## 3 Proposed Chaos-Based Stream Ciphers Based on Secure PRNGs of Chaotic Sequences
### Statistical analysis  
To analyze the resilience of the proposed chaos-based stream cipher against most statistical attacks, we perform the following statistical analysis: histogram, chi-square, entropy, and correlation. **Histogram and chi-square tests**  
The histogram of the ciphered image plays a crucial role in determining the effectiveness of the encryption process. This indicates the pattern in which the gray levels of pixels in an encrypted image are arranged, and it should be highly similar to a uniform distribution. Figures 13 and 14 show the results for Barbara and Boats of size \(512\times 512\times 3\) as an example, in (a) the plain images, (b) the cipher images by LSP-SC, (c) the cipher images by LST-SC, (d) the cipher images by LSPT-SC, and in (e-h) their histograms, respectively. We can see that the histogram of the ciphered image is highly similar to a uniform distribution and vastly different from the original image. Using Equation (6), we perform a chi-square test on ciphered images to establish their uniformity statistically, where here: \(N_{c}\) is the number of levels, which is equal to \(2^{8}=256\). As demonstrated in Table 9, the chi-square test results indicate that the histograms of the encrypted images tested are distributed uniformly, as he theoretical values are higher than the experimental ones. **Entropy analysis**  
The randomness of the encrypted image can be quantitatively measured through the entropy information provided by [PERSON] (1958):  
\[H(C)=-\sum_{i=0}^{N_{c}-1}P(c_{i})\times log_{2}(P(c_{i})) \tag{44}\]  
with \(H(C)\) represents the entropy of the ciphered image and \(P(c_{i})\) denotes the probability of occurrence of each gray level (\(c_{i}=0,1,\ldots,255\)). When the probability is the same, the entropy reaches its peak value (=8). The encryption algorithm is more robust when the experimental entropy value is closer to the maximum value. The results of the entropy test on the original and ciphered images are displayed in Table 10.  
Figure 14: Results of Boats image. (**a**) Plain image. (**b**) Encrypted image by LSP-SC. (**c**) Encrypted image by LST-SC. (**d**) Encrypted image by LSPT-SC. (**e**) Histogram of plain image. (**f**) Histogram of the ciphered image by LSP-SC. (**g**) Histogram of the ciphered image by LST-SC. (**h**) Histogram of the ciphered image by LSPT-SC. \begin{table}
\begin{tabular}{l l c c c c c} \hline \hline \multicolumn{2}{l}{**Chi-Square Test Value**} & \multicolumn{1}{l}{**Lena**} & \multicolumn{1}{l}{**[PERSON]**} & \multicolumn{1}{l}{**[PERSON]**} & \multicolumn{1}{l}{**[PERSON]**} & \multicolumn{1}{l}{**Boats**} \\ \hline \(\chi^{2}_{th}\) & & 293.2478 & 293.2478 & 293.2478 & 293.2478 & 293.2478 & 293.2478 \\ \hline \multirow{3}{*}{\(\chi^{2}_{exp}\)} & LSP-SC & 240.8893 & 251.3581 & 257.1510 & 243.0104 & 240.6367 \\  & LST-SC & 221.4082 & 265.3704 & 241.5293 & 266.8132 & 228.1328 \\ \cline{1-1}  & LSPT-SC & 264.9030 & 244.7487 & 260.0397 & 280.6595 & 254.4102 \\ \hline \end{tabular}
\end{table} 
Table 9: Chi-square test. The entropies of the encrypted images are close to the optimal value; therefore, the stream cipher proposed in this study has a high level of robustness, as demonstrated by these results. **Correlation analysis**  
In an original image, correlation analysis can be used to measure the relationship between adjacent pixels in horizontal, vertical, and diagonal directions. This can be useful in identifying patterns and features in the image, such as edges and textures. An effective encryption process should result in ciphered images with minimal correlation and redundancy between adjacent pixels (as close to zero as possible). We carry out the following steps to evaluate the correlation: First, 8000 pairs of adjacent pixels are chosen at random from the test image, and then the correlation coefficients are calculated using the formula below.