Injectivity w.r.t. Distribution of Elements in the Compressed Sequences Derived from Primitive Sequences over $Z\mathrm{/}{p}^{e}Z$

Let $p\ge 3$ be a prime and $e\ge 2$ an integer. Let $\sigma (x)$ be a primitive polynomial of degree $n$ over $Z\mathrm{/}{p}^{e}Z$ and $G^{\mathrm{\prime }}(\sigma (x),p^e)$ the set of primitive linear recurring sequences generated by $\sigma (x)$. A compressing map $\phi$ on $Z\mathrm{/}{p}^{e}Z$ naturally induces a map $\widehat{\phi }$ on $G^{\mathrm{\prime }}(\sigma (x),p^e)$. For a subset $D$ of the image of $\phi$,$\widehat{\phi }$ 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 $1-2(p-1)\mathrm{/}(p^n-1)$ of primitive polynomials of degree $n$, a clear criterion on $\phi$ is obtained to decide whether $\widehat{\phi }$ is injective w.r.t. $D$-uniformity, and the majority of maps on $Z\mathrm{/}{p}^{e}Z$ induce injective maps on $G^{\mathrm{\prime }}(\sigma (x),p^e)$. Furthermore, a sufficient condition on $\phi$ is given to ensure injectivity of $\widehat{\phi }$ w.r.t. $D$-uniformity. It follows from the sufficient condition that if $\sigma (x)$ is strongly primitive and the compressing map $\phi (x)=f(x_{e-1})$, where $f(x_{e-1})$ is a permutation polynomial over ${\mathbb{F}}_p$, then $\widehat{\phi }$ 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 $G^{\mathrm{\prime }}(\sigma (x),p^e)$.