# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 3 Proposed Types and Approaches
### Using the Linear-Equations Approach to Design Types 1 and 2  
The linear-equations approach can be mapped from the polynomial approach. Because we already introduced the polynomial approach, we only need to know how to map from the polynomial approach to the linear-equations approach. In the polynomial approach, every participant has a shadow file comprising several shares, whereas in the linear-equations approach, every participant has a shadow file comprising several equations. For both approaches, the threshold \(t_{\textit{artificial}}\) must be met to unveil the secret. Specifically, in the polynomial approach, participants attending a meeting must have at least \(t_{\textit{artificial}}\) distinct shares, whereas in the linear-equations approach, participant attendees must have at least \(t_{\textit{artificial}}\) independent equations. The linear-equations approach thus proceeds as follows. ```
1:Create an \(n_{\textit{artificial}}\)-by-\(t_{\textit{artificial}}\) matrix \(A\) so that any \(t_{\textit{artificial}}\) of its \(n_{\textit{artificial}}\) rows are independent. 2:Partition secret image to non-overlapping segments of \(t_{\textit{artificial}}\) pixel values each. 3:for each secret segment, \(D_{\textit{secret}}\), do
4:for\(k=1\) to \(n_{\textit{artificial}}\)do
5: Calculate the inner product value \(IP_{k}=Row_{k}\cdot D_{\textit{secret}}\). 6:endfor
7:for\(i=1\) to \(n_{\textit{articipants}}\)do
8:for\(j=1\) to \(n_{\textit{artificial}}\)do
9:if Participant \(P_{i}\) gets the share \(S_{j}\) in the polynomial approach then
10: Participant \(P_{i}\) gets the pair \((Row_{j},IP_{j})\) in linear equations approach. 11:endif
12:endfor
13:endfor
14:endfor
15:return the rows and inner product values collected by each participant. ```  
**Algorithm 2** General procedure to derive the linear-equations approach from the (\(t_{\textit{artificial}}\), \(n_{\textit{artificial}}\)) polynomial approach  
Notably, every share \(S_{j}\) in the polynomial approach is replaced by a corresponding equation \(Eq_{j}\) in the linear-equations approach. Here, each equation \(Eq_{j}\) is formed of two parts: firstly, the \(t_{\textit{artificial}}\) left-hand side coefficients of equation \(Eq_{j}\) are the \(t_{\textit{artificial}}\)-dimensional vector \(Row_{j}\); secondly, the right-hand side of equation \(Eq_{j}\) is the inner product value \(IP_{j}\). Hence, any \(t_{\textit{artificial}}\) independent equations can uniquely solve for the \(t_{\textit{artificial}}\) values of the \(t_{\textit{artificial}}\)-dimensional vector \(D_{\textit{secret}}\). If a participant \(P\) gets \(\{S_{1}\), \(S_{2}\), \(S_{5}\), \(S_{7}\}\) in the polynomial approach, then that participant \(P\) also gets \(\{Eq_{1}\), \(Eq_{2}\), \(Eq_{5}\), \(Eq_{7}\}\), i.e. \(\{Row_{j}\ \&\ Ip_{j}\}_{j}=\) 1, 2, 5, 7 in the linear-equations approach. Thus, regardless of whether a Type 1 (the forbidden type) or Type 2 (the cross-department support type) system is used, just map the method from polynomial approach to linear-equations approach, by letting the shares held by each participant be replaced by the corresponding equations, or vice versa. For example, if the polynomial approach is used, where participant \(P_{a}\) gets \(\{S_{1}\), \(S_{2}\), \(S_{5}\), \(S_{7}\}\) and participant \(P_{b}\) gets \(\{S_{3}\), \(S_{4}\), \(S_{6}\), \(S_{8}\}\), to obtain the corresponding instance of the linear-equations approach, we let participant \(P_{a}\) get \(\{Row_{j}\); \(IP_{j}\}_{j}=\) 1, 2, 5, 7 and participant \(P_{b}\) get \(\{Row_{j}\); \(IP_{j}\}_{j}=\) 3, 4, 6, 8. Moreover, if rows are created using a specified process or algorithm, as discussed in Section 4.1, then each participant does not need to store the rows; for example, \(P_{a}\) only stores \(\{IP_{j}\}_{j}=\) 1, 2, 5, 7 and \(P_{b}\) only stores \(\{IP_{j}\}_{j}=\) 3, 4, 6, 8. Types 1 and 2 are detailed as follows.