## 3 Proposed Method  
In this section, a new dual image based reversible data hiding method using most significant bits and the center shifting technique is proposed. During the data hiding process, the proposed method securely hides data by generating two different stego images from a single cover image. During the data extraction process, the secret data and the original cover image are obtained from these two stego images with lossless. The flowchart of data hiding process is presented in Figure 3. In the first step of data hiding process, the number \(n\), which represents how many of the most significant bits of the pixel value will be used, is determined. The size of secret data or the maximum acceptable rates of distortion are decisive factors in choosing the \(n\) number. In the second step, the secret data is converted into a \((n+1)\) bit binary series, and each element of the series is converted to a decimal system to obtain the s series, which represents the symbols to be hidden. Briefly, \((n+1)\) bit hiding is performed for the selected value of \(n\). Therefore, \((n+1)/2\) bits of secret data per pixel (bpp) in the original cover image is hidden. In this way, the payload is always greater than 1 (bpp). For the determined number of \(n\), the initial  
Figure 2: The example of k = 3.  
series is created by using the most significant \(n\) bits of the pixel values, and the range of this series is between \([0,\ 2^{n}-1]\). Equation (8) presents the formula used to calculate this initial series. \[d=\left\lfloor\frac{X_{i,j}}{2^{8-n}}\right\rfloor \tag{8}\]  
The difference between the initial series values and the secret data is calculated to aim to reduce the average of the secret data to be embedded in the stego image. This operation causes the range of \([0,2^{n+1}-1]\) coming from secret data to expand and take values in the range of \(\left[-2^{n}+1,2^{n+1}-1\right]\). Equation (9) shows the computation of these secret data. \[d^{\prime}=s_{i}-d_{i} \tag{9}\]  
This difference needs to be shifted in the opposite direction by half the interval of the initial series to minimize mean square error. To achieve this, Equation (10) is used, and the master record series is obtained, which will be hidden in the cover images. Subtracting the value \(2^{n-1}\) from the obtained hidden symbols series, the interval of change \(\left[-2^{n}-2^{(n-1)}+1,\ 2^{(n+1)}-2^{(n-1)}-1\right]\). This range expression, when simplified, becomes \(\left[-3\times 2^{(n-1)}+1,\ 3\times 2^{(n-1)}-1\right]\). \[d^{\prime\prime}=d^{\prime}-2^{(n-1)} \tag{10}\]  
The change of the first and second stego images are obtained by dividing the master record series into 2 and \(-2\), respectively. These changes are achieved using Equation (11). \[d_{1}=\left\lfloor\frac{d^{\prime\prime}}{2}\right\rfloor,\ d_{2}=\left\lfloor \frac{d^{\prime\prime}}{-2}\right\rfloor \tag{11}\]  
By adding these differences to the cover image as in Equation (12), two stego images are obtained and the data hiding process is completed. \[X^{\prime}=X+d_{1}\ \ \ and\ \ \ X^{\prime\prime}=X+d_{2} \tag{12}\]  
During data hiding process, the change range is also divided into two, as the master record series is divided into two. Therefore, the range of change from \(\left[-3\times 2^{(n-1)}+1,\ 3\times 2^{(n-1)}-1\right]\) to \(\left[-3\times 2^{(n-2)}\ +\ 1/2,\ 3\times 2^{(n-2)}\ -\ 1/2\right]\). By rounding down the obtained stego pixel values, the change intervals are obtained as \(\left[-3\times 2^{(n-2)}\,\ 3\times 2^{(n-2)}-1\right]\). To avoid FOBP, the data hiding process is performed only  
Figure 3: Data hiding process. for pixels in the range of \(\left[3\times 2^{(n-2)}\,\ 256-3\times 2^{(n-2)}\right]\). Figure 4 shows how the change difference ranges with the changes made in the secret data for \(n\) = 4. During the data extraction process, if any of the pixel values in the same position of the two stego images are within this range and are not equal, it means that secret data is hidden there.