## 4 Experiments and Analysis of Results
### Security
#### 4.4.2 Extraction Correctness  
The legitimate receiver can successfully extract the encrypted secret data \(\mathbf{m}_{i}^{\prime}\) from the marked encrypted image using the authorized key \(K_{si}\), while an attacker remains unable to extract the secret data accurately. Assuming an attacker attempts to decrypt \(\mathbf{m}_{i}^{\prime}\) using a forged key, the extracted secret is compared bit by bit with \(\mathbf{m}_{i}\). If they match, it is marked as 0; otherwise, it is marked as 1. Experimental results in Figure 9(a) show that occurrences of 1 and 0 have equal probabilities \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & **Type** & **Encryption Methods** & \
\begin{tabular}{c} **Relative Expansion** \\ **Rate** \\ \end{tabular} &
\begin{tabular}{c} **Total Expansion Rate** \\ \end{tabular} \\ \hline [PERSON] et al. [14] & VRBE & Stream cipher & 1 & 1 \\ [PERSON] et al. [12] & VRIE & Homomorphic & 256 & 256 \\ [PERSON] et al. [21] & VRAE & Secret sharing & 1 & n \\ [PERSON] et al. [22] & VRAE & Secret sharing & \(1/(r-1)\) & \(n/(r-1)\) \\ [PERSON] et al. [23] & VRAE & Secret sharing & 1 & n \\ [PERSON] et al. [24] & VRAE & Secret sharing & 1 & n \\ Proposed & VRIE + VRAE & Secret sharing & 1 & n \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Comparison of data expansion rates for different algorithms. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Algorithms** & **Encryption Methods** & \multicolumn{3}{c}{**Computational Complexity**} \\ \cline{3-5}  & & **Data Embedding** & **Data Extraction** & **Image Recovery** \\ \hline [PERSON] et al. [4] & AES encryption & \(O(1)\) & \(O(1)\) & \(O(1)\) \\ [PERSON] et al. [8] & LWE encryption & \(O(N^{2})\) & \(O(1)\) & \(O(N^{2})\) \\ [PERSON] et al. [13] & Paillier encryption & \(O(N^{3})\) & \(O(1)\) & \(O(N^{3})\) \\ [PERSON] et al. [19] & [PERSON] Secret sharing & \(O(N)\) & \(O(1)\) & \(O(N\text{log}^{2}N)\) \\ [PERSON] et al. [22] & CFSS Secret sharing & \(O(N)\) & \(O(1)\) & \(O(N\text{log}^{2}N)\) \\ Proposed & MSS Secret sharing & \(O(N)\) & \(O(N^{2}\text{log}^{2}N)\) & \(O(N\text{log}^{2}N)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Computational complexities of different RHD-EI algorithms. of 0.5 and are entirely independent. When the receiver uses the data hiding key \(K_{si}\) for decryption and extraction, the extracted data match the pre-embedding data, demonstrated in Figure 10b. This underscores that without \(K_{si}\), attackers cannot extract valid information from the marked encrypted image, thus fortifying the algorithm against unauthorized access. Only with a valid key can the receiver accurately retrieve the secret data. If an attacker tampers with a marked encrypted image, the algorithm detects such alterations by comparing identifying information before and after embedding. When the algorithm's threshold is set to (3, 4), we forge a random image and perform identifying information extraction alongside any two legitimate marked encrypted images. Subsequently, we compare the extracted data with the pre-embedding identifying information from each image. The result, as shown in Figure 10c, reveals significant differences, indicating the receiver's inability to extract correct identifying information. Choosing any three marked encrypted images for decryption, the experiment demonstrates that the extracted identifying information aligns with the pre-embedding identifying information, as depicted in Figure 10d. When an attack takes place during the transmission of encrypted images from the image owner to the data hider, we manipulate any encrypted image to replicate this process, and involve this image in the subsequent process of embedding and extracting secret data. By analyzing identifying information, we can determine the occurrence of the attack, as illustrated in Figure 10e. Conversely, when no attack is present, the experimental results are depicted in Figure 10f.