## 2 Our Proposed Methods
### Image Encryption
In the third iteration (\(i=3\)), we generate a 6-ary random number from 1 to 5, say \(k=1_{6}\); we exchange the first index 1 and the last index, which is 5, thus leading to \(\{\pi\}=\{6,\ 2,\ 8,\ 7,\ 5,\ 1,\ 4,\ 3\}\). The final permuted index \(\{\pi\}=\{5,\ 8,\ 2,\ 6,\ 7,\ 1,\ 4,\ 3\}\), implying that the bit-level permutation produces \(\{D\}=\{1,\ 1,\ 1,\ 0,\ 1,\ 0,\ 1,\ 0\}\) representing a decimal value 234. In our implementation, we align a total of \(4\times H\times V\) components (each pixel contains the R-, G-, B- and E-channel components). We then apply the bit-level random permutation, described above, to all of the bits. We remark that the random permutation shuffles the indices in place rather than producing a shuffled copy. Along with the increase of iterations, the number of indices in the shuffled set increases. Consequently, the random permutation algorithm has the time complexity of \(O(N)\) and the space complexity of \(O(1)\). * **Step 4:** Pixel diffusion. The final step is to encrypt the shuffled pixels using exclusive-OR operator, \(\oplus\), with respect to the pseudo-random sequence, \(\boldsymbol{R}\boldsymbol{\
u}=\{R_{4},R_{4+1},\ldots,R_{4+k\times H\times V-1 }\}\). Without loss of generality, we list the shuffled pixel as \(I^{\prime}_{5}=\{P_{1,R^{\prime}},\ P_{1,G^{\prime}},\ P_{1,\ B^{\prime}},\ P_{1,E^{ \prime}},\ldots,\ P_{H\times V,R^{\prime}},\ P_{H\times V,G^{\prime}},\ P_{H \times V,B^{\prime}},\ P_{H\times V,E^{\prime}}\}\). The pixel encryption contains two sub-steps. First, we generate five initial values, \(\{P_{0,R^{\prime}},\ P_{0,C^{\prime}},\ P_{0,B^{\prime}},\ P_{0,E^{\prime}},\ C_{0,E^{\prime}}\}\), from the encryption \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline \(i\) & \(k\) & Exchange & **1 st** & **2 nd** & **3 rd** & **4 th** & **5 th** & **6 th** & **7 th** & **8 th** \\ \hline \multicolumn{8}{c}{Original index \(\{\pi\}\)} & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
1 & \(3_{8}\) & \((3,8)\) & 1 & 2 & 8 & 4 & 5 & 6 & 7 & 3 \\
2 & \(4_{7}\) & \((4,7)\) & 1 & 2 & 8 & 7 & 5 & 6 & 4 & 3 \\
3 & \(1_{6}\) & \((1,6)\) & 6 & 2 & 8 & 7 & 5 & 1 & 4 & 3 \\
4 & \(4_{5}\) & \((5,7)\) & 6 & 2 & 8 & 5 & 7 & 1 & 4 & 3 \\
5 & \(1_{4}\) & \((5,6)\) & 5 & 2 & 8 & 16 & 7 & 1 & 4 & 3 \\
6 & \(2_{3}\) & \((2,8)\) & 5 & 8 & 2 & 6 & 7 & 1 & 4 & 3 \\
7 & \(0_{2}\) & \(\begin{array}{c}\text{No}\\ \text{change}\\ \text{Permuted index}\ \{\pi\}\end{array}\) & 5 & 8 & 2 & 6 & 7 & 1 & 4 & 3 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: An example of random index permutation, where \(\{\pi\}=\{1,\ 2,\ 3,\ 4,\ 5,\ 6,\ 7,\ 8\}\) represents the original index order corresponding to an 8-bit pixel. Figure 3: Trajectories of 2D-SLMM with an initial value \((x_{0},\ y_{0})=(0.15874,\ 0.38797)\) and the control values \((\alpha,\ \beta)=(1,\ 3)\). key. Next, we utilize Equation (11) to cipher four components. Note that since we have provided five initial values, the index ranges from 0 to \(H\times V\). \[\left\{ \begin{array}{l}C_{i+1,\ R^{\prime}}=R^{\prime}_{4i}\oplus P_{i+1,R^{ \prime}}\oplus P_{i,R^{\prime}}\ \oplus C_{i,E^{\prime}}\\ C_{i+1,G^{\prime}}=R^{\prime}_{4i+1}\oplus P_{i+1,G^{\prime}}\oplus P_{i,G^{ \prime}}\oplus C_{i+1,R^{\prime}}\\ C_{i+1,B^{\prime}}=R^{\prime}_{4i+2}\oplus P_{i+1,B^{\prime}}\oplus P_{i,B^{ \prime}}\oplus C_{i+1,G^{\prime}}\\ C_{i+1,E^{\prime}}=R^{\prime}_{4i+3}\oplus P_{i+1,E^{\prime}}\oplus P_{i,E^{ \prime}}\oplus C_{i+1,B^{\prime}}\end{array} \right. \tag{11}\]  
As an example, let \(\mathbf{P}^{\prime}_{1}=\left(P_{1,R^{\prime}},\ P_{1,G^{\prime}},\ P_{1,\ B^{ \prime}},\ P_{1,E^{\prime}}\right)=\left(164,\ 87,\ 17,\ 126\right)\) represent the first stego pixel and pseudo-random sequence \(\mathbf{R}^{\prime}=\left\{38,\ 39,\ 246,\ 133,\ 151,\ldots\right\}\). Let \(\left(P_{0,R^{\prime}},\ P_{0,G^{\prime}},\ P_{0,\ B^{\prime}},\ P_{0,E^{ \prime}},\ C_{0,E^{\prime}}\right)=\left(196,\ 118,\ 25,\ 57,\ 49\right)\) be the initial five values.