2025/10/16 by Nikolaos Verykios, Verykios, Nikolaos, Christos Gogos +1
Computer Science · #Coding theory and cryptography #Cryptographic Implementations and Security #Cryptography and Residue Arithmetic #Cryptography and Security (cs.CR) #FOS: Computer and information sciences #FOS: Mathematics #Group Theory (math.GR) #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2510.15108
openalex publication_date 2025/10/16 · openalex created_date 2025/10/21 · openalex updated_date 2026/07/28
This paper investigates the algebraic and graphical structure of the ring ℤsp, with a focus on its decomposition into finite fields, kernels, and special subsets. We establish classical isomorphisms between \mathbbFs and p\mathbbFs, as well as p\mathbbFs⋆ and p\mathbbFs+1,⋆. We introduce the notion of arcs and rooted trees to describe the pre-periodic structure of ℤsp, 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 \mathbbDsp, 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 ℤsp contains inner cycles that are derived predictably from the cycles of the finite fields p\mathbbFs and s\mathbbFp, and we discuss the cryptographic relevance of \mathbbDsp, highlighting its potential for analyzing cyclic attacks and factorization methods.