When Six Gates are Not Enough
Michael Codish, Lu\'is Cruz-Filipe, Michael Frank, Peter, Schneider-Kamp

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.
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