A Note on Generalized Bent Criteria for Boolean Functions

In this paper, we consider the spectra of Boolean functions with respect to the action of unitary transforms obtained by taking tensor products of the Hadamard kernel, denoted by H, and the nega-Hadamard kernel, denoted by N. The set of all such transforms is denoted by {H, N}ⁿ. A Boolean function is said to be bent₄ if its spectrum with respect to at least one unitary transform in {H, N}ⁿ is flat. We obtain a relationship between bent, semibent, and bent₄ functions, which is a generalization of the relationship between bent and negabent Boolean functions proved by Parker and Pott [cf., LNCS 4893 (2007), 9-23]. As a corollary to this result, we prove that the maximum possible algebraic degree of a bent₄ function on n variables is [n/2] and, hence, solve an open problem posed by Riera and Parker [cf., IEEE-TIT 52:9 (2006), 4142-4159].

I. Introduction

Let us denote the set of integers, real numbers, and complex numbers by , , and , respectively, and let the ring of integers modulo be denoted by . The vector space is the space of all -tuples of elements from with the standard operations. By “+” we denote the addition over , , and , whereas “” denotes the addition over for all . Addition modulo is denoted by “+” and it is understood from the context. If and are in , we define the scalar (or inner) product by In , let and denote the zero vector and the all-one vector, respectively. The cardinality of a set is denoted by . If , then denotes the absolute value of , and denotes the complex conjugate of , where and .