A new class of negabent functions

TL;DR

This paper introduces a new class of negabent functions called $2q$-negabent functions, extending the concept to functions from $ ext{Z}_q^n$ to $ ext{Z}_{2q}$, and provides constructions and properties of these functions.

Contribution

The paper extends negabent functions to a broader class called $2q$-negabent functions, introduces the nega-Hadamard transform, and offers new constructions for these functions.

Findings

Defined the $2q$-negabent functions and the nega-Hadamard transform.

Provided two constructions for $2q$-negabent functions for different cases.

Presented examples illustrating the properties of $2q$-negabent functions.

Abstract

Negabent functions were introduced as a generalization of bent functions, which have applications in coding theory and cryptography. In this paper, we have extended the notion of negabent functions to the functions defined from $\mathbb{Z}_q^n$ to $\mathbb{Z}_{2q}$ ($2q$-negabent), where $q \geq 2$ is a positive integer and $\mathbb{Z}_q$ is the ring of integers modulo $q$. For this, a new unitary transform (the nega-Hadamard transform) is introduced in the current set up, and some of its properties are discussed. Some results related to $2q$-negabent functions are presented. We present two constructions of $2q$-negabent functions. In the first construction, $2q$-negabent functions on $n$ variables are constructed when $q$ is an even positive integer. In the second construction, $2q$-negabent functions on two variables are constructed for arbitrary positive integer $q \ge 2$. Some examples of $2q$-negabent functions for different values of $q$ and $n$ are also presented.