## 3 Experimental Results and Analysis
### Security Analysis
#### 3.2.5 Entropy Analysis  
Entropy is a physical quantity to express the degree of uncertainty in a system. Since [PERSON] [39] introduced the concept to information theory, it is referred to as the Shannon entropy or information entropy.  
For an 8-bit grayscale image, \(Z\), the definition of Shannon entropy is shown in Equation (16), where \(P(z_{i})\) represents the probability of the \(i\)-th gray level \(z_{i}\) occurring in the image:  
\[H(Z)=-\sum_{i=0}^{255}P(z_{i})log_{2}[P(z_{i})] \tag{16}\]  
If \(Z\) is an ideal random image, \(P(z_{i})=1/256\) and \(H(Z)=8\). Therefore, the Shannon entropy for an encrypted 8-bit image is targeted to 8, representing the success of image encryption. Table 8 lists the Shannon entropy for the plaintext image and the encrypted one. The statistics demonstrate that our scheme is so effective that it can produce Shannon entropy for the encrypted RGBE image close to 8 in all channels.  
Furthermore, [PERSON] et al. [40] proposed the local Shannon Entropy (LSE) to overcome the weakness of the (global) Shannon entropy and introduced a process to conduct the statistical test. For an 8-bit grayscale image, \(Z\), the LSE can be computed in Equation (17), which averages the information entropy on \(k\) randomly chosen non-overlapping blocks  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Channel**} & \multicolumn{3}{c}{**Plaintext Image**} & \multicolumn{3}{c}{**Encrypted Image**} \\ \cline{2-7}  & \(\mathbf{H}\) & \(\mathbf{V}\) & \(\mathbf{D}\) & \(\mathbf{H}\) & \(\mathbf{V}\) & \(\mathbf{D}\) \\ \hline R & 0.869519 & 0.886546 & 0.806847 & \(-\)0.001245 & 0.000738 & \(-\)0.001069 \\ G & 0.849231 & 0.871814 & 0.780838 & 0.001059 & 0.000908 & \(-\)0.001039 \\ B & 0.857073 & 0.877966 & 0.788200 & 0.001848 & 0.003866 & \(-\)0.002014 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Adjacent correlation coefficients for the test image “memorial”.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Channel** & **Plaintext Image** & **Encrypted Image** \\ \hline Red & 6.96112745 & 7.99949461 \\ Green & 6.91334510 & 7.99950031 \\ Blue & 5.46477440 & 7.99948285 \\ Exponent & 2.85423907 & 7.99949461 \\ \hline \hline \end{tabular}
\end{table}
Table 8: Shannon entropy values of the image “memorial” in the plain and encrypted images.  
Figure 9: Adjacent correlation coefficients are close to zero for all 20 stego test images in the (**a**) horizontal, (**b**) vertical, and (**c**) diagonal directions.  
each of which contains \(n\) pixels. If \((k,\ n)=(30,\ 1936)\), the ideal value of LSE is 7.9025. If the significance level, \(\alpha\), is set as 0.05, the LSE of a ciphered image should be within the range of [7.9019, 7.9030] in order to pass the statistical test:  
\[\overline{H_{k,\ n}}(Z)=\sum_{i=1}^{k}\frac{H(B_{i})}{k} \tag{17}\]  
Figure 10 shows the results of LSE for all 20 stego ciphered images. The figure indicates that only two local Shannon entropy values are not within the range to pass the test, indicating that the pass rate is as high as 97.5%. The global and the local Shannon entropy test confirm that our algorithm generates stego encrypted images exhibiting both global and local randomness; thereby, they are capable of resisting the entropy attacks.