## 3 Chaos Synchronization Attribution-Based Encryption Scheme  
In this section, we give the detailed algorithm of our brand-new system, which combines PF-CP-ABE with chaos synchronization. The synchronization characteristic in PF-CP-ABE is utilized to give another approach to get the secret number. Although PF-CP-ABE greatly reduces the complexity and computation of CP-ABE, if there are a lot of data and they keep being sent, the users should also employ the algorithm every time when receiving the data. This is quite ineffective, so we applied another algorithm to improve it. Due to it combining the concept of ABE with synchronization characteristics of the chaos system, we have given it the name Chaos Synchronization Attribute-Based Encryption (CS-ABE), which can be divided into four parts, as shown below. Set up. An elliptic curve \(E_{p}(a,b)\) over a finite field \(GF(p)\) of order \(r\) with generator \(g\) is agreed upon. The point \(g\) generates a cyclic subgroup in \(E_{p}(a,b)\). In addition, a hash function \(\{0,1\}^{*}\to Z_{r}^{*}\) is defined to map every user's \(ID\) to \(Z_{r}\), and every user's \(ID\) is unique, which means \(H(ID)_{user1}\
eq H(ID)_{user2}\). The trust authority, which can be a broker, randomly selects an integer \(t\in Z_{r}\) as the authority's main private key and then computes \(tg\) to be the main public key. At the same time, the authority randomly selects an integer \(k_{i}\in Z_{r}\) for every attribute \(i\) and computes \(PK_{i}=k_{i}g\) to be its public key. The data owner and user hold a chaos master and slave system, respectively, and select \(\alpha\) satisfying \(|1-\alpha|<1\); \(c\) satisfying \(|0.99-0.01\alpha|<1\); and initial conditions such as \(x_{1m}\), \(x_{2m}\), \(x_{3m}\), \(x_{1s}\), \(x_{2s}\), and \(x_{3s}\). The master data owner hold system is as below:  
\[ \begin{array}{l}x_{1m}(k+1)=0.99x_{1m}(k)+0.01x_{2m}(k)\\ x_{2m}(k+1)=0.028x_{1m}(k)+0.999x_{2m}(k)-0.001x_{1m}(k)x_{3m}(k)\\ x_{3m}(k+1)=0.997x_{3m}(k)+0.001x_{1m}(k)x_{2m}(k).\end{array}  \tag{17}\]  
The slave data user hold system is as below:  
\[ \begin{array}{l}x_{1s}(k+1)=0.99x_{1s}(k)+0.01x_{2s}(k)\\ x_{2s}(k+1)=0.028x_{1s}(k)+0.999x_{2s}(k)+u(k)\\ x_{3s}(k+1)=0.997x_{3s}(k)+0.001x_{1s}(k)x_{2s}(k).\end{array}  \tag{18}\]  
(1) Key generation  
A private key of each attribute (topic name) \(i\) is generated for each user \(ID\), and the authority computes the formulas below to get their corresponding private keys and record them. \[SK_{i,ID}=\ k_{i}+H(ID)t \tag{19}\]  
Figure 3: The response of the error function of the master –slave chaos system. 2 Encryption  
The plain text is first mapped to a point \(M\) on the elliptic curve \(E_{p}(a,b)\). State variables of the chaos system are selected to generate an integer \(s\in Z_{r}\) called a secret number, and then compute  
\[C_{0}=M+sg. \tag{20}\]  
The encryption algorithm is associated with an access structure and does not need any polynomials. The data owner defines the access structure and transforms it to an \(n\times l\) LSSS matrix, and then randomly selects a vector \(v\in Z_{r}^{l}\) with \(s\) as its first entry and lets \(\lambda_{x}\) denote \(N_{x}\cdot v\), where \(N_{x}\) is row \(x\) of \(N\). They also randomly select a vector \(u\in Z_{r}^{l}\) with \(0\) as its first entry and let \(\omega_{x}\) denote \(N_{x}\cdot u\). The ciphertext would be  
\[C_{1,x}=\lambda_{x}g+\omega_{x}PK_{p(x)},\ C_{2,x}=\omega_{x}g,\ \forall x \tag{21}\]  
(3) Decryption  
The user transmits their \(ID\) and \((C_{2,x},\rho(x))\) to the authority, and lets the authority verify its identity. If the authority confirms that the user is valid, it secretly sends back a result according to each \((C_{2,x},\rho(x))\). The result is computed as below:  
\[\sum C_{2,x}SK_{\rho(x),ID}=\sum\bigl{(}\omega_{x}g\bigl{(}k_{\rho(x)}+H(ID)t \bigr{)}\bigr{)}=\sum\bigl{(}\omega_{x}k_{\rho(x)}g+\omega_{x}H(ID)tg\bigr{)}. \tag{22}\]  
With the above result, the user can then compute  
\[