## 4 Simulation Results and Attack Test
### Statistical Analysis
#### 4.2.2 Correlation Analysis  
In general, the coefficient correlation between two adjacent different pixels in a variety of plaintext images are quite strong, which means it is necessary to eliminate such correlation by using an efficient and secure encryption algorithm. The aim of a low coefficient correlation among the neighboring pixels should be achieved in the corresponding cipher image after the encryption process for the plaintext image. The formula for calculating the correlation between two different contiguous pixels is displayed below as (5). We choose the pairs of pixels in the plaintext image and the cipher image in three different directions: horizontal, vertical, and diagonal. The coefficient correlation of the plaintext image and the corresponding cipher image in three directions were obtained by using Formula (5). The test results of the correlation in three directions for the plaintext images and the images after encryption are shown in Table 1. The consequences of the correlation coefficients  
Figure 11: Histograms. (**a,b**) are histograms of the Lena image and the encrypted Lena image. (**c,d**) are histograms of the Cameraman image and the encrypted Cameraman image. (**e,f**) are histograms of the Pepper image and the encrypted Pepper image. (**g,h**) are histograms of the Baboon image and the encrypted Baboon image.  
of the comparison among the schemes in [20; 34; 35] are shown in Table 2. From the displayed data, we know the results of the correlation are close to zero in the cipher image, which means the correlations are significantly reduced. Moreover, it is obvious that our encryption scheme performed better compared with the other schemes mentioned before.  
\[p_{xy}=\frac{E[x-E(x)][y-E(y)]}{D(x)D(y)} \tag{5}\]  
The distribution of the contiguous pixels for the plaintext image and the encrypted image are revealed in Figure 12, to avouch the randomness of the simulation test, we stochastically select nearly 10,000 pairs of pixels from the image in three directions. Through the analysis of the figure which shows the distribution, it is clear that the values of the adjacent pixels in all three directions are completely randomized after the encryption. This illustrates the reliable defensiveness under the statistical attack, which means for the attackers, their attempts to gain statistical information from the cipher images are futile. In conclusion, our scheme performed well through the statistical analysis.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Horizontal** & **Vertical** & **Diagonal** \\ \hline Lena & 0.9449 & 0.9728 & 0.9210 \\ Encrypted Lena & \(-\)0.0023 & \(-\)0.0017 & \(-\)0.0022 \\ Cameraman & 0.9330 & 0.9592 & 0.9074 \\ Encrypted Cameraman & \(-\)0.0023 & 0.0035 & \(-\)0.0053 \\ Pepper & 0.9633 & 0.9706 & 0.9360 \\ Encrypted Pepper & 0.0044 & \(-\)0.0008 & 0.0019 \\ Baboon & 0.7985 & 0.7419 & 0.6893 \\ Encrypted Baboon & \(-\)0.0042 & 0.0010 & \(-\)0.0010 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Correlation coefficients of different images.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & **Horizontal** & **Vertical** & **Diagonal** \\ \hline Lena & 0.9449 & 0.9728 & 0.9210 \\ Our Encrypted Lena & \(\mathbf{-0.0023}\) & \(\mathbf{-0.0017}\) & \(\mathbf{-0.0022}\) \\ [20] & 0.0064 & 0.0035 & 0.0035 \\ [34] & 0.0034 & \(-\)0.0076 & 0.0062 \\ [35] & 0.0024 & \(-\)0.0086 & 0.0402 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Correlation coefficients of the Lena images encrypted with diverse schemes.  
Figure 12: _Cont._