Cryptography and Communications

, Volume 11, Issue 1, pp 93–107 | Cite as

The multiplicative complexity of 6-variable Boolean functions

  • Çağdaş ÇalıkEmail author
  • Meltem Sönmez Turan
  • René Peralta
Part of the following topical collections:
  1. Special Issue on Boolean Functions and Their Applications


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.


Affine equivalence Boolean functions Circuit complexity Cryptography Multiplicative complexity 

Mathematics Subject Classifications (2010)

94A60 06E30 



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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Copyright information

© This is a U.S. Government work and not under copyright protection in the US; foreign copyright protection may apply 2018

Authors and Affiliations

  1. 1.National Institute of Standards and TechnologyGaithersburgUSA
  2. 2.Dakota Consulting Inc.Silver SpringUSA

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