# Secret Image Sharing Revisited: Forbidden Type, Support Type, and Their Two Approaches
## 4 Discussion: Other Details of the Design
### Sharing that Requires All Departments
Step 2: Because there are 3 departments, we partition these 15 shares into 3 equal parts of 5 shares each. Next, assign shares \(S_{1}\)-\(S_{5}\) to \(DEP_{L}\), \(S_{6}\)-\(S_{10}\) to \(DEP_{M}\), and \(S_{11}\)-\(S_{15}\) to \(DEP_{H}\). Step 3: Every department can use its 5 shares to create \(C_{3}^{4}\) = 5 possible combinations of 4 shares each. Distribute these 5 combinations to the 5 people in the department. For example,  
\(DEP_{L}\): \(P_{1}\) = {\(S_{2}\), \(S_{3}\), \(S_{4}\), \(S_{5}\)}, \(P_{2}\) = {\(S_{1}\), \(S_{3}\), \(S_{4}\), \(S_{5}\)}, \(P_{3}\) = {\(S_{1}\), \(S_{2}\), \(S_{4}\), \(S_{5}\)}, \(P_{4}\) = {\(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{5}\)}, \(P_{5}\) = {\(S_{1}\), \(S_{2}\), \(S_{3}\), \(S_{4}\)}. \(DEP_{Mobtained by doctors \(\{D_{1},D_{3}\}\) and \(\{D_{2},D_{4}\}\), respectively. Both Figure 4b,c are identical to Figure 4a. However, if \(\{D_{1},D_{2}\}\) try to disclose the image, they only get Figure 4d. In the above, the experiment is designed as follows. Split the original \(512\times 1024\) image in Figure 4a to two \(512\times 512\) images: one is the lung image, and the other is the medical history image. To show the mixed use of the two approaches, the lung image and the medical history image are shared using distinct approaches. The \(512\times 512\) lung image is shared using linear-equations approach with \((t_{participants},n_{participants},f)=(2,4,0)\). To achieve this, we need a 2-by-4 matrixA such that each row has 2 elements, and any 2 of the 4 rows are independent. There are infinitely many ways to design this matrix. One such way is to let \(Row_{i}=(I,i+1)\) for \(I=1,\ldots,4\). Then, for each two-pixels pair of the X-ray image, let doctor \(D_{i}\)store an integer \(IP_{i}\) which is the inner product value of \(Row_{i}\) and a 2-dimensional vector formed of the two pixel values. Therefore, for the \(512\times 512\) X-ray image, each doctor will store \(512\times 512/2=512\times 256\) integers. On the other hand, the right half of Figure 4a, i.e., the \(512\times 512\) medical history image, is shared using polynomial approach with \((t_{participants},n_{participants},f)=(2,4,1)\). To achieve this, we use \((t_{artificial},n_{artificial})=(4,7)\) in traditional polynomial-based sharing to share the \(512\times 512\) medical history image and obtain \(n_{artificial}=7\) shares \(\{S_{1},\ldots,S_{7}\}\) such that any \(t_{artificial}=4\) of them can recover the medical history. Then we distribute these 7 shares to the 4 doctors. The distribution is that \(D_{1}\) gets \(\{S_{1},\)\(S_{2}\}\), \(D_{2}\) gets \(\{S_{2},\)\(S_{3}\}\), \(D_{3}\) gets \(\{S_{4},\)\(S_{5}\}\), \(D_{4}\) gets \(\{S_{6},\)\(S_{7}\}\). Note that we deliberately let the share \(S_{2}\) appear in both \(D_{1}\) and \(D_{2}\). By doing this, the total number of different shares in the combination \(\{D_{1},\)\(D_{2}\}\) is only \(2+1=3\), which is less than the threshold value 4, meaning that the medical history cannot be revealed. Conversely, any other combination \(\{D_{i},\)\(D_{j}\}\) yields 4 distinct shares, and hence, can reveal the medical history. Notably, for the \(512\times 512\) medical history image, every doctor holds a record which is constituted by 2 shares; and each share is 4 times smaller in size than the original \(512\times 512\) medical history image in the \((4,7)\) threshold scheme. Therefore, the medical history data held by each doctor has \(512\times 256\) bytes which is \(2\times(1/4)=\frac{1}{2}\) of the size of the \(512\times 512\) medical history image. Now, combining the two results, we can see that each doctor holds \(512\times 512/2=512\times 256\) integers for the X-ray image and also holds \(512\times 256\) bytes for the medical history image. Since each integer is an inner product value of a 2-element row and a vector formed of two pixel values, the integer is between 0 and  
Figure 4: An application example of the sharing with forbidden combination. (**a**) The original \(512\times 1024\) lung \(x\)-ray image of a patient.