## 2 Key Technologies  
### MED Pixel Prediction  
Pixel prediction techniques use known pixel values or contextual information to predict the value of a target pixel. When the predicted value closely matches the target pixel, it can be combined with coding strategies to achieve storage compression. This compression creates extra space that allows for the embedding of high-capacity secret data. Since its introduction by [PERSON] et al. (2020), the MED predictor has been widely adopted in various data hiding algorithms due to its exceptional prediction accuracy ([PERSON] et al., 2020; [PERSON] et al., 2020). In Figure 2, the MED predictor uses three neighboring pixel values around the target pixel \(x(i,j)\) to estimate its value, resulting in the predicted value \(px(i,j)\). The process of pixel prediction is detailed in the Equation (1).  
\[  
2. Produce n distinct positive integers \(p_{1}\), \(p_{2}\),..., \(p_{n}\), with each \(p_{k}\) (\(1\leq k\leq n\)) also less than \(F\).
3. Initialize an \(n\times r\) matrix where the first column is filled with the \(n\) integers \(q_{k}\), and the elements of columns 2 through \(r\) are calculated using \(x(i,j)=x(i,j-1)\times(pi+j-2)\).  
For partitioning the secret \(s\), the sharing vector is constructed as \(\mathbf{a}=[s,a_{1},\ldots,a_{r-1}]^{T}\), where \(a_{1},\ldots,a_{r-1}\) are \((r-2)\) randomly chosen values. If the secret being shared is the first in a set, \(a_{1}\) can be any value; otherwise, \(a_{1}\) is a randomly chosen sub-secret from the previous secret, to incorporate ciphertext feedback for enhanced security. Secret sharing is executed according to the following equation, resulting in a vector \(\mathbf{f}=[f_{1},f_{2},\ldots,f_{n}]^{T}\), which comprises \(n\) sub-secrets.  
\[\mathbf{f}=\mathbf{X}\times\mathbf{a}\;modF \tag{2}\]  
The \(i\)-th sub-secret can also be computed from the following equation:  
\[f_{i}=s\times x_{i1}+\sum_{j=2}^{r}x_{ij}a_{j-1}\;\text{mod}\;F \tag{3}\]  
Here, \(x_{ij}\) represents the \(j\)-th element of the \(i\)-th row of the coefficient matrix \(\mathbf{X}\).  
We assume that the receiver \(P_{i}\) holds the sub-secret (\(ID_{i},f_{j}\)), where \(i\) is the identification serial number. When \(r\) shares (\(ID_{1},\mathbf{f}_{1}\)), (\(ID_{2},\mathbf{f}_{2}\)),..., (\(ID_{r},\mathbf{f}_{r}\)) are collected, the first step involves generating the \(n\times r\) coefficient matrix \(\mathbf{X}\). Then, the 1 st, 2 nd,..., \(r\)-th rows of matrix \(\mathbf{X}\) are selected to construct the matrix \(\mathbf{X}\mathbf{r}\). The det(\(\mathbf{X}\mathbf{r}\)) and \(F\) are mutually prime. The secret \(s\) can be obtained by the following equation:  
\[\mathbf{a}=\mathbf{X}\mathbf{r}^{-1}\times f^{(r)}\;\text{mod}\;F \tag{4}\]  
Here, \(\mathbf{f}^{(r)}=[f_{1},f_{2},\ldots,f_{r}]^{T}\) represents a reconstructed vector of \(r\) sub-secrets. The first element of vector \(\mathbf{a}\) is extracted to retrieve the secret \(s\). In the context of an \((r,n)\)-threshold MSS, any party with fewer than \(r\) sub-secrets cannot extract valid information, underscoring the algorithm's robust tolerance and heightened security. The algorithm's resilience and security are fortified by introducing random parameters during the encryption process for data embedding via data replacement. This embedding mechanism is incorporated into the proposed algorithm, augmenting its disaster resistance and enabling secure distributed data storage in cloud service environments.