Algorithm 3: Data extracting and image decryption

Input: Marked sharing images ${C^{\prime }}^{\left(1\right)}, \dots ,{C^{\prime }}^{\left(k\right)}$, Decryption keys $X_1, \dots , X_k$

Output: Data S(1), S(2),…, S(k), Original image M

Step 1. Take part B of the image ${C^{\prime }}^{\left(1\right)}, \dots ,{C^{\prime }}^{\left(k\right)}$ to obtain the secret information S(1), ..., S(k).

Step 2. For sharing a set of pixels of the image ${C^{\prime }}^{\left(\mathrm{t}\right)}$(${C^{\prime }}^{\left(\mathrm{t}\right)}$i,j, ..., ${C^{\prime }}^{\left(\mathrm{t}\right)}$i,j+k−1), extract the P part of the pixel set and merge it into the encrypted pixel values y(t)i,j, $t=1, 2, \dots , k$.

Step 3. Bring the encrypted pixel values y(1)i,j,, y(2)i,j, ..., y(k)i,j with the decryption keys $X_1, \dots , X_k$ into Lagrange polynomial of Equation (4), build the polynomial $P_{i,j+k-1}{x}^{k-1}+P_{i,j+k-2}{x}^{k-2}+\dots +P_{i,j}$.

Step 4. The coefficients of the polynomial $P_{i,j+k-1}{x}^{k-1}+P_{i,j+k-2}{x}^{k-2}+\dots +P_{i,j}$ are used to obtain the pixel sets (Pi,j, …, Pi,j+k−1) of the original images.

Step 5. Repeat Step 2 to Step 4 until all pixel groups (Pi,j, …, Pi,j+k−1) have been processed.

Step 6. Combine all pixel groups (Pi,j, …, Pi,j+k−1) into the original image M.

Step 7. Output the original image M and the secret information S(1), S(2),..., S(k).