A Confidential Transmission Method for High-Speed Power Line Carrier Communications Based on Generalized Two-Dimensional Polynomial Chaotic Mapping

Abstract

The deep integration of smart grid and Internet of Things technologies has made high-speed power line carrier communication a key communication technology in energy management, industrial monitoring, and smart home applications, owing to its advantages of requiring no additional wiring and offering wide coverage. However, the inherent characteristics of power line channels, such as strong noise, multipath fading, and time-varying properties, pose challenges to traditional encryption algorithms, including low key distribution efficiency and weak anti-interference capabilities. These issues become particularly pronounced in high-speed transmission scenarios, where the conflict between data security and communication reliability is more acute. To address this problem, a secure transmission method for high-speed power line carrier communication based on generalized two-dimensional polynomial chaotic mapping is proposed. A high-speed power line carrier communication network is established using a power line carrier routing algorithm based on the minimal connected dominating set. The autoregressive moving average model is employed to determine the degree of transmission fluctuation deviation in the high-speed power line carrier communication network. Leveraging the complex dynamic behavior and anti-decoding capability of generalized two-dimensional polynomial chaotic mapping, combined with the deviation, the communication key is generated. This process yields encrypted high-speed power line carrier communication ciphertext that can resist power line noise interference and signal attenuation, thereby enhancing communication confidentiality and stability. By applying reference modulation differential chaotic shift keying and integrating the ciphertext of high-speed power line carrier communication, a secure transmission scheme is designed to achieve secure transmission in high-speed power line carrier communication. The experimental results demonstrate that this method can effectively establish a high-speed power line carrier communication network and encrypt information. The maximum error rate obtained by this method is 0.051, and the minimum error rate is 0.010, confirming its ability to ensure secure transmission in high-speed power line carrier communication while improving communication confidentiality.

1. Introduction

With the ongoing advancement of smart grid infrastructure, high-speed power line carrier communication has emerged as a predominant information transmission technology in power systems [1]. This approach utilizes existing power lines as transmission media, achieving cost efficiency and power–communication integration while supporting power system automation and intelligent operations [2,3]. However, this technology faces significant challenges, particularly regarding environmental complexity and data security [4]. The complexity stems from two primary factors: the inherent physical characteristics of power lines that degrade communication signals, and electromagnetic interference generated by diverse electrical equipment within the power system, both of which substantially increase communication difficulties [5]. Furthermore, the exposed nature of power lines renders them susceptible to external interference and malicious attacks, posing substantial risks to data security.

The generalized two-dimensional polynomial chaotic mapping generates chaotic sequences with complex behaviors. It uses configurable control parameters and adjustable polynomial degrees. These sequences serve as highly secure keys. They enhance communication confidentiality. The mapping adapts to diverse environments and requirements. Addressing power line carrier communication vulnerability to noise and signal fading, this mapping’s inherent complexity provides inherent resistance to such disturbances, thereby improving transmission reliability. This paper develops a secure transmission method incorporating (1) a minimal connected dominating set-based routing algorithm that intelligently selects optimal key nodes for efficient and stable paths, and (2) an autoregressive moving average model that precisely analyzes transmission fluctuations by quantifying deviation degrees to capture dynamic channel characteristics. For encryption, the method innovatively combines the mapping’s dynamic properties with real-time fluctuation data to generate adaptive keys, ensuring robust security against power line impairments. The encrypted ciphertext demonstrates strong resistance to channel noise and attenuation, significantly enhancing communication confidentiality and stability. The complete solution implements reference modulation differential chaotic shift keying to guarantee information integrity during transmission. The experimental results validate the method’s effectiveness in establishing secure high-speed power line carrier networks, offering crucial technical support for smart grid development.

