# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 4 Discussion: Other Details of the Design
### Sharing that Requires All Departments  
In some circumstances, representatives from every department must be present. One example is a labor union meeting, where every department of the union must have at least one attendee to disclose a secret. We thus apply our method to this type of sharing, which we call \"all-department\" sharing. Without loss of generality, we still consider \(3\) departments: \(DEP_{H}\), \(DEP_{M}\), and \(DEP_{L}\). For simplicity, we also assume that every department has \(4\) people (i.e., \(4\) participants). Hence,  
\(DEP_{L}=\{P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\}\), \(DEP_{M}=\{P_{5}\), \(P_{6}\), \(P_{7}\), \(P_{8}\}\), and \(DEP_{H}=\{P_{9}\), \(P_{10}\), \(P_{11}\), \(P_{12}\}\). As with the preceding examples, every participant owns a shadow. We now demonstrate how shadows can be created for these participants. Figure 3: Six of the almost infinitely many examples of sharing a secret image by mixing the two approaches. (**a**) means that only the top quarter can use linear-equations approach (LEA); (**b**) means that only the leftmost quarter can use LEA; (**c**) means that only the blocks (i,j) whose \(i+j\) are even can use LEA; (**d**) means that only blocks \(\{1,2\}\) of each odd row and blocks \(\{3,4\}\) of each even row can use LEA; (**e**) means that only the top-leftmost, top-rightmost, bottom-leftmost, bottom-rightmost, and the \(4\) central blocks can use LEA; (**f**) means that only the \(4\) central blocks can use LEA. Step 1: Use traditional (\(t_{\textit{artificial}}\), \(n_{\textit{artificial}}\)) = (9, 12) sharing to share the secret and obtain \(n_{\textit{artificial}}\) = 12 shares {\(S_{1}\),..., \(S_{12}\)} so that any \(t_{\textit{artificial}}\) = 9 of the shares can unveil the secret. Step 2: Partition these 12 shares into 3 equal parts of 4 shares each. Next, assign shares \(S_{1}\)-\(S_{4}\) to \(DEP_{L}\), \(S_{5}\)-\(S_{8}\) to \(DEP_{M}\), and \(S_{9}\)-\(S_{12}\) to \(DEP_{H}\). Note that each share appears in exactly one department. Step 3: Every department can use its 4 shares to create \(C_{3}^{4}\) = 4 combinations of 3 shares each. Distribute these 4 combinations to the 4 people in the department. For example,  
\(DEP_{L}\): \(P_{1}\) = {\(S_{1}\), \(S_{2}\), \(S_{3}\)}, \(P_{2}\) = {\(S_{1}\), \(S_{2}\), \(S_{4}\)}, \(P_{3}\) = {\(S_{1}\), \(S_{3}\), \(S_{4}\)}, \(P_{4}\) = {\(S_{2}\), \(S_{3}\), \(S_{4}\)}. \(DEP_{M}\): \(P_{5}\) = {\(S_{5}\), \(S_{6}\), \(S_{7}\)}, \(P_{6}\) = {\(S_{5}\), \(S_{6}\), \(S_{8}\)}, \(P_{7}\) = {\(S_{5}\), \(S_{7}\), \(S_{8}\)}, \(P_{8}\) = {\(S_{6}\), \(S_{7}\), \(S_{8}\)}. \(DEP_{H}\): \(P_{9}\) = {\(S_{9}\), \(S_{10}\), \(S_{11}\)}, \(P_{10}\) = {\(S_{9}\), \(S_{10}\), \(S_{12}\)}, \(P_{11}\) = {\(S_{9}\), \(S_{11}\), \(S_{12}\)}, \(P_{12}\) = {\(S_{10}\), \(S_{11}\), \(S_{12}\)}. Every department has 4 shadows. Any 2 departments together can only have 8 different shares, which are still less than \(t_{\textit{artificial}}\) = 9. Therefore, the secret can be revealed only when participants come from all 3 departments. In case the departments have more members, \(t_{\textit{artificial}}\) and \(n_{\textit{artificial}}\) can be increased to overcome the problem. For example, if the number of people in every department increases from 4 to 5, where \(DEP_{L}\) = {\(P_{1}\), \(P_{2}\), \(P_{3}\), \(P_{4}\), \(P_{5}\)}, \(DEP_{M}\) = {\(P_{6}\), \(P_{7}\), \(P_{8}\), \(P_{9}\), \(P_{10}\)}, and \(DEP_{H}\) = {\(P_{11}\), \(P_{12}\), \(P_{13}\), \(P_{14}\), \(P_{15}\)}, we can simply increase \(t_{\textit{artificial}}\) and \(n_{\textit{artificial}}\), for example, 12 and 15, respectively. The method for creating shadows in this case is detailed as follows:  
Step 1: Use (\(t_{\textit{artificial}}\), \(n_{\textit{artificial}}\)) = (12, 15) in traditional sharing to share the secret and obtain \(n_{\textit{artificial}}\) = 15 shares {\(S_{1}\),..., \(S_{15}\)}, where any \(t_{\textit{artificial}}\) = 12 of the shares can be used to recover the secret.