Landscape Boolean functions

Abstract

In this paper we define a class of Boolean and generalized Boolean functions defined on F n 2 with values in 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.

Root-Hadamard transforms and complementary sequences

“…Further, by Corollary 1 of [14], we see that, if g : F n 2 → Z 2 k is a function given by g(x) = a 0 (x) + 2a 1 (x) + • • • + 2 k−1 a k−1 , and s ≥ 0 is an integer, then, g is s-gplateaued if and only if, for each c ∈ F k−1 2 , the Boolean function g c defined as g c (x) = c • (a 0 (x), . .…”

“…Using Theorem 3.2 of [14], this argument can be also extended to landscape functions, in a similar way as in the plateaued case.…”

“…For more on (generalized) Boolean functions, the reader can consult [2,3,10,14,18,20] and the references therein.…”

“…Two of us introduced the next concept in [14]. We call a function f ∈ GB q n a landscape function if there exist t ≥ 1,…”