2. Literature Review

To address the secure transmission challenges in high-speed power line carrier communication, numerous scholars and research institutions have conducted extensive investigations. Bouzinis, P. S. et al. first quantize federated learning model parameters to generate bit streams, divide them into substreams, and perform channel coding to generate protected substreams for transmission [6]. This method improves the accuracy and speed of parameter transmission under noisy channels but increases the complexity and computational overhead. Vishal Sharad, H. et al. use an optimized routing algorithm to find optimal data transmission paths in wireless sensor networks (WSNs), considering factors like node energy consumption, transmission distance, and link quality. A multipath transmission strategy is used, where multiple paths transmit data simultaneously [7]. In dynamic WSN environments, this method requires frequent updates to adapt to new network conditions, reducing communication stability. Zenalden, F. et al. utilize social attributes between devices (e.g., user behavioral patterns and social relationships) to optimize D2D communication, dynamically adjusting the strategy to adapt to changing network environments and user requirements [8]. This method is limited by user privacy and security concerns. Alizadeh, M. et al. construct a network control system model using local state information on subsystems, establishing distributed trigger events through stability analysis. In practical applications, it can intelligently select the best communication path and transmission strategy according to network conditions and data transmission requirements to realize efficient and reliable data exchange [9]. This method can be affected by incomplete or delayed information. Nizampatnam, S. et al. dynamically adjust the contention window size based on data traffic and transmission demand, balancing network load to avoid congestion and delays [10]. However, this can lead to network transmission instability, particularly during drastic changes in network load. Ercan, S.U. proposes a revolution in data transmission for distribution networks based on power line communication. Power line communication is a new technology that uses existing power infrastructure for data transmission, supports communication through power lines, and allows devices to exchange information and access the Internet through the power grid. This approach offers many advantages, including wide availability, cost-effectiveness, and easy deployment without additional cabling. PLC has applications in various fields, including smart grid management, home automation, industrial control systems, and Internet of Things connectivity. This paper presents two-way communication from the power supply side to the load side using the BPSK and QPSK modulation techniques. Monte Carlo simulation is used to predict the theoretical channel of a transformer and to compare the efficiency of the proposed method. Randomly selected data from 10 k to 50 k are used to compare performance curves. In addition, test sound data are sent from the source side to the load side, and it is observed that using QPSK modulation techniques results in an almost zero bit error rate [11]. However, it is difficult to dynamically adjust the fixed modulation mode according to channel conditions, which leads to a significantly higher bit error rate in the high-frequency band than in the low-frequency band. Ayele, E.D. et al. propose a method to enhance the network security of distributed microgrids based on a summary of communication protocols and standards. The effective operation of distributed energy depends heavily on the communication system adopted in the microgrid. This paper discusses the basic communication requirements, structures, and protocols needed to establish secure connections in microgrids. The integration of various security methods is also evaluated, and a case study is shown to illustrate the implementation of a distributed network security communication system in a microgrid environment. Finally, the research emphasizes ongoing work and proposes potential future research directions in the field of microgrid communication [12]. However, narrowband interference in the power line channel makes it impossible to perceive the interference position and adjust the carrier frequency in real time, leading to low spectrum utilization.

In smart grid and wireless sensor network (WSN) environments, chaotic encryption methods have gained significant attention due to their high security and strong anti-interference capabilities. The proposed secure transmission method for high-speed power line carrier communication, based on generalized two-dimensional polynomial chaotic mapping, demonstrates distinct advantages and differences when compared with existing chaotic encryption approaches employed in smart grids and WSNs.

(1)

Compared with the chaotic encryption method in smart grid

Complexity Enhancement: Existing chaotic encryption methods in smart grids typically employ one-dimensional or basic two-dimensional chaotic systems (e.g., logistic mapping or two-dimensional logistic mapping). While demonstrating chaotic properties, these systems exhibit relatively limited dynamic behavior complexity and key space. The proposed method substantially enhances system complexity and unpredictability through generalized two-dimensional polynomial chaotic mapping with configurable control parameters and adjustable maximum polynomial degrees, thereby strengthening encryption key security and crack resistance.

Enhanced Anti-interference Capability: Conventional chaotic encryption methods often prove ineffective against power line channel impairments, including strong noise, multipath fading, and time-varying characteristics, frequently resulting in elevated bit error rates. This work addresses these limitations by integrating an autoregressive moving average model to analyze transmission fluctuations, enabling dynamic key adaptation that effectively mitigates channel noise and signal attenuation while enhancing communication stability and reliability.

Network Construction and Optimization: Traditional smart grid communication networks frequently suffer from suboptimal connectivity and transmission efficiency considerations, leading to increased communication latency. The proposed solution employs a power line carrier routing algorithm based on minimal connected dominating sets to intelligently select critical nodes for optimal path formation, significantly improving network transmission efficiency and stability.

(2)

Compared with chaotic encryption methods in WSN environment

Adaptability and Flexibility: Existing chaotic encryption methods in WSN environments primarily emphasize low power consumption and simplicity, often proving inadequate for complex, dynamic network environments requiring adaptive encryption strategies. The proposed method addresses both encryption security and dynamic adaptation, automatically adjusting encryption keys and routing algorithms according to power line channel variations, thereby enhancing communication flexibility and environmental adaptability.

Key Generation and Management: Traditional WSN chaotic encryption methods frequently employ static or predefined key generation mechanisms, struggling to accommodate network topology changes and channel condition variations. This work introduces transmission fluctuation deviation-based dynamic key generation, ensuring both real-time key updates and robust security while effectively preventing key cracking or reuse.

Transmission Efficiency and Reliability: Conventional chaotic encryption approaches in WSNs often sacrifice transmission quality due to energy and computational constraints. The proposed solution simultaneously achieves high-volume, real-time data transmission and maintains communication reliability/integrity through optimized routing algorithms and encryption strategies.

