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

Let $n⩾3$ be an odd integer and $u$ an element in the finite field ${\text{gf}}_{3^n}$. The Ness-Helleseth function is the binomial $f_u(x)=u{x}^{d_1}+x^{d_2}$ over ${\text{gf}}_{3^n}$, 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 $\chi (u+1)=\chi (u-1)\mathrm{\ne }\chi (u)$, and has differential uniformity at most 4 if $\chi (u+1)\mathrm{\ne }\chi (u-1)$ and $u\mathrm{\notin }{\text{gf}}_3$. Here $\chi (\cdot )$ denotes the quadratic character on ${\text{gf}}_{3^n}$. 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 $u\in {\text{gf}}_3$. The remaining problem is the differential spectrum of $f_u$ with $\chi (u+1)\mathrm{\ne }\chi (u-1)$ and $u\mathrm{\notin }{\text{gf}}_3$. 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.