## 2 Related Works
### The EMD Method and Applications Based on It  
The proposed steganographic technique by [PERSON] [15] employs the \((2n+1)\) number system for encoding secret numbers into groups of \(n\) pixels. In this scheme, altering a single pixel's value within each group by either increasing or decreasing by one hides the data, offering \(2n\) correction methods. A secret message, represented as a binary data stream, is segmented into parts containing \(L=\lfloor K\cdot log_{2}(2n+1)\rfloor\) bits each, with these segments then expressed in base \((2n+1)\) as \(K\). Utilizing the \((2n+1)\) system, the method embeds a secret number across \(n\) pixels after a pseudorandom permutation of all hidden pixels, guided by a secret key. This reorganization results in pixel groups, each comprising \(n\) pixels labeled as \(x_{1},x_{2},\ldots,x_{n}\). [PERSON] specifies the extraction function \(f\)--a weighted sum modulo \((2n+1)\) based on the grayscale values of these pixels (Equation (3))--streamlining data embedding and retrieval. \[f(x_{1},x_{2},\ldots,x_{n})=\left[\sum_{i=1}^{n}(x_{i}\cdot i)\right]\,mod\,(2n +1) \tag{3}\]  
If the secret number (let us denote it as \(m\)) matches the output of the extraction function for the original pixel set, there is no need to change the pixel value. However, if \(m_{i}\) does not align with the extraction function's output, we compute the difference, denoted as \(s\), between \(m_{i}\) and the extraction function output, utilizing modulo \((2n+1)\). If \(s\) is less than or equal to \(n\), then the value of the pixel at index \(s\) is incremented by one. Conversely, if \(s\) exceeds \(n\), the value of the pixel at index \(2n+1-s\) is decremented by one. Consider, for instance, an original pixel group and secret message represented as \([136,140]\) and \(m=2\). In this scenario, the computed value of the extraction function \(f=[(136\times 1+140\times 2)]\,\,mod\,5\) equals one, corresponding to a given secret number in the quintal system. To determine \(s\), one calculates the difference between the secret number \(m\) and the output of \(f\), applying modulo \(5\) for the operation. Hence, \(s=m-f=(2-1)\bmod 5=1\). Should \(s\) be less than \(n=2\), the pixel value at the index corresponding to \(s\) is incremented by one. Consequently, this adjustment increases the first pixel's value by one, leading to an updated stego pixel group of \([137,140]\). In cases where the intended secret number for embedding  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \multirow{2}{*}{**Rule**} & \multirow{2}{*}{**Case**} & \multicolumn{3}{c}{**Pixel Value Modification**} & \multicolumn{6}{c}{**The Final Modified Pixel Values**} \\ \cline{3-11}  & & \(x_{i}^{1}\) & \(x_{i+1}^{1}\) & \(x_{i}^{2}\) & \(x_{i+1}^{2}\) & \(x_{i}^{2}\) & \(x_{i+1}^{1}\) & \(x_{i}^{2}\) & \(x_{i+1}^{2}\) \\ \hline
1 & 3 & 0 & 0 & \(-1\) & 0 & \(x_{i}^{1}+2\) & \(x_{i+1}^{1}+1\) & \(x_{i}^{2}-1\) & \(x_{i+1}^{2}\) \\
2 & 6 & 0 & 1 & 0 & 1 & \(x_{i}^{1}\) & \is '0', \(s\) is determined as \((0-1)\) mod \(5=4\). Given that \(s\) exceeds \(n\), the procedure calls for decreasing the value of the pixel at index \(2n+1-s=2\times 2+1-4=1\), resulting in a modified stego pixel group of \([135,140]\). On the receiving end, the extraction of the secret number is facilitated through the computation of the extraction function applied to the stego pixel group. [PERSON] et al. (2015) [18] performed the embedding process through the following two steps: the first step involved generating a displayed image using an EMD algorithm, and the second step involved generating a subsequent displayed image using an adaptive method, which referenced the first image. The quality of the resulting images was asymmetric, with the first image exhibiting a higher quality than the second. [PERSON] et al. (2019) [20] introduced a new dual-image-based RDH method leveraging the EMD matrix. This method identifies the four vertices of a \(3\times 3\) block and may shift two stego-pixel pairs to these vertices under certain conditions.