## 3 Confidential Transmission Methods for High-Speed Power Line Carrier Communications
### 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  
\[H=-\int_{0}^{1}p(x)\log_{2}p(x)dx=\log_{2}(1-0)=\infty \tag{1}\]  
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.