# Verifiable (2, \(n\)) Image Secret Sharing Scheme Using Sudoku Matrix
## 3 Research Methods
### The Procedure of Hiding Secret Image and Authentication Codes to Cover Images  
We adopt Equations (3) and (4), to describe the secret image divided into \(n\) shadows and the corresponding authentication codes hidden back into the shadows, respectively. There is a secret image of size \(n\times n\) to be hidden, while there can be several cover images of size \(2n\times 2n\) depending on the numbers of shareholders participating in hiding the secret data. There are two square pixels of each cover image to hide two pixels of secret data values and the corresponding authentication codes, as shown in Figure 4a. Each square pixel consists of four pixels. The first square pixel, denoted by \(B_{1}\), and the second square pixel, denoted by \(B_{2}\), are used to embed the first pixel, denoted by \(p_{1}\), and the second secret pixel, denoted by \(p_{2}\), of the secret image in different colors, as shown in Figure 4b. Here,  
Figure 3: An example of our schematic diagram. (**a**) Real share 1. (**b**) Real share 2. (**c**) Authentication result—fully recovered. (**d**) Real share 1. (**e**) Fake share 2. (**f**) Authentication result—not recovered completely. for \(B_{j}\) (\(j\in\{1,2\}\)), the original pixel values of the four pixels, located in the upper-left pixel, the upper-right pixel, the bottom-left pixel, and the bottom-right pixel, are represented as \(B_{j}^{1}\), \(B_{j}^{2}\), \(B_{j}^{3}\), and \(B_{j}^{4}\), respectively. During the secret sharing procedure, first, the pixel value in the secret image needs to be converted to a base value of 9, denoted by \((\beta_{j}^{1}\beta_{j}^{2}\beta_{j}^{3})_{9}\), where \(\beta_{j}^{1}\) is in the range of 0 and 3, and \(\beta_{j}^{2}\) and \(\beta_{j}^{3}\) are in the range of 0 and 8, and \(j\) are equal to 1 and 2 if the digits are generated by pixels \(p_{1}\) and \(p_{2}\), respectively. Secondly, the \(\beta_{j}^{2}\) and \(\beta_{j}^{3}\) are embedded into the top two pixels (\(B_{j}^{1}\) and \(B_{j}^{2}\)) of the cover images with Equation (3). Here, the \(x_{i}\) represents the ID number of the cover image. \[y_{i,j}=\beta_{j}^{2}x_{i}+\beta_{j}^{3}\mod 9 \tag{3}\]  
\[z_{1}=\beta_{1}^{1}x_{i}+f_{2}\mod 9 \tag{4}\]  
Figure 4: Schematic diagram of secret data embedded into a cover image. **(a)** Schematic diagram. **(b)** An embedding example. An embedding example is shown in Figure 4b, prior to undergoing the embedding procedure. We suppose that the first pixel value \(p_{1}\) in the pixel pair retrieved by the secret image is 127, for which the ID of the cover image is \(x_{i}=1\), and the converted value is \((151)_{9}\), indicated as \(\beta_{1}^{1}\), \(\beta_{1}^{2}\), \(\beta_{1}^{3}\), from left to right, respectively. With Equation (3), the \(y_{i}\) is 6 because the equation is generated as \(y_{i,1}=5x_{i}+1\) mod 9 if \(x_{i}=1\). Suppose that the original pixel values of \(B_{1}^{1}\) and \(B_{1}^{2}\) are 100 and 120, respectively. To hide the \(y_{i,1}\) value, the pixels \(B_{1}^{1}\) and \(B_{1}^{2}\) are changed as stego pixels \(B_{1}^{1\prime}\) and \(B_{1}^{2\prime}\), respectively. After embedding \(y_{i,1}\) (\(y_{i,1}=6\) in the example), the values \(B_{1}^{1\prime}\) and \(B_{1}^{2}\) will be 101 and 121, respectively, as shown in Figure 5. In the same way, the digits \(\beta_{2}^{2}\) and \(\beta_{2}^{3}\) generated by the secret pixel \(p_{2}\) can be hidden as \(B_{2}^{1}\) and \(B_{2}^{3}\) to generate the stego pixels \(B_{2}^{1\prime}\) and \(B_{2}^{2\prime}\). For example, if \(p_{2}\) is equal to 126, the values of \(\beta_{2}^{2}\) and \(\beta_{2}^{3}\) are 5 and 0, respectively. After the hiding procedure, the stego pixels \(B_{2}^{1\prime}\) and \(B_{2}^{2\prime}\) are 100 and 121, respectively. The authentication codes, denoted by \(f_{1}\), \(f_{2}\), \(f_{3}\), and \(f_{4}\), are the average of pixels \(p_{1}\) and \(p_{2}\) and transformed into \((f_{1}f_{2}/f_{3}f_{4})_{2,9,9,2}\) via a multiple notation system.