Further Result of Compressing Maps on Primitive Sequences Modulo Odd Prime Powers
Xuan-Yong Zhu, Wen‐Feng Qi · IEEE Transactions on Information Theory · 2007
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.