## 3 Proposed Method
### Image Encryption
\[Dim=D(i),\;2^{i}=dim(i=0\ldots 7) \tag{5}\]4 PSNR of recovered image comparison between the proposed method and previous methods when Tifany is cover image [5; 8; 10; 16; 17; 19; 24]. From Table 1, the Shannon entropy values of the two methods are similar, both close to 8. Based on our experimental results, except for image F16, the proposed method in this paper demonstrates superior Shannon entropy compared to [14], thereby enhancing algorithm security. 2. Histogram analysis

Figure 16a-c present the histograms of the original images of Tiffany, Peppers, and Barbara, respectively. Figure 16d-f display the histograms of their encrypted versions, while Figure 16g-i show the histograms of the marked images. As we can see from Figure 16a-c, the pixels are unevenly distributed, the information of the images can be easily analyzed, and the correlation between pixels is strong. As can be easily seen in Figure 16d-i, it is difficult to analyze the related information of the images, the correlation between the pixels is weak, and the security of the image is strong. It can be concluded that the pixel correlation between the encrypted image obtained by the proposed method and the marked encrypted image is weak; that is, the proposed scheme can resist statistical analysis attacks. 3. KPA attack

This part of the experiment was used to check whether the encrypted image in the method could intuitively access the main content of the original image after being subjected \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline
**Image** & **Barbara** & **Boat** & **F16** & **Goldhill** & **House** & **Lake** & **Peppers** & **Splash** & **Tiffany** \\ \hline Original image & 7.4664 & 7.1914 & 6.7140 & 7.4778 & 7.2336 & 7.4845 & 7.5925 & 7.2541 & 6.6009 \\ Method [14] & 7.9965 & 7.9947 & 7.9915 & 7.9977 & 7.9861 & 7.9981 & 7.9914 & 7.9901 & 7.9899 \\ Proposed method & 7.9979 & 7.9956 & 7.9912 & 7.9985 & 7.9869 & 7.9983 & 7.9944 & 7.9924 & 7.9913 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of Shannon entropy of encrypted images of two methods. Figure 16: Histogram of different images obtained in this paper. to the attack method proposed in [26]. From Figure 17, it can be seen that the encrypted images of the eight standard test images examined can visually reveal the main content of the image contour after the KPA attack, which indicates that the BPCX encryption method has difficulty resisting some of the KPA. To demonstrate the proposed method's efficacy in resisting KPA attack, the attack method proposed in [26] was evaluated on the encrypted image of the same original images. As shown in Figure 18, the encrypted images after the attack are all noise-like images that do not reveal any information about the content of the images. This suggests that the proposed method can withstand the KPA attack proposed by [26]. ### Embedded Capacity

The embedding capacity of the proposed method is calculated as follows:

\[Capacity=\sum_{i=1}^{7}Cap(2^{i}), \tag{9}\]

Figure 17: The results of the attack test on eight standard test images encrypted by the BPCX method without pixel bit modulation: (**a**) Tiffany, (**b**) Splash, (**c**) Barbara, (**d**) F16, (**e**) Goldhill, (**f**) House, (**g**) Lake, (**h**) Peppers. Figure 18: The results of the proposed method in this paper for the attack test after encrypting eight standard test images: (**a**) Tiffany, (**b**) Splash, (**c**) Barbara, (**d**) F16, (**e**) Goldhill, (**f**) House, (**g**) Lake, (**h**) Peppers. where \(Cap(2^{i})\) (\(i=1,2,3,4,5,6,7\)) denotes the embedding capacity provided by each partition category of the cover image. \(Cap(2^{i})\) is calculated in two cases:

Case 1: When \(2^{i}>2\), the embedded capacity vacated by each block of size \(dim\) is calculated as follows:

\[Cap(2^{i})=\sum_{j=1}^{n}(2^{i}\times 2^{i}-1)\times md_{j}-6 \tag{10}\]

where \(n\) denotes the number of partition in the cover image with \(dim\) (\(dim=2^{i}\)) size, and where \(md_{j}\) is the shared MSBs in the \(j\)th \(dim\ \times\ dim\) sized pixel block.