A note on generalization of bent boolean functions

Bimal Mandal; Aditi Kar Gangopadhyay

doi:10.3934/amc.2020069

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2021, Volume 15, Issue 2: 329-346. Doi: 10.3934/amc.2020069

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A note on generalization of bent boolean functions

Bimal Mandal^1, and
Aditi Kar Gangopadhyay^2, ,

1.
CARAMBA, INRIA Nancy-Grand Est., 54600, France
2.
Department of Mathematics, Indian Institute of Technology Roorkee, 247667, India

^* Corresponding author: Aditi Kar Gangopadhyay
^* Corresponding author: Aditi Kar Gangopadhyay

Received: April 2019

Revised: February 2020

Early access: April 8, 2020

Published online: April 8, 2020

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Abstract

Suppose that $ \mu_p $ is a probability measure defined on the input space of Boolean functions. We consider a generalization of Walsh–Hadamard transform on Boolean functions to $ \mu_p $-Walsh–Hadamard transforms. In this paper, first, we derive the properties of $ \mu_p $-Walsh–Hadamard transformation for some classes of Boolean functions and specify a class of nonsingular affine transformations that preserve the $ \mu_p $-bent property. We further derive the results on $ \mu_p $-Walsh–Hadamard transform of concatenation of Boolean functions and provide some secondary constructions of $ \mu_p $-bent functions. Finally, we discuss the $ \mu_p $-bentness for Maiorana–McFarland class of bent functions.

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Mathematics Subject Classification: Primary: 06E30, 94C10; Secondary: 05A05.

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