# A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography
## 2 Formulation of the Problem
### Histogram, Covariance, and Correlation Computations Based on the Weighted Data in the Image Pixels  
When considering cryptographic image data processing, there are two important tools used to analyze image processing--i.e., the gradient filter and the gradient orientation filter. The gradient filter provides an image corresponding to the image gradient's magnitude and is computed using the discrete derivatives of a Gaussian of the pixel radius. Meanwhile, the gradient orientation filter provides an image corresponding to the local orientation that is parallel to the image gradient and is computed using the discrete derivatives of a Gaussian of the pixel radius, thus returning values between \(\frac{-\pi}{2}\) and \(\frac{\pi}{2}\)[32].  
The important analyses for image cryptography via encryption algorithms are the histogram, covariance, and correlation analyses. Histogram analysis is conducted to determine the pixel distribution. The distribution of an image's pixel values prior to encryption is usually concentrated on the pixels' highest space value. Therefore, the best encryption spreads the pixel values out over the pixel space. Another tool, the adjacent pixel correlation, was used to test the correlations between pairs of adjacent pixels.  
Mathematica was used to produce the histogram distributions for the ACM encryption in Equation (24), 2D linear map in Equation (23), and 2D linear map in Equation (31). Four histogram distributions of the weighted data for the images shown in Figures 2, 4 and 9 are shown in Figure 13. There are clear differences between the histograms for the plain image and those for the cipher images.  
Generally, the histograms generated by the cipher images have flat distributions (uniform). In contrast, the histogram generated by the plain image focuses on some space pixel values. The covariances and correlations for the image data for \(N=110\) iterations are shown in Table 4.  
\begin{table}
\begin{tabular}{c c c} \hline \hline  & **Covariance** & **Correlation** \\ \hline Plain Image & 0.001048070 & 0.01704610 \\ Cipher Image Generated by Equation (24) & 0.000202934 & 0.00928261 \\ Cipher Image Generated by Equation (23) & 0.000042265 & 0.00187458 \\ Cipher Image Generated by Equation (31) & \(-0.00000098\) & \(-0.00041965\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: The covariances and correlations of the image data based on the weighted data (gradient orientation filter, gradient filter) of the image for \(N=110\) iterations.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **2D Linear Map** & **(\(\alpha_{0}\),\(y_{0}\))** & **LE: \(\lambda^{\text{max}}(x_{0})\)** & **LN: \(e^{\lambda^{\text{max}}(x_{0})}\)** \\ \hline
1. & ACM & (0.001, 0.001) & 0.9533117189 & 2.59428699560 \\
2. & sine-Gordon & (0.001, 0.001) & 6.6821101678 & 798.001253131 \\
3. & mKdV & (0.001, 0.001) & 0.7638455683 & 2.14651493883 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The LE number and LN number for the 2D linear ACM map (Equation (24)), the 2D linear sine-Gordon map (Equation (23)), and the 2D linear mKdV map (Equation (31)) closed fixed points \((x^{*},y^{*})=(0,0)\).