# MPI-I-92-120

## Separating the communication complexities of MOD m and MOD p circuits

### Grolmusz, Vince

**MPI-I-92-120**. March** **1992, 16 pages. | Status:** **available - back from printing | Next --> Entry | Previous <-- Entry

Abstract in LaTeX format:

We prove in this paper that it is much harder to evaluate depth--2,

size--$N$ circuits with MOD $m$ gates than with MOD $p$ gates by

$k$--party communication protocols: we show a $k$--party protocol

which communicates $O(1)$ bits to evaluate circuits with MOD $p$ gates,

while evaluating circuits with MOD $m$ gates needs $\Omega(N)$ bits,

where $p$ denotes a prime, and $m$ a composite, non-prime power number.

Let us note that using $k$--party protocols with $k\geq p$ is crucial

here, since there are depth--2, size--$N$ circuits with MOD $p$ gates

with $p>k$, whose $k$--party evaluation needs $\Omega(N)$ bits. As a

corollary, for all $m$, we show a function, computable with a depth--2

circuit with MOD $m$ gates, but not with any depth--2 circuit with MOD

$p$ gates.

It is easy to see that the $k$--party protocols are not weaker than the

$k'$--party protocols, for $k'>k$. Our results imply that if there is a

prime $p$ between $k$ and $k'$: $k<p\leq k'$, then there exists a

function which can be computed by a $k'$--party

protocol with a constant number of communicated bits, while any

$k$--party protocol needs linearly many bits of communication. This

result gives a hierarchy theorem for multi--party protocols.

Acknowledgement:** **

References to related material:

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**URL to this document: **http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1992-120

**BibTeX**
`@TECHREPORT{``Grolmusz92b``,`

` AUTHOR = {Grolmusz, Vince},`

` TITLE = {Separating the communication complexities of MOD m and MOD p circuits},`

` TYPE = {Research Report},`

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

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

` NUMBER = {MPI-I-92-120},`

` MONTH = {March},`

` YEAR = {1992},`

` ISSN = {0946-011X},`

`}`