## 2 Proposed Scheme
### Secret Extraction and Decryption Phases  
This phase consists of three steps: the decryption step, data extraction step, and recovery step. In the decryption step, the keys \(s_{1}\) and \(s_{2}\) will generate random bits \(r_{1}r_{2}r_{3}\) and \(r(i,j)\). Using Equation (9), the encrypted pixel \(p^{*k}(i,j)\) can be decrypted as \(p^{\prime k}(i,j)\), where \(r^{k}(i,j)\) is the random bit used to decrypt the \(k\) bits of pixel \(p^{k}(i,j)\). Thus, the \(p^{\prime k}(i,j)\) can be reconstructed as \(10111101\oplus 01011001=1110010\). In the data extraction step, the value of \(w_{1}^{*}w_{2}^{*}w_{3}^{*}\) can be obtained through Equation (8) key \(s_{1}\). In the same example, the values of \(w_{1}^{\prime}w_{2}^{*}w_{3}^{*}\) and \(r_{1}r_{2}r_{3}\) are \(111\) and \(001\), which output \(w_{1}^{*}w_{2}^{*}w_{3}^{*}\) as \(111\oplus 001=110\). After that, the first 3 MSBs of the decrypted image are integrated as \(w_{1}^{*}w_{2}^{*}w_{3}^{*}\) with predicted pixels \(\hat{p}^{5}(i,j)\), \(\hat{p}^{6}(i,j)\), \(\hat{p}^{7}(i,j)\) and \(\hat{p}^{8}(i,j)\) using Equation (7) to calculate \(mC\). \(mC\) determines \(w_{1}w_{2}w_{3}\). The decimal of \(w_{1}w_{2}w_{3}\) represents \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Label**} & \multicolumn{3}{c}{**Smooth Image**} & \multicolumn{3}{c}{**Complex Image**} \\ \cline{2-7}  & **Capacity** & **Code Length** & **Payload** & **Capacity** & **Code Length** & **Payload** \\ \hline
0 & 0 & 3 & \(-3\) & 1 & 3 & \(-2\) \\
1 & 3 & 3 & 0 & 2 & 3 & \(-1\) \\
2 & 4 & 3 & 1 & 3 & 3 & 0 \\
3 & 5 & 3 & 2 & 4 & 3 & 1 \\
4 & 6 & 3 & 3 & 5 & 3 & 2 \\
6 & 8 & 3 & 5 & 7 & 3 & 4 \\
7 & 8 & 3 & 5 & 7 & 3 & 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The pure payloads of the smooth image and complex image. Figure 2: Three pieces of the embedded pixel. the indicator in column \"Label\" of Table 1. Mapping to a fallen case, if the corresponding value of \(payload(i,j)\) is larger than 0, we can obtain the hidden data from the 4 th MSB to the \(3+payload(i,j)\)th MSB of \(p^{\prime}(i,j)\) as shown in the second piece of Figure 2. In contrast, if \(payload(i,j)\) is less than 1, no secret data are extracted. In the previous example, the predicted pixel is \(\hat{p}(2,2)=106\); thus, the values of \(\hat{p}^{5}(i,j)\), \(\hat{p}^{6}(i,j)\), \(\hat{p}^{7}(i,j)\) and \(\hat{p}^{8}(i,j)\) are 0, 1, 1, 0, respectively. The value of \(m\) is constructed as \([1100110]\) by Equation (6). With Equation (7), the value of \(w_{1}w_{2}w_{3}\) is 101. The decimal value of \(w_{1}w_{2}w_{3}\) is equal to 5, which indicates \(payload(i,j)\) equal to 3. In the example, the hidden data are extracted as 001 (from the 4 th MSB to the 6 th MSB of \(p^{\prime}(2,2)\)). The algorithm for the recovery step is shown in Algorithm 1. The value of \"Capacity\" in Table 1 (for clarity, the value depicted by \(payload(i,j)\), represents the number of bits that can be recovered from the predicted pixel). Furthermore, two recovery situations are for smooth images and complex images. For the smooth image, if \(payload(i,j)\) equals 0, no bits of the predicted pixel \(\hat{p}(i,j)\) can be used to recover the original pixel \(p(i,j)\), and the original 3 MSBs of \(p(i,j)\) are recovered through the hidden data extracted by other pixels. As for a complex image, if \(payload(i,j)\) is less than 1, the \(|payload(i,j)|\) MSBs are also recovered by the hidden data extracted by other pixels. The 4 th MSB onward to the last bit of \(p(i,j)\) is recovered by that of \(p^{\prime}(i,j)\). If \(payload(i,j)\) is larger than 0, the first \(payload(i,j)-1\) MSBs of the predicated pixel are the same as the first \(payload(i,j)-1\) MSBs of the original pixel. In addition, the \(payload(i,j)\)th MSB of \(p(i,j)\) is recovered by the flipped value of \(\hat{p}(i,j)\). The last \(5-payload(i,j)\) bits of \(p^{\prime}(i,j)\) can be used to recover those of \(p(i,j)\). That is, the \(payload(i,j)\)th MSB of \(p(i,j)\) is equal to \(1-\hat{p}^{8-payload(i,j)}(i,j)\).