Generalized Bent Functions and Their Gray Images

Abstract

In this paper we prove that generalized bent (gbent) functions defined on Z n 2 with values in Z 2 k are regular, and find connections between the (generalized) Walsh spectrum of these functions and their components. We comprehensively characterize generalized bent and semibent functions with values in Z 16 , which extends earlier results on gbent functions with values in Z 4 and Z 8 . We also show that the Gray images of gbent functions with values in Z 2 k are semibent/plateaued when k = 3, 4.

Partial spread and vectorial generalized bent functions

“…Hence regular gbent functions always appear in pairs. First note that for y ∈ V n we have We finally remark that as shown in [8], gbent functions from V n to Z 2 t , t ≥ 1, which are the functions in which we are most interested in this article, are always regular. Therefore the dual of a gbent function is always defined and it is a gbent function, as well.…”

“…We will follow our notations from [8] and denote the set of all generalized Boolean functions by GB q n and when q = 2, by B n . A function f ∈ GB q n is called generalized bent (gbent) if |H (q) f (u)| = 2 n/2 for all u ∈ V n .…”

Generalized bent Boolean functions and strongly regular Cayley graphs

“…In turn, f (x) = 2 −n w W f (w)(−1) u·x . We use the notation as in [10,11,12,15,16] (see also [14,17]) and denote the set of all generalized Boolean functions by GB q n and when q = 2, by B n . A function f : V n → Z q is called generalized bent (gbent) if |H f (u)| = 2 n/2 for all u ∈ V n .…”

Landscape Boolean functions

“…By modifying a method of Kumar, Scholtz and Welch [6], in [9] it was shown that all gbent functions f ∈ GB 2 k n are regular, except for n odd and k = 2, in which case one has H f (u) = 2 n−1 2 (±1 ± i). We observe that with our definition of regularity, a function cannot be regular unless the moduli of all nonzero Walsh-Hadamard coefficients of f are of the form 2…”

“…Generalized Boolean functions have become an active area of research [4,5,6,8,9,10,13,15,16,18], with most of these papers dealing with descriptions/constructions of generalized bent/plateaued functions. We show here that in fact these characterizations of generalized bent/plateaued in terms of the components of the function are in fact particular instances of the more general case of generalized landscape functions, which are introduced in this paper.…”