## 1 Introduction  
Chaotic system is a complex nonlinear system. In recent years, it has attracted extensive attention in the field of engineering research. The reason is that chaotic systems have very rich dynamic behavior, unpredictable trajectory, white noise-like broadband and the initial value sensitivity of the butterfly effect. The chaos dynamics in most mechanical engineering systems is undesirable and needs to be suppressed because it will affect the performance or damage the mechanical structure of systems. In 1990, [PERSON] ([PERSON], [PERSON] and [PERSON]) [1] proposed the study of chaos control. Researchers pointed out that, when parameter values of chaotic systems are slightly modulated, the chaotic behavior will produce huge and unpredictable changes. Several feedback control methods to suppress chaos behavior have been proposed in the literature [2; 3]. Although the chaos behavior might be undesired in most mechanical systems, the noise-like behavior of chaos is useful to the application of image encryption and secure communication [4; 5; 6; 7; 8]. Due to the practical application of chaos synchronization, since the pioneering research of [PERSON] and [PERSON] [9], chaos synchronization has become an interesting research field. Many control methods have been proposed to solve the synchronization problem for chaotic systems, such as \(H\infty\) control [10; 11], fuzzy sliding mode control [12; 13], adaptive control [14; 15] and so forth. Among proposed control designs, sliding mode control method is often the first choice for researchers because it is not sensitive to system parameters and external disturbances and with good robustness. In these two decades, chaotic systems have been widely applied in the areas of communication, medicine and biology to solve several important engineering problems, especially in the security field of communication. In the previous reports [4; 5; 6], the authors used the continuous chaotic systems with well-designed synchronization controllers to achieve the secure communication. However, in practical circuit implementations for continuous chaotic systems, analog components, such as operational amplifier (OPA), resistors, capacitors are necessary. Unfortunately, these analog components are sensitive to environmental conditions and often result in circuit instability. On the other hand, owing to the progress of microcontrollers with digital signal processing (DSP) technology, using microcontrollers to replace the analog circuits has become more and more popular and important. Discrete chaotic systems implemented by the microcontroller are more robust and less susceptible to temperature variation and component aging but also at a lower price. Therefore, to solve the problem of instability of analog circuits, we need to study the discrete chaotic systems realized by microcontrollers. Until now, the applications of discrete chaotic systems to solve image encryption have been proposed in References [7; 8]. However, the synchronization control problem of discrete chaotic systems has not been well solved in these studies. In addition, to the best of our knowledge, using discrete chaotic systems to design the synchronized large-scale CRNGs has not been well discussed. Furthermore, recently, due to the advent of quantum computers, many traditional encryption methods may become unsafe. The encryption can be quickly cracked by quantum computers. Because the chaos system has rich dynamic characteristics, it can generate a large amount of random signals in a very short time and many scholars have pointed out that chaotic encryption method may be one of the solutions against the attack of quantum computers [16; 17]. For the above reasons, in this paper, we study the design of synchronized large-scale CRNGs and its application to secure communication. In order to simplify the presentation of the paper, we choose Duffing map for this study. However, the proposed modulation approach can also effectively applied to other classes of discrete chaotic systems.