## 1 Introduction  
In recent decades, there has been increasing interest in the study of dynamical systems, particularly chaotic systems. Chaotic systems are traditional nonlinear dynamical systems that are sensitive to initial conditions, a phenomenon often referred to as the butterfly effect. This effect implies that when two identical chaotic systems start with nearly the same initial conditions, their trajectories diverge exponentially over time, leading to significantly different outcomes. The development of chaotic systems started in 1963 when [PERSON] discovered a three-dimensional (3D) chaotic system for weather modeling [1]. In 1979, [PERSON] introduced a four-dimensional (4D) system describing the dynamics of chemical reactions, known as a hyperchaotic system, which is chaos with more than one positive Lyapunov exponent [2]. Hyperchaos has been studied intensively in different fields due to its interesting properties such as high capacity, enhanced security, and increased efficiency [3]. Many types of chaotic/hyperchaotic systems have been introduced in both continuous-time and discrete-time domains. Examples of continuous-time systems include Chua's circuit [4, 5], Duffin, and Lu and Chen systems. In contrast, the Logistic map, Henon map [6], Henon 3D map, and Lorenz discrete map [7] are examples of discrete-time systems. Previous research in this area has explored various techniques for implementing secure communication systems, including electronic circuits [8, 9], FPGA implementation [10], and memristor-based circuits [11, 12]. In addition, secure communication systems can also be realized through post-quantum encryption algorithms [13], quantum-securecommunications [14], and physical layer security [15]. While these approaches focus on providing robustness against quantum attacks or ensuring security at the communication layer, the proposed chaotic-based encryption offers an alternative based on the intrinsic unpredictability of chaotic systems, making it an effective method for data masking and encryption. The study of chaos is very important because of its applications in robotics [16; 17], cryptography [18], neural network [19], secure communication [20], etc. The protection of multimedia files against copying or attacking across public channels has become a research subject in recent years [21; 22]. The first chaos-based secure communication system introduced in 1993 is additive chaos masking. This method involves using identical chaotic oscillators at both the transmitter and the receiver [23; 24; 25; 26]. One state variable of the oscillator on the transmitter side, known as the chaotic mask signal, is added to the message signal giving the transmitted signal. Since the chaotic signal is very complex, the information message cannot be extracted without knowing the exact value of the mask signal. This scheme has drawbacks such as sensitivity to noise in the channel and parameter mismatches between the identical chaotic oscillators in the transmitter and the receiver [27]. Moreover, as demonstrated in [28], it has a relatively low degree of security, as researchers were able to predict the carrier behavior and break the system by subtracting the predicted values from the carrier, thereby reconstructing the message signal through some processing operations. To date, many attack methods have been developed to break the chaotic communication system and schemes. Therefore, there remains a requirement and necessity to develop a scheme that can be potentially employed against attacks. In the literature, many papers have used the chaos cascade concept to construct new chaotic systems, which increases the value of Lyapunov exponents and expands the parameter space [29; 30]. [PERSON] [31] proposed a chaos-based cryptosystem called the cascaded chaotic masking technique, which improves security by using two chaotic signals to increase the complexity of the encrypted signal. In [26], the authors used multiple double levels of chaotic masking to encrypt a speech signal.