## 3 Experiment Results and Analysis
### Security Analysis
Therefore, both the encrypted and marked-encrypted images exhibit distinct statistical characteristics from the original image. Similar results for the histogram analysis were found for all the test images. 3. _The \(\chi^{2}\) test_ The \(\chi^{2}\) values are computed for the original, the encrypted, and the marked-encrypted images. The \(\chi^{2}\) value is described as follows: \[\chi^{2}=\sum_{i}\left(f_{i}-\frac{n\times m}{256}\right)^{2}/\frac{n\times m }{256},\] (11) where \(i\) is the number of gray levels \((0,\dots,255)\), \(f_{i}\) is the observed frequency count of each gray level, and \(n\times m\) is the image size. It should be noted that the \(\chi^{2}\) test is a measure of how much the observed frequencies deviate from the expected frequencies, which is, in our case, the uniform distribution. Assuming a significant level of 0.05, then the critical value for the given significance level and degrees of freedom is \(\chi^{2}(255,0.05)=293\). The \(\chi^{2}\) values of the different images for Lena and Jetplane are presented in Table 1. These values, the \(\chi^{2}\) statistics, are compared to the critical value to determine whether to reject the null hypothesis that the image is uniformly distributed or not. If the calculated \(\chi^{2}\) statistic for an image is greater than the critical value, the null hypothesis is rejected, indicating that the image is not uniformly distributed. If the calculated \(\chi^{2}\) statistic is less than or equal to the critical value, the null hypothesis is not rejected, and the image is uniformly distributed. From Table 1, we observe that for the original Lena and Jetplane images the \(\chi^{2}\) values are very high--242,173 and 715,669, respectively.However, the \(\chi^{2}\) values obtained by the encrypted and the marked-encrypted versions of Lena and Jetplane images are less than the critical value 293, which implies that the distributions of these images are uniform, thus indicating that the encryption scheme is of good quality. 4. _The correlation test_ Real images typically exhibit a high correlation between adjacent pixels in the horizontal, vertical, or diagonal directions. However, an effective encryption scheme should produce a ciphertext with a low correlation between adjacent pixel values [41]. Table 1 presents the correlation coefficients of the original images, Lena and Jetplane, as well as their respective encrypted and marked-encrypted versions obtained using our proposed RDHEI method. The correlation coefficients of the original images are close to 1, indicating a high correlation among adjacent pixels. Specifically, the correlation coefficients for Lena and Jetplane are 0.9710 and 0.9357, respectively. However, the encrypted and marked-encrypted images exhibit significantly low correlation coefficients near 0, indicating effective suppression of the correlation between adjacent pixel values. Similar results were obtained for all the test images. Furthermore, the correlation distributions [42] in the horizontal direction of the original images, Lena and Jetplane, along with their respective encrypted and marked-encrypted images, are illustrated in Figure 18. In this figure, the \(x\)-axis corresponds to the pixel intensity at coordinates \((x,y)\) within the image, while the \(y\)-axis represents the pixel intensity of the horizontally adjacent pixel at coordinates \((x+1,y)\). Each point on the plot reflects the relationship between the intensity of a pixel and the intensity of its immediate neighbor in the horizontal direction. A higher density of points around a particular region on the plot means a stronger correlation between the pixel at \((x,y)\) and its adjacent counterpart at \((x+1,y)\). As depicted in Figure 18, the original images exhibit a high correlation between horizontal pixels, whereas the encrypted and marked-encrypted images show no correlation between adjacent pixels. It is worth noting that the correlation distribution is used to assess the degree of similarity between pixel values at nearby locations in the image.