# An Improved Multi-Chaotic Public Key Algorithm Based on Chebyshev Polynomials
## 1 Introduction
### Trend
\begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline
**Year** & **Author** & **Description** & **Evaluation** \\ \hline
1987 & [PERSON] [22] & A public key cryptography algorithm based on cellular automation, which can be seen as a discrete dynamical system. & It has significant limitations, such as poorly in processing large-scale data, with slow computational processes that cannot meet real-time requirements. \\ \hline
2003 & [PERSON] [23; 24] & A chaotic public key cryptography algorithm based on Chebyshev polynomials, which is extended to finite fields due to security flaws and deployed in applications such as exchange protocols and digital signatures. & It represents a primary focus within the realm of chaotic public key cryptography, marking the inaugural application of Chebyshev polynomials in a more extensive manner to public key encryption algorithms. \\ \hline
2004 & [PERSON] [26] & A public key cryptography algorithm based on toroidal self-isomorphism, which can be seen as an extension of public key cryptography based on Chebyshev polynomials. & The algorithm’s security is claimed to be consistent with the RSA algorithm and is purportedly applicable to the ElGamal and Rabin policy algorithms. However, only a simple example is provided, and the conclusion awaits confirmation. \\ \hline
2005 & [PERSON] [27] & A public key cryptography algorithm based on multiple chaotic systems. & It has been proven to have security flaws, such as its susceptibility to brute force attacks. \\ \hline
2009 & [PERSON]. [29] & A new public key cryptography algorithm constructed using the \(AA_{\theta}\) chaotic system, [PERSON] [30] & By solving the discrete logarithm problem modulo 1, it has been demonstrated that this cryptographic system is not sufficiently secure. \\ \hline
2010 & [PERSON] [31] & An identity-based encryption scheme is proposed based on Chebyshev chaotic mapping (IBE). & It has security flaws. People can more easily attack by exploiting the semi-group property of extended Chebyshev polynomials on \(Z_{p}\). \\ \hline
2014 & [PERSON]. [33] & New identity-based encryption and signature schemes are proposed based on Chebyshev polynomials. & The security and efficiency still need to be verified. \\ \hline
2014 & [PERSON] [34] & A three-party key exchange protocol is proposed based on Chebyshev polynomials and proven to be secure in the standard model. & It has promoted the progress of chaotic public key cryptography research as the protocol is the first provably secure 3 PAKE protocol using Chebyshev chaotic maps under the standard model. \\ \hline
2015 & [PERSON] [35] & A three-party key exchange protocol based on Chebyshev chaotic mapping is proposed based on password authentication. & It has promoted the progress of chaotic public key cryptography research as the protocol is the first provably secure 3 PAKE protocol using Chebyshev chaotic maps under the standard model. \\ \hline
2018 & [PERSON] [36] & An efficient cryptosystem is based on the Logistic-Chebyshev map. & Its security has not been verified yet. \\ \hline
2021 & [PERSON] [37] & A secure dual-factor key agreement protocol based on Chebyshev polynomials. & Preliminarily addressed the issue of user unlinkability in authentication key protocols based on chaotic maps. \\ \hline
2022 & [PERSON] [38] & Secure client authentication based on Chebyshev chaotic mapping. & It has higher security and computational performance. \\ \hline
2023 & [PERSON] [39] & An efficient three-party authenticated key exchange process (AKEP) is proposed using Chebyshev chaotic mapping. & It presents a novel type of key exchange process, based on an efficient three-party authenticated key exchange procedure (AKEP) using Chebyshev chaotic maps with client anonymity. \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Major research history of chaotic public key cryptography.