# A Two-Dimensional mKdV Linear Map and Its Application in Digital Image Cryptography
## 2 Formulation of the Problem
### Construction of a Two-Dimensional Linear Map (Case Study: Sine-Gordon Map)  
Consider a partial discrete sine-Gordon equation on a two-dimensional lattice \(\mathbb{Z}^{2}\) is defined as follows (see [25; 26] for the construction of a generalized sine-Gordon map):  
\[\theta_{1}(V_{l,m+1}V_{l+1,m}-V_{l+1,m+1}V_{l,m})+\theta_{2}V_{l+1,m+1}V_{l,m+ 1}V_{l+1,m}V_{l,m}=\theta_{3} \tag{14}\]  
where \(V,\theta_{1},\theta_{2},\theta_{3}\in\mathbb{R}\). A travelling wave solution of Equation (14) was obtained by the ansatz equation (Equation (7)). Therefore, from Equation (14) we then obtained  
\[\theta_{1}(V_{n+z_{2}}V_{n+z_{1}}-V_{n+z_{1}+z_{2}}V_{n})+\theta_{2}V_{n+z_{1} +z_{2}}V_{n+z_{2}}V_{n+z_{1}}V_{n}=\theta_{3} \tag{15}\]  
Let us take \(z_{1}=z_{2}=1\) and choose \(\theta_{1}=\mu\theta_{2}\) and \(\theta_{3}=\lambda\theta_{2}\); we then have the following equation  
\[\mu\Big{(}V_{n+1}^{2}-V_{n+2}V_{n}\Big{)}+\Big{(}V_{n}V_{n+1}^{2}V_{n+2}\Big{)}=\lambda \tag{16}\]  
Equation (16) is equivalent to the mapping  
\[\left. \begin{array}{l}V_{1}^{\prime}=\frac{\left(\lambda-\mu V_{1}^{2} \right)}{V_{0}\left(V_{1}^{2}-\mu\right)}\\ V_{1}^{\prime}=V_{1}\end{array} \right\} \tag{17}\]  
According to Equation (17), we denote by \(\mathbf{x}\) the sequence in \(\mathbb{R}^{2}\), which is defined as \(\mathbf{x}=(x_{1}=V_{1},x_{2}=V_{0})\). Therefore, a general mapping of the plane \(\mathbf{f}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\), which assigns a point \(\mathbf{x}=(x_{1},x_{2})\) and an image point \(\mathbf{x}^{\prime}=\big{(}x_{1}^{\prime},x_{2}^{\prime}\big{)}\), can be written as  
\[\mathbf{x}^{\prime}=\mathbf{f}_{(\mu,\lambda)}(\mathbf{x}) \tag{18}\]  
where  
\[\mathbf{f}_{(\mu,\lambda)}:\qquad\mathbb{R}^{2} \rightarrow \mathbb{R}^{2}\] \[(x_{1},x_{2}) \mapsto \left(\frac{\left(\lambda-\mu x_{1}^{2}\right)}{x_{2}\left(x_{1} ^{2}-\mu\right)},x\right).\]  
It is easy to check that the mapping (18) has such integrals as follows. \[F_{(\mu,\lambda)}(x_{1},x_{2})=\mu\bigg{(}\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{ 1}}\bigg{)}-\bigg{(}x_{1}x_{2}+\lambda\frac{1}{x_{1}x_{2}}\bigg{)}. \tag{19}\]  
For all \(n\in\mathbb{N}\), the solution of Equation (16) is on a level set of \(F_{(\mu,\lambda)}(x_{1},x_{2})\). Let us fix \(\lambda=1\) to obtain a special case of Equation (18)--namely,  
\[\mathbf{f}_{\mu,1}:(x_{1},x_{2})\mapsto\Bigg{(}\frac{\left(1-\mu x_{1}^{2} \right)}{x_{2}\big{(}x_{1}^{2}-\mu\big{)}},x_{1}\Bigg{)} \tag{20}\]  
It is easy to check that the mapping in Equation (20) has the integral:  
\[\hat{F}(x_{1},x_{2};\mu)=\mu\bigg{(}\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}} \bigg{)}-\bigg{(}x_{1}x_{2}+\frac{1}{x_{1}x_{2}}\bigg{)} \tag{21}\]  
The mapping in Equation (20) has several properties:  
* The mapping \(\mathbf{f}_{\mu,1}\) has an integral equation (Equation (21)). * \(\mathbf{f}_{\mu,1}\) is measure-preserving--i.e., \[\left|D\mathbf{f}_{\mu,1}\right|=\frac{\rho(x_{1},x_{2})}{\rho\left(\frac{\left(1 -\mu x_{1}^{2}\right)}{x_{2}\left(x_{1}^{2}-\mu\right)},x_{1}\right)}\] where \[\rho(x_{1},x_{2})=\frac{\left(x_{1}+x_{2}\right)}{\left(x_{1}x_{2}\right)^{2}} \left[\partial_{\mu}F(x_{1},x_{2};\mu)\right]^{-1}=\frac{1}{x_{1}x_{2}}\]
* There is a reversing symmetry \(G(x_{1},x_{2})=(x_{2},x_{1})\) such that \(G^{\circ}\mathbf{f}_{\mu,1}{}^{\circ}G^{-1}=\mathbf{f}_{\mu,1}^{-1}\) (small circle denoting the symbol for composition). This means that \(\mathbf{f}_{\mu,1}\) is reversible (\(\mathbf{f}_{\mu,1}{}^{\circ}G^{\circ}\mathbf{f}_{\mu,1}=G\)). Note that one of the fixed points on the map is (1,1) [13; 26]. Thus, a linear map can be constructed based on the map form in Equation (20) at a closed fixed point (1,1) for which the Jacobian matrix \(A\) is written as follows:  
\[A=\left( \begin{array}{cc}\frac{2\left(\mu^{2}-1\right)}{\left(1-\mu\right)^ {2}}&-1\\ 1&0\end{array} \right) \tag{22}\]  
From Equation (20) and Equation (22), for \(\mu\in\mathbb{R}\backslash\{1\}\), we have the following linear map:  
\[\left( \begin{array}{c}{x_{1}}^{\prime}\\ {x_{2}}^{\prime}\end{array} \right)=\left(