The multiplicative complexity of 6-variable Boolean functions

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 wi...

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Bibliographic Details
Published in:Cryptography and communications Vol. 11; no. 1; pp. 93 - 107
Main Authors: Çalık, Çağdaş, Sönmez Turan, Meltem, Peralta, René
Format: Journal Article
Language:English
Published: New York Springer US 01.01.2019
Springer Nature B.V
Subjects:
Affine transformations
Boolean algebra
Boolean functions
Boolean Functions and their Applications II
Circuits
Codes
Coding and Information Theory
Communications Engineering
Complexity
Computer Science
Data Structures and Information Theory
Equivalence
Gates
Information and Communication
Mathematical analysis
Mathematics of Computing
Networks
06E30
Boolean functions
Multiplicative complexity
94A60
Affine equivalence
Cryptography
Circuit complexity
ISSN:1936-2447, 1936-2455
Online Access:Get full text
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Summary: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 ≤ 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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ISSN:1936-2447
1936-2455
DOI:10.1007/s12095-018-0297-2