## 2 Optical Hyperspectral Image Cryptosystem
### D Arnold Map and Block Mobile Method  
Arnold transform, called cat mapping, is a nonlinear discrete system often used in image encryption. The 3D Arnold transform evolves from the 2D Arnold in order to meet the higher security requirements as encrypted objects change from lower to higher dimensions. The mathematical definition [36; 37] of a 3D Arnold map is as follows:  
\[B=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array}\right] \tag{1}\]  
\[\left\{\begin{array}{l}a_{11}=1+ace,\;\;a_{12}=c,\;\;\;a_{13}=c+ac+abce\\ a_{21}=f+ae+acef,\;\;\;a_{22}=cf+1\\ a_{23}=bf+abcef+acf+abe+a\\ a_{31}=ade+e,\;\;a_{32}=d\\ a_{33}=abde+ad+be+1\end{array}\right. \tag{2}\]  
\[\left(\begin{array}{l}x_{n+1}\\ y_{n+1}\\ h_{n+1}\end{array}\right)=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array}\right]\left[\begin{array}{l}x_{n}\\ y_{n}\\ h_{n}\end{array}\right](\text{mod}L) \tag{3}\]  
As shown in Equation (1), \(B\) is the coefficient matrix of 3D Arnold and the parameters must satisfy the relationship in Equation (2). The variation range of these variables is a positive integer in Equation (2). As shown in Equation (3), the period of the transformation is dependent on \(L\), which is the size of image; before the transformation, the position coordinates of the pixels is \((x_{n},y_{n},h_{n})\) and \((x_{n+1},y_{n+1},h_{n+1})\) is the position coordinates of pixels after the transformation. The pixel value of the image changes according to the position.  
However, in the specific application, the premise of using 3D is that the object of the transformation has to be a square matrix, such as 256 \(\times\) 256 \(\times\) 256. To solve this problem, the block mobile method, as an innovative use of 3D Arnold, is proposed in this paper.  
\[a=\left\lceil\
icefrac{{x}}{{h}}\right\rceil \tag{4}\]  
In Figure 2a, the hyperspectral images have three dimensions. In Equation (4), \(x\) and \(y\) are the side length of a single band image, then the minimum number of square moves is \(a^{2}\). Then, follow a certain rule to move the square horizontally, longitudinally and diagonally until the entire image area is covered, as shown in Figure 2b. Firstly, set the side length of the spatial mobile module as \(h\). From the \(x\)-\(y\) plane, there is a square moving with a side length of \(h\), and it can cover the whole plane at least four times. In this article, the step length we take is long enough to make the moving module move twice along the \(x\) axis and twice along the \(y\) axis, so that the whole hyperspectral image can be covered. From the perspective of permutation and combination, there are 24 kinds of this movement mode. The schematic diagram shows that the module moves to \((0,0)\) points first, and then to \((256,0)\), then to \((0,256)\), then to \((256,256)\) finally. After moving four times, in order to avoid periodic cycles, the 3D Arnold scrambling blocks can also be selected to be executed \(2^{108}\) times in different positions.  
The process and results of the block mobile method are shown in Figure 2b,c. As the block steps, the overlapping areas will appear. The end result is that 3D Arnold performs different times in different areas of the image, so that a higher level of security is achieved. More importantly, this method is easy to implement and can be applied to the vast majority of hyperspectral images.