Further investigation on differential properties of the generalized Ness–Helleseth function

Let n be an odd positive integer, p be an odd prime with \(p\equiv 3\pmod 4\), \(d_{1} = {{p^{n}-1}\over {2}} -1 \) 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 \(p^n \ge 7\), 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 \({\mathbb {F}}_{p^n}\).