## 3 Proposed 2D Chaotic Sine Cosine Logistic Map  
Starting from the idea that a chaotic system with a large number of parameters can offer a good security level if the parameters are very sensitive to any small variation, we propose a 2D version of the MLM, which is called a 2D sine cosine logistic map (2 DSCLM). The mathematical model of the 2 DSCLM is given by:  
\[\begin{cases}x_{i}=\alpha(1-\beta|\cos(j)|)x_{i-1}(1-x_{i-1})\text{ with }i=1,\dots,n\text{ and }0<x_{0}<1\\ y_{j}=\delta(1-\varphi|\sin(x_{i})|)y_{j-1}(1-y_{j-1})\text{ with }j=1,\dots,n \text{ and }0<y_{0}<1\end{cases} \tag{10}\]  
where \(\alpha,\delta\in[3.54\dots 4]\) and \(\beta,\varphi\in[0\dots 0.25]\).  
In Figure 1, the bifurcation diagram of the 2 DSCLM is shown. For this purpose, the values of \(\alpha\) and \(\delta\) parameters were increased by a step of \(10^{-4}\) with \(\alpha,\delta\in[3.54\dots 4]\), \(\beta=\varphi=0.02\), and \(x_{0}=y_{0}=0.5\). Then, the produced \(x\) and \(y\) sequences were displayed for 5000 iterations. The results in Figure 1 illustrate that \(x\) and \(y\) sequences are characterized by bifurcations \(\alpha\in[3.57\dots 4]\) and \(\delta\in[3.61\dots 4]\). These results indicated that the 2 DSCLM exhibits chaotic behavior.  
Figure 1: The bifurcation diagram of \(x\) (**a**) and \(y\) (**b**) sequences produced by 2 DSCLM for \(\alpha,\delta\in[3.54\dots 4]\), \(\beta=\varphi=0.02\), and \(x_{0}=y_{0}=0.5\).  
To further validate the chaotic behavior of the 2 DSCLM, the Lyapunov exponent (LE) of 2 DSCLM sequences (\(x\) and \(y\)) can be computed via the following formula [37]:  
\[LE=\lim_{n\to\infty}\left[\frac{1}{n}\sum_{i=1}^{n}\log_{2}\left|\frac{dx_{i+1}}{ dx_{i}}\right|\right] \tag{11}\]  
where \(\log_{2}\) represents the log base 2 function.  
Figure 2 shows the LE values of \(x\) and \(y\) sequences produced via the 2 DSCLM. From this figure, we can see that the values of LE became positive for \(\alpha\in[3.57\ldots 4]\) and \(\delta\in[3.61\ldots 4]\), indicating chaos in 2 DSCLM sequences. In other words, the 2 DSCLM model strongly depends on its control parameters and initial conditions. This property is fundamental for any chaotic system.  
To further validate the chaotic behavior of the 2 DSCLM, \(x\) and \(y\) sequences were computed for specified control parameters and initial values. Then, we slightly modified only one parameter by \(10^{-15}\), and then \(x^{*}\) and \(y^{*}\) sequences were computed using (9). Finally, the original sequences (\(x\) and \(y\)) with (\(x^{*}\) and \(y^{*}\)) were generated using the varied parameter, and the difference in absolute value between these sequences are displayed in Figure 3, which shows 500 iterations of \(x\) and \(y\) sequences obtained using the parameters \(\{\alpha,\beta,\delta,\varphi,x_{0},y_{0}\}=\{3.99,0.021,3.99,0.021,0.5,0.5\}\) as well as \(x^{*}\) and \(y^{*}\) sequences obtained when varying the parameter \(\beta\) by \(\Delta=+10^{-15}\). The results displayed in this figure clearly indicate the high sensitivity of the 2 DSCLM to a slight variation of its control parameters, which confirms the chaotic character of the proposed system. It can also be noticed that the first 50 iterations of the original (\(x\) and \(y\)) and modified sequences (\(y\) and \(y^{*}\)) are identical. Therefore, this shortcoming of the 2 DSCLM should be considered when integrating the proposed chaotic map into information security systems.  
After validating the chaotic behavior of the 2 DSCLM, it can be used in a robust security scheme designed to ensure a high level of security when communicating EPFs via an insecure communication channel in IoMTs.