Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function
<|reference_start|>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 $\chi(u+1) = \chi(u-1)$, 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 $\mathbb{F}_{p^n}$.<|reference_end|>
@article{xia2024further,
    title={Further Investigation on Differential Properties of the Generalized
  Ness-Helleseth Function},
    author={Yongbo Xia, Chunlei Li, Furong Bao, Shaoping Chen, Tor Helleseth},
    journal={arXiv preprint arXiv:2408.17272},
    year={2024},
    archivePrefix={arXiv},
    eprint={2408.17272},
    primaryClass={cs.CR cs.DM cs.IT math.IT math.NT}
}