Partitioning $\mathbb{Z}_{sp}$ in finite fields and groups of trees and cycles

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.