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Abstract

We present a multi--party protocol for computing certain functions of an $n\times k$ $0-1$ matrix $A$. The protocol is for $k$ players, where player $i$ knows every column of $A$, except column $i$. {\it Babai, Nisan} and {\it Szegedy} proved that to compute $GIP(A)$ needs $\Omega (n/4^k)$ bits to communicate. We show that players can count those rows of matrix $A$ which sum is divisible by $m$, with communicating only $O(mk\log n)$ bits, while counting the rows with sum congruent to 1 $\pmod m$ needs $\Omega (n/4^k)$ bits of communication (with an odd $m$ and $k\equiv m\pmod {2m}$). $\Omega(n/4^k)$ communication is needed also to count the rows of $A$ with sum in any congruence class modulo an {\it even} $m$. The exponential gap in communication complexities allows us to prove exponential lower bounds for the sizes of some bounded--depth circuits with MAJORITY, SYMMETRIC and MOD$_m$ gates, where $m$ is an odd -- prime or composite -- number.