(Not) weakly regular univariate bent functions

TL;DR

This paper introduces a new method for constructing univariate bent functions from plateaued functions, resulting in the first explicit class of non-weakly regular univariate bent functions with simple binomial representations.

Contribution

It presents the first direct construction of univariate non-weakly regular bent functions using a novel merging procedure of plateaued functions.

Findings

Constructed infinite classes of bent functions with simple monomial and binomial forms.

Provided the first explicit example of non-weakly regular univariate bent functions.

Demonstrated a new method to generate bent functions from plateaued functions on subspaces.

Abstract

In this article a procedure to construct bent functions from $\F_{p^n}$ to $\F_p$ by merging plateaued functions which are bent on ($n-2$)-dimensional subspaces of $\F_{p^n}$ is presented. Taking advantage of such classes of plateaued functions with a simple representation as monomials and binomials, we obtain infinite classes of bent functions with a fairly simple representation. In particular we present the first direct construction of univariate not weakly regular bent functions, and give one class explicitly in a simple representation with binomials.