In summary, the generalized two-dimensional polynomial chaotic mapping-based secure transmission method outperforms existing smart grid/WSN chaotic encryption techniques across multiple dimensions: system complexity, anti-interference capability, network optimization, adaptability/flexibility, key management, and transmission efficiency/reliability.

3. Confidential Transmission Methods for High-Speed Power Line Carrier Communications

3.1. High-Speed Power Line Carrier Communication Network Formation

In high-speed power line carrier communication, establishing a stable link requires careful selection of appropriate power lines and nodes, forming the essential foundation for subsequent data transmission. A stable and reliable network guarantees that chaotic sequences generated by chaotic mapping maintain their complexity and unpredictability without causing transmission interference, thereby improving communication confidentiality and security. The minimal connected dominating set method enables the identification and construction of key node sets that preserve full network connectivity while minimizing node count, effectively reducing data transmission redundancy and delays [13]. This approach is implemented through a power line carrier routing algorithm to construct the high-speed power line carrier communication network.

The concept of vertex domination in graph theory can be defined as a vertex dominating other vertices. In an undirected graph $G\left(V,F\right)$, for a vertex subset $V^{\prime }\subseteq V$, if for $u\in \left(V-V^{\prime }\right)$, there exist $u\in V^{\prime }$ such that $\left(u,v\right)\in F$, then $u$ is said to dominate $v$, and $V^{\prime }$ is considered a dominating set of $G$. If any two vertices in an undirected graph $G$ are connected by a path, then $G$ is considered a connected graph. Let $S$ be a dominating set of $G\left(V,F\right)$. If there exists an edge set $F^{\prime }\subseteq F$ that makes $G$ a connected graph, then $S$ is considered a connected dominating set of graph $G\left(V,F\right)$. If for $\forall v\in S$, $S-\left\{v\right\}$ is not a connected dominating set, then $S$ is considered a minimal connected dominating set of graph $G\left(V,F\right)$.

The primary objective of the high-speed power line carrier routing and networking algorithm is to construct an optimal network structure among the carrier nodes. A high-speed power line carrier communication network with bidirectional communication can be modeled as an undirected connected graph $G\left(V,F\right)$, where $V$ denotes the set of all carrier nodes, and $F$ is the set of edges, with each edge of $f\in F$ indicating the existence of a communication path between two carrier nodes.

The research on high-speed power line carrier communication networking strategies primarily centers on dividing the network into logically interconnected subnetworks and ensuring reliable communication between devices [14]. Within a given area containing $n$ carrier devices, maximizing network utilization efficiency requires selecting a minimal set of $m$ devices with superior communication capabilities to serve as routing nodes, thereby guaranteeing reliable network-wide communication. This challenge is mathematically formulated as a minimal connected dominating set problem in graph theory.

From a graph theory perspective, this work establishes a weighted network analysis model employing the minimal connected dominating set concept. Building upon this model, we propose a novel high-speed power line carrier routing algorithm that executes the following procedure: First, the algorithm selects initial routing nodes via maximum weight priority determination. Specifically, for each carrier node, it verifies whether the node exhibits the largest weight among its neighbors, thus ensuring the selection of the highest signal-to-noise ratio node as a dominating node. Second, the method performs network connectivity re-optimization on preliminary routing nodes to derive the minimal connected dominating set solution. During optimization, when any neighbor remains unconnected through existing dominating nodes, the algorithm designates it a new dominating node.

Therefore, in the algorithm implementation, each carrier device is classified into one of three states: undetermined, a dominating node, or a dominated node. The undetermined state occurs during initial network formation when a device’s role remains undefined. The dominating node and dominated node states represent operational roles adopted during or after network formation, where dominating nodes function as routing nodes within the network.

The specific steps for the formation of a high-speed power line carrier communication network are as follows:

Step 1: Each carrier device in the network acquires a unique node identifier ($ID$) and is assigned a communication performance weight $v\left(w\right)$ based on its signal-to-noise ratio. During initialization, all carrier devices set their state to “undetermined” ($A_v=0$), where $A_v$ represents the three states of carrier equipment in the process of networking. When the value is 0, it means that the carrier equipment is undetermined; when the values are 1 and 2, it corresponds to the state of the dominant node and dominated node of the carrier equipment, respectively.

Step 2: A carrier device initiates a networking request by broadcasting at maximum power [15], collecting the $ID$ and weight information of other devices within its communication range. If device $v$ has the highest weight among all one-hop neighbors in set $M_v$, proceed to Step 3; otherwise, proceed to Step 4.

Step 3: Update the device’s state to “dominant node” ($A_v=1$) and broadcast its node number and state value to all one-hop neighbors.

Step 4: The device updates its state based on received broadcast messages [16], categorized as follows:

(1)

If any carrier device $u$ in the network receives broadcast information $B_v=1$ from the carrier device $v$, and its current state is “undetermined”, it updates its status to “dominated node” ($A_v=2$) and broadcasts its status information to devices within its communication range.

