MPI-INF/SWS Research Reports 1991-2021

2. Number - only D1


Circuits and multi-party protocols

Grolmusz, Vince

January 1992, 16 pages.

Status: available - back from printing

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.

URL to this document:

Hide details for BibTeXBibTeX
  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},