# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 3 Proposed Types and Approaches
### Using the Polynomial Approach to Design Types 1 and 2
Because the threshold for \"number of shares\" is 9 and \(P_{1}=\{S_{1}\), \(S_{2}\), \(S_{3}\)}, \(P_{2}=\{S_{4}\), \(S_{5}\), \(S_{6}\}\), and \(P_{3}=\{S_{7}\), \(S_{8}\), \(S_{3}\), \(S_{6}\}\), the combination {\(P_{1}\), \(P_{2}\), \(P_{3}\)} yields only 8 shares {\(S_{1}\),..., \(S_{8}\)}, which is still fewer than 9, and the secret cannot be unveiled. Likewise, {\(P_{4}\), \(P_{5}\), \(P_{6}\)} yields only 8 shares {\(S_{9}\),..., \(S_{16}\)} because \(P_{4}=\{S_{9}\), \(S_{10}\), \(S_{11}\)}, \(P_{5}=\{S_{12}\), \(S_{13}\), \(S_{14}\}\), and \(P_{6}=\{S_{15}\), \(S_{16}\), \(S_{11}\), \(S_{14}\}\). Thus, the secret cannot be unveiled under the combination {\(P_{4}\), \(P_{5}\), \(P_{6}\)}. As for the other combinations {\(P_{i}\), \(P_{j}\), \(P_{k}\)}, members in the combination can view the secret because at least 9 shares are available. The proof of the aforementioned claims is routine and thus not presented for brevity. Table 1 presents some other examples. Notably, because more constraints must be considered, the design is likely to become more difficult when the number of forbidden combinations increases. \begin{table}
\begin{tabular}{c c c} \hline \hline  & & **The Shares {\(S_{i}\)} Appeared in Each Participant’s Shadow** \\ (\(t_{\text{participants}}\), \(n_{\text{participants}}\), \(f\)) & **The Assigned / Forbidden Combinations** & **The Shares {\(S_{i}\)} Appeared in Each Participant’s Shadow** \\  & **Combinations** & **(Underline Means This Share in Participant \(P_{i}\))** \\  & & **Also Appears in Some Other Participant \(P_{j}\)** \\  & & **Let \(t_{\text{artificial}}=12\).** \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Some examples of sharing with forbidden combinations. **Example 3** (3, \(n_{participants}\),f). : _Sharing whose forbidden combinations always contain a trouble-making couple \(P_{1}\) and \(P_{2}\). See the Appendix A for the design._  
Type 2 (Sharing with cross-department support):  
Let each participant belong to one of several departments, for example, 3 departments {\(DEP_{H}\), \(DEP_{M}\), \(DEP_{L}\)}. Without loss of generality, assume the parameter pair (\(t\) = threshold, \(n\) = number of people in this department) for the 3 departments are, respectively, (\(t_{H}\) = 3, \(n_{H}\) = 4) for \(DEP_{H}\), (\(t_{M}\) = 5, \(n_{M}\) = 6) for \(DEP_{M}\), and (\(t_{L}\) = 7, \(n_{L}\) = 7) for \(DEP_{L}\). As the first step in the design, we shall assign a value to an artificial threshold \(t_{artificial}\), which is larger than any given local threshold. Subsequently, we use this artificial threshold \(t_{artificial}\) to share the secret and thus obtain several shares such that any \(t_{artificial}\) shares can recover the secret. We then distribute these shares to each person of each department. In this cross-department support type, we assume that shares are not repeatedly distributed. Let each  
Figure 1: Sharing and decryption experiment of Example 2, which is a (\(t_{participants}\) = 3, \(n_{participants}\) = 6, \(f\) = 2) example with 2 forbidden combinations. (**a**) is original image; (**b**) uses (\(t_{artificial}\), \(n_{artificial}\)) = (9, 16) in traditional sharing to share (**a**) and create 16 shares; (**c**) creates 6 shadows for the 6 participants \(P_{1}\) to \(P_{6}\) by assigning the 16 shares in (**b**) to them; (**d**) shows that each of the 18 combinations can reveal the secret. Here, the combination {\(P_{1}\), \(P_{2}\), \(P_{6}\)} is used as example; (**e**) shows that the 2 forbidden combinations cannot reveal the secret because each combination gets less than 9 shares. person in \(DEP_{H}\), \(DEP_{M}\), and \(DEP_{L}\) obtain, respectively, \(Q_{H}\), \(Q_{M}\), and \(Q_{L}\) shares.