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

2026-05-20

AI Summary

This study investigates the functional graph structure of Chebyshev permutation polynomials over the composite ring ℤ_{2^{k₁}3^{k₂}}, where elements serve as vertices and the polynomial mapping defines directed edges, thereby characterizing the topology of cycles and tails. By integrating tools from algebraic number theory and graph theory, and leveraging newly established properties of Chebyshev polynomials modulo powers of 2 and 3, the work reveals a striking regularity in the functional graphs when binary and ternary components coexist: the number of cycles of any fixed length remains constant, while branching patterns grow predictably with k₁ and k₂. Extending prior results on prime-power rings, this research demonstrates that despite their seemingly complex behavior, such nonlinear mappings exhibit a highly structured nature, offering a theoretical foundation for analyzing the security of cryptographic systems based on these mappings.

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.

Problem

• Chebyshev polynomials
• functional graph
• permutation polynomials
• adic rings
• graph structure

Innovation

• Chebyshev permutation polynomials
• functional graph
• binary-ternary adic rings
• cycle structure
• cryptographic dynamics