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

Nikolaos Verykios, Christos Gogos

TL;DR

This work analyzes the ring $\mathbb{Z}_{sp}$ for distinct primes $s$ and $p$ by decomposing it into substructures tied to the finite fields $\mathbb{F}_s$ and $\mathbb{F}_p$ and their multiples $s\mathbb{F}_p$ and $p\mathbb{F}_s$. It constructs a Cartesian product ring $s\mathbb{F}_p \times p\mathbb{F}_s$ and proves an isomorphism $h$ with $\mathbb{Z}_{sp}$, enabling any element to be written uniquely as a combination of $s$ and $p$ components. The paper then analyzes the dynamics of the quadratic map $f(x)=x^2$ on these spaces, showing that the kernel $\mathbb{K}_{sp}$ forms a quad-tree of height $n=\max\{k,l\}$ and that cycles in $\mathbb{Z}_{sp}$ decompose into cycles from $s\mathbb{F}_p$ and $p\mathbb{F}_s$, with cycle lengths given by $\mathrm{lcm}$ of component lengths. It introduces arcs and trees and a levelwise product to generate rooted trees from cyclic elements, and defines the cryptographic set $\mathbb{D}_{sp}$ as the portion of $\mathbb{Z}_{sp}$ that is neither a multiple nor off-by-one, showing its cycles and trees are governed by the corresponding structures in the component fields. The work highlights cryptographic relevance, linking these structural insights to cyclic attacks, RSA security considerations, and potential implications for factorization methods such as the quadratic sieve, suggesting new avenues for cryptanalytic analysis based on the dynamics of $\mathbb{Z}_{sp}$.

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.

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

TL;DR

This work analyzes the ring for distinct primes and by decomposing it into substructures tied to the finite fields and and their multiples and . It constructs a Cartesian product ring and proves an isomorphism with , enabling any element to be written uniquely as a combination of and components. The paper then analyzes the dynamics of the quadratic map on these spaces, showing that the kernel forms a quad-tree of height and that cycles in decompose into cycles from and , with cycle lengths given by of component lengths. It introduces arcs and trees and a levelwise product to generate rooted trees from cyclic elements, and defines the cryptographic set as the portion of that is neither a multiple nor off-by-one, showing its cycles and trees are governed by the corresponding structures in the component fields. The work highlights cryptographic relevance, linking these structural insights to cyclic attacks, RSA security considerations, and potential implications for factorization methods such as the quadratic sieve, suggesting new avenues for cryptanalytic analysis based on the dynamics of .

Abstract

This paper investigates the algebraic and graphical structure of the ring , with a focus on its decomposition into finite fields, kernels, and special subsets. We establish classical isomorphisms between and , as well as and . We introduce the notion of arcs and rooted trees to describe the pre-periodic structure of , and prove that trees rooted at elements not divisible by or can be generated from the tree of unity via multiplication by cyclic arcs. Furthermore, we define and analyze the set , consisting of elements that are neither multiples of or nor "off-by-one" elements, and show that its graph decomposes into cycles and pre-periodic trees. Finally, we demonstrate that every cycle in contains inner cycles that are derived predictably from the cycles of the finite fields and , and we discuss the cryptographic relevance of , highlighting its potential for analyzing cyclic attacks and factorization methods.
Paper Structure (11 sections, 20 theorems, 29 equations, 4 figures, 1 table)

This paper contains 11 sections, 20 theorems, 29 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Related work
  3. The fields sFp, pFs and the groups Fp+-1, Fs+-1
  4. A ring of a Cartesian product
  5. The tree structures of Zsp
  6. Cycles in Zsp
  7. Arcs and Trees
  8. Isomorphism of sKp x pFs
  9. The set of cryptography Dsp
  10. Cycles, Trees, and the Cryptographic Set Dsp
  11. Conclusion

Key Result

Theorem 2.1

Let $p$ and $s$ be distinct prime numbers. Consider the ring $\mathbb{Z}_{sp}$ and define the subset Then $p\mathbb{F}_s$ admits a field structure isomorphic to the finite field $\mathbb{F}_s$, via the map where $\beta \in \mathbb{Z}$ satisfies $\beta p \equiv 1 \mod s$.

Figures (4)

  • Figure 1: Subfigure \ref{['Fig_TreeKsp_NumExample_1']} depicts the binary tree corresponding to ${41}\mathbb{K}_{29}$, while subfigure \ref{['Fig_TreeKsp_NumExample_2']} depicts the binary tree corresponding to ${29}\mathbb{K}_{41}$. In \ref{['Fig_TreeKsp_NumExample_3']}, the quad tree corresponding to $\mathbb{K}_{29\times 41}$, that results from the binary trees in \ref{['Fig_TreeKsp_NumExample_1']} and \ref{['Fig_TreeKsp_NumExample_2']}, is depicted. Note that these figures constitute the graphical representation of Table \ref{['Fig_levelsKsp']}.
  • Figure 2: Subfigure \ref{['fig:cycles3_in_1_a']} shows the cycles $C_4$ of $23\mathbb{F}_{11}$ and $C_{10}$ of $11\mathbb{F}_{23}$, which based on Theorem \ref{['Theorem_Cycles_Zsp_Correspondence']} create the external cycle $C_{20}$ of the ring $\mathbb{Z}_{11 \times 23}$. In \ref{['fig:cycles3_in_1_b']}, the 5 inner $C_4$ cycles of $C_{20}$ are depicted. In \ref{['fig:cycles3_in_1_c']}, the 2 inner $C_{10}$ cycles are shown.
  • Figure 3: In \ref{['Fig_TreeKsp_1']}, the quad tree representing the kernel is $\mathbb{F}_{29\times89}$. In \ref{['Fig_TreeKsp_2']} there is a typical cycle of $29\mathbb{F}_{89}$ which is isomorphic to $\mathbb{F}_{89}$. The trees rooted at its cyclic nodes are the binary trees isomorphic to $\mathbb{K}_{89}$. In \ref{['Fig_TreeKsp_3']}, the combined set $h(29\mathbb{F}_{89}^e)=(29\mathbb{F}_{89},89\mathbb{K}_{29})$, consisting of the cycle $29\mathbb{F}_{89}$, where each cyclic element is rooted by a tree isomorphic to the kernel tree shown in \ref{['Fig_TreeKsp_1']}. For each cyclic element $w$ of \ref{['Fig_TreeKsp_3']} it holds that $w=29t+1$.
  • Figure 4: The three boxes at the top row represent the three subsets of $s\mathbb{F}_p$, while the three boxes at the leftmost column represent the three subsets of $p\mathbb{F}_s$. Each remaining box corresponds to a subset of $\mathbb{Z}_{sp}$ formed by combining the sets of its row and column.

Theorems & Definitions (29)

  • Theorem 2.1
  • Theorem 2.2
  • Lemma 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Definition 3.1
  • Theorem 3.2
  • Theorem 3.3
  • Corollary 3.4
  • ...and 19 more