Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function

TL;DR

This paper investigates the differential properties of a generalized Ness-Helleseth function over finite fields, providing conditions for it to be APN and analyzing its differential spectrum using quadratic character sums.

Contribution

It offers necessary and sufficient conditions for the function to be APN and characterizes its differential spectrum in terms of quadratic character sums, extending prior work beyond the ternary case.

Findings

Characterization of APN conditions for the function.

Expression of differential spectrum via quadratic character sums.

Extension of Ness-Helleseth function analysis to broader finite fields.

Abstract

Let $n$ be an odd positive integer, $p$ be a prime with $p\equiv3\pmod4$, $d_{1} = {{p^{n}-1}\over {2}} -1 $ and $d_{2} =p^{n}-2$. The function defined by $f_u(x)=ux^{d_{1}}+x^{d_{2}}$ is called the generalized Ness-Helleseth function over $\mathbb{F}_{p^n}$, where $u\in\mathbb{F}_{p^n}$. It was initially studied by Ness and Helleseth in the ternary case. In this paper, for $p^n \equiv 3 \pmod 4$ and $p^n \ge7$, we provide the necessary and sufficient condition for $f_u(x)$ to be an APN function. In addition, for each $u$ satisfying $\chi(u+1) = \chi(u-1)$, the differential spectrum of $f_u(x)$ is investigated, and it is expressed in terms of some quadratic character sums of cubic polynomials, where $\chi(\cdot)$ denotes the quadratic character of $\mathbb{F}_{p^n}$.