(2)

If any carrier device $u$ receives broadcast information $B_v=2$ from the carrier device $v$, and its current state is “undetermined”, it evaluates the weights of other devices in the received broadcast set $M_v$. Specifically, it checks whether there exists a dominant node in subset $Q$, (where $Q$ represents devices with weights greater than $v$). If such a node exists, $u$ updates its status to “dominating node” ($A_v=1$) and broadcasts its node number and status to devices within its communication range; otherwise, it continues waiting.

Step 5: If all carrier devices in the network have determined their states (i.e., $G^{\prime }=0$, where $G^{\prime }$ denotes the set of undetermined nodes), the system enters a waiting time $T$ before proceeding to the connectivity optimization phase.

Step 6: To ensure network reliability and minimize the number of routing nodes, the following optimization rules are applied:

(1)

For any dominated node $u$, if two devices $u_1$ and $u_2$ within its communication range cannot connect via existing dominating nodes, $u$ updates its status to “dominating node” ($A_u=1$).

(2)

For any dominant node $u$, if two carrier devices $u_1$ and $u_2$ within its communication range can be connected through other dominant nodes (excluding $u$ itself), the state of $u$ is updated to a dominated node ($A_u=2$). Simultaneously, $u$ identifies the highest-weight dominating node within its neighbor set $M_u$ as its own dominating node.

This algorithm determines the state of each carrier node in the high-speed power line carrier network. Each dominating node establishes connections with its corresponding dominated nodes to form subnetworks. Furthermore, any two dominated nodes are directly connected, creating a complete network where the carrier nodes act as cluster heads. Ultimately, the dominating nodes interconnect to form the high-speed power line carrier communication network.

3.2. Cryptographic Security Analysis

In the secure transmission method for high-speed power line carrier communication, cryptographic security represents one of the core performance evaluation metrics. This paper conducts a detailed security analysis focusing on three key aspects: key space size, resistance against known attacks, and entropy measurement, thereby demonstrating the method’s high security characteristics.

The key space represents the set of all possible keys, and its size directly influences the encryption system’s resistance to brute-force attacks.

In the generalized two-dimensional polynomial chaotic mapping system, the cryptographic key is determined by three components: the chaotic system’s initial conditions, the control parameters, and the polynomial’s highest degree. This configuration ensures that even with supercomputing capabilities, an attacker would require a computationally impractical time frame (exceeding the universe’s age) to exhaust all possible key combinations, thereby guaranteeing the encryption system’s theoretical security.

The system’s resistance to known attacks manifests in four key aspects:

(1)

Statistical attacks: The quasi-random sequences generated by chaotic mapping exhibit white-noise-like statistical properties (e.g., autocorrelation and cross-correlation). The generalized two-dimensional polynomial chaotic mapping ensures uniform and unpredictable sequence distributions, preventing attackers from extracting meaningful information through statistical analysis techniques like differential or linear cryptanalysis.

(2)

Chosen-plaintext/ciphertext attacks: Extreme sensitivity to initial conditions and parameters makes deducing the chaotic system’s internal state impossible even with known plaintext–ciphertext pairs. Dynamic parameter adjustment (e.g., transmission fluctuation-based key updates) further strengthens this resistance.

(3)

Side-channel attacks: Hardware-based implementation inherently decouples side-channel information (power consumption, electromagnetic radiation) from cryptographic keys. Combined with dynamic parameter adaptation, this effectively mitigates power analysis and timing attacks.

(4)

Known-key attacks: The chaotic mapping’s complexity and parameter diversity ensure security even with partial key exposure. Regular key updates through the system’s built-in mechanism provide additional protection against such compromises.

Entropy serves as a quantitative measure of information uncertainty, where higher entropy values correspond to greater system security. For chaotic sequences, entropy can be quantified as the number of bits per symbol. Considering a chaotic sequence with a uniformly distributed output range of [0, 1], the per-sample entropy is given by

$$H=-{\displaystyle {\int }_0^{1}p\left(x\right)}{\log }_{2}p\left(x\right)dx={\log }_2\left(1-0\right)=\infty$$

(1)

Leveraging the long-periodicity and aperiodicity characteristics of chaotic sequences, the total entropy value exhibits linear growth with increasing sequence length, surpassing traditional encryption algorithms significantly. This high-entropy property ensures that attackers cannot predict the complete sequence from partial information, thereby maintaining data confidentiality and integrity.

3.3. Confidential Transmission Method Based on Generalized Two-Dimensional Polynomial Chaotic Mapping

3.3.1. 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 ${\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_t$. The anomalous deviation of the actual network transmission time series is calculated as

$$d_t=e^{{\displaystyle \frac{\left|{\epsilon }_t-{\overline{\epsilon }}_{t-1}\right|}{{\delta }_{t-1}}}}$$

(2)

where $d_t$ indicates the degree of deviation at the moment $t$; ${\overline{\epsilon }}_{t-1}$ 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.

