## 3 Experiment Results and Analysis
### Security Analysis  
Figures 15 and 16 illustrate the proposed RDHEI method for two test images, Lena and Jetplane, respectively. Figure 15a showcases the original Lena image. After reordering the bit planes in the mapped prediction error image, the allocation of space for data embedding, and the insertion of auxiliary information in the corresponding bit planes, the resulting image displayed in Figure 15b is obtained. The encrypted image, obtained by the content-owner using their encryption key with a net payload of 3.6 bpp, is depicted in Figure 15c. Subsequently, the data-hider incorporates additional data into the encrypted image, resulting in the creation of a marked encrypted image shown in Figure 15d. The disparity between the  
Figure 14: The test images: (**a**) Lena, (**b**) Baboon, (**c**) Jetplane, (**d**) Man, and (**e**) Tiffany. Figure 13: The three possible scenarios depending on the receiver’s possession of the different keys for data extraction and image recovery. encrypted image and the marked-encrypted image is presented in Figure 15e. Finally, the recovered Lena image is showcased in Figure 15f, with a Mean Square Error (MSE) value of 0, affirming its matching with the original one. The same outcomes are observed for the Jetplane image illustrated in Figure 16.  
Figure 16: The proposed RDHEI method for the Jetplane image is depicted as follows: (**a**) the original Jetplane image, (**b**) the reordered mapped prediction error image, (**c**) the encrypted image, (**d**) the marked encrypted image with a net payload of ER = 3.64 bpp, (**e**) the bitwise XOR operation between the marked and encrypted image, (**f**) and the reconstructed image with an MSE value of 0. Figure 15: The proposed RDHEI method for the Lena image is depicted as follows: (**a**) the original Lena image, (**b**) the reordered mapped prediction error image, (**c**) the encrypted image, (**d**) the marked encrypted image with a net payload of ER = 3.6 bpp, (**e**) the bitwise XOR operation between the marked and encrypted image, (**f**) and the reconstructed image with an MSE value of 0. Additionally, this section aims to demonstrate the robustness of the proposed encryption scheme against different attacks. To achieve this goal, several metrics are employed, namely the histogram, the \(\chi^{2}\) statistic, the correlation coefficient, and the information entropy. These metrics serve as evidence that an ideal encryption algorithm should be resilient to different types of attacks. 1. _The key space analysis_ The key space denotes the total count of distinct keys available for utilization in the encryption process. Within our proposed algorithm, the secret keys encompass the control parameters and the initial values of the two used PWLCM maps \(k_{\varepsilon}=(x_{0},\eta_{1},y_{0},\eta_{2})\). Assuming a computational precision equivalent to double-precision numbers, which is approximately \(10^{16}\), then the key space \(H_{x_{0}}\) and \(H_{y_{0}}\) is approximately \(10^{16}\), while \(H_{\eta_{1}}\) and \(H_{\eta_{2}}\) are around \(0.5\times 10^{16}\). Consequently, the total key space \(H\) is calculated as \(H=H_{x_{0}}\times H_{y_{0}}\times H_{\eta_{1}}\times H_{\eta_{2}}\), yielding a value of approximately \(0.25\times 10^{64}\). The key space equates to approximately \(1.519\times 2^{210}\). For an encryption system to be considered effective and resilient against brute-force attacks, the size of the key space should not fall below \(2^{100}\), making such attacks infeasible [40]. Consequently, the key space of our encryption scheme is large enough to ensure robust protection against all kinds of brute-force attacks. 2. _The histogram analysis_ We have analyzed the histograms of several test images and their corresponding encrypted and marked-encrypted images. Figure 17 presents the histograms of the encrypted and marked-encrypted images for Lena and Jetplane. The histograms shown in Figure 17 indicate that the encrypted and the marked-encrypted images follow a uniform distribution.