## 3 Experimental Results and Analysis
### Security Analysis
#### 3.2.6 Image Sensitivity  
A good algorithm designed for image encryption must be sensitive to any tiny difference between two input images, even though there is only a one-bit difference because a hacker can try to find the relationship between two ciphered images by modifying one bit of the plaintext image. Thus, image sensitivity is also an indicator to show the ability of resisting differential attacks. The number of pixel change rates (NPCR) and the unified averaged changed intensity (UACI) are the two most common quantities used to evaluate the strength of image encryption algorithms with respect to differential attacks [41; 42; 43]. NPCR and UACI are defined in Equations (18)-(20), where \(I_{1}(i,j)\) and \(I_{2}(i,j)\) are the pixel values in images \(I_{1}\) and \(I_{2}\):  
\[D(i,j)=\left\{\begin{array}{ll}0&if\ I_{1}(i,j)=I_{2}(i,j)\\ 1&if\ I_{1}(i,j)\
eq I_{2}(i,j)\end{array}\right. \tag{18}\]  
\[NPCR(I_{1},\ I_{2})=\frac{\sum_{i,j}D(i,\ j)}{H\times V}\times 100\% \tag{19}\]  
\[UACI(I_{1},\ I_{2})=\frac{1}{H\times V}\left(\sum_{i,j}\frac{|I_{1}(i,j)-I_{2}( i,j)|}{255}\right)\times 100\% \tag{20}\]  
The variance of NPCR and UACI is related to the resolutions of the test images. In addition, it is unclear how high NPCR/UACI is such that the image cipher does actually have a high security level able to resist malicious attacks. To this end, [PERSON] et al. [42] proposed a mathematical model for ideally encrypting images and then they derived expectations and variances of NPCR and UACI used to form statistical hypothesis tests. Their findings indicate that the ideal values of NPCR and UACI are 99.6094 and 33.4635, respectively.  
Table 9 reports the results of NPCR and UACI for the test image \"memorial\". We remark that E-channel has a narrow histogram band in the plaintext image (see Figure 7a). However, our encryption performs so effectively that it shows near uniform distortion after the image encryption, as shown in Figure 7b. Consequently, all NPCR values, including the  
Figure 10: The local Shannon Entropy (LSE) results for all 20 stego ciphered images.  
E channel, are close to the ideal value (99.6094), while the UACI statistics are also close to the ideal value (33.4635), reflecting that the stego encrypted image, \"memorial\", can resist the differential attack.  
We conducted a statistical hypothesis test, suggested in [42] for all 20 stego encrypted images. We first flipped the least significant bit (LSB) of four corner pixels and the center pixel in R-channel, generating five respective stego encrypted images, each of which is one-bit different from its original. We then calculated the NPCR and UACI values for five pairs of images (original vs. the image with a flipping pixel) for analysis.  
Figure 11 shows the NPCR and UACI results averaged from five pair of images for 20 stego encrypted images. The NPCR results show that their values are larger than the threshold, thus passing the hypothesis test. The UACI results indicate that their values are within the range formed by the lower bound and upper bound, implying that these images pass the UACI hypothesis test. According to Table 9 and Figure 11, we conclude that our encryption scheme produces good performance in resisting the differential attacks.