## 4 Experiments and Analysis of Results
### Security
#### 4.4.3 Uncertainty
This outcome results from the algorithm's process of embedding secret data into the image, disrupting its pixel correlation, and subsequently performing XOR encryption on the encrypted images, further diminishing the pixel correlation coefficient to nearly zero. Figure 11: Results of pixel distribution in experimental images. (**a**) Histogram of the original image. (**b**) 3D pixel distribution of the original image. (**c**) Histogram of the encrypted image. (**d**) 3D pixel distribution of the encrypted image. (**e–j**) Pixel distribution and of the marked encrypted images I, II, III. (**k**) Histogram of the original image. (**l**) 3D pixel distribution of the recovered image. In practical application scenarios, algorithms often face unknown attacks. Therefore, we use Spearman's rank correlation coefficient (SRCC) and Kendall rank correlation coefficient (KRCC) [38] to test the monotonicity of images, further assessing whether attackers can extract information about the original images. Experimental results are shown in Table 11, where [38] indicates the following: when SRCC and KRCC are close to \(1\) or \(-1\), there is a high correlation between images, while values close to \(0\) indicate low correlation. In our algorithm, both the SRCC and KRCC values of the marked encrypted images \(\mathbf{IM}_{i}\) are close to \(0\). This can be attributed to the encrypted secret data displaying pseudo-random characteristics before embedding, while the algorithm maintains the ciphertext feedback properties of the MSS secret sharing scheme, along with certain scalability attributes. As a result, our algorithm demonstrates robust security. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Direction**} & \multirow{2}{*}{**Original Image**} & \multirow{2}{*}{**Encrypted**} & \multirow{2}{*}{**Marked Encrypted**} & \multirow{2}{*}{**Marked Encrypted**} & \multirow{2}{*}{**Marked Encrypted**} & \multirow{2}{*}{**Anveraged of the Marked Encrypted**} \\  & & & **Image I** & & **Image II** & **Image III** \\ \hline horizontal & 0.9582 & 0.0015 & 0.0019 & 0.0027 & 0.0022 & 0.0023 \\ vertical & 0.9656 & 0.0016 & 0.0033 & 0.0025 & 0.0026 & 0.0028 \\ diagonal & 0.9382 & 0.0021 & 0.0015 & 0.0019 & 0.0027 & 0.0021 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Correlation coefficient of test images. Figure 12: Scatter plot of correlation coefficient of test images. (**a**) Horizontal distribution of the original image. (**b**) Horizontal distribution of the marked encrypted image. (**c**) Vertical distribution of the original image. (**d**) Vertical distribution of the marked encrypted image. (**e**) Diagonal distribution of the original image. (**f**) Diagonal distribution of the marked encrypted image. The information entropy of an image represents the balance in the occurrence frequency of each grayscale level of its pixels. A higher information entropy value indicates a greater amount of information contained in the image and better defense against statistically based attacks. The information entropy is calculated using the following equation:  
\[H=-\sum_{i=1}^{N}\Pr\left(\sigma_{i}\right)\log\Pr\left(\sigma_{i}\right) \tag{23}\]  
where \(N\) represents the number of pixel values, \(\sigma_{i}\) denotes the \(i\)-th pixel value, and \(\Pr(\sigma_{i})\) signifies the probability of the \(i\)-th pixel value occurring. For a grayscale image with an 8-bit depth, where the pixel values \(\sigma_{i}\) range from 1 to 255, the information entropy reaches its peak when the probability \(\Pr(\sigma_{i})\) equals 1/256, resulting in a maximum entropy value of 8. In our experiment, we assessed the information entropy of the pertinent images, with the results detailed in Table 12. The average information entropy of the marked encrypted images was determined to be 7.9914, closely approaching the theoretical maximum of 8, indicating a uniform distribution of pixel values. This algorithm effectively shields the image and secret data from statistic-based attacks, demonstrating robust security.