## 4 Evaluation and Discussion
### Keystream Analysis
#### 4.3.1 Key Sensitivity Analysis  
The key, denoted as the initial value \(x_{0}\) in the chaos calculation, plays a crucial role in this study. This experiment is designed to explore the impact of small differences between the key used for embedding and the one used for extraction on the similarity of the original secret and extracted images. A notable dissimilarity between the two images arising from a minor variance in the embedding and extraction keys signifies a more resilient system. Essentially, a random guess for the extraction key that is not identical to the one used for embedding, even if it is close, will yield a completely different extracted image compared to the original secret image.  
The experiment commences by embedding the secret image into the cover image with the parameters