发表机构
Faculty of Computer Science, Ruhr University Bochum（鲁尔大学波鸿分校计算机学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文针对APN函数，证明了其线性度在偶数维和奇数维分别有更紧的上界，并给出了基于非平坦分量数量的线性度界，改进了现有结果。
AI 中文摘要
对于$n\ge5$，设$F\colon\mathbb{F}_2^n\to \mathbb{F}_2^n$为几乎完全非线性函数，并记$N=2^n$。证明了$F$的线性度$\mathcal{L}(F)$（即非零分量最大的绝对Walsh系数）在偶数维中至多为$N-10$，在奇数维中至多为$N-6$。这改进了偶数维中$N-6$和奇数维中$N-4$的一般上界。进一步证明，对于每个固定的$k$，当$n\to\infty$时，非零分量中按重数计的第$k$大的绝对Walsh系数至多为$(1+O_k(2^{-n}))N/\sqrt{k}$。第二和第四大的系数分别至多为$2\lfloor N/3\rfloor$和$N/2$。最后，推导了线性度关于非平坦非零分量数量$q$的界。在奇数维中，对于$q>0$，有$\mathcal{L}(F)^2\le N(1+\sqrt{q(N-1)})$，因此$\mathcal{L}(F)/N\to1$蕴含$q/N\to1$。在偶数维中，对于每个$1/2\le C<1$，条件$q\le(4C-C^2-1)N/4+1$蕴含$\mathcal{L}(F)\le CN$。特别地，$q\le3N/16+2$蕴含$\mathcal{L}(F)\le N/2$。
英文摘要
For $n\ge5$, let $F\colon\mathbb{F}_2^n\to \mathbb{F}_2^n$ be almost perfect nonlinear and write $N=2^n$. It is proven that the linearity $\mathcal{L}(F)$ of $F$, i.e., the largest absolute Walsh coefficient of a nonzero component, is at most $N-10$ in even dimension and at most $N-6$ in odd dimension. This improves the general upper bound of $N-6$ in even dimension and $N-4$ in odd dimension. It is further proven that, for each fixed $k$, the $k$-th largest absolute Walsh coefficient among nonzero components, counted with multiplicity, is at most $(1+O_k(2^{-n}))N/\sqrt{k}$ as $n\to\infty$. The second and fourth largest coefficients are at most $2\lfloor N/3\rfloor$ and $N/2$, respectively. Finally, a bound on the linearity in terms of the number $q$ of nonplateaued nonzero components is derived. In odd dimension, for $q>0$, we have $\mathcal{L}(F)^2\le N(1+\sqrt{q(N-1)})$, so that $\mathcal{L}(F)/N\to1$ implies $q/N\to1$. In even dimension, for every $1/2\le C<1$, the condition $q\le(4C-C^2-1)N/4+1$ implies $\mathcal{L}(F)\le CN$. In particular, $q\le3N/16+2$ implies $\mathcal{L}(F)\le N/2$.
