The communication complexity of addition

Suppose each of ka parts per thousand currency signn (o(1)) players holds an n-bit number x (i) in its hand. The players wish to determine if a (ia parts per thousand currency signk) x (i) =s. We give a public-coin protocol with error 1% and communication O(k logk). The communication bound is independent of n, and for ka parts per thousand yen3 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) Omega(k logk) lower bound for various m. We also obtain some lower bounds over the integers, including Omega (k log logk) for protocols that are one-way, like ours. We give a protocol to determine if ax (i) > s with error 1% and communication O(k logk) log n. For ka parts per thousand yen3 this improves on Nisan's O(k log(2) 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) Omega(logn) lower bound to determine if x (1) > x (2). This improves on the Omega(aelogn) bound by Smirnov (1988). Troy Lee informed us in January 2013 that an Omega(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 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).