## 2 Algorithms
### Chaos Synchronization  
In order to reduce the time required for the transmission of certification data and maintain its security of PF-CP-ABE, we propose Chaos Synchronization Attribute-Based Encryption (CS-ABE). As is well-known, the chaos system is a complex nonlinear system and possesses properties such as a broadband noise-like waveform, and is sensitive to initial values like a butterfly effect, etc. The state response does not converge and does not diverge and is limited to strange attractors with a random-like characteristic. These properties offer some advantages for applications in secure communication. Since the concept of the master-slave system emerged and research on the synchronization controller of the chaos system was launched [17], there has been extensive research on it. There are many different design methodologies for synchronization controllers, such as the sliding mode controller [18], adaptive controller [19], etc., but most of them applied in secure communication are analog chaos systems. In this study, since we dealt with digital information, it was necessary to transform it to a discrete  
Figure 1: Access structure. chaos system so that we could discuss and implement a relevant control method, and then apply it to the design of our CS-ABE algorithm. There are many types of continuous chaos systems, the most well-known of which is the Lorentz system. This paper uses the Lorenz equation and its dynamics can be described as follows:  
\[ \begin{array}{c}\dot{x_{1}}(t)=-ax_{1}(t)+ax_{2}(t)\\ \dot{x_{2}}(t)=\ cx_{1}(t)-x_{2}(t)-x_{1}(t)x_{3}(t)\\ \dot{x_{3}}(t)=\ -bx_{3}(t)+x_{1}(t)x_{2}(t).\end{array}  \tag{5}\]  
In consequence, in this paper, we will describe the approach for transforming a continuous system model to a discrete system one. The dynamic equation of the continuous system can be described as follows:  
\[\dot{x}(t)=\ Ax+Bg(x(t)), \tag{6}\]  
where \(g(x(t))\in R^{n}\) is a nonlinear vector. \(B\in R^{n\times m}\), \(A\), and \(B\) are controllable, so the optimal discrete time system can be matched with System (5) as follows:  
\[x_{d}(k+1)T=\ Gx_{d}(kT)+Hg(x_{d}(kT)), \tag{7}\]  
where \(G=e^{AT}\), \(H=[G-I_{n}]A^{-1}B\)[20], and \(T\) is the sampling time. We can rearrange (5) with a matrix representation satisfying (6):  
\[\dot{x}(t)=\left[ \begin{array}{c}\dot{x_{1}}(t)\\ \dot{x_{2}}(t)\\ \dot{x_{3}}(t)\end{array} \right]=\left[ \begin{array}{ccc}-a&a&0\\ c&-1&0\\ 0&0&-b\end{array} \right]x(t)+\left[ \begin{array}{ccc}0&0\\ 1&0\\ 0&1\end{array} \right]\left[ \begin{array}{c}-x_{1}x_{3}\\ x_{1}x_{2}\end{array} \right]. \tag{8}\]  
According to (7), by selecting the sampling time \(T=0.0001s\), \(a=10\), \(b=\frac{8}{3}\), and \(c=28\), we get the discrete system of (5) as  
\[ \begin{array}{c}x_{d}(k+1)T=\ \left[\begin{array}{c}x_{d1}(k+1)T\\ x_{d2}(k+1)T\\ x_{d3}(k+1)T\end{array}\right]=\left[\begin{array}{ccc}0.990&0.010&0\\ 0.028&0.999&0\\ 0&0&0.997\end{array}\right]x_{d}(kT)+\\ \left[\begin{array}{ccc}0&0\\ 0.001&0\\ 0&0.001\end{array}\right]\left[\begin{array}{c}-x_{d1}(kT)x_{d3}(kT)\\ x_{d1}(kT)x_{d2}(kT)\end{array}\right].\end{array}  \tag{9}\]  
The simulation of the discrete chaos system (9) is shown in Figure 2. From this, we can identify some characteristics, such as strange attractors and the unpredictability of its random-like signal. Figure 2: Strange attractors of the discrete Lorenz system (9). For secure communication application, a synchronization controller is needed to synchronize master and slave chaos systems. When both systems are synchronized with each other, they can be applied to our cryptographic algorithm. We used a sliding mode controller, because it has a better robustness and fewer control parameters, to ensure the synchronization. Similar to [21], master and slave systems were designed as follows:  
\[ \begin{array}{c}x_{1m}(k+1)=0.99x_{1m}(k)+0.01x_{2m}(k)\\ x_{2m}(k+1)=0.028x_{1m}(k)+0.999x_{2m}(k)-0.001x_{1m}(k)x_{3m}(k)\\ x_{3m}(k+1)=0.997x_{3m}(k)+0.001x_{1m}(k)x_{2m}(k)\end{array}  \tag{10}\]  
\[