3.3.2. 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:

$$\left\{\begin{array}{l}{x}_{\eta +1}=4{\alpha }_1\times x_{\eta }\left(1-x_{\eta }\right)+\mu \times y_{\eta }\\ y_{\eta +1}=4{\alpha }_2\times y_{\eta }\left(1-y_{\eta }\right)+\mu \times x_{\eta }\end{array}\right.$$

(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

$$\left\{\begin{array}{l}{x}_{\eta +1}=\left({\alpha }_1{x}_{\eta }+\mu y_{\eta }^{\rho }\right)\mathrm{mod}\theta \\ y_{\eta +1}=\left({\alpha }_2{x}_{\eta }+r_{\eta }\right)\mathrm{mod}\theta \end{array}\right.$$

(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 $\left(x_0,y_0\right)$ 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.

The system architecture of the encrypted high-speed power line carrier communication network, implementing generalized two-dimensional polynomial chaotic mapping for information transmission, is shown in Figure 1.

The operational process of the encrypted high-speed power line carrier communication network implementing generalized two-dimensional polynomial chaotic mapping is as follows:

Step 1: Load the key $K_x=x_1{d}_1{x}_2{d}_2\cdots x_{\tau }{d}_{\tau }$ generated by the generalized two-dimensional polynomial chaotic mapping equations.

Step 2: Encrypt the transmission information $\chi$ of the high-speed power line carrier communication network using the following encryption formula:

$$P_{K_x}\left(\chi \right)=\cdots \left(x_1{d}_1\oplus {\chi }_1\right)\cdots \left(x_{\tau }{d}_{\tau }\oplus {\chi }_{\tau }\right)\cdots \left(x_1{d}_1\oplus {\chi }_{\tau +1}\right)\cdots \left(x_{\tau }{d}_{\tau }\oplus {\chi }_{2\tau }\right)\cdots$$

(5)

Step 3: Generate the temporary ciphertext ${\chi }^{\prime }={\chi }_1^{\prime }{\chi }_2^{\prime }\cdots {\chi }_{\tau }^{\prime }\cdots {\chi }_{\tau +1}^{\prime }{\chi }_{\tau +2}^{\prime }\cdots {\chi }_{2\tau }^{\prime }$.

Step 4: Load the key $K_y=y_1{d}_1{y}_2{d}_2\cdots y_{\tau }{d}_{\tau }$ generated by the generalized two-dimensional polynomial chaotic mapping equations.

Step 5: Perform secondary encryption on the temporary ciphertext ${\chi }^{\prime }$ using the following formula:

$$P_{K_y}\left({\chi }^{\prime }\right)=\cdots \left(y_1{d}_1\oplus {\chi }_1^{\prime }\right)\cdots \left(y_{\tau }{d}_{\tau }\oplus {\chi }_{\tau }^{\prime }\right)\cdots \left(y_1{d}_1\oplus {\chi }_{\tau +1}^{\prime }\right)\cdots \left(y_{\tau }{d}_{\tau }\oplus {\chi }_{2\tau }^{\prime }\right)\cdots$$

(6)

Step 6: Generate the final ciphertext ${\chi }^{″}={\chi }_1^{″}{\chi }_2^{″}\cdots {\chi }_{\tau }^{″}\cdots {\chi }_{\tau +1}^{″}{\chi }_{\tau +2}^{″}\cdots {\chi }_{2\tau }^{″}$.

The decryption flow of the information transmitted by a high-speed power line carrier communication network using generalized two-dimensional polynomial chaotic mapping is shown in Figure 2.

The decryption process using two-dimensional polynomial chaotic mapping for high-speed power line carrier communication networks is as follows:

Step 1: Extract the encrypted ciphertext ${\chi }^{″}={\chi }_1^{″}{\chi }_2^{″}\cdots {\chi }_{\tau }^{″}\cdots {\chi }_{\tau +1}^{″}{\chi }_{\tau +2}^{″}\cdots {\chi }_{2\tau }^{″}$ through sequential reading [20].

Step 2: Load the decryption key $K_y=y_1{d}_1{y}_2{d}_2\cdots y_{\tau }{d}_{\tau }$ generated by the generalized two-dimensional polynomial chaotic mapping equations.

Step 3: Decrypt the ciphertext ${\chi }^{″}$ transmitted by the high-speed power line carrier communication network, using the following formula:

$$P_{K_y}\left({\chi }^{\prime }\right)=\cdots \left(y_1{d}_1\oplus {\chi }_1^{″}\right)\cdots \left(y_{\tau }{d}_{\tau }\oplus {\chi }_{\tau }^{″}\right)\cdots \left(y_1{d}_1\oplus {\chi }_{\tau +1}^{″}\right)\cdots \left(y_{\tau }{d}_{\tau }\oplus {\chi }_{2\tau }^{″}\right)\cdots$$

(7)

Step 4: Generate the decrypted temporary ciphertext ${\chi }^{\prime }={\chi }_1^{\prime }{\chi }_2^{\prime }\cdots {\chi }_{\tau }^{\prime }\cdots {\chi }_{\tau +1}^{\prime }{\chi }_{\tau +2}^{\prime }\cdots {\chi }_{2\tau }^{\prime }$.

