# A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography
## 2 Formulation of the Problem
### The CAC Properties of \(P\Delta E\ mKdV\)  
The partial discrete mKdV equation (P\(\Delta\)E mKdV) can be studied through the quadrilateral equation. The quadrilateral equation is used to study the dependent variables in discrete integrated systems via a two-dimensional regular lattice quadratic approach. The method uses a four-sided planar graph (planar quadgraph) that represents the cell decomposition of a surface (see Figure 1a). As shown in Figure 1b, a geometric shape constructed with quadrilaterals is known as a simple cube. A quadrilateral equation is said to have CAC properties if it has the properties of a tetrahedron.  
Suppose that a point in a cell is referred to as \(u_{p}=u(i\alpha_{1},j\alpha_{2})=u_{i,j}\). Meanwhile, suppose the other three points are \(u_{1}=u_{i+1,j}\),\(u_{2}=u_{i,j+1}\), and \(u_{12}=u_{i+1,j+1}\). A simple quadrilateral (orange block in Figure 1) represents an equation that is expressed as  
\[\Xi(u,u_{1},u_{2},u_{12};\alpha_{1},\alpha_{2})=0 \tag{1}\]  
Figure 1: Simple quadrilateral (a) and simple cube (b).  
In Equation (1), \(\Xi\) is affine-linear for all arguments. Thus, for a quadrilateral, as in Figure 1a, Equation (1) is equivalent to  
\[B(u,u_{2},\alpha_{2},\lambda)B(2u,u_{12},\alpha_{1},\lambda)B(u_{12},u_{1}, \alpha_{2},\lambda)B(u_{1},u,\alpha_{1},\lambda)=I\]  
or  
\[B(u_{12},u_{1},\alpha_{2},\lambda)B(u_{1},u,\alpha_{1},\lambda)=B(u_{2},u, \alpha_{2},\lambda)B(u_{12},u_{2},\alpha_{1},\lambda) \tag{2}\]  
where \(B\) is a matrix depending on an arbitrary parameter \(\lambda\).  
In [23], the mathematicians [PERSON], [PERSON], and [PERSON] (ABS) classified all quadrilateral equations as having CAC properties (tetrahedron properties). For a simple cube, these properties mean that variable \(u_{123}\) depends only on \(u_{1}\), \(u_{2}\), and \(u_{3}\). One of the quadrilateral equations classified as having CAC properties by ABS classification is the P\(\Delta\)E mKdV. On the other hand, the P\(\Delta\)E sine-Gordon is not classified as having CAC properties.  
Consider the standard P\(\Delta\)E mKdV equation defined as in [24]:  
\[p(V_{l+1,m}V_{l+1,m+1}-V_{l,m}V_{l,m+1})=q(V_{l,m+1}V_{l+1,m+1}-V_{l,m}V_{l+1, m}) \tag{3}\]  
for the fields \(V_{l,m}\) defined at the sites \((l,m)\) of a two-dimensional lattice \(\mathbb{Z}^{2}\). In Equation (3), the subscripts denote the values of the independent variable, \(V\) is the dependent variable, and \(p\) and \(q\) are arbitrary constants.  
Suppose \(\theta_{3}=\theta_{4}=p\) and \(\theta_{1}=\theta_{2}=q\). Then, Equation (3) can be written as follows:  
\[\theta_{1}V_{l,m}V_{l,m+1}-\theta_{2}V_{l+1,m}V_{l+1,m+1}-\theta_{3}V_{l,m}V_{ l+1,m}+\theta_{4}V_{l,m+1}V_{l+1,m+1}=0 \tag{4}\]  
Let \(V_{l,m}=u,V_{l+1,m}=u_{1},V_{l,m+1}=u_{2}\), and \(V_{l+1,m+1}=u_{12}\). Then, Equation (3) can be expressed with a quadrilateral equation--i.e.,  
\[p(uu_{2}-u_{1}u_{12})-q(uu_{1}-u_{2}u_{12})=0 \tag{5}\]  
According to the cube in Figure 1b, several quadrilateral equations can be derived in terms of the down side \(\left(u_{12}=\frac{u(pu_{2}-qu_{1})}{pu_{1}-qu_{2}}\right)\), left side \(\left(u_{13}=\frac{u(pu_{3}-ku_{1})}{pu_{1}-ku_{3}}\right)\), back side \(\left(u_{23}=\frac{u(qu_{3}-ku_{2})}{qu_{2}-ku_{3}}\right)\), top side \(u_{123}=\frac{u_{3}(pu_{32}-qu_{13})}{pu_{13}-qq_{23}}\), right side \(\left(u_{123}=\frac{u_{2}(pu_{32}-ku_{12})}{pu_{12}-ku_{23}}\right)\), and front side \(\left(u_{123}=\frac{u_{1}(qu_{13}-ku_{12})}{qu_{12}-ku_{13}}\right)\). Substituting these quadrilateral equations into Equation (5) and solving for the variable \(u_{123}\), we arrive at the same quadrilateral equation for the right-top-back side lattices--i.e.,  
\[u_{123}=\frac{u_{1}\left(qu_{3}\big{(}p^{2}-k^{2}\big{)}+ku_{2}\big{(}q^{2}-p ^{2}\big{)}\right)+pu_{2}u_{3}\big{(}k^{2}-q^{2}\big{)}}{qu_{2}(p^{2}-k^{2})+ pu_{1}(k^{2}-q^{2})+ku_{3}(q^{2}-p^{2})} \tag{6}\]  
In Equation (6), the variable \(u_{123}\) depends on \(u_{1}\), \(u_{2}\), and \(u_{3}\) only, which means that the P\(\Delta\)E mKdV has CAC properties.