Injectivity w.r.t. Distribution of Elements in the Compressed Sequences Derived from Primitive Sequences over $Z/p^eZ$

Abstract

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 $G'(σ(x),p^e)$.

42 pages, updated version