# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 3 Proposed Types and Approaches  
In this section, we introduce two extended types (Types 1 and 2) of image sharing; we also demonstrate how they can be implemented by using either the polynomial approach or linear-equations approach. Type 1 (Sharing with forbidden combinations):  
To understand what Type 1 image sharing is, without loss of generality, consider 6 people {\(P_{1}\),..., \(P_{6}\)} who share together a secret of the company. The secret is such that 3 or more people must be gathered to unveil it. Hence, the secret can be recovered by \((\frac{6!}{(6-3)!3! })=20\) possible combinations of personnel if no combination is forbidden. However, according to a security check, \(P_{1}\), \(P_{3}\), and \(P_{5}\) worked for a rival company before working in ours, making them less trustworthy. The combination of these 3 employees should thus be excluded as a combination that allows access to the secret. This is an example of Type 1 image sharing with a single forbidden combination, namely, {\(P_{1}\), \(P_{3}\), \(P_{5}\)}. Of course, if the boss of our company wishes to be more careful, he can even forbid more combinations such as {\(P_{1}\), \(P_{3}\), \(x\)}, {\(P_{1}\), \(P_{5}\), \(x\)}, and/or {\(P_{3}\), \(P_{5}\), \(x\)}, where \(x\) can be any of the other employees (i.e., any other \(P_{i}\)). In the preceding example, some combinations cannot be used to gain access to the secret. We call it \"sharing with forbidden combinations\" and denote it as {\(t_{\mathit{purticipants}}\), \(n_{\mathit{purticipants}}\), \(f\)} sharing. Here,  
\(n_{\mathit{participants}}\): The number of participants who share the secret. Each participant will hold a so-called \"shadow\" file. \(t_{\mathit{purticipants}}\): The minimum number of people required to recover the secret. Any \(t_{\mathit{participants}}\) of the \(n_{\mathit{participants}}\) participants can recover the secret by using the their shadow files, unless the \(t_{\mathit{participants}}\) people constitute a forbidden combination. \(f\): The number of forbidden combinations. For example, the preceding example is a (3, 6, 1) scheme if there is only one forbidden combination {\(P_{1}\), \(P_{3}\), \(P_{5}\)} and a (3, 6, 4) scheme if all {\(P_{1}\), \(P_{3}\), \(x\)} combinations are forbidden. Traditional sharing, which has no forbidden combinations, can be denoted as {\(t_{\mathit{participants}}\), \(n_{\mathit{participants}}\), \(f\)} = (3, 6, 0), and it is thus treated as a special case of Type 1 image sharing. Type 2 (Sharing with cross-department support):  
Without loss of generality, assume there are 3 departments, namely {\(DEP_{H}\), \(DEP_{M}\), \(DEP_{L}\)}. For officers in department \(DEP_{H}\), assume any 3 of them can recover the secret. For officers in department \(DEP_{M}\), assume any 4 of them can recover the same secret. For officers in department \(DEP_{L}\), assume any 5 of them can recover the same secret. Notice that this one secret has 3 thresholds: \(t_{H}\) = 3, \(t_{M}\) = 4, \(t_{L}\) = 5. Hence, each department has its own threshold. This system can be easily implemented by repeating the traditional sharing system thrice: first, use a (\(t\), \(n\)) = (3, \(n_{H}\)) sharing system to share the secret among the \(n_{H}\) officers of department \(DEP_{H}\), then use a (\(t\), \(n\)) = (4, \(n_{M}\)) sharing system to share the \"same\" secret among the \(n_{M}\) officers of department \(DEP_{M}\), and then use a (\(t\), \(n\)) = (5, \(n_{L}\)) sharing system to share the \"same\" secret among the \(n_{L}\) officers of department \(DEP_{L}\). However, due to illness, a terrorist attack, or business trips abroad, not every officer of the same department comes to the office every day. It is thus quite possible that on some workdays, the department has insufficient personnel to unveil the secret. For the company or government to still function, an auxiliary system must be designed to allow the secret to be unveiled in such a circumstance.