A further study on the Ness-Helleseth function

Let $F_{p^n}$ be a finite field with $p^n$ elements. Ness and Helleseth in [29] first studied a class of functions over $F_{p^n}$ with the form $f\left(x\right)=u{x}^{\frac{p^n-3}{2}}+x^{p^n-2},u\in F_{p^n}^{⁎}$, which is called the Ness-Helleseth function. The $f\left(x\right)$ has been proved to be an almost perfect nonlinear (APN) function by Ness and Helleseth for $p=3$ in [29] and by Zeng et al. for any odd prime p in [43] under the condition $p^n\equiv 3\left(\mathrm{mod}4\right)$ and $\eta (1+u)=\eta (u-1)$. In this paper, we continue to study the Ness-Helleseth functions under the condition that $p^n\equiv 3\left(\mathrm{mod}4\right)$ and $\eta (1+u)\ne \eta (u-1)$. Firstly, we prove that $f\left(x\right)$ is a permutation polynomial with differential uniformity not more than 4 if $\eta (1+u)=\eta (1-u)$. Moreover, for some more special u, f is an involution with differential uniformity at most 3. Secondly, we show that $f\left(x\right)$ is a locally-APN function for $u=\pm 1$. In addition, the differential spectrum and boomerang spectrum of $f\left(x\right)$ are obtained via judging the number of solutions of some special equations. We obtain the first non-PN function that its boomerang uniformity can attain 0 or 1.