The Differential Spectrum of the Power Mapping x pn −3
Haode Yan, Yongbo Xia, Chunlei Li, Tor Helleseth, Maosheng Xiong, Jinquan Luo · IEEE Transactions on Information Theory · 2022
Let$n$be a positive integer and$p$a prime. The power mapping$x^{p^{n}-3}$over${\mathbb {F}}_{p^{n}}$has desirable differential properties, and its differential spectra for$p=2,\,3$have been determined. In this paper, for any odd prime$p$, by investigating certain quadratic character sums and some equations over${\mathbb {F}}_{p^{n}}$, we determine the differential spectrum of$x^{p^{n}-3}$with a unified approach. The obtained result shows that for any given odd prime$p$, the differential spectrum can be expressed explicitly in terms of$n$. Compared with previous results, a special elliptic curve over${\mathbb {F}}_{p}$plays an important role in our computation for the general case$p \ge 5$.