{"ID":2854871,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2510.15108","arxiv_id":"2510.15108","title":"Partitioning $\\mathbb{Z}_{sp}$ in finite fields and groups of trees and cycles","abstract":"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.","short_abstract":"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...","url_abs":"https://arxiv.org/abs/2510.15108","url_pdf":"https://arxiv.org/pdf/2510.15108v1","authors":"[\"Nikolaos Verykios\",\"Christos Gogos\"]","published":"2025-10-16T19:59:36Z","proceeding":"cs.CR","tasks":"[\"cs.CR\",\"math.GR\",\"math.NT\"]","methods":"[]","has_code":false}