论文标题
完美非线性地图的图像集
Image sets of perfectly nonlinear maps
论文作者
论文摘要
我们考虑有限字段的差异$ d $均匀地图的图像集。我们通过扩展用于平面图的方法，对此类地图的图像大小进行了下限，并研究其前图分布。我们将结果应用于研究$ d $均匀的dembowski-oStrom多项式。此外，我们专注于在二进制字段上特别有趣的APN地图案例。我们表明，具有最小图像大小的APN地图必须具有非常特殊的预映射分布。我们证明，对于$ n $，几个apn地图家庭的图像集都很少。我们提出结果，将特殊地图的图像集与它们的Walsh频谱连接起来。尤其是，我们表明，几个大类的APN地图具有经典的WALSH频谱，这一事实是通过其图像集的最小性来解释的。最后，我们在APN地图的图像大小上显示上限。
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.
