# An Improved Multi-Chaotic Public Key Algorithm Based on Chebyshev Polynomials
## 1 Introduction  
With the advent of cloud computing, the Internet of Things, the mobile internet, and the big data era, security threats brought by the deployment of new technologies are constantly expanding, and designing new high-performance encryption schemes has become an important problem to be urgently solved. Compared with traditional encryption systems, chaotic public key cryptography has higher security, better resistance to attacks, a longer validity period, and more application areas [1, 2]. As a new direction in the field of public key cryptography research, it is still in the initial stage and has very important exploratory value.  
Cryptography is a very active research field, with traditional ciphers mainly including DES, AES, RSA, and other algorithms [3]. In recent years, new ideas have stimulatedmany new forms of cipher systems, primarily including quantum cryptography [4], visual cryptography [5], biometric cryptography [6], DNA cryptography [7], homomorphic cryptography [8], attribute-based encryption [9], and Post-Quantum Cryptography [10; 11].  
Chaos theory is considered the third revolution in physics in the 20 th century, following relativity and quantum mechanics. It is the seemingly random behavior generated by deterministic systems. It has extreme sensitivity to initial conditions and parameters, and long-term unpredictability, which conforms to some characteristics of cryptography, opening a new field of research in chaotic cryptography. In 1949, [PERSON] [12] pointed out the confusion and diffusion principles in cipher design, which correspond precisely to the pseudo-randomness and sensitivity to initial conditions of chaotic systems: the ergodicity and topological mixing of chaotic systems correspond to the confusion properties of ciphers; on the other hand, the extreme sensitivity of chaotic systems to initial conditions and control parameters corresponds to the diffusion properties of ciphers. Diffusion can create an avalanche effect, producing completely different outputs from tiny changes to the initial input. This suggests that chaos can be applied to the field of cryptography. [PERSON] et al. [13] summarized some relationships between chaotic systems and cryptography. Chaotic cryptography theory emerged in the mathematics and physics fields in the early 1960s, and it is known as one of the three major scientific revolutions of the 20 th century along with quantum theory and relativity. In 1989, [PERSON] proposed the chaotic cipher algorithm, which provided a chaotic sequence cipher scheme based on modified logistic mappings [14]. In particular, [PERSON] and [PERSON] published an article on chaotic synchronized secure communication in 1990 [15], sparking a research wave in chaotic cryptography. At the same time, analysis of chaotic cipher systems has also been carried out, proving the insecurity of some chaotic cipher systems and proposing improvement measures. In recent years, significant progress has been obtained in the research of chaotic cryptography [16; 17; 18; 19; 20; 21], but there are still some challenging problems to be solved, such as the problem of finite precision digitization of chaotic systems, the security and efficiency of chaotic cipher design, etc. Therefore, compared with traditional cryptography, the development of chaotic cryptography is still not mature and perfect, lacking sufficient persuasiveness. On the other hand, compared with symmetric cryptography, the study of chaotic public key cryptography is relatively weak. Designing public key cryptographic systems based on chaotic systems is a new scientific exploration attempt, which has important research value and significance.