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

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 $χ(u+1) = χ(u-1)$, the differential spectrum of $f_u(x)$ is investigated, and it is expressed in terms of some quadratic character sums of cubic polynomials, where $χ(\cdot)$ denotes the quadratic character of $\mathbb{F}_{p^n}$.