Boolean functions with multiplicative complexity 3 and 4

Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta ( 2002 ), and Find et al. (IJICoT 4 (4), 222–236, 2017 ), respe...

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Published in:Cryptography and communications Vol. 12; no. 5; pp. 935 - 946
Main Authors: Çalık, Çağdaş, Turan, Meltem Sönmez, Peralta, René
Format: Journal Article
Language:English
Published: New York Springer US 01.09.2020
Springer Nature B.V
Subjects:
Affine transformations
Boolean
Boolean algebra
Boolean functions
Circuits
Codes
Coding and Information Theory
Communications Engineering
Complexity
Computer Science
Data Structures and Information Theory
Equivalence
Gates (circuits)
Information and Communication
Linearity
Lower bounds
Mathematics of Computing
Networks
06E30
Boolean functions
94A60
Affine equivalence
Multiplicative complexity
ISSN:1936-2447, 1936-2455
Online Access:Get full text
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Summary:Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta ( 2002 ), and Find et al. (IJICoT 4 (4), 222–236, 2017 ), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimension d i m ( f ) of a Boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity of f is at least ⌈ d i m ( f )/2⌉. For MC 3, this implies that there are no equivalence classes other than those 24 identified in Çalık et al. ( 2018 ). Using the techniques from Çalık et al. and the new relation between the dimension and MC, we identify all 1277 equivalence classes having MC 4. We also provide a closed formula for the number of n -variable functions with MC 3 and 4. These results allow us to construct AND-optimal circuits for Boolean functions that have MC 4 or less, independent of the number of variables they are defined on.
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ISSN:1936-2447
1936-2455
DOI:10.1007/s12095-020-00445-z