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Cryptography and Communications

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03-04-2018

The multiplicative complexity of 6-variable Boolean functions

Authors: Çağdaş Çalık, Meltem Sönmez Turan, René Peralta

Published in: Cryptography and Communications | Issue 1/2019

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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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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.

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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Title: The multiplicative complexity of 6-variable Boolean functions
Authors: Çağdaş Çalık
Meltem Sönmez Turan
René Peralta
Publication date: 03-04-2018
Publisher: Springer US
Published in: Cryptography and Communications / Issue 1/2019
Print ISSN: 1936-2447
Electronic ISSN: 1936-2455
DOI: https://doi.org/10.1007/s12095-018-0297-2

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