Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function

• URL: http://arxiv.org/abs/2408.17272v1
• Date: Fri, 30 Aug 2024 13:18:23 GMT
• Title: Further Investigation on Differential Properties of the Generalized Ness-Helleseth Function
• Authors: Yongbo Xia, Chunlei Li, Furong Bao, Shaoping Chen, Tor Helleseth,
• Abstract summary: The function defined by $f_u(x)=uxd_1+xd_2$ is called the generalized Ness-Helleseth function over $mathbbF_pn$. For each $u$ satisfying $chi(u+1) = chi(u-1)$, the differential spectrum of $f_u(x)$ is investigated.
• Score: 13.67029767623542
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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 $\mathbb{F}_{p^n}$.

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