# An Improved Multi-Chaotic Public Key Algorithm Based on Chebyshev Polynomials
## 6 Conclusions and Future Work  
Due to the similar characteristics between chaotic systems and cryptography, chaos-based public key encryption is currently a hot topic. This paper improves the performance and security of the Bose Multi-Chaotic Public Key Cryptographic Algorithm (BMPKC) from the aspects of efficiency and reliability. The proposed algorithm is named CMPKC-\(k_{i}\), which overcomes the shortcomings of previous algorithms by using selective coefficient \(k_{i}\), chaotic hash functions, and logistic mapping to generate pseudo-random sequences. CMPKC-\(k_{i}\) provides better security because it leads to more complex algorithms and the process of public key generation is sent in parallel, making it difficult for attackers to obtain much information about the key or decrypt the message. Compared to the standard CEPKC and BMPKC, CMPKC-\(k_{i}\) takes longer to break private keys. Meanwhile, we also explore the combination of the improved chaotic public key encryption system with the AES encryption method, which provides more possibilities for practical applications of the system. In summary, this solution has the characteristics of strong practicality and high security, making contributions to the field of information encryption. We believe that its advantage lies in the fact that even if attackers guess the chaotic sequence \(m\) within the same period, they still need to obtain the specific \(k_{i}\) from the selective coefficient and break the chaotic hash function, which is almost impossible within the limited time. Therefore, the security of the algorithm is greatly improved. Although the overall design of this solution is satisfactory, it requires more time for key generation due to parallel computation and the occurrence of selective coefficients through several generation stages. In the case of large prime numbers, the efficiency of the algorithm will decrease, so the selective coefficient cannot be too large or it will have a negative impact on the required time for the algorithm, as seen in Figure 10, which can be considered as a drawback and an issue that needs continuous improvement. In conclusion, after performance and security analysis, the proposed CMPKC-\(k_{i}\) algorithm demonstrates superior results with improved security. As future research work, we plan to compare the proposed public key encryption system with other common advanced public key encryption schemes in different application scenarios for performance analysis and conduct a more detailed assessment of security aspects. The algorithm will also be extended to parallel machines so that multiple related operations can be parallelly computed on multi-core processors. Compared to similar public key encryption systems, this will result in lower computational costs and wider acceptability in IoT devices. Additionally, we can explore the potential of combining other chaotic mappings, fractals, and traditional encryption methods (such as AES, DES, and Blowfish) to achieve higher efficiency and security in the field of encryption. Interested researchers can implement this algorithm on hardware platforms such as field-programmable gate arrays (FPGAs) and application-specific integrated circuits (ASICs) to optimize its performance for practical applications. In conclusion, this research brings possibilities for continual exploration and advancement in the digital security field. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Prime Length** & **CEPKC** & **BMPKC** & **CMPKC** & **CMPKC-\(k_{i}\)** \\
**(in bits)** & **(in ms)** & **(in ms)** & **(in ms)** & **(in ms)** \\ \hline
4 & 12.42 & 5.27 & 20.66 & 25.16 \\
5 & 40.84 & 16.16 & 260.85 & 335.96 \\
6 & 263.38 & 129.09 & 3288.19 & 4438.59 \\
7 & 1286.35 & 653.69 & 49,154.35 & 65,516.34 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Brute force attack time comparison in milliseconds. **Author Contributions:** Conceptualization, C.Z., Y. L, A.T. and L.W. ; methodology, C.Z. and L.W. ; software, C.Z., J.B., T.G. and S.P. ; validation, Y.L., A.T. and L.W.