Step 5: Load the decryption key $K_x=x_1{d}_1{x}_2{d}_2\cdots x_{\tau }{d}_{\tau }$ generated by the generalized two-dimensional polynomial chaotic mapping equations.

Step 6: Perform secondary decryption on the intermediate ciphertext ${\chi }^{\prime }$ using the following formula:

$$P_{K_y}\left(\chi \right)=\cdots \left(x_1{d}_1\oplus {\chi }_1^{\prime }\right)\cdots \left(x_{\tau }{d}_{\tau }\oplus {\chi }_{\tau }^{\prime }\right)\cdots \left(x_1{d}_1\oplus {\chi }_{\tau +1}^{\prime }\right)\cdots \left(x_{\tau }{d}_{\tau }\oplus {\chi }_{2\tau }^{\prime }\right)\cdots$$

(8)

Step 7: The high-speed power line carrier communication network transmits information $\chi$.

3.3.3. Confidential Transmission Scheme for High-Speed Power Line Carrier Communications Based on Generalized Two-Dimensional Polynomial Chaotic Mapping

The inherent unpredictability and ergodicity of chaotic mapping make it particularly suitable for secure data transmission in networks. When ciphertext encrypted via the generalized two-dimensional polynomial chaotic mapping (Section 3.3.2) is employed for data transmission, its output distribution critically impacts the bit error rate (BER) performance. The generalized 2D polynomial chaotic mapping proposed in this work achieves a near-uniform output distribution across the entire value space, thereby enhancing secure communication performance. By integrating the reference-modulated differential chaotic shift keying (RM-DCSK) with the ciphertext encrypted via the generalized 2D polynomial chaotic mapping (Section 3.3.2), the RM-DCSK secure transmission scheme ensures robust and secure high-speed power line carrier communication.

The schematic diagram of the ciphertext interface of chaotic map and RM-DCSK modulation is shown in Figure 3.

The advantages of a high-speed power line carrier communication secure transmission scheme based on generalized two-dimensional polynomial chaotic mapping over traditional secure transmission methods are as follows:

(1)

Confidentiality enhancement

The scheme leverages the complex dynamic behavior of generalized two-dimensional polynomial chaotic mapping to generate cryptographically strong key sequences with high randomness and unpredictability. This significantly elevates the confidentiality level of transmitted data and substantially increases the computational complexity for potential attackers. The expanded key space in two-dimensional space provides superior anti-deciphering capability compared to traditional methods, effectively mitigating the risks of information leakage and malicious attacks.

(2)

Anti-Interference performance

The proposed encryption scheme demonstrates exceptional resistance to power line channel impairments including electromagnetic interference, equipment noise, and signal attenuation. The inherent randomness and complexity of the chaotic mapping ensure consistent communication stability and reliability, while preserving data integrity and accuracy during transmission. This enhanced robustness enables reliable operation in highly dynamic communication environments.

(3)

Transmission efficiency

The scheme fully exploits the high-bandwidth characteristics of power line carrier communication to support high-throughput, low-latency data transmission. Through optimized modulation/demodulation algorithms, the system achieves superior spectral efficiency and overall performance. This meets and exceeds the stringent requirements of smart grid applications, while providing scalable infrastructure for future power system advancements.

The RM-DCSK secure transmission scheme consists of transmitter and receiver components. The transmitter first encrypts the message bits with a key generated by the generalized 2D polynomial chaotic mapping in Section 3.3.2, produces a high-speed power line carrier communication ciphertext, and then transmits the transmitter and receiver components. The receiver decodes the received ciphertext using the key generated by the generalized 2D polynomial chaotic mapping in Section 3.3.2 to recover the message bits.

