# A New One-Dimensional Compound Chaotic System and Its Application in High-Speed Image Encryption
## 4 Experimental Test and Security Analysis
### Sensitivity Analysis  
(1) Sensitivity to plaintext  
The robustness of encrypted images against chosen plaintext attacks and differential cryptanalysis is usually measured by the number of pixels change rate (NPCR) and the unified average change intensity (UACI) of pixel values. These metrics quantify the gap between encrypted images obtained from plaintext images and ciphertext images obtained from slightly modified plaintext images. NPCR and UACI are defined as follows:  
\[D(i,j)=\left\{\begin{array}{l}1,ifC(i,j)\
eq C(i,j),\\ 0,ifC(i,j)=C(i,j).\end{array}\right. \tag{35}\]  
\[NPCR=\frac{1}{M\times N}\sum_{i=1}^{M}\sum_{j=1}^{N}D(i,j)\times 100\%, \tag{36}\]  
\[UACI=\frac{1}{M\times N}(\sum_{i=1}^{M}\sum_{j=1}^{N}\frac{|C(i,j)-C(i,j)|}{25 5})\times 100\%. \tag{37}\]  
where \(C(I,j)\) represents the pixel value of the ciphertext image obtained from the original plaintext image at coordinates \((i,j)\), where it represents the pixel value of the ciphertext image obtained by encrypting after changing the lowest bit of a pixel of the original plaintext image at coordinates \((i,j)\); \(M\) and \(N\) represent the height and width of the image, respectively. The definition of \(D(i,j)\) is as follows: if \(C(i,j)=C(i,j)\), then \(D(i,j)=1\), otherwise,  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & **Plain** & \multirow{2}{*}{**Ours**} & **Ref. [33]** & **Ref. [34]** & **Ref. [35]** & **Ref. [36]** \\ \hline Lena & 7.5683 & 7.9978 & 7.9970 & 7.9975 & 7.9977 & 7.9976 \\ Baboon & 7.3385 & 7.9974 & / & 7.9971 & 7.9970 & / \\ Peppers & 7.5251 & 7.9973 & 7.9971 & 7.9970 & 7.9973 & 7.9974 \\ Cameraman & 7.0097 & 7.9971 & 7.9971 & / & 7.9969 & / \\ \hline \hline \end{tabular}
\end{table}
Table 4: Information entropy comparison of ciphertext images encrypted by different algorithms.  
\(D(i,j)=0\). NPCR = 99.6094% and UACI = 33.4635% are the ideal expectations [13]. The larger the values of NPCR and UACI, the greater the difference between ciphertext images obtained from two plaintext images with slight difference, and the more sensitive (better) the algorithm is to plaintext.  
In this experiment, first a plaintext image is encrypted for one round and the ciphertext image is recorded as **C**; then a pixel is randomly selected from the original image and is changed significantly, then the modified image is encrypted to obtain the ciphertext image **C** and 100 experiments are conducted. Finally, calculate the NPCR and UACI values between **C** and each **C**'. The average values of NPCR and UACI of 100 experiments are shown in Tables 5 and 6, respectively.  
(2) Sensitivity to secret key  
Key sensitivity is also one of the most basic security features of any cryptographic algorithm. On the one hand, the distribution of pixel values of encrypted images should sensitively change with the change of encryption keys, therefore, tiny differences in encryption keys must generate completely different password images. We encrypted a pair of the same plaintext images with randomly selected keys and slightly modified keys, and then tested the NPCR value between the two ciphertext images. The result showed that the NPCR value is very close to 99.61%, which indicates that the encryption process is sensitive to the key. On the other hand, the decryption of the ciphertext image must not use a slightly different key. Figure 9 shows the images decrypted with the correct key and the wrong keys which slightly deviated from the correct key, respectively. The results show that even if the decryption key is slightly different from the correct key, the original image cannot be restored, indicating that the decryption process is sensitive to keys. Therefore, the new cryptographic scheme proposed in this paper is sensitive to the key in the process of encryption and decryption.