Let${\BBZ}/(p^{e})$be the integer residue ring with odd prime$p$and integer$e\geq 2$. 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],$$2\leq \deg g(x)\leq p-1, \eta \in $GF$\,(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.

Let Z/(p/sup e/) be the integer residue ring with odd prime p/spl ges/5 and integer e/spl ges/2. For a sequence a_ over Z/(p/sup e/), there is a unique p-adic expansion a_=a_/sub 0/+a_/spl middot/p+...+a_/sub e-1//spl middot/p/sup e-1/, where each a_/sub i/ is a sequence over {0,1,...,p-1}, and can be regarded as a sequence over the finite field GF(p) naturally. Let f(x) be a primitive polynomial over Z/(p/sup e/), and G'(f(x),p/sup e/) the set of all primitive sequences generated by f(x) over Z/(p/sup e/). Set /spl phi//sub e-1/ (x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,1/(x/sub 0/, x/sub 1/,...,x/sub e-2/) /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) = x/sub e-1//sup k/ + /spl eta//sub e-2,2/(x/sub 0/,x/sub 1/,...,x/sub e-2/) where /spl eta//sub e-2,1/ and /spl eta//sub e-2,2/ are arbitrary functions of e-1 variables over GF(p) and 2/spl les/k/spl les/p-1. Then the compression mapping /spl phi//sub e-1/:{G'(f(x),p/sup e/) /spl rarr/ GF(p)/sup /spl infin// a_ /spl rarr/ /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) is injective, that is, a_ = b_ if and only if /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl phi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/). Furthermore, if f(x) is a strongly primitive polynomial over Z/(p/sup e/), then /spl phi//sub e-1/(a_/sub 0/,...,a_/sub e-1/) = /spl psi//sub e-1/(b_/sub 0/,...,b_/sub e-1/) if and only if a_ = b_ and /spl phi//sub e-1/(x/sub 0/,...,x/sub e-1/) = /spl psi//sub e-1/(x/sub 0/,...,x/sub e-1/) for a_,b_ /spl isin/ G'(f(x),p/sup e/).