论文标题
二次APN功能的新实例
New Instances of Quadratic APN Functions
论文作者
论文摘要
在最近的一项工作中，Beierle，Brinkmann和Leander提出了递归树的搜索，以查找小小的线性自我等量的APN排列。在本文中，我们描述了如何适应此搜索以找到许多二次APN函数的新实例。特别是，我们在维度八的尺寸中发现了12,921个新的二次APN函数，在尺寸为9的35个新的二次APN函数和五个新的二次APN函数的维度为十个尺寸为10到CCZ等效性。值得注意的是，尺寸为9的35个新APN函数中有两个是APN排列。 Among the 8-bit APN functions, there are three extended Walsh spectra that do not correspond to any of the previously-known quadratic 8-bit APN functions and, surprisingly, there exist at least four CCZ-inequivalent 8-bit APN functions with linearity $2^7$, i.e., the highest possible non-trivial linearity for quadratic functions in dimension eight.
In a recent work, Beierle, Brinkmann and Leander presented a recursive tree search for finding APN permutations with linear self-equivalences in small dimensions. In this paper, we describe how this search can be adapted to find many new instances of quadratic APN functions. In particular, we found 12,921 new quadratic APN functions in dimension eight, 35 new quadratic APN functions in dimension nine and five new quadratic APN functions in dimension ten up to CCZ-equivalence. Remarkably, two of the 35 new APN functions in dimension nine are APN permutations. Among the 8-bit APN functions, there are three extended Walsh spectra that do not correspond to any of the previously-known quadratic 8-bit APN functions and, surprisingly, there exist at least four CCZ-inequivalent 8-bit APN functions with linearity $2^7$, i.e., the highest possible non-trivial linearity for quadratic functions in dimension eight.
