When Six Gates are Not Enough

TL;DR

This paper uses the pigeonhole principle to prove that certain Boolean functions on 7 inputs require at least 7 multiplications in GF(2), demonstrating a lower bound on multiplicative complexity.

Contribution

It establishes a new lower bound on the multiplicative complexity of Boolean functions on 7 inputs using combinatorial reasoning.

Findings

Boolean functions on 7 inputs need at least 7 multiplications

Lower bound on multiplicative complexity established

Uses pigeonhole principle for proof

Abstract

We apply the pigeonhole principle to show that there must exist Boolean functions on 7 inputs with a multiplicative complexity of at least 7, i.e., that cannot be computed with only 6 multiplications in the Galois field with two elements.