## 3 Experimental Results and Analysis
### Security Analysis
#### 3.2.3 Histogram Analysis  
The histogram contains much statistical information about an image, and the histogram of a plain image contains obvious distribution features. An eligible encryption algorithm should completely eliminate these features. Figure 7 shows histograms of the plaintext image and the ciphered one. Our encryption algorithm makes the distribution more uniform and completely different from the histograms of the plaintext in four channels. No meaningful information can be obtained from the histogram of the stego encrypted HDR image. Therefore, our scheme can effectively prevent statistical attacks.  
Variance of histogram (VOH) quantitatively reflects the randomness of the image. VOH is calculated by Equation (14), where \(Z=\{z_{0},\ z_{1},\ \ldots,\ z_{255}\}\) denotes the number of counts at each bin in the histogram and \(\mu\) represents the average value over all bins. The smaller the VOH, the more randomness the test image holds:  
\[Var(Z)=\sum_{i=0}^{255}E\Big{[}\big{(}z_{i}-\mu\big{)}^{2}\Big{]} \tag{14}\]  
Apart from VOH, we utilize the Chi-square (\(\chi^{2}\)) test. When the significance level, \(\alpha\), is set as 0.05 and the degree of freedom is 255; the threshold of the \(\chi^{2}\) test is 293.25. If the \(\chi^{2}\) value of a histogram is under this threshold, the histogram can be considered to perform with uniform distribution statistically.  
Table 6 lists VOH and \(\chi^{2}\) values of the \"memorial\" test image before and after encryption. The results show that encryption reduces the VOH values dramatically, from over several millions to as small as several thousands. In addition, the \(\chi^{2}\) values also show a significant decrease in the encrypted image, from one hundred thousand to several hundreds. Finally, \(\chi^{2}\) values in four channels are below the threshold, thereby all 20 stego encrypted test images pass the \(\chi^{2}\) test.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No** & **Test Type** & \(p\)**-Value** & **Result** & **No** & **Test Type** & \(p\)**-Value** & **Result** \\ \hline
1 & Frequency & 0.931466 & Pass & 9 & Maurer’s universal statistical & 0.307327 & Pass \\
2 & Block frequency & 0.926042 & Pass & 10 & Linear complexity & 0.917021 & Pass \\
3 & Runs & 0.652705 & Pass & 11 & Serial & 0.880/5.08 & Pass \\
4 & Longest run of ones in a block & 0.966102 & Pass & 12 & Approximate entropy & 0.422853 & Pass \\
5 & Binary matrix rank & 0.483810 & Pass & 13 & Cummulative sums (forward) & 0.969889 & Pass \\
6 & Spectral & 0.308930 & Pass & 14 & Cummulative sums (reverse) & 0.952027 & Pass \\
7 & Non-overlapping template matching & 0.055807 & Pass & 15 & Random excursions & 0.650265 & Pass \\
8 & Overlapping template matching & 0.477088 & Pass & 16 & Random excursions variant & 0.288640 & Pass \\ \hline \hline \end{tabular}
\end{table}
Table 5: NIST test results for 2D-SLMM with the initial values \((x_{0},\ y_{0})=(0.15874,\ 0.38797)\).  
Figure 7: Histograms of the stego HDR RGBE image, “memorial”; (**a**) the histogram of the plain stego image; (**b**) the histogram of the stego encrypted image.