The communication complexity of addition

Abstract

Suppose each of kn o(1) players holds an n-bit number x i in its hand. The players wish to determine if ∑ ik 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 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 ∑x i > s with error 1% and communication O(k logk) log n. For k≥3 this improves on Nisan’s O(k log2 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(logn) lower bound to determine if x 1 > x 2. 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).

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Correspondence to Emanuele Viola.

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Supported by NSF grants CCF-0845003, CCF-1319206.

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Viola, E. The communication complexity of addition. Combinatorica 35, 703–747 (2015). https://doi.org/10.1007/s00493-014-3078-3

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Mathematics Subject Classification (2000)

  • 68Q17
  • 68Q10