New explicit injective compressing mappings on primitive sequences over $\mathbb {Z}_{p^e}$

Abstract

Linear feedback shift registers over residue rings play a vital role in communication theory and cryptography. Let p be an odd prime and e ≥ 2 an integer. For any integer N ≥ 2, let $\mathbb {Z}_{N}$ denote the residue ring modulo N. Let σ(x) be a primitive polynomial over $\mathbb {Z}_{p^e}$, and G^′(σ(x),p^e) the set of primitive linear recurring sequences generated by σ(x). Given a mapping $\varphi :\mathbb {Z}_{p^e}\rightarrow \mathbb {Z}_{N}$, its induced mapping $\widehat {\varphi }$ transforms a sequence (…,u_i− 1,u_i,u_i+ 1,… ) to (…,φ(u_i− 1),φ(u_i),φ(u_i+ 1),… ). Then φ is called an injective compressing mapping (w.r.t. s-uniformity) if for any two distinct sequences $\underline {u},\underline {v}\in G^{\prime }(\sigma (x),p^e)$, at least one element of $\mathbb {Z}_{N}$ ($s\in \mathbb {Z}_{N}$) is distributed in $\widehat {\varphi }(\underline {u})$ differently from in $\widehat {\varphi }(\underline {v})$. It has been desirable to construct explicit injective compressing mappings (w.r.t. s-uniformity). Let the i-th coordinate a_i of $a\in \mathbb {Z}_{p^e}$ be given by $a={\sum }_{i = 0}^{e-1}a_{i}p^{i}$, $a_{i}\in \mathbb {Z}_{p}$. In this correspondence, it is proved that any permutation polynomial in the (e − 1)-th coordinate is an injective compressing mapping w.r.t. s-uniformity for all (but one) $s\in \mathbb {Z}_{p}$, and the efficiently implemented bitwise right-shift operator is an injective compressing mapping. Furthermore, two families of new injective compressing mappings are given in the form of coordinate polynomials.