(1)

Transmitting end: ${\chi }_{2l}$ serves as the information bits for $2l$, and ${\chi }_l^{″}$ represents the high-speed power line carrier communication ciphertext encrypted using the generalized 2D polynomial chaotic mapping described in Section 3.3.2, with a line length of $H$. This is expressed as ${\chi }_l^{″}=\left\{{\chi }_i^{″}\left|2lH<i<\left(2l+1\right)H\right.\right\}$, where $H$ denotes the spreading index. The first time slot within the high-speed power line carrier communication network, established in Section 3.1, is designated as the reference time slot. The ciphertext ${\chi }^{″}$ acts as an information carrier, transmitting the bits ${\chi }_{2l}$ in the first time slot. The second time slot consists of two parts: the first part is the information transmitted in the previous time slot, ${\chi }_{2l}{\chi }_{i-H}^{″}$, serving as the message carrier, followed by the transmission of the bits ${\chi }_{2l+1}$ in the second time slot. The latter part is the high-speed power line carrier communication ciphertext of the current time slot, ${\chi }_i^{″}$, which satisfies

$${\chi }_{\left(2l+1\right)H}^{″}={\chi }_{2\left(l+1\right)H}^{″},\forall l\in \left\{0,\pm 1,\cdots \right\}$$

(9)

(2)

Receiving end: When the receiver obtains the high-speed power line carrier communication ciphertext, it decrypts and recovers the original information bits using the key derived from the generalized two-dimensional polynomial chaotic mapping in Section 3.3.2. The RM-DCSK confidential transmission scheme primarily relies on chaotic delay characteristics and non-coherent demodulation techniques. The received ciphertext deviates from the original due to noise interference during transmission across different high-speed power line carrier communication networks. In this paper, the additive white Gaussian noise (AWGN) channel model is adopted as the primary communication medium, and the channel interference is modeled as additive white Gaussian noise ${\phi }_i$. Therefore, the received ciphertext $c_i$ is expressed as

$$c_i={\chi }_i^{″}+{\phi }_i$$

(10)

Decryption is carried out using the key pairs generated by the generalized 2D polynomial chaotic mapping of Section 3.3.2 to decode $c_i$, thereby recovering the original information bits and completing the confidential transmission of high-speed power line carrier communication.

4. Experimental Analysis

A typical building power system constitutes the experimental environment. The power line follows a typical tree topology model with a 500 m distribution radius. The power area includes a collection terminal, collectors, single-phase smart meters, and three-phase smart meters from different manufacturers. All carrier devices possess unique identification numbers and are interconnected via power lines. To account for differences in communication performance, devices are parameterized according to their characteristics. Variables are incorporated to characterize the effective communication radius under different channel conditions, capturing time-varying characteristics. The experiment employs the BPSK coding scheme $L$ for carrier communication. MATLAB software 2025 edition is utilized to simulate the secure transmission method of high-speed power line carrier communication.

The experimental simulation utilizes a communication network comprising 38 carrier devices distributed within a 500 m radius. Figure 4 illustrates the physical connections between the meters. The network consists of 1 collection terminal, 3 collectors, 30 single-phase smart meters, and 4 three-phase smart meters from different manufacturers. The carrier communication employs BPSK modulation, with an average effective communication distance of $L=100 \mathrm{m}$. The main reason for employing BPSK modulation technology lies in its straightforward implementation and excellent noise immunity, making it particularly suitable for power line carrier communication systems operating in complex noise environments. However, alternative modulation techniques, such as quadrature phase shift keying (QPSK), can enhance the data transmission rate. Nevertheless, in high-frequency bands, the bit error rate may increase significantly due to the inherent channel characteristics of power lines, potentially compromising communication reliability and stability.

The network formation results, presented in Figure 4, are derived by implementing the proposed technique on the carrier equipment depicted in Figure 5 to establish a high-speed power line carrier communication network.

Figure 5 demonstrates that the proposed method successfully establishes interconnections among all carrier nodes within the high-speed power line carrier network. Carrier devices numbered 38, 16, 17, 28, 31, and 37 are designated as cluster head nodes due to their superior communication performance. This selection criterion ensures network stability by minimizing frequent reconfigurations when channel conditions vary, thereby enhancing the overall reliability of the high-speed power line carrier communication system.

For a chaotic mapping, its bifurcation diagram illustrates the trajectories under different parameters. The diagram visually demonstrates the traversal and non-periodicity properties of the chaotic mapping under various control parameters. Figure 6 shows the 3D view of the bifurcation diagram for the generalized 2D polynomial chaotic mapping employed in this study, with two parameters (taking parameters ${\alpha }_1$ and $\mu$ as examples).

As shown in Figure 5, under the configuration of control parameters ${\alpha }_1$ and $\mu$, the state variables are uniformly distributed throughout the entire space, demonstrating that the generalized two-dimensional polynomial chaotic mapping technique adopted significantly enhances the security performance of high-speed power line carrier communication. This mapping method makes it extremely difficult to intercept information from transmitted signals, greatly improving the protection capabilities of the high-speed power line carrier communication network and ensuring the confidentiality and integrity of the transmitted data.

A critical metric for evaluating the encryption performance of the generalized two-dimensional polynomial chaotic mapping involves analyzing the distribution characteristics of plaintext and ciphertext, as well as the randomness of ciphertext binary sequences (0–1). Non-random or non-uniform ciphertext distributions could enable attackers to compromise the encrypted data. To rigorously evaluate this method’s encryption performance, we encrypted an 8 KB English text, with the resulting plaintext and ciphertext distributions illustrated in Figure 7 and Figure 8, respectively.

