## 3 Confidential Transmission Methods for High-Speed Power Line Carrier Communications
### Confidential Transmission Method Based on Generalized Two-Dimensional Polynomial Chaotic Mapping
\begin{array}{l}x_{\eta+1}=4\alpha_{1}\times x_{\eta}\big{(}1-x_{ \eta}\big{)}+\mu\times y_{\eta}\\ y_{\eta+1}=4\alpha_{2}\times y_{\eta}\big{(}1-y_{\eta}\big{)}+\mu\times x_{ \eta}\end{array} \right. \tag{3}\]  
where \(x_{\eta}\) and \(y_{\eta}\) represent the chaotic mapping state variables at time \(\eta\); \(x_{\eta+1}\) and \(y_{\eta+1}\) denote the state variables at time \(\eta+1\); \(\alpha_{1}\), \(\alpha_{2}\), and \(\mu\) denote the control parameters of the two-dimensional logistic mapping. In standard two-dimensional logistic mapping, these parameters adjust the system's nonlinear characteristics and dynamic behavior. For generalized two-dimensional polynomial chaotic mapping, they serve as key parameters governing the system's complexity and unpredictability. The parameter selection criteria ensure system stability while enabling the generation of diverse chaotic sequences through appropriate choices of \(\alpha_{1}\), \(\alpha_{2}\), and \(\mu\), thereby significantly expanding the key space. To enhance chaotic mapping dynamics, we propose a generalized two-dimensional polynomial chaotic mapping based on Formula (3), incorporating different control parameters and the maximum polynomial degree. The mathematical expression is  
\[\left\{ \begin{array}{l}x_{\eta+1}=\left(\alpha_{1}x_{\eta}+\mu y_{\eta}^{\rho} \right)\text{mod}\theta\\ y_{\eta+1}=\left(\alpha_{2}x_{\eta}+r_{\eta}\right)\text{mod}\theta\end{array} \right. \tag{4}\]  
where \(r\) denotes the control parameters of the two-dimensional polynomial chaotic mapping; \(\rho\) denotes the highest polynomial degree; and \(\theta\) denotes the modal extraction factor. Specifically, \(r\) determines the mapping's structural complexity, analogous to the highest polynomial degree, which governs the chaotic mapping's complexity and topological characteristics. The parameter selection follows these criteria: increasing \(r\) enhances the mapping complexity, generating more intricate and unpredictable chaotic sequences, thereby strengthening encryption robustness. Similarly, \(\rho\) functions as a polynomial coefficient or supplementary adjustment parameter to fine-tune the chaotic mapping's properties. The selection criterion requires choosing \(\rho\) values that maximize sensitivity to initial conditions and parameter variations, thus improving the encryption system's resistance to minor perturbations. Through the dynamic adjustment of \(r\) and \(\rho\), the encryption system adapts to evolving communication environments and requirements, significantly enhancing its robustness and reliability. The generalized two-dimensional polynomial chaotic mapping model defined in Formula (4) generates a chaotic sequence for information transmission in the high-speed power line carrier communication network. The encryption and decryption keys are produced by integrating this chaotic sequence with the communication network's deviation degree from Section 3.3.1, implemented through the following procedure:  
Step 1: Select an appropriate generalized two-dimensional polynomial chaotic system equation as the chaotic sequence generation algorithm for high-speed power line carrier communication networks [18]. Step 2: Initialize the iteration with values \((x_{0},y_{0})\) to generate the initial chaotic sequence for the high-speed power line carrier communication network. Step 3: Perform iterative computations from the current trajectory position to produce the chaotic sequence for transmitted information [19]. Step 4: Generate encryption and decryption keys \(K_{x}=x_{1}d_{1}x_{2}d_{2}\cdots x_{\tau}d_{\tau}\) and \(K_{y}=y_{1}d_{1}y_{2}d_{2}\cdots y_{\tau}d_{\tau}\) by applying the communication network deviation \(d\) (from Section 3.3.1) through \(\tau\) mapping and iteration cycles.