1. The Grey Histogram Analysis We compare the worldwide distributors grey histogram of the image before and after encryption to analyze the statistical performance. Figure 5(a) shows the grey histogram of the original image and Figure 5(b) shows the grey histogram of the encrypted image. From the two figures, we can see that the original pixel grey values are concentrated on some value, but the pixel grey values after the encryption are scattering in the entire pixel value space, namely, two images have lower similarity. Clearly, it is difficult to use the statistical performance of the pixel grey value to recover the original image. Thereby, our algorithm has strong ability of resisting statistical attack.Figure 5The grey histogram of the original image and the encrypted image. (a) The grey histogram of the original image.

(b) The grey histogram of the encrypted image.4.4.2. Correlation Coefficient Analysis The correlation of the adjacent pixels in original image is very high, an effective encryption algorithm can reduce the correlation of between adjacent pixels. Here, we randomly select 3000 pairs (horizontal, vertical and diagonal) of adjacent pixels from the original image and the encrypted image, then by using the following formulas to calculate the correlation coefficient:E(x)=1N��i=1Nxi,D(x)=1N��i=1N(xi?E(x))2,cov?(x,y)=1N��i=1N(xi?E(x))(yi?E(y)),rxy=cov?(x,y)D(x)��D(y),(10)where x and y are grey value of two adjacent pixels in the image.Figures 6(a) and 6(b) show the correlation of two horizontally adjacent pixels in the original image and that in the encrypted image, where the correlation coefficients are 0.

9432 and 0.1366, respectively. Other results are shown in Table 1. From Figure 6(b) and Table 1, we can see that the correlation coefficient of the adjacent pixels in encrypted image is low, which is close to 0. It follows from Figure 6(b) and Table 1 that the proposed image encryption algorithm has strong ability of resisting statistical attack.Figure 6Correlation of two horizontally adjacent pixels in the original image and in the encrypted image.Table 1Correlation coefficients of two adjacent pixels in two image.4.4.3. Information Entropy It is well known that information entropy can measure the distribution of grey value in the image. We can make sure that the bigger information entropy the more uniform for the distribution of grey value.

The definition of information entropy is as follows:H(m)=?��i=0LP(mi)log2P(mi),(11)where mi is the ith grey value for L level grey image and P(mi) is the emergence probability of mi. The information entropy of an idea random image is 8. For the proposed algorithm, the information entropy is 7.9975. It is very close Carfilzomib to 8. 5. Comparison with Other Encryption Algorithms In this section, we will compare our proposed algorithm with existing chaos-based and DNA-based encryption algorithms.