# Fault-Tolerant Visual Secret Sharing Schemes without Pixel Expansion
## 3 Proposed Scheme  
The main concept of our algorithm is as follows. First, taking \(n\times n\) pixels as a unit, the image is divided into several units. In the first generated share, each unit is randomly chosen from the patterns we designed. For the generation of the second share, the number of black and white pixels in each unit on the original secret image needs to be counted individually. This will be used to select the suitable pattern according to the pattern of the first share for the same unit. Run the steps sequentially and repeatedly until the second share is generated. Taking the idea of [PERSON] and [PERSON]'s scheme as a reference [14], we design the special patterns for the main encryption scheme and apply them to the Algorithm KK1 [2]. We design the fault-tolerant VSS schemes by taking \(n\times n\) pixels as a unit, for \(n=3\), \(4\), \(5\) or \(6\). The final design of the patterns for encryption schemes when \(n\) is \(3\), \(4\), \(5\) and \(6\) is shown in Tables 3-6 respectively. In the following, we analyze the transmittance when two shares are not stacked correctly. Because the pattern of each unit is symmetric, the analysis results are all equal when two units are stacked by shifting one pixel to the right, left, up or down. Here, we only show the result for the \(n=4\) case where \(G_{1}\) shifts one pixel right. All combinations for such a case when the pixel in the secret image is white or black are shown in Tables 7 and 8 respectively. The way of stacking has been shown as Figure 3, where the red square represents the stacked position of the unit in \(G_{1}\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Image** & \(G_{1}\) & \(G_{2}\) & **Stack** & **Image** & \(G_{1}\) & \(G_{2}\) & **Stack** \\ \hline \hline \multirow{2}{*}{\(\square\)} & & & & & & & \\  & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The designed patterns for \(n=3\). Figure 2: The patterns in [PERSON] and [PERSON]’s scheme [14]. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & \(G_{1}\) & \(G_{2}\) & **Stack** & **Image** & \(G_{1}\) & \(G_{2}\) & **Stack** \\ \hline  & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 6: The designed patterns for \(n=6\). Figure 3: Stacked image with one pixel right shifted. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & \(G_{1}\) & \(G_{2}\) & **Stack** & **Image** & \(G_{1}\) & \(G_{2}\) & **Stack** \\ \hline  & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \(\Box\) & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The designed patterns for \(n=4\). \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Image** & \(G_{1}\) & \(G_{2}\) & **Stack** & **Image** & \(G_{1}\) & \(G_{2}\) & **Stack** \\ \hline \hline \end{tabular}
\end{table} 
Table 7: The analysis of stacked units with one pixel shift when the pixel in the secret image is white. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Image** & \(G_{1}\) & \(G_{2}\) & **Stack** & **Image** & \(G_{1}\) & \(G_{2}\) & **Stack** \\ \hline \hline \end{tabular}
\end{table} 
Table 8: The analysis of stacked units with one pixel shift when the pixel in the secret image is black. In Tables 7 and 8 each little square represents a pixel, and the red square is the area of the unit in \(G_{1}\) that has been stacked with \(G_{2}\) with one pixel shift. In this case, the ration of the number of white pixels to the total pixels of the stacked unit when the pixel in the secret image is white is \((32+36+8+4)/256=80/256=5/16\). Similarly, when the pixel in the secret image is black, this ratio is \((16+12+8+12)/256=48/256=3/16\). In summary, Table 9 gives the transmittance analysis for stacking two units for \(n=3,4,5\) or \(6\).