Partial spread and vectorial generalized bent functions

Abstract

In this paper we generalize the partial spread class and completely describe it for generalized Boolean functions from F n 2 to Z 2 t . Explicitly, we describe gbent functions from F n 2 to Z 2 t , which can be seen as a gbent version of Dillon's P S ap class. For the first time, we also introduce the concept of a vectorial gbent function from F n 2 to Z m q , and determine the maximal value which m can attain for the case q = 2 t . Finally we point to a relation between vectorial gbent functions and relative difference sets.

Landscape Boolean functions

“…It is not very difficult to show that landscape functions exist for every dimension, as our next proposition shows, which adapts some classical inductive plateaued construction (see, for instance, [6,9,10,13,16] for the construction of generalized Boolean bent functions), as well as the paper [14], which contains some constructions of semibent and even more general plateaued in the spirit of Maiorana-McFarland construction of bent functions.…”

“…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.…”

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 .…”

Decomposing Generalized Bent and Hyperbent Functions

“…In [2] Carlet and Gaborit proved that all functions in the class of P S ap are hyperbent. We proceed similarly for a class of gbent functions from GB 2 k 2n presented in [9], which can be seen as a function in a generalized P S ap class. We recall the functions in the next proposition.…”