A note on generalization of bent boolean functions

Abstract

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

A Characterization of Generalized Boolean Functions Employed in CDMA Communications

“…These generalizations of Boolean functions given by the various researchers are helpful in the construction of some efficient cryptographic algorithms for smooth digital communication. For Boolean functions in cryptography and error correcting codes, we may refer to MacWilliams (1977), Carlet (2010), Singh et al (2013), Zhuo and Chong (2015), Gangopadhyay et al (2019), Singh and Paul (2019), Mandal and Gangopadhyay (2021), Mandal et al (2022), Singh et al (2022), Tiwari and Sharma (2023). In the present article, by considering the Schmidt's generalization, we discuss some more results of generalized Boolean functions in respect of GNT.…”