Circuits and multi-party protocols

Using multi-party communication techniques, we prove that depth-3 circuits with a threshold gate at the top, arbitrary symmetric gates at the next level, and fan-in k MODm gates at the bottom need exponential size to compute the k-wise inner product function of Babai, Nisan and Szegedy, where m is an odd positive integer satisfying \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $ m \equiv k \bmod 2m $\end{document}. This is one of the rare lower-bound results involving MODm gates with non-prime power moduli.¶Exponential gap theorems are also presented between the multi-party communication complexities of closely related functions.