## 3 Numerical Simulation
### Test the Sensitivity of Keys  
This paper will firstly test the sensitivity of some of the keys in protecting the secret images. Some attackers are designed to have a complete encryption and decryption system, but with partial keys which are incorrect.  
Suppose the attackers wants to gain the three keys, one is the number of the 3D Arnold transformation (\(k=3\)), one is the number of the triangular association encryptions (\(a=3\)) and one is the order of the block mobile method (\(O=``1234\"\)).  
First, attacker-1 has obtained all of the keys, except the number of the 3D Arnold transformations. He assumes that \(k=2\) and Figure 7 shows the decrypted image. Attacker-1 cannot obtain any information about the original text from Figure 7d-g.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Key Space** \\ \hline Ref. [41] & \(3.41\times 10^{38}\) \\ Ref. [42] & \(10^{45}\) \\ Ref. [43] & \(10^{56}\) \\ Ref. [44] & \(>\)\(10^{77}\) \\ Our cryptosystem & \(>\)\(10^{87}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Comparison of key space among several algorithms.  
Figure 7: _Cont._  
Attacker-2 has obtained all of the keys, except the number of triangular association encryptions. He assumes that \(a=2\) and Figure 8 shows the decrypted image. Attacker-2 cannot obtain any information about the original text from the Figure 8a-d.  
Figure 8: Experimental data: (**a–c**) corresponding decrypted image of 30 th, 70 th and 100 th band with \(a=2\); (**d**) decrypted image combined of (**a–c**).  
Figure 7: Experimental data: (**a–c**) original image of 30 th, 70 th and 100 th band; (**d–f**) corresponding decrypted image with k = 2; (**g**) decrypted image combined of (**d–f**).  
Attacker-3 has obtained all of the keys, except the order of the block mobile method. He assumes that \(O\) = \"1324\" and Figure 9 shows the decrypted image. Attacker-3 cannot obtain any information about the original text from the Figure 9a-d.  
Besides, the keys held by the recipient and the three attackers are listed in Table 2, with the SSIM values by calculating the difference between the original and the tampered value. The results in Table 2 show that any small change in the keys can make a huge difference in the results, demonstrating the high sensitivity and ability of the keys to protect information.  
The keys above can only be integers, then this paper will further test the sensitivity of the angle of the Gyrator transformation, whose range of change is real. Figure 10 shows the corresponding SSIM curve with the angle of the Gyrator transformation changing near the right value, under the condition that the correct key and decryption method are known. In this test, the correct angle and the sampling step are set to 0.5 and 0.005. When the step is +1 or -1, that is, when the angle is 0.495 and 0.505, two decrypted images are completely unrecognizable. In fact, it would be worse if the angle value deviates more from the set value. Therefore, in terms of the effect of protecting secret images, the angle is a good additional key.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(k\) & \(a\) & \(O\) & SSIM–30 th & SSIM–70 th & SSIM–100 th \\ \hline
3 & 3 & 1234 & 1 & 1 & 1 \\
2 & 3 & 1234 & 0.0387 & 0.0371 & 0.0355 \\
3 & 2 & 1234 & 0.0091 & 0.0109 & 0.0086 \\
3 & 3 & 1324 & 0.0286 & 0.0308 & 0.0268 \\ \hline \hline \end{tabular}
\end{table}
Table 2: The results by using different initial conditions.  
Figure 9: Experimental data: (**a–c**) corresponding decrypted image of 30 th, 70 th and 100 th band with \(O\) = “1324”; (**d**) decrypted image combined of (**a–c**).