## 2 Synchronization Design and Dynamic Chaos-Based Random Key Generators  
### Chaos Synchronization Design  
In this research, a novel CAES cryptosystem was considered with a synchronized chaos-based dynamic random key generator. Before generating the new dynamic chaos-based random keys, the first problem to be well solved was the chaos synchronization. Therefore, the synchronization problem and the controller design were first formulated so as to achieve synchronization for master-slave hyperchaotic Henon maps [22]. Master hyperchaotic Henon map:  
\[ \begin{split} x_{1}(k+1)&=1.76-x_{2}^{2}(k)-0.1\ x_{ 3}(k)\\ x_{2}(k+1)&=x_{1}(k)\\ x_{3}(k+1)&=x_{2}(k)\\ \end{split}  \tag{1}\]  
Slave hyperchaotic Henon map:  
\[ \begin{split} y_{1}(k+1)&=1.76-y_{2}^{2}(k)-0.1y_{ 3}(k)+u(k)\\ y_{2}(k+1)&=y_{1}(k)\\ y_{3}(k+1)&=y_{2}(k)\\ \end{split} , \tag{2}\]  
where \(x=\left[ \begin{array}{cccc}x_{1}&x_{2}&x_{3}\\ \end{array} \right]^{T}\) and \(y=\left[ \begin{array}{cccc}y_{1}&y_{2}&y_{3}\\ \end{array} \right]^{T}\) denote the state vectors of the master and slave chaotic Henon maps, respectively. \(u(k)\) is the added controller in the (2) slave system to guarantee the random state vector synchronized with the (1) master system. Strange attractors of the master Henon map (1) are shown in Figure 1. Figure 1 shows some of the characteristics of chaotic systems, such as strange attractors and unpredictability to the random-like signal response. To discuss the synchronization controller design, the error state was defined as \(e_{i}(k)=y_{i}(k)-x_{i}(k),i=1,2,3\). This resulted in the following error system:  
\[ \begin{split} e_{1}(k+1)&=-e_{2}(k)(x_{2}(k)+y_{2}( k))-0.1e_{3}(k)+u(k)\\ e_{2}(k+1)&=e_{1}(k)\\ e_{3}(k+1)&=e_{2}(k)\end{split}  \tag{3}\]  
Here, the aim was to design a controller, \(u(k)\), to result in \(\underset{k\rightarrow\infty}{lim}\|e_{i}(k)\|=0,i=1,2,3\). In other words, the states of master-slave hyperchaotic systems described in Equations (1) and (2) could be fully synchronized with random chaos dynamics. To obtain a good robustness, similar to that of the literature [23], the sliding mode control was introduced to design the controller (\(u(k)\)) in order to ensure the exact synchronization. Generally, there are two steps for completing the design of the sliding mode controller (\(u(k)\)). First, it is necessary to design a proper sliding surface (s(\(k\))) and to ensure the state synchronization of the controlled system under this assumption, such that the controlled trajectory of the systems can hit and operate in the sliding mode (i.e., \(s(k)=0\)). Then, it is necessary to design the sliding mode controller (\(u(k)\)) to guarantee that the state trajectory of the controlled states can be forced to the sliding manifold as expected. In order to complete the above design steps, the sliding surface for the sliding manifold was selected as follows:  
\[s(k)=e_{1}(k)+\alpha e_{2}(k)+\beta e_{3}(k) \tag{4}\]  
where \(s(k)\in R\) and \(\alpha,\beta\) are the chosen parameters. If the controlled system can be forced to enter the sliding motion under the effect of \(u(k)\) (in other words, \(s(k)=0\)), then \(e_{1}(k)=-\alpha e_{2}(k)-\beta e_{3}(k)\). By substituting the relation of \(e_{1}(k)=-\alpha e_{2}(k)-\beta e_{3}(k)\) into the error Equation (3) of the master-slave system, it can be rewritten as follows:  
\[\left[ \begin{array}{c}e_{2}(k+1)\\ e_{3}(k+1)\end{array} \right]=\left[ \begin{array}{cc}-\alpha&-\beta\\ 1&0\end{array} \right]\left[ \begin{array}{c}e_{2}(k)\\ e_{3}(k)\end{array} \right]=AE(k) \tag{5}\]  
From Equation (5), obviously, if \(\alpha\) and \(\beta\) are chosen to ensure \(|\lambda_{i}(A)|<1,i=1,2\), then \(E(k)=\left[ \begin{array}{cc}e_{2}(k)&e_{3}(k)\end{array} \right]^{T}\) can converge to zero. Furthermore, as the controlled systems operate in the sliding mode, \(e_{1}(k)=-\alpha e_{2}(k)-\beta e_{3}(k)\) and \(e_{1}(k)=0\) are ensured when \(E(k)=0\).