幂映射$x^{p^n-3}$的微分谱
The Differential Spectrum of the Power Mapping $x^{p^n-3}$
摘要：
设$n$为正整数，$p$为素数。$x^{p^n-3}$在$\mathbb{F}_{p^n}$上的幂映射具有理想的微分性质，并确定了它对$p=2,\,3$的微分谱.本文对任意奇素数$p$，通过研究$\mathbb{F}_{p^n}$上的二次特征和及方程，用统一的方法确定了$x^{p^n-3}$的微分谱.结果表明，对于任意给定的奇素数$p$，微分谱都可以用$n$显式表示.与以前的结果相比，一个特殊的椭圆曲线$\mathbb{F}_{p}$在我们的计算中起着重要作用的一般情况$p \ge 5$。
英文摘要：
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 $p \ge 5$.
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