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.