## 3 Proposed Multi-Cascaded Chaotic Modular Fractional Sine Map (MCC-MF Sine Map)  
The cascade chaotic system (CCS) is a general 1D chaotic framework for creating new nonlinear chaotic systems using any two 1D chaotic maps as seed maps; it was first introduced in [66]. [PERSON] et al. also suggested a dynamic parameter-control chaotic system (DPCCS) [67] based on the concept of the CCS. The DPCCS has a simple architecture that uses the control map's output to dynamically modify the seed map's parameters. CCS and DPCCS have straightforward hardware implementation, simple structures, and wildly unpredictable behavior. In this section, a new MCC-MF sine map is introduced and analyzed. The development of discrete fractional calculus allowed for the effective incorporation and capture of memory effects in nonlinear discrete temporal systems. Complex features are seen in chaotic systems with a fractional order. Assume that a sequence \(\rho(n)\) is given and the isolated time scale \(\aleph_{\alpha}\) is represented in terms of the real valued constant \(\tau\) as \(\{\tau,\tau+1,\tau+2,\dots,\}\) such that \(\rho:\aleph_{\tau}\rightarrow\mathbb{R}\). The difference operator is denoted by \(\Delta\), where \(\Delta\rho(n)=\rho(n+1)-\rho(n)\). Then, we summarize some of the basic definitions related to discrete fractional calculus as follows:  
The fractional sum of order \(\alpha\) (\(\alpha>0\)) is given by [68]:  
\[\Delta_{\tau}^{-\alpha}\rho(t)=\frac{1}{\Gamma(\alpha)}{\sum}_{m=\tau}^{t- \alpha}\frac{\Gamma(t-m)}{\Gamma(t-m-\alpha+1)}\rho(m),\ t\in\aleph_{\tau+ \alpha}. \tag{9}\]  
The Caputo-like delta difference of order \(\alpha\) is defined by [68]:  
\[{}^{C}\Delta_{\tau}^{\alpha}\rho(t)=\Delta_{\tau}^{-(n-\alpha)}\Delta^{n}\rho (t)=\frac{1}{\Gamma(n-\alpha)}{\sum}_{m=\tau}^{t-(n-\alpha)}\frac{\Gamma(t-m )}{\Gamma(t-m-n+\alpha+1)}\Delta^{n}\rho(m),\ t\in\aleph_{\tau+n-\alpha},\ n= \lfloor\alpha\rfloor+1. \tag{10}\]  
The delta fractional difference equation of order \(\alpha\) is represented by [69]:  
\[{}^{C}\Delta_{\tau}^{\alpha}\rho(t)=f(t+\alpha-1,\ \rho(t+\alpha-1)), \tag{11}\]  
The equivalent discrete fractional integral is given by [70]:  
\[y(l)=\rho_{0}(t)+\frac{1}{\Gamma(\alpha)}{\sum}_{m=\tau+n-\alpha}^{t-\alpha} \frac{\Gamma(t-m)}{\Gamma(t-m-\alpha+1)}\times f(m+\alpha-1,\ \rho(m+\alpha-1)),\ t\in\aleph_{\tau+n}. \tag{12}\]  
Note that the initial iteration in this case is [71]:  
\[\rho_{0}(t)={\sum}_{k=0}^{n-1}\frac{\Gamma(t-\tau+1)}{k!\Gamma(t-\tau-k+1)} \Delta^{k}\rho(\tau). \tag{13}\]  
The non-modular fractional sine chaotic map is given by [72]:  
\[x(i)=x(0)+\frac{1}{\Gamma\
u}{\sum}_{j=1}^{n}\frac{\Gamma(i-j+v)}{\Gamma(i-j+ 1)}r_{1}sin(\pi r_{1}) \tag{14}\]  
The proposed MCC-MF sine map is designed based on the concept of a cascade chaotic system. The fractional chaotic map is given by [68; 69; 70; 71; 72] and the final mathematical model is given by:\[x(i)=x(0)+\frac{1}{\Gamma\,v_{i}}{\sum_{j=1}^{n}}\frac{\Gamma(i-j+v_{i})}{\Gamma( i-j+1)}r_{1}sin(\pi r_{2}sin(\pi r_{3}sin(\pi r_{4}sin(\pi x(j-1))))) \tag{15}\]  
where \(r_{1},r_{2},r_{3}\) and \(r_{4}\) are the control parameters and \(x(0)\) is the initial condition of the proposed map. Using more than one parameter of the sine map gives a high Lyapunov exponent (LE) value, high chaotic range and a large key space. The block diagram of the proposed MCC-MF sine map is shown in Figure 2. The proposed MCC-MF sine map consists of four fractional chaotic sine maps connected in concatenated form with different secret parameters. The modular function is used to improve the chaotic property based on the continuity of the map output. The effect of the fractional order on the chaotic map can be shown in Figure 3, where Figure 3a-d describe the output series, the bifurcation diagram (BD), and the Lyapunov value. Figure 3: _Cont._  
Figure 2: Proposed MCC-MF sine map block diagram. The NIST test suit, which consists of 16 statistical tests, is used to determine the randomness of the proposed MCC-MF sine map.