## 2 Optical Hyperspectral Image Cryptosystem
### Triangular Association Encryption Algorithm (TAEA)  
For grayscale and color images, as a key part of the image encryption diffusion, the diffusion is often used in-band. However, for hyperspectral images, the realization of diffusion in-band and inter-band can not only improve the efficiency, but also improve the security level. The TAEA designed in this paper can achieve the above goals and requirements perfectly. The model is as follows:  
\[\begin{split}& I[x_{n+1}(i),y_{n+1}(i),h_{n+1}(i)]\\ &=g\{I[x_{n}(i),y_{n}(i),h_{n}(i)]\}+g\{I[x_{n}(i+a_{0}),y_{n}(i+ a_{0}),h_{n}(i+a_{0})]\}\\ &+g\{I[x_{n}(i),y_{n}(i),N+1]\}\mod N\end{split} \tag{5}\]  
Figure 2: The block mobile method of 3D Arnold. (**a**) 3D model of hyperspectral image; (**b**) The movement schematic diagram of 3D Arnold transform; (**c**) The result of transformation and the number of transformations in different regions.  
where, \([x_{n+1}(i),y_{n+1}(i),h_{n+1}(i)]\) is the pixel coordinates after transformation; and \(I[x_{n+1}(i),y_{n+1}(i),h_{n+1}(i)]\) is the corresponding pixel value; \([x_{n}(i),y_{n}(i),h_{n}(i)]\) is the pixel coordinates before transformation; and \(I[x_{n}(i),y_{n}(i),h_{n}(i)]\) is the corresponding pixel value; \([x_{n}(i),y_{n}(i),N+1]\) is the pixel coordinates at layer \(N+1\) corresponding to \([x_{n}(i),y_{n}(i),h_{n}(i)]\), and \(I[x_{n}(i),y_{n}(i),N+1]\) is the corresponding pixel value; \(n\) is steps of iteration; \(i=1,2,\cdots,N\) is number of transformation; \(N\) is system grid points. \([x_{n}(i+a_{0}),y_{n}(i+a_{0}),h_{n}(i+a_{0})]\) are determined by the model as follows:  
\[\left(\begin{array}{c}x_{n}(i+a_{0})\\ y_{n}(i+a_{0})\\ h_{n}(i+a_{0})\end{array}\right)=\left[\begin{array}{ccc}a_{11}&a_{12}&a_{13} \\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\end{array}\right]^{a_{0}}\left[\begin{array}{c}x_{n}(i) \\ y_{n}(i)\\ h_{n}(i)\end{array}\right](\text{mod}N) \tag{6}\]  
where \(a_{0}\) is a positive integer. According to the reference [36], the period of 3D Arnold is related to the size of image, and the period of the 64 \(\times\) 64 \(\times\) 64 image is 112. The object size of this study is 256 \(\times\) 256 \(\times\) 189, and the period is much larger than 112. Therefore, to be on the safe side, we set the variation range of \(a_{0}\) as 112.  
The mathematical form of function \(g\), called dynamic integer tent mapping [38], is as follows. The tent map has good ergodic uniformity and can improve the optimization speed of the algorithm. Moreover, it can be used together with Arnold transform to give better play to the chaotic characteristics of the system:  
\[g(x_{k})=\left\{\begin{array}{ll}\mu(f_{k}+\frac{1}{2}),&f_{k}\in[0,2^{b-1} )\\ \mu(2^{b}-1-f_{k}),&f_{k}\in[2^{b-1},2^{b}-1]\end{array}\right. \tag{7}\]  
\[f_{k}=(x_{k}+j_{k})\text{mod}(2^{b}) \tag{8}\]  
where \(\mu\in(0,2]\) and \(b\) is digit of the system; and \(j_{k}\) represents the displacement of the mapping. Multi-dimensional chaotic integer sequences with good pseudo randomness can be generated in this model rapidly and in parallel.  
In space, three points \([x_{n}(i),y_{n}(i),h_{n}(i)]\), \([x_{n}(i),y_{n}(i),N+1]\) and \([x_{n}(i+a_{0}),y_{n}(i+a_{0}),h_{n}(i+a_{0})]\) form a triangle, as shown in Figure 3, so the algorithm model is called the triangular association encryption algorithm (TAEA).  
Figure 3: A triangle in space formed by three points of the image from the algorithm (TAEA).