Some Results on Bent-Negabent Boolean Functions over Finite Fields

TL;DR

This paper characterizes quadratic negabent Boolean functions with trace representation, explores their relation to bent functions, and constructs infinite classes of negabent functions with maximum degree over finite fields.

Contribution

It provides a complete characterization of quadratic negabent monomial functions and introduces the first constructions of trace-based negabent functions with optimal degree.

Findings

Complete characterization of quadratic negabent monomials

Relation established between negabent and bent functions via quadratic functions

Infinite classes of negabent functions with maximum degree over finite fields

Abstract

We consider negabent Boolean functions that have Trace representation. We completely characterize quadratic negabent monomial functions. We show the relation between negabent functions and bent functions via a quadratic function. Using this characterization, we give infinite classes of bent-negabent Boolean functions over the finite field $\F_{2^n}$, with the maximum possible degree, $n \over 2$. These are the first ever constructions of negabent functions with trace representation that have optimal degree.