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

Computer Science > Cryptography and Security [Submitted on 20 May 2026] Graph Structure of Chebyshev Permutation Polynomials over Binary and Ternary Adic Rings Xiaoxiong Lu, Yuling Dai, Chengqing Li 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. Subjects: Cryptography and Security (cs.CR) MSC classes: 94A55, 11T06, 37p25 Cite as: arXiv:2605.21819 [cs.CR]   (or arXiv:2605.21819v1 [cs.CR] for this version)   https://doi.org/10.48550/arXiv.2605.21819 Focus to learn more Submission history From: Lu Xiaoxiong [view email] [v1] Wed, 20 May 2026 23:39:55 UTC (189 KB) Access Paper: HTML (experimental) view license Current browse context: cs.CR < prev   |   next > new | recent | 2026-05 Change to browse by: cs References & Citations NASA ADS Google Scholar Semantic Scholar Export BibTeX Citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer (What is the Explorer?) Connected Papers Toggle Connected Papers (What is Connected Papers?) Litmaps Toggle Litmaps (What is Litmaps?) scite.ai Toggle scite Smart Citations (What are Smart Citations?) Code, Data, Media Demos Related Papers About arXivLabs Which authors of this paper are endorsers? | Disable MathJax (What is MathJax?)