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

Abstract

Let $n$ be an odd positive integer, $p$ be a prime with $p\equiv 3(\mathrm{mod}4)$, $d_1=\frac{p^n-1}{2}-1$ and $d_2=p^n-2$. The function defined by $f_u(x)=u{x}^{d_1}+x^{d_2}$ 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 $p^n\equiv 3(\mathrm{mod}4)$ and $p^n\ge 7$, 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 ${\mathbb{F}}_{p^n}$.

34 pages