## 3 Confidential Transmission Methods for High-Speed Power Line Carrier Communications
### Confidential Transmission Method Based on Generalized Two-Dimensional Polynomial Chaotic Mapping  
3.1 Analysis of Transmission Fluctuation Deviations in High-Speed Power Line Carrier Communication Networks  
The autoregressive moving average (ARMA) model analyzes transmission fluctuation deviations within the high-speed power line carrier communication network established in Section 3.1. When abnormal fluctuations occur during data transmission, the ARMA model's transmission characteristics undergo modifications. Consequently, deviation analysis of the ARMA model's transmission time series is required to identify network abnormalities. In this network, the residual quantifies the transmission deviation and serves as the deviation measure. The residual sequence of network transmission characteristics is analyzed, which is based on the deviation between the current high-speed power line carrier communication transmission and normal data. At the same time, the network transmission residual is updated periodically. Since the residual series exhibits non-stationary behavior, differencing (first-order or higher) is applied to achieve stationarity before modeling. Following smoothing, the ARMA model simulates and captures the series fluctuations, enabling predictive analysis. The implementation process comprises the following steps:  
First, the autocorrelation coefficient (ACF) and partial autocorrelation coefficient (PACF) are computed. Second, by analyzing the autocorrelation and partial autocorrelation properties of the samples, an appropriate series is selected to fit the ARMA model, and its unknown parameters are estimated. Subsequently, the model validity is verified; if validation fails, the model is re-selected for fitting. Upon successful validation, the model is optimized, and the best-performing model is selected from the validated candidates through comprehensive evaluation. Finally, the optimal model is employed to forecast future time series. After completing the fitting process, let the residual sequence of the high-speed power line carrier communication network transmission at time \(t\) be \(\varepsilon_{1},\varepsilon_{2},\ldots,\varepsilon_{t}\). The anomalous deviation of the actual network transmission time series is calculated as  
\[d_{t}=e^{\frac{|t-\overline{t}_{t-1}|}{d_{t-1}}} \tag{2}\]  
where \(d_{t}\) indicates the degree of deviation at the moment \(t\); \(\overline{\varepsilon}_{t-1}\) indicates the mean value of the residuals at the moment \(t-1\); and \(\delta_{t-1}\) denotes standard deviation. Among them, the deviation degree quantifies the abnormality between the actual network transmission time series and the expected or fitted values. A larger deviation indicates greater discrepancy between the transmission time series at time t and the expected/fitted value. The residual mean, calculated as the average of all residuals (differences between actual and fitted values) over a specified period, reflects the overall deviation level between actual transmission performance and expected/fitted values. The standard deviation measures data dispersion, where a higher value indicates greater fluctuations in the residual sequence, corresponding to increased divergence between actual and expected transmission time series. Conversely, a smaller standard deviation signifies reduced fluctuations and minimal differences. 3.2 Encryption and Decryption for High-Speed Power Line Carrier Communication Based on Generalized Two-Dimensional Polynomial Chaotic Mapping  
Using a generalized two-dimensional polynomial chaotic mapping, the deviation \(d_{t}\) of the acquired high-speed power line carrier communication network is computed in conjunction with Section 3.3.1. The key challenge to high-speed power line carrier communication [17] is to resist the interference of power line noise and signal attenuation while improving the confidentiality and stability of communication. Two-dimensional (2D) logistic mapping models:  
\[\left\{