A note on the $ c $-differential spectrum of an AP$ c $N function
Tingting Pang, Nian Li, Xiangyong Zeng, Haiying Zhu · Advances in Mathematics of Communications · 2022
Motivated by a recent work of Zhang and Yan on the \begin{document}$ c $\end{document} -differential spectrum of some power functions over finite fields, we further study an AP \begin{document}$ c $\end{document} N function and express its \begin{document}$ c $\end{document} -differential spectrum in terms of \begin{document}$ (i, j, k)_2 $\end{document} , i.e., the cardinality of the intersection \begin{document}$ (\mathcal{C}^{(2)}_i+1)\cap\mathcal{C}^{(2)}_j\cap(\mathcal{C}^{(2)}_k-1) $\end{document} for \begin{document}$ i, j, k\in\{0, 1\} $\end{document} , where \begin{document}$ \mathcal{C}^{(2)}_0, \mathcal{C}^{(2)}_1 $\end{document} are the cyclotomic classes of order two over the finite field \begin{document}$ \mathbb{F}_{p^n} $\end{document} , \begin{document}$ p $\end{document} is an odd prime and \begin{document}$ n $\end{document} is a positive integer. By virtue of the cyclotomic numbers of orders two and four, we determine the values of \begin{document}$ (i, j, k)_2 $\end{document} for \begin{document}$ i, j, k\in\{0, 1\} $\end{document} , which may be of independent interest. As an application, we give another proof of the \begin{document}$ c $\end{document} -differential spectrum of an AP \begin{document}$ c $\end{document} N function over finite fields with characteristic \begin{document}$ 5 $\end{document} . Further, we refine the result of Zhang and Yan in the sense that we completely characterize the conditions when the \begin{document}$ c $\end{document} -differential equation of the AP \begin{document}$ c $\end{document} N function has one solution and two solutions, respectively.