# MPI-I-92-104

## Circuits and multi-party protocols

### Grolmusz, Vince

**MPI-I-92-104**. January** **1992, 16 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

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.

Acknowledgement:** **

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**URL to this document: **http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1992-104

**BibTeX**
`@TECHREPORT{``Grolmusz92a``,`

` AUTHOR = {Grolmusz, Vince},`

` TITLE = {Circuits and multi-party protocols},`

` TYPE = {Research Report},`

` INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},`

` ADDRESS = {Im Stadtwald, D-66123 Saarbr{\"u}cken, Germany},`

` NUMBER = {MPI-I-92-104},`

` MONTH = {January},`

` YEAR = {1992},`

` ISSN = {0946-011X},`

`}`