article

The communication complexity of addition

Author:

Emanuele ViolaAuthors Info & Claims

Combinatorica, Volume 35, Issue 6

Pages 703 - 747

https://doi.org/10.1007/s00493-014-3078-3

Published: 01 December 2015 Publication History

Abstract

Suppose each of k≤no(1) players holds an n-bit number xi in its hand. The players wish to determine if i≤kxi=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 xi >s with error 1% and communication O(k logk) log n. For k 3 this improves on Nisan's O(k log2n) 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 x1 >x2. 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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Published In

cover image Combinatorica

Combinatorica Volume 35, Issue 6

December 2015

105 pages

ISSN:0209-9683

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Copyright © Copyright © 2015 J$#225;nos Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg.

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 01 December 2015

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