending the result about AC(0) by Linial, Mansour, and Nisan (J. ACM; 1993).