# The Multiplicative Complexity of 6-variable Boolean Functions

Published: April 03, 2018

### Author(s)

Cagdas Calik, Meltem Sonmez Turan, Rene C. Peralta

### Abstract

The multiplicative complexity of a Boolean function is the minimum number of AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. \cite{TuranSonmez2015} showed that $n$-variable Boolean functions can be implemented with at most $n-1$ AND gates for $n\leq 5$. A counting argument can be used to show that, for $n \geq 7$, there exists $n$-variable Boolean functions with multiplicative complexity of at least $n$. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing all circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150\,357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.
Citation: Cryptography and Communication
Pub Type: Journals