Further Result of Compressing Maps on Primitive Sequences Modulo Odd Prime Powers

摘要

Let ${BBZ}/(p^{e})$ be the integer residue ring with odd prime $p$ and integer $egeq 2$. For a sequence $underline{a}$ over ${BBZ}/(p^{e})$, there is an unique $p$-adic expansion $underline{a}=underline{a}_{0}+ underline{a}_{1}cdot p+cdots +underline{a}_{e-1}cdot p^{e-1}$, where each $underline{a}_{i}$ is a sequence over ${0,1,ldots,p-1}$, and can be regarded as a sequence over the prime field GF $,(p)$ naturally. Let $f(x)$ be a strongly primitive polynomial over ${BBZ}/(p^{e})$ , and $G^{prime }(f(x),p^{e})$ the set of all primitive sequences generated by $ f(x)$ over ${BBZ}/(p^{e})$. Suppose that $Gamma ={g(x_{e-1})+eta (x_{0},ldots,x_{e-2}),vert, g(x)in$GF $,(p)[x],$ $2leq deg g(x)leq p-1, eta i-n $ GF $,(p)[x_{0},ldots,x_{e-2}]}$. It is shown that any function in $Gamma$ is an injective map from $G^{prime }(f(x),p^{e})$ to GF$,(p)^{infty }$, and the derived sequences of different functions are also different. That is, $varphi (underline{a}_{0},ldots, underline{a}_{e-1})=psi (underline{b}_{0},ldots,underline{b}_{e-1})$ if and only if $underline{a}=underline{b}$ and $varphi =psi$ for $varphi,psi in Gamma$ and $underline{a},underline{b}in G^{prime}(f(x),p^{e})$ . These injective functions in $Gamma$ can be considered as good candidates for the keys of a stream cipher.