The Graph Structure of a Class of Permutation Maps over Ring $mathbb{Z}_{p^k}$

2025-06-25

AI Summary

A systematic analytical framework for characterizing the cycle structure of permutation mappings over $mathbb{Z}_{p^k}$ remains lacking. Method: This paper establishes a unified algebraic dynamical framework to rigorously describe cycle-length distributions and local structural evolution, integrating generating functions, minimal polynomials, and Hensel lifting theory to derive analytic methods for cycle graph structures under increasing modulus $p^k$. Contribution/Results: It provides the first complete characterization of the exact period patterns and fractal-like nested cycle structures of the Cat map over $mathbb{Z}_{p^k}$. The framework yields rigorous algebraic foundations and computationally tractable tools for statistical randomness assessment of pseudorandom sequences, verification of controllable periodicity, and design of cryptographically secure systems.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Applications of cryptography
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Understanding the periodic and structural properties of permutation maps over residue rings such as $mathbb{Z}_{p^k}$ is a foundational challenge in algebraic dynamics and pseudorandom sequence analysis. Despite notable progress in characterizing global periods, a critical bottleneck remains: the lack of explicit tools to analyze local cycle structures and their evolution with increasing arithmetic precision. In this work, we propose a unified analytical framework to systematically derive the distribution of cycle lengths for a class of permutation maps over $mathbb{Z}_{p^k}$. The approach combines techniques from generating functions, minimal polynomials, and lifting theory to track how the cycle structure adapts as the modulus $p^k$ changes. To validate the generality and effectiveness of our method, we apply it to the well-known Cat map as a canonical example, revealing the exact patterns governing its cycle formation and transition. This analysis not only provides rigorous explanations for experimentally observed regularities in fixed-point implementations of such maps but also lays a theoretical foundation for evaluating the randomness and dynamical behavior of pseudorandom number sequences generated by other nonlinear maps. The results have broad implications for secure system design, computational number theory, and symbolic dynamics.

Problem

• Analyze local cycle structures of permutation maps over Z_p^k
• Develop tools to track cycle evolution with increasing precision
• Understand randomness and dynamics of pseudorandom sequences

Innovation

• Generating functions analyze cycle length distribution
• Minimal polynomials track cycle structure evolution
• Lifting theory adapts to modulus changes