**The Multiplicative Complexity of 6-variable Boolean Functions**

*Cagdas Calik and Meltem Sonmez Turan and Rene 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. 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 exist $n$-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 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 150357 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.

**Category / Keywords: **Affine equivalence and Boolean functions and Circuit complexity and Cryptography and Multiplicative complexity

**Date: **received 27 Dec 2017, last revised 2 Jan 2018

**Contact author: **meltemsturan at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20180102:174523 (All versions of this report)

**Short URL: **ia.cr/2018/002

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