论文标题
Chebyshev置换式多项式的图形结构$ \ Mathbb {z} _ {p^k} $
The Graph Structure of Chebyshev Permutation Polynomials over Ring $\mathbb{Z}_{p^k}$
论文作者
论文摘要
了解特定域上非线性图的基础图结构对于评估其实际应用的潜力至关重要。在本文中，我们调查了Chebyshev置换式多项式上的相关\ textIt {函数图}的结构，$ \ mathbb {z} _ {p^k} $，$ p $是三个大于三个的素数，其中每个数字都被认为是一个顶点和现有的映射关系，并且在两个绘图之间是指的范围。根据Chebyshev多项式及其衍生物的一些新属性，我们披露了功能图的基本结构如何相对于参数$ k $的演变。首先，我们提出从任何给定顶点开始的路径长度的完整而明确的形式。然后，我们表明，功能图的强模式表明，随着$ k $的增加，任何给定长度的循环数始终保持恒定。此外，我们严格地证明了功能图的优雅结构的规则，并通过实验验证它们。我们的结果对于研究数字计算机中非线性图的复杂性以及其加密应用的安全性分析可能很有用。
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.
