# Reversible Data Hiding in Encrypted Image Based on Multi-MSB Embedding Strategy
## 4 Experimental Results and Analysis
### A Detailed Example for the Proposed Method  
In this part, we apply the common image Lena to introduce the performance of the proposed method. Figure 8 shows the results. For the experiment results, the given embedding rate are 0.5 bpp and 0.9 bpp, and the optimal parameters (\(m^{*}\), \(T^{*}\)) are (7, 5) and (6, 4), respectively. Figure 8a is the original image Lena. In Figure 8b, under (\(m^{*}\), \(T^{*}\)) = (7, 5), the upper quarter of the rearranged image is composed of black pixels, and the appearance of the original image can be seen. The other three-quarters of the rearranged image are composed of various white pixels and edged pixels, which looks disordered. Figure 8c is the rearranged image under (\(m^{*}\), \(T^{*}\)) = (6, 4). The rearranged images are encrypted to become noise-like, as shown in Figure 8d,e. It is well-known that each original image has its own histogram characteristic which can be used for retrieving original image. As shown in Figure 8f,g, the histograms of the encrypted images are nearly uniformly distributed, which can well protect the content of the image to withstand the statistical attack. In addition, we utilize correlation of adjacent pixels and information entropy to illustrate  
Figure 8: Experimental results for the proposed method. (**a**) Original image Lena. (**b**) Rearranged image with (\(m\), \(T\)) = (7,5). (**c**) Rearranged image with (\(m\), \(T\)) = (6,4). (**d**) The encrypted image of Figure 8b. (**e**) The encrypted image of Figure 8c. (**f**) Histogram of Figure 8d. (**g**) Histogram of Figure 8e. (**h**) Marked image with 0.5 bpp. (**i**) Marked image with 0.9 bpp. (**j**) Reconstructed image under 0.5 bpp with only encryption key. (**k**) Reconstructed image under 0.9 bpp with only encryption key. (**l**) Reconstructed image with both encryption key and data-hiding key. image encryption security further. To test the correlation between two adjacent pixels, 10,000 pairs of two horizontally, vertically, and diagonally adjacent pixels from an image are randomly selected, respectively, and then the corresponding correlation coefficient \(r_{xy}\) of each pair is calculated using the following equations:  
\[cov(x,y)=E[(x-E(x))(y-E(y))] \tag{19}\]  
\[r_{xy}=\frac{cov(x,y)}{\sqrt{D(x)}\sqrt{D(y)}} \tag{20}\]  
where \(x\) and \(y\) are values of the two adjacent pixels in the image, \(E(x)\) is the mean value of \(x\), and \(D(x)\) is the variance of \(x\). If \(r_{xy}\) is close to 1, \(x\) and \(y\) have a strong correlation. Otherwise, if \(r_{xy}\) is close to 0, it indicates there is no correlation between \(x\) and \(y\). Table 2 shows the comparison of adjacent pixel correlation between the original image and Figure 8d,e. It is obvious that the values of pixel correlation are extremely close to 0. It indicates encryption in our method is able to highly disorganize correlation of pixels in three directions. The entropy is the most outstanding feature of the randomness. The information entropy \(H(X)\) of a message source \(X\) can be calculated as  
\[H(X)=-\sum_{i=0}^{L-1}P(x_{i})\log_{2}P(x_{i}) \tag{21}\]  
where \(X\)={\(x_{0}\), \(x_{1}\),..., \(x_{L-1}\)} and \(P(x_{i})\) is the probability of \(x_{i}\). If the information entropy is close to the maximum value, it means the encrypted image acquires excellent properties of randomness. For a grayscale image, \(x_{i}\) is an integer with the range of [0,255]. More specifically, each pixel in gray scale image can be encoded by 8 binary bits, and the ideal value of information entropy is 8. Information entropies of Figure 8d,e are 7.9831 and 7.9754, respectively. In summary, excellent histogram distribution, correlation coefficient, and entropy of encrypted images indicate the encryption method is secure in the proposed method. After image encryption, secret bits are embedded into the multi-MSB of pixels in the proposed method, it will greatly change the value of pixels, but there is no need to consider the visual damage of the image due to data embedding in the encrypted image.