Abstract
The multiplicative complexity of a Boolean function is the minimum number of two-input 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. [1] 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 ≥ 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 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.
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Notes
We abuse notation here, identifying a node with the function it computes.
The Gaussian binomial coefficient \(\binom {m}{r}_{q}\) is defined to be \(\frac {(1-q^{m})(1-q^{m-1}){\cdots } (1-q^{m-r + 1})}{(1-q)(1-q^{2})\cdots (1-q^{r})}\) for \(r\leq m\), and zero otherwise.
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Acknowledgments
We thank Ray Perlner for his suggestions on enumerating the subspaces of a vector space. We also thank Luís Brandão and the anonymous reviewers for helpful comments and suggestions.
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This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications
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Çalık, Ç., Sönmez Turan, M. & Peralta, R. The multiplicative complexity of 6-variable Boolean functions. Cryptogr. Commun. 11, 93–107 (2019). https://doi.org/10.1007/s12095-018-0297-2
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DOI: https://doi.org/10.1007/s12095-018-0297-2