Constructing bent functions and bent idempotents of any possible algebraic degrees

TL;DR

This paper introduces new infinite families of bent functions and constructs bent idempotents of any algebraic degree, solving longstanding open problems and expanding the understanding of these optimal combinatorial objects.

Contribution

It presents the first univariate construction of infinite bent idempotents of any algebraic degree over finite fields, extending previous work and solving open problems.

Findings

New infinite families of bent functions constructed.

First univariate construction of bent idempotents of any algebraic degree.

Anti-self-dual bent functions and their sums are characterized.

Abstract

Bent functions as optimal combinatorial objects are difficult to characterize and construct. In the literature, bent idempotents are a special class of bent functions and few constructions have been presented, which are restricted by the degree of finite fields and have algebraic degree no more than 4. In this paper, several new infinite families of bent functions are obtained by adding the the algebraic combination of linear functions to some known bent functions and their duals are calculated. These bent functions contain some previous work on infinite families of bent functions by Mesnager \cite{M2014} and Xu et al. \cite{XCX2015}. Further, infinite families of bent idempotents of any possible algebraic degree are constructed from any quadratic bent idempotent. To our knowledge, it is the first univariate representation construction of infinite families of bent idempotents over $\mathbb{F}_{2^{2m}}$ of algebraic degree between 2 and $m$, which solves the open problem on bent idempotents proposed by Carlet \cite{C2014}. And an infinite family of anti-self-dual bent functions are obtained. The sum of three anti-self-dual bent functions in such a family is also anti-self-dual bent and belongs to this family. This solves the open problem proposed by Mesnager \cite{M2014}.