解决关于二次APN函数的一个猜想及与弯曲函数相关的新二次$(n,n)$-函数
Resolving a conjecture on quadratic APN functions and a new quadratic $(n,n)$-function associated to crooked functions
- Universities of Paris 8, France（巴黎第八大学）
- Bergen, Norway（卑尔根大学）
- Department of Mathematics, Vanderbilt University, USA（范德堡大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该研究解决了Gorodilova关于二次APN函数正交导数非零分量代数次数的猜想，证明了弯曲函数正交导数的相关性质，还得到了$m$-序列问题的同余结果及特定布尔函数的精确代数次数。
AI中文摘要：
我们称一个$(n,n)$-函数$F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$为弯曲函数，当且仅当对任意非零的$a \in \mathbb{F}_2^n$，差分映射$D_aF(x)=F(x)+F(x+a)$的像为一个仿射超平面。目前已知的所有弯曲函数例子均为二次几乎完美非线性（APN）函数，等价地，对所有已知的弯曲函数，对任意$a \in \mathbb{F}_2^n$，$D_aF$都是仿射映射。弯曲函数$F$的正交导数$\pi_F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$是满足以下条件的函数：$\pi_F(0)=0$，且对任意非零的$a$，集合$\{0,\pi_F(a)\}^\perp$是$\mathrm{Im}(D_aF)$的基础向量空间。我们证明，当$n \geq 4$且$F$为弯曲函数时，若$k$为非负整数且$F$有$2^k$个二次分量函数，则$\pi_F$至少有$2^n-2^{n-k}$个非零分量的代数次数为$n-2$。特别地，我们解决了Gorodilova的猜想：当$F$为二次APN函数时，$\pi_F$的每个非零分量的代数次数均为$n-2$。作为推论，我们证明对任意偶数$n \geq 4$，任何至少有一个二次分量的弯曲$(n,n)$-函数都至少有5个半弯曲分量。作为第二个主要结果，对$n \geq 4$，我们关联于弯曲函数$F$得到一个二次函数$\varepsilon_F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$，它满足关于$F$在二维线性子空间上求和的强几何-组合条件。此外，我们得到了关于Johansen、Helleseth和Kholosha提出的$m$-序列问题的一个同余结果，并确定了与特定类 plateau 向量函数的弯曲和近弯曲分量相关的某些布尔函数的精确代数次数。
英文摘要：
We say an $(n,n)$-function $F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ is a crooked function if for any nonzero $a \in \mathbb{F}_2^n$, the image of $D_aF(x)=F(x)+F(x+a)$ is an affine hyperplane. The only known examples of crooked functions are all quadratic almost perfect nonlinear (APN), or equivalently, for every known crooked function, $D_aF$ is affine for all $a \in \mathbb{F}_2^n$. The ortho-derivative $π_F \colon\mathbb{F}_2^n \to \mathbb{F}_2^n$ of a crooked function $F$ is the function such that $π_F(0)=0$, and for any nonzero $a$, the set $\{0,π_F(a)\}^\perp$ is the underlying vector space of $\mathrm{Im}(D_aF)$. We prove that for $n \geq 4$ and a crooked function $F$, if $k$ is a non-negative integer such that $F$ has $2^k-1$ quadratic component functions, $π_F$ has at least $2^n-2^{n-k}$ component functions of algebraic degree $n-2$. In particular, we resolve Gorodilova's conjecture that every component function of $π_F$ has algebraic degree $n-2$ when $F$ is quadratic APN. As a second main result, for $n \geq 4$, we associate to a crooked function $F$ a quadratic function $\varepsilon_F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n$ that satisfies a strong geometric-combinatorial condition regarding the sums of $F$ over $2$-dimensional linear subspaces. As a corollary to both of our main results, we prove that for any even $n \geq 4$, any quadratic APN $(n,n)$-function has at least $n$ semi-bent components. Furthermore, we obtain a congruence result on a problem on $m$-sequences introduced by Johansen, Helleseth, and Kholosha, and we determine the exact algebraic degrees of some Boolean functions associated to the bent and near-bent components of particular classes of plateaued vectorial functions.
