Graph Structure of Chebyshev Permutation Polynomials over Binary and Ternary Adic Rings

TL;DR

This paper analyzes the graph structure of Chebyshev permutation polynomials over binary and ternary rings, revealing regularities and cycle patterns crucial for cryptography and pseudorandom generation.

Contribution

It provides an explicit characterization of path lengths and cycle structures of these polynomials over composite rings, extending prior prime-power ring analyses.

Findings

Graph exhibits strong regularities despite complexities

Constant number of cycles of given length

Predictable branching patterns as parameters increase

Abstract

Understanding the functional graph of a nonlinear map over a finite domain is crucial for analyzing its dynamical complexity and potential applications in cryptography and pseudorandom generation. In this paper, we investigate the graph structure of Chebyshev permutation polynomials over the ring $\mathbb{Z}_{2^{k_1}3^{k_2}}$, where $k_1$ and $k_2$ are positive integers and $0\in\{k_1, k_2\}$. Each element of the ring is regarded as a vertex, and the mapping relation defined by the polynomial corresponds to a directed edge. Building on new properties of Chebyshev polynomials modulo powers of $2$ and $3$, we provide an explicit characterization of path lengths and cycle structures in the functional graph. We show that, despite the complexities introduced by the binary and ternary components, the graph exhibits strong regularities, including a constant number of cycles of a given length and predictable branching patterns as $k_1$ and $k_2$ increase. Our results extend previous studies over prime-power rings, offering insights into the emergence of complexity in digital nonlinear maps and supporting the security analysis of their cryptographic applications.