Landscape Boolean Functions

TL;DR

This paper introduces landscape Boolean functions, a broad class encompassing bent, semibent, and plateaued functions, providing their complete characterization, construction methods, and insights into their spectral properties.

Contribution

It defines landscape functions on $\

Findings

Complete characterization of landscape functions in terms of components.

Unified framework for generalized bent, semibent, and plateaued functions.

Inductive construction method for functions with any number of nonzero Walsh-Hadamard coefficients.

Abstract

In this paper we define a class of Boolean and generalized Boolean functions defined on $\mathbb{F}_2^n$ with values in $\mathbb{Z}_q$ (mostly, we consider $q=2^k$), which we call landscape functions (whose class containing generalized bent, semibent, and plateaued) and find their complete characterization in terms of their components. In particular, we show that the previously published characterizations of generalized bent and plateaued Boolean functions are in fact particular cases of this more general setting. Furthermore, we provide an inductive construction of landscape functions, having any number of nonzero Walsh-Hadamard coefficients. We also completely characterize generalized plateaued functions in terms of the second derivatives and fourth moments.