A note on the differential spectrum of the Ness-Helleseth function

Let $n\geqslant3$ be an odd integer and $u$ an element in the finite field $\gf_{3^n}$. The Ness-Helleseth function is the binomial $f_u(x)=ux^{d_1}+x^{d_2}$ over $\gf_{3^n}$, where $d_1=\frac{3^n-1}{2}-1$ and $d_2=3^n-2$. In 2007, Ness and Helleseth showed that $f_u$ is an APN function when $\chi(u+1)=\chi(u-1)=\chi(u)$, is differentially $3$-uniform when $\chi(u+1)=\chi(u-1)\neq\chi(u)$, and has differential uniformity at most 4 if $\chi(u+1)\neq\chi(u-1)$ and $u\notin\gf_3$. Here $\chi(\cdot)$ denotes the quadratic character on $\gf_{3^n}$. Recently, Xia et al. determined the differential uniformity of $f_u$ for all $u$ and computed the differential spectrum of $f_u$ for $u$ satisfying $\chi(u+1)=\chi(u-1)$ or $u\in\gf_3$. The remaining problem is the differential spectrum of $f_u$ with $\chi(u+1)\neq\chi(u-1)$ and $u\notin\gf_3$. In this paper, we fill in the gap. By studying differential equations arising from the Ness-Helleseth function $f_u$ more carefully, we express the differential spectrum of $f_u$ for such $u$ in terms of two quadratic character sums. This complements the previous work of Xia et al.