Lower bound for monotone Boolean convolution

TL;DR

This paper establishes a tight lower bound of $n^2$ and-gates for any monotone Boolean circuit computing the n-dimensional Boolean convolution, matching the known upper bound.

Contribution

It proves a tight lower bound for the complexity of monotone Boolean convolution circuits, resolving a fundamental question in circuit complexity.

Findings

Any monotone Boolean circuit for n-dimensional convolution needs at least n^2 and-gates.

The lower bound matches the trivial upper bound, confirming optimality.

Provides a precise complexity characterization for monotone Boolean convolution circuits.

Abstract

Any monotone Boolean circuit computing the $n$-dimensional Boolean convolution requires at least $n^2$ and-gates. This precisely matches the obvious upper bound.