## 3 Our Proposed Scheme
### Encryption Scheme
\[\text{NPCR}_{E}=\left(1-\frac{1}{2^{\log 2L}}\right)\times 100\%\quad, \tag{9}\]

\[\text{UACI}_{E}=\frac{1}{L^{2}}\left(\frac{\sum_{i=1}^{L-1}x(x+1)}{L-1}\right) \times 100\%\quad, \tag{10}\]

where \(L\) is the gray level of test images. If the test images are 8-bit gray levels, the expectation of NPCR is 99.6094%, and the expectation of UACI is 33.4635%. In Table 5, the average of NPCR is higher than 99.6094% and the average of UACI is close to 33.4635%. The test result means that our proposed image cryptosystem is secure enough to resist the differential attack. ### Statistical Analysis

#### 4.3.1 Histogram Analysis

Statistical indiscernibility is one of the important indexes to measure the quality of a cipher image. A good cipher image, which can resist statistical analysis, should have a uniform histogram at each gray level. The histogram analysis results are shown in Figure 5. The test plain images of 'Lena', 'Baboon', 'Pepper', 'Airplane' and 'Lake' are shown in Figure 5a-e, and the corresponding histograms in Figure 5f-j. The corresponding ciphers are in Figure 5k-o, and the corresponding histograms of these cipher images are in Figure 5p-t. We use chi-squared test to evaluate the uniformity of encrypted image's histogram. When we set the significance level \(\alpha=0.05\), chi-squared test results of cipher images are given in Table 6. According to our histogram analysis results, our proposed cryptosystem has good diffused property to resist the statistical attack. #### 4.3.2 Correlation Coefficient

Correlation coefficient analysis is a measurement of the correlation among the adjacent pixels in the image. The encryption process should break the correlation of adjacent pixels; therefore, the less correlation among adjacent pixels in the cipher image, the better the security. The correlation coefficient is given in Equation (11). \[r_{ab}=\frac{\text{cov}(a,b)}{\sqrt{D(a)D(b)}}\;, \tag{11}\]

where \(a\) and \(b\) are two gray values of adjacent pixels, and

\[\text{E}(a)=\frac{1}{N}\sum_{i=1}^{N}a_{i}, \tag{12}\]

\[\text{D}(a)=\frac{1}{N}\sum_{i=1}^{N}\left(a_{i}-\text{E}(a)\right)^{2}, \tag{13}\]

\[\sigma_{u}=\sqrt{\frac{(Q+2)\left(Q^{2}+2Q+3\right)}{18(Q+1)^{2}QH}}. \tag{8}\]0 \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**NPCR(\%)**} & \multicolumn{4}{c}{**UACI(\%)**} \\ \cline{2-10}
**Image** & **Max** & **Min** & **Average** & \
\begin{tabular}{c} **\(\mathcal{N}_{\mathbf{0.05}}^{*}\)** \\ **99.5893\%** \\ \end{tabular} & **Max** & **Min** & **Average** &
\begin{tabular}{c} **\(\mathcal{U}_{\mathbf{0.05}}^{*-}=\mathbf{33.3730\%}\)** \\ **\(\mathcal{U}_{\mathbf{0.05}}^{*+}=\mathbf{33.5541\%}\)** \\ \end{tabular} \\ \hline Lena & 99.6437 & 99.5731 & 99.6159 & Pass & 33.5589 & 33.4246 & 33.4846 & Pass \\ Baboon & 99.6407 & 99.5816 & 99.6168 & Pass & 33.6188 & 33.4176 & 33.4808 & Pass \\ Pepper & 99.6453 & 99.5777 & 99.6217 & Pass & 33.5854 & 33.4302 & 33.5017 & Pass \\ Airplane & 99.6425 & 99.5926 & 99.6138 & Pass & 33.5315 & 33.4212 & 33.4963 & Pass \\ Lake & 99.6488 & 99.5853 & 99.6170 & Pass & 33.5369 & 33.4102 & 33.4725 & Pass \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The results of number of pixels change rate (NPCR) and unified average changing intensity (UACI). For this test, 10,000 pairs of adjacent pixels have been randomly selected from plain images and corresponding cipher images. The test results are shown in Table 7. The correlation distributions are shown in Figure 6 and the rows (a)-(e) corresponds to images of 'Lena', 'Baboon', 'Pepper', 'Airplane' and 'Lake', respectively; the column (1) shows corresponding plain images; columns (2)-(4) \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Image**} & \multirow{2}{*}{**\(p\)-Value**} & \multicolumn{2}{c}{**Decision (\(H=0\) _or \(H=1\)_)**} \\  & & \(\alpha=0.05\) \\ \hline Lena & 0.25502 & Pass \\ Baboon & 0.7198 & Pass \\ Pepper & 0.97584 & Pass \\ Airplane & 0.67795 & Pass \\ Lake & 0.15236 & Pass \\ \hline \hline \end{tabular}
\end{table}