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

TL;DR

This paper investigates conditions under which compressing maps on primitive sequences over $Z/p^eZ$ preserve all information, providing criteria and specific families of maps that ensure injectivity related to distribution uniformity.

Contribution

It offers a criterion for injectivity of compressing maps on primitive sequences, including a sufficient condition involving permutation polynomials and three specific families of such maps.

Findings

Most primitive polynomials induce injective maps.

A criterion for injectivity w.r.t. D-uniformity is established.

Certain families of maps guarantee injectivity.

Abstract

Let $p\geq3$ be a prime and $e\geq2$ an integer. Let $\sigma(x)$ be a primitive polynomial of degree $n$ over $Z/p^eZ$ and $G'(\sigma(x),p^e)$ the set of primitive linear recurring sequences generated by $\sigma(x)$. A compressing map $\varphi$ on $Z/p^eZ$ naturally induces a map $\hat{\varphi}$ on $G'(\sigma(x),p^e)$. For a subset $D$ of the image of $\varphi$,$\hat{\varphi}$ 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 $\varphi$ is obtained to decide whether $\hat{\varphi}$ is injective w.r.t. $D$-uniformity, and the majority of maps on $Z/p^eZ$ induce injective maps on $G'(\sigma(x),p^e)$. Furthermore, a sufficient condition on $\varphi$ is given to ensure injectivity of $\hat{\varphi}$ w.r.t. $D$-uniformity. It follows from the sufficient condition that if $\sigma(x)$ is strongly primitive and the compressing map $\varphi(x)=f(x_{e-1})$, where $f(x_{e-1})$ is a permutation polynomial over $\mathbb{F}_{p}$, then $\hat{\varphi}$ 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 $G'(\sigma(x),p^e)$.