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 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) Ω(lg n) lower bound to determine if x1> x2. This improves on the Ω ( √ lg n) bound by Smirnov (1988).

We highlight the challenge of proving correlation bounds between boolean functions and integer-valued polynomials, where any non-boolean output counts against correlation. We prove that integer-valued polynomials of degree 1 2 lg2 lg2 n have zero correlation with parity. Such a result is false for modular and threshold polynomials. Its proof is based on a strengthening of an anti-concentration result by Costello, Tao, and Vu (Duke Math. J. 2006). 1