This paper investigates the algebraic and graphical structure of the ring $\mathbb{Z}_{sp}$, with a focus on its decomposition into finite fields, kernels, and special subsets. We establish classical isomorphisms between $\mathbb{F}_s$ and $p\mathbb{F}_s$, as well as $p\mathbb{F}_s^{\star}$ and $p\mathbb{F}_s^{+1,\star}$. We introduce the notion of arcs and rooted trees to describe the pre-periodic structure of $\mathbb{Z}_{sp}$, and prove that trees rooted at elements not divisible by $s$ or $p$ can be generated from the tree of unity via multiplication by cyclic arcs. Furthermore, we define and analyze the set $\mathbb{D}_{sp}$, consisting of elements that are neither multiples of $s$ or $p$ nor "off-by-one" elements, and show that its graph decomposes into cycles and pre-periodic trees. Finally, we demonstrate that every cycle in $\mathbb{Z}_{sp}$ contains inner cycles that are derived predictably from the cycles of the finite fields $p\mathbb{F}_s$ and $s\mathbb{F}_p$, and we discuss the cryptographic relevance of $\mathbb{D}_{sp}$, highlighting its potential for analyzing cyclic attacks and factorization methods.

翻译：本文研究环 $\mathbb{Z}_{sp}$ 的代数与图结构，重点探讨其分解为有限域、核及特殊子集的方式。我们建立了 $\mathbb{F}_s$ 与 $p\mathbb{F}_s$ 之间、以及 $p\mathbb{F}_s^{\star}$ 与 $p\mathbb{F}_s^{+1,\star}$ 之间的经典同构关系。引入弧与有根树的概念来描述 $\mathbb{Z}_{sp}$ 的预周期结构，并证明以不可被 $s$ 或 $p$ 整除的元素为根的树，可通过乘以循环弧从单位元树生成。此外，我们定义并分析了集合 $\mathbb{D}_{sp}$，该集合由既非 $s$ 或 $p$ 的倍数也非“偏差一”元素的元素组成，并证明其图可分解为循环与预周期树。最后，我们证明 $\mathbb{Z}_{sp}$ 中的每个循环均包含可从有限域 $p\mathbb{F}_s$ 与 $s\mathbb{F}_p$ 的循环可预测导出的内部循环，并讨论了 $\mathbb{D}_{sp}$ 在密码学中的相关性，强调其在分析循环攻击与因子分解方法方面的潜力。