The Graph Structure of Chebyshev Permutation Polynomials over Ring $\mathbb{Z}_{p^k}$

TL;DR

This paper analyzes the structure of the functional graph of Chebyshev permutation polynomials over rings of the form $ ext{Z}_{p^k}$, revealing how graph properties evolve with parameter $k$ and implications for cryptography.

Contribution

It provides explicit formulas for path lengths and cycle counts, and uncovers stable patterns in the graph structure as $k$ varies, advancing understanding of nonlinear maps over rings.

Findings

Path lengths from any vertex are explicitly characterized.

Number of cycles of any given length remains constant as $k$ increases.

Structural rules of the functional graph are rigorously proven and experimentally verified.

Abstract

Understanding the underlying graph structure of a nonlinear map over a particular domain is essential in evaluating its potential for real applications. In this paper, we investigate the structure of the associated \textit{functional graph} of Chebyshev permutation polynomials over a ring $\mathbb{Z}_{p^k}$, with $p$ being a prime number greater than three, where every number in the ring is considered as a vertex and the existing mapping relation between two vertices is regarded as a directed edge. Based on some new properties of Chebyshev polynomials and their derivatives, we disclose how the basic structure of the functional graph evolves with respect to parameter $k$. First, we present a complete and explicit form of the length of a path starting from any given vertex. Then, we show that the strong patterns of the functional graph that the number of cycles of any given length always remains constant as $k$ increases. Moreover, we rigorously prove the rules on the elegant structure of the functional graph and verify them experimentally. Our results could be useful for studying the emergence of the complexity of a nonlinear map in digital computers and security analysis of its cryptographic applications.