## 2 Our Proposed Methods
### Image Decryption  
The decryption process is the inverse operation of the encryption process. The input is the stego encrypted HDR RGBE image, \(I_{SE}\), and the output is the deciphered stego HDR RGBE image, \(I_{S}\), which contains a hidden message.  
The image decryption contains the following three steps:  
* **Step 1:** We determine parameters, \(h,\ q,\ x_{K},\ y_{K}\) from the secret decryption key. Then, we input them as initial values in the 2D-SLMM to generate the pseudo-random sequence \(\mathbf{R}=\left\{R_{0},R_{1},\ldots,R_{q},\ R_{q+1},\ \ldots,\ R_{q+4\times H \times V-1}\right\}\). We discard the first \(q\) items so the resultant \(\mathbf{R}^{\prime}\) is exactly the same as that produced in the encryption process.
* **Step 2:** We determine five setting values, \(\left(P_{0,R^{\prime}},\ P_{0,G^{\prime}},\ P_{0,B^{\prime}},\ P_{0,E^{\prime}},\ C_{0,E^{\prime}}\right)\) from the decryption key. Then, we utilize Equation (12) to decipher stego encrypted pixels in four components: \[\left\{\begin{array}{l}P_{i+1,R^{\prime}}=R^{\prime}_{4i}\oplus C_{i+1,\ R^{\prime}}\oplus P_{i,R^{\prime}}\oplus C_{i,E^{\prime}}\\ P_{i+1,G^{\prime}}=R^{\prime}_{4i+1}\oplus C_{i+1,G^{\prime}}\oplus P_{i,G^{ \prime}}\oplus C_{i+1,R^{\prime}}\\ P_{i+1,B^{\prime}}=R^{\prime}_{4i+2}\oplus C_{i+1,B^{\prime}}\oplus P_{i,B^{ \prime}}\oplus C_{i+1,G^{\prime}}\\ P_{i+1,E^{\prime}}=R^{\prime}_{4i+3}\oplus C_{i+1,E^{\prime}}\oplus P_{i,E^{ \prime}}\oplus C_{i+1,B^{\prime}}\end{array}\right.\] (12)  
Following up on our previous example, we present \(\mathbf{R}^{\prime}=\left\{38,\ 39,\ 246,\ 133,\ 151,\ldots\right\}\) and the setting values \(\left(P_{0,R^{\prime}},\ P_{0,G^{\prime}},\ P_{0,B^{\prime}},\ P_{0,E^{\prime}},\ C_{0,E^{\prime}} \right)=\left(196,\ 118,\ 25,\ 57,\ 49\right)\). Let \(\mathbf{C}^{\prime}_{1}=\left(119,\ 113,\ 143,\ 77\right)\) be the first stego encrypted pixel. Referring to Equation (12), we produce \(\mathbf{P}_{1,R^{\prime}}=38\oplus 119\oplus 196\oplus 49=164\). We can apply the same manner to decipher the other three components. Thus, the decrypted stego pixel becomes \(P^{\prime}_{1}=\left(164,\ 87,\ 17,\ 126\right)\).  
* **Step 3:** Pixel bit-level inverse permutation and diffusion. The recipient can adopt the secret encryption key, \(k_{e}\), to generate pseudo-random numbers to accomplish the inverse bit-level permutation. We follow up on the previous example. Let \(\{\pi t\}=\{5,\ 8,\ 2,\ 6,\ 7,\ 1,\ 4,\ 3\}\) represent the index of position in an input pixel. When utilizing the inverse version of the permutation scheme, we also generate a \((9-i)\)-ary random number \(k\) for the \(i\)-th iteration. However, we apply these random numbers in a reverse order, i.e., let \(j=8-i\) and \(j\) be the actual index we used in the inverse version. Then, we exchange index \(j\)+1 with index \(k\). The process continues 7 times until the indices in \(\{\pi t\}\) are fully permuted.  
Table 3 continues the example for the inverse index permutation, where the shaded color indicates an index to be exchanged between itself and the index \(k\), which is determined by the generated random number. In the first round (\(j=1\)), the random number produced is \(k=0_{2}\), and no exchange is needed because it is in the 1 st index. In the second iteration (\(j=2\)), the random number produced is \(k=2_{3}\); we exchange the 2 nd index, 8, and the \(j+1=3\)rd index, which is 2, thus leading to the current \(\{\pi t\}=\{5,\ 2,\ 8,\ 6,\ 7,\ 1,\ 4,\ 3\}\). In the third iteration (\(j=3\)), the random number produced is \(k=14\); we exchange the 1 st index, which is 5, and the 4 th index, which is 6, thus updating \(\{\pi t\}\) to become \(\{\pi t\}=\{6,\ 2,\ 8,\ 5,\ 7,\ 1,\ 4,\ 3\}\). The final inverse permuted index \(\{\pi\}=\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8\}\), indicating that \(\{D\}=\{0,\ 1,\ 0,\ 1,\ 1,\ 0,\ 1,\ 1\}\) represents the inverse bit-level permutation, which is the same as the original decimal pixel value 91.