## 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

- 1.
We abuse notation here, identifying a node with the function it computes.

- 2.
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.

- 3.
Commercial equipment and software referred to in this paper are identified for informational purposes only, and does not imply recommendation of or endorsement by the National Institute of Standards and Technology, nor does it imply that the products so identified are necessarily the best available for the purpose.

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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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## Additional information

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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### Keywords

- Affine equivalence
- Boolean functions
- Circuit complexity
- Cryptography
- Multiplicative complexity

### Mathematics Subject Classifications (2010)

- 94A60
- 06E30