## 3 Our Proposed Scheme
### Encryption Scheme

Table 6: Histogram uniformity evaluation by chi-squared test. Figure 5: Histograms. (**a–e**) The plain images of ‘Lena’, ‘Baboon’, ‘Pepper’, ‘Airplane’ and ‘Lake’; (**f-j**) the corresponding histograms of these plain images; (**k–o**) the corresponding cipher images by encryption; (**p–t**) the corresponding histograms of these cipher images. correspond to the plain images' distributions of 'horizontal direction','vertical direction' and 'diagonal direction', respectively; column (5) is the corresponding cipher images by encryption, and columns (6)-(8) correspond to the cipher images' distributions of 'horizontal direction','vertical direction' and 'diagonal direction', respectively. ### Key Sensitivity Analysis

Key sensitivity is the basic requirement of image cryptosystem security, and it can reflect the performance of diffusion. For this test, we select 'Lena' to be the test image. Firstly, we encrypt 'Lena' using initial key, which was declared at the beginning of Section 4, and denote the cipher image as C1. C1 was used as the benchmark for comparison experiments, then we modify a parameter or initial value of the chaotic system which are the real keys of our proposed scheme. In the first test, we modify a parameter of PWLCM which is used in generating the rules selection control sequence as \(p=p+10^{-15}\), and denote the corresponding cipher image as \(C2\). In the second test, we modify the initial value of PWLCM which is used in generating key sequence as \(x=x+10^{-15}\), and denote the corresponding cipher image as C3. In the third test, we modify the initial value of hyper chaotic Lorenz system which is used in permutation as \(x_{0}=x_{0}+10^{-15}\), and denote the corresponding cipher image as C4. In the fourth test, we modify the initial value of hyper chaotic Lorenz system as \(y_{0}=y_{0}+10^{-15}\), and denote the corresponding cipher image as C5. The test result is shown in Figure 7. The difference \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Image**} & \multicolumn{2}{c}{**Horizontal**} & \multicolumn{2}{c}{**Vertical**} & \multicolumn{2}{c}{**Diagonal**} \\ \cline{2-7}  & **Plain** & **Cipher** & **Plain** & **Cipher** & **Plain** & **Cipher** \\ \hline Lena & 0.9839 & 0.00025 & 0.9717 & 0.0019 & 0.9589 & 0.00033 \\ Baboon & 0.7670 & 0.0029 & 0.8728 & 0.0021 & 0.7369 & 0.0036 \\ Pepper & 0.9782 & 0.0035 & 0.9765 & \(-0.00028\) & 0.9615 & \(-0.0059\) \\ Airplane & 0.9642 & \(-0.0082\) & 0.9666 & 0.0086 & 0.9380 & 0.0056 \\ Lake & 0.9723 & 0.0043 & 0.9754 & \(-0.0052\) & 0.9573 & 0.0056 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Correlation coefficients. Figure 6: Correlation distributions. Rows (**a**)–(**c**) correspond to images of ‘Lena’, ‘Baboon’, ‘Pepper’, ‘Airplane’ and ‘Lake’, respectively; Column (1) shows the corresponding plain images; Columns (2)–(4) correspond to the plain images’ distributions of ‘horizontal direction’, ‘vertical direction’ and ‘diagonal direction’, respectively; Column (5) is the corresponding cipher images by encryption; Columns (6)–(8) correspond to the cipher images’ distributions of ‘horizontal direction’, ‘vertical direction’ and ‘diagonal direction’, respectively. between C1 and C2 is shown in Figure 7g, the difference between C1 and C3 is shown in Figure 7h, the difference between C1 and C4 is shown in Figure 7i and the difference between C1 and C5 is shown in Figure 7j. We also use NPCR and UACI to quantitatively measure the difference between cipher images. The result is shown in Table 8. According to this test result, our proposed image cryptosystem has enough key sensitivity. ### Information Entropy

#### 4.5.1 Global Shannon Entropy

The degree of uncertainty of a cipher image can reflect the diffusion performance of an image cryptosystem, and the global information entropy is a measurement to indicate the uncertainty degree of the whole image information. The global information entropy is given in Equation (15).