## 3 Proposed Method
### Image Encryption
\[C_{i} = Trunc(P,md)+e_{i}\] (6) \[C_{i} = Trunc(tag,P,md)+e_{i}\] (7)

Figure 12 shows an example of reconstructed encrypted image pixels, where \(Dim\) = \(D\)(2) in a, \(Dim\) = \(D\)(1) in b, and \(Dim\) = \(D\)(0) in c. By performing the above operations until the end tag is retrieved, the receiver will reconstruct the encrypted image, which previously embeds the data, and the remaining data in \(Q\) is the encrypted embedded data; then, the corresponding data is decrypted based on the type of key employed. Figure 11: Pixel bit reconstruction when all blocks are usable. ## 4 Experimental Result and Analysis

In order to assess the efficacy of the proposed method, a set of experiments was conducted, encompassing evaluations of reversibility, security analysis, and comparison of embedding capacity. For the purpose of comparison of embedding capacity, 10,000 images used in the experiments were downloaded from the BOSSbase1.01 image database [30]. The experiments were carried out on a computer equipped with an Intel i9-10980 HK 2.40 GHz CPU, 64 GB RAM, and MATLAB R2023a. ### Separability and Reversibility

In the experiment, the grayscale image of size 512 \(\times\) 512 shown in Figure 13 was taken as embedded data. Taking the image Tiffany as the cover image, Figure 14 shows the original image, original encrypted image, encrypted image after pixel modulation, marked image, extracted data, and recovered image, respectively. From Figure 14e,\(f\), it can be seen that the extracted data are consistent with the embedded data while the recovered image is identical to the original image. As shown in Figure 14d,b, even if the encrypted original image and embedded data were obtained from the marked image without the corresponding key, the corresponding data could not be decrypted. This indicates that the proposed method achieves separability. Figure 15 shows the embedding rate-PSNR (peak signal-to-noise ratio) comparison on the image Peppers for the existing methods [5; 8; 10; 16; 17; 19; 24]. For any given embedding rate, the PSNR of the proposed method was \(\infty\), which indicates that the proposed method achieves reversibility in recovering the original image. Notably, in the contrasting papers, due to the performance of some papers, which did not reach the corresponding embedding rate, we only drew a few short line segments for the representation of the paper in Figure 15.

\[Dim=D(i),\;2^{i}=dim(i=0\ldots 7) \tag{5}\]0 _Cont._

\[Dim=D(i),\;2^{i}=dim(i=0\ldots 7) \tag{5}\]1 Embedded data. ### Security

1. Shannon entropy

In the RDH-EI method, Shannon entropy is one of the important indexes used to evaluate the performance security of the scheme, and it is usually used to evaluate the amount of information about the randomness of the pixel distribution of the encrypted image. It is calculated through (8):

\[Dim=D(i),\;2^{i}=dim(i=0\ldots 7) \tag{5}\]2

where \(h\) denotes the total number of different symbols in the source, \(\beta\) denotes the \(i\)th symbol, and \(P(\beta_{i})\) denotes the probability of occurrence of the pixel-valued symbol \(\beta_{i}\). For an image with a grayscale of 256 and eight bits depth, the theoretical maximum information entropy is 8. The closer the Shannon entropy of the encrypted image approaches 8, the more uniformly distributed are the pixels, indicating the higher source security and improved encryption performance of the algorithm. Table 1 demonstrates the comparison of Shannon entropy between the encrypted image under the BPCX encryption method presented in [14] and the encryption approach based on pixel bit modulation in this paper. \[Dim=D(i),\;2^{i}=dim(i=0\ldots 7) \tag{5}\]3 Image demonstration of several key stages of the proposed method: (**a**) original image, (**b**) original encrypted image, (**c**) pixel-modulated encrypted image, (**d**) marked image, (**e**) extracted data, (**f**) recovered image.