A Construction of Weakly and Non-Weakly Regular Bent Functions

TL;DR

This paper introduces a new technique for constructing p-ary bent functions from near-bent functions, resulting in classes of both weakly regular and non-weakly regular bent functions, including the first infinite class of non-weakly regular bent functions.

Contribution

It presents a novel construction method for p-ary bent functions from near-bent functions, expanding the known classes to include non-quadratic and non-weakly regular bent functions.

Findings

Constructed classes of non-quadratic bent functions.

Produced the first infinite class of non-weakly regular bent functions.

Demonstrated that some constructions yield both weakly regular 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 non-quadratic 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.