论文标题
消失的公寓：关于功能平面性及其应用的组合观点
Vanishing Flats: A Combinatorial Viewpoint on the Planarity of Functions and Their Application
论文作者
论文摘要
对于$ \ mathbb {f} _2^n $到$ \ mathbb {f} _2^n $的函数$ f $，$ f $的平面度通常是通过其微分均匀性和差异光谱来衡量的。在本文中，我们提出了消失的公寓的概念，该概念提供了平面性的组合观点。首先，$ f $的消失公寓数量可以视为$ f $和几乎完美的非线性功能之间的距离的量度。在某些情况下，消失的公寓数量是差异均匀性和差异频谱之间的“中间”概念，其中包含的信息比差异均匀性更多，但是小于差异光谱。其次，一组消失的公寓形成了一种称为部分四倍系统的组合配置，因为它传达了有关$ f $的详细结构信息。我们通过考虑消失的公寓数量以及与单一和Dembowski-Ostrom多项式相关的部分四倍系统来启动这项研究。此外，我们将消失的平台应用于矢量空间的分区中的应用中，将其用于差异等等的仿射空间。我们总结了本文的其他几个问题和挑战。
For a function $f$ from $\mathbb{F}_2^n$ to $\mathbb{F}_2^n$, the planarity of $f$ is usually measured by its differential uniformity and differential spectrum. In this paper, we propose the concept of vanishing flats, which supplies a combinatorial viewpoint on the planarity. First, the number of vanishing flats of $f$ can be regarded as a measure of the distance between $f$ and the set of almost perfect nonlinear functions. In some cases, the number of vanishing flats serves as an "intermediate" concept between differential uniformity and differential spectrum, which contains more information than differential uniformity, however less than the differential spectrum. Secondly, the set of vanishing flats forms a combinatorial configuration called partial quadruple system, since it convey detailed structural information about $f$. We initiate this study by considering the number of vanishing flats and the partial quadruple systems associated with monomials and Dembowski-Ostrom polynomials. In addition, we present an application of vanishing flats to the partition of a vector space into disjoint equidimensional affine spaces. We conclude the paper with several further questions and challenges.
