完全非线性映射的像集
Image sets of perfectly nonlinear maps
摘要：
我们考虑有限域的差分$d$ -一致映射的图像集。通过推广平面映射的方法，给出了这类映射的像大小的下界，并研究了它们的原像分布。我们应用这些结果研究了$d$ -一致Dembowski-Ostrom多项式.此外，我们专注于一个特别有趣的情况下，APN映射二进制字段。我们表明，APN地图的最小图像的大小必须有一个非常特殊的原像分布。我们证明，即使$n$的图像集的几个研究家庭的APN地图是最小的。我们提出的结果连接图像集的特殊地图与沃尔什谱。特别是，我们表明，事实上，几个大类的APN地图有经典的沃尔什谱解释了他们的图像集的极小。最后，我们提出了APN映射的图像大小的上界。
英文摘要：
We consider image sets of differentially $d$-uniform maps of finite fields. We present a lower bound on the image size of such maps and study their preimage distribution, by extending methods used for planar maps. We apply the results to study $d$-uniform Dembowski-Ostrom polynomials. Further, we focus on a particularly interesting case of APN maps on binary fields. We show that APN maps with the minimal image size must have a very special preimage distribution. We prove that for an even $n$ the image sets of several well-studied families of APN maps are minimal. We present results connecting the image sets of special maps with their Walsh spectrum. Especially, we show that the fact that several large classes of APN maps have the classical Walsh spectrum is explained by the minimality of their image sets. Finally, we present upper bounds on the image size of APN maps.
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