注射率w.r.t. $Z/p^eZ$上本原序列压缩序列中元素的分布
Injectivity w.r.t. Distribution of Elements in the Compressed Sequences Derived from Primitive Sequences over $Z/p^eZ$
摘要：
设$p\geq3$为素数，$e\geq2$为整数。设$σ(x)$是$Z/p^eZ$上的一个次数为$n$的本原多项式，$G'(σ(x),p^e)$是由$σ(x)$生成的本原线性递归序列的集合。$Z/p^eZ$上的压缩映射$φ$自然地导出$G'(σ(x),p^e)$上的映射$\hatφ$。对于$φ$的图像的子集$D$，$\hatφ$被称为是内射w.r.t. $D$ -均匀性，如果在压缩序列中$D$的元素的分布暗示了原始基元序列的所有信息。在这一对应关系中，对于至少$1-2(p-1)/(p^n-1)$的$n$次本原多项式，给出了$φ$的一个明确的判别准则，以判定$\hatφ$是否为相对于t的单射多项式. $D$ -均匀性，并且$Z/p^eZ$上的大多数映射在$G'(σ(x),p^e)$上诱导内射映射。进一步给出了$φ$的一个充分条件，保证了$\hatφ$的可内射性. $D$ -均匀性。由充分条件得出：若$σ(x)$是强本原的，且压缩映射$φ(x)=f(x_{e-1})$，其中$f(x_{e-1})$是$\mathbb{F}_{p}$上的置换多项式，则$\hatφ$是关于r. $D$ -$\emptyset\neq D\subset\mathbb{F}_{p}$的均匀性。此外，我们还给出了$G'(σ(x),p^e)$上诱导内射映射的三类特殊压缩映射。
英文摘要：
Let $p\geq3$ be a prime and $e\geq2$ an integer. Let $σ(x)$ be a primitive polynomial of degree $n$ over $Z/p^eZ$ and $G'(σ(x),p^e)$ the set of primitive linear recurring sequences generated by $σ(x)$. A compressing map $φ$ on $Z/p^eZ$ naturally induces a map $\hatφ$ on $G'(σ(x),p^e)$. For a subset $D$ of the image of $φ$,$\hatφ$ is called to be injective w.r.t. $D$-uniformity if the distribution of elements of $D$ in the compressed sequence implies all information of the original primitive sequence. In this correspondence, for at least $1-2(p-1)/(p^n-1)$ of primitive polynomials of degree $n$, a clear criterion on $φ$ is obtained to decide whether $\hatφ$ is injective w.r.t. $D$-uniformity, and the majority of maps on $Z/p^eZ$ induce injective maps on $G'(σ(x),p^e)$. Furthermore, a sufficient condition on $φ$ is given to ensure injectivity of $\hatφ$ w.r.t. $D$-uniformity. It follows from the sufficient condition that if $σ(x)$ is strongly primitive and the compressing map $φ(x)=f(x_{e-1})$, where $f(x_{e-1})$ is a permutation polynomial over $\mathbb{F}_{p}$, then $\hatφ$ is injective w.r.t. $D$-uniformity for $\emptyset\neq D\subset\mathbb{F}_{p}$. Moreover, we give three specific families of compressing maps which induce injective maps on $G'(σ(x),p^e)$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
