# Hybrid Chaotic-Based PRNG for Secure Cryptography Applications
## 5 Results and Analysis
### Encryption Quality Analysis
#### 5.1.6 Information Entropy  
The benefit of using entropy as a measure of information security in cryptography is that it provides a quantitative measure of the uncertainty or randomness in the data distribution, which is directly related to the security of the ciphering algorithm. A higher entropy value indicates a more uniform distribution of data and less information to be  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & & & \multicolumn{3}{c}{_Original Image_} & & & \multicolumn{3}{c}{_Encrypted Image_} \\ \cline{4-10} _File Name_ & _Size_ & _V-_ & \multicolumn{1}{c}{**H-**} & \multicolumn{1}{c}{**D-**} & \multicolumn{1}{c}{_Correlation_} & \multicolumn{1}{c}{\(\chi^{2}\)} & \multicolumn{1}{c}{_V-_} & \multicolumn{1}{c}{**H-**} & \multicolumn{1}{c}{**D-**} & \\  & & _Correlation_ & _Correlation_ & _Correlation_ & _Correlation_ & _Correlation_ & _Correlation_ & \(\chi^{2}\) \\ \hline bird.png & 2625 \(\times\) 2250 & 0.9876 & 0.9736 & 0.9706 & 37,036 \(\times\) 10\({}^{2}\) & \(-6.46\times 10^{-4}\) & 0.0188 & \(-\)0.0061 & 271.7370 \\ lion.png & 2362 \(\times\) 3047 & 0.971 & 0.9764 & 0.957 & 42,846 \(\times\) 10\({}^{4}\) & 0.0189 & \(-\)0.0164 & 0.0198 & 249.2168 \\ pyramids.png & 4755 \(\times\) 3090 & 0.9894 & 0.9908 & 0.9841 & 62,090 \(\times\) 10\({}^{3}\) & \(-\)0.0033 & \(-\)0.038 & 0.0143 & 230.8222 \\ sphinx.png & 1704 \(\times\) 2272 & 0.9793 & 0.9839 & 0.9733 & 55,280 \(\times\) 10\({}^{2}\) & 0.0253 & 0.0126 & 0.0058 & 235.8652 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Vertical, horizontal, and diagonal correlation of some images and their encrypted versions. In addition to the chi-square analysis.  
Figure 9: Correlation analysis of 3000 adjacent pixels in an original and encrypted image: (**a**) the original image; (**b**) its diagonal correlation; (**c**) its horizontal correlation; (**d**) its vertical correlation; (**e**) the encrypted image; (**f**) its diagonal correlation; (**g**) its horizontal correlation; and (**h**) its vertical correlation.  
extracted from it, resulting in a more secure ciphering algorithm. Entropy is a measure of the overall randomness of an image or a system and is as expressed by Equation (19):  
\[H=-\sum_{i=1}^{2^{n}}P(m_{i})\log_{2}P(m_{i}) \tag{19}\]  
where \(n\) represents the number of bits of color intensity, and \(P(m_{i})\) is the probability of a color intensity, \(m_{i}\), in the image. The higher the entropy value, the more uniform the distribution of the encrypted image will be, i.e., almost nothing is left as information that can be extracted from the image. Local Shannon entropy, on the other hand, analyzes the randomness of the image at a local scale. The local entropy of nonoverlapping blocks of an image is calculated using Equation (20):  
\[LSE=\frac{1}{k}\sum_{i=1}^{k}H(I_{B_{i}}) \tag{20}\]  
where \(k\) is the number of blocks, and \(H(I_{B_{i}})\) is the Shannon entropy of the block. In this paper, the significant values are \(\alpha=0.05\), \(\alpha=0.01\), \(\alpha=0.001\), \(k=30\), and a block size of 1936 pixels. In cryptography, the higher the entropy of an encrypted image, the more secure it is considered to be, as it contains less information that can be extracted by an attacker. Table 3 confirms that the proposed scheme falls within this critical interval of different image sizes.