Figure 7 and Figure 8 reveal significant spatial distribution differences in ASCII values between plaintext and ciphertext. While the original plaintext displays distinct statistical patterns, the ciphertext processed by the proposed encryption method demonstrates uniform distribution characteristics. This transformation effectively obscures the original data features, preventing potential ciphertext-only attacks through statistical analysis and consequently enhancing data security.

Using this method, the information for this typical building power system is transmitted securely by high-speed power line carrier communication, with secure transmission results shown in

Table 1. Secure transmission results of high-speed power line carrier communication.

Serial Number	Transmission Information Class	Encryption Time (ms)	Transmission Delay (ms)	Transmission Result
1	Real-time power data	12.5	8.2	The data was transmitted successfully, the encryption effect was good, and the data could not be analyzed directly.
2	Voltage monitoring data	10.7	7.5	The data is transmitted safely and no leakage occurs.
3	Device remote control instruction	9.8	6.8	Instructions were successfully executed without being detected or tampered with during the transmission.
4	Energy consumption comparison report	15.2	9.1	The report is successfully transmitted, and the confidentiality of the content is guaranteed.
5	Safety warning information	8.9	6.2	Early warning information was communicated in a timely manner, and relevant personnel responded quickly.
6	Equipment maintenance reminder	11.3	7.9	A reminder message was successfully sent, and the maintenance plan was successfully executed.

.

As indicated in Table 1, this method effectively employs the generalized two-dimensional polynomial chaotic mapping technique to achieve deep information encryption, significantly enhancing the confidentiality level during transmission. Moreover, this method has been successfully implemented in high-speed power line carrier communication to ensure secure data transmission, guaranteeing that the received information remains intact and free from decryption or tampering, thereby further strengthening the system’s security. The encryption time varies between 8.9 ms and 15.2 ms, averaging approximately 11.4 ms, while the transmission delay ranges from 6.2 ms to 9.1 ms, with an average of 7.9 ms. All information categories are transmitted successfully, and the security performance is confirmed (including leakage prevention, tamper resistance, and timely response). The generalized two-dimensional polynomial chaotic mapping encryption significantly enhances data security and effectively resists ciphertext-only attacks. The encryption time and transmission delay, measured in milliseconds, meet the real-time requirements of high-speed power line carrier communication. The proposed method demonstrates excellent performance in security, real-time capability, and reliability, making it suitable for applications with strict data confidentiality requirements, such as smart grids, industrial monitoring, and smart home systems.

The bit error rate (BER) between the received data and the original data is calculated to verify the secure transmission performance of this method in high-speed power line carrier communication. Furthermore, the secure transmission performance of this method is compared with that of two-dimensional logistic mapping, the method in reference [6], and the method in reference [7] under different noise levels. The analysis results are shown in Figure 9.

As can be seen from Figure 9, when the signal-to-noise ratio is small and different chaotic maps are used, the method in this paper can obtain almost the same bit error rate, with a maximum value of 0.051. However, with the increase in signal-to-noise ratio, using the generalized two-dimensional polynomial chaotic mapping proposed in this paper can obtain a smaller bit error rate than using the two-dimensional logistic mapping, reference [6]’s method, and reference [7]’s method. The minimum bit error rate obtained by this method is 0.010, and the minimum bit error rate obtained by two-dimensional logistic mapping is 0.020. It shows that generalized two-dimensional polynomial chaotic mapping is more suitable for secure information transmission than two-dimensional logistic mapping, using reference [6]’s method and reference [7]’s method, which can effectively resist the interference of power line noise and signal attenuation and improve the confidentiality and stability of communication.

5. Conclusions

As a new communication technology, high-speed power line carrier communication offers the advantages of fast transmission speed, wide coverage, and low cost. However, with the continuous development of communication technologies, information security has become a critical concern. Therefore, we study a secure transmission method for high-speed power line carrier communication based on generalized two-dimensional polynomial chaotic mapping. Using generalized two-dimensional polynomial chaotic mapping for information encryption can effectively enhance the security performance of power line carrier communication and ensure that information remains protected against leakage or tampering during transmission, which is crucial for ensuring the safe and stable operation of power systems. The experimental results demonstrate that the proposed method exhibits significant advantages in terms of bit error rate and security performance, making it suitable for various communication scenarios with high security requirements. Specifically, the carrier devices numbered 38, 16, 17, 28, 31, and 37 are selected as cluster head nodes, thereby improving the stability and reliability of the high-speed power line carrier communication network. This method significantly enhances the security performance of high-speed power line carrier communication and improves the protection of encrypted data. The proposed method achieves comparable bit error rates, with a maximum of 0.051 and a minimum of 0.010, demonstrating its ability to effectively resist power line noise interference and signal attenuation while improving stability.