A construction of weakly and non-weakly regular bent functions

Abstract

In this article a technique for constructing p-ary bent functions from near-bent functions is presented. Two classes of quadratic p-ary functions are shown to be near-bent. Applying the construction of bent functions to these classes of near-bent functions yields classes of nonquadratic bent functions. We show that one construction in even dimension yields weakly regular bent functions. For other constructions, we obtain both weakly regular and non-weakly regular bent functions. In particular we present the first known infinite class of non-weakly regular bent functions.

Quadratic functions and maximal Artin–Schreier curves

“…As is well known, quadratic functions are partially bent functions, see [5]. Their Walsh spectrum is described as follows, see [5,6,8]:…”

A construction of bent functions from plateaued functions

“…Versions of the above corollaries for the special case that s = 1 were used in our work [3] to present the first infinite classes of non-weakly regular bent functions. These bent functions are in odd dimension, their algebraic degree is p + 1.…”

“…The case s = 0 corresponds to bent functions by definition. For 1-plateaued functions the term near-bent function is common (see [3,7]), binary 1-plateaued and 2-plateaued functions are referred to as semi-bent functions in [4].…”

Partial spread and vectorial generalized bent functions

“…f (u) = 2 n/2 ζ ju q for some 0 ≤ j u < q, then -following the notation for bent functions in odd characteristic (see [1,6]) -we call f a regular gbent function. Similar as for bent functions we call f * the dual of f , if 2 n/2 ζ f * (u) q = H (q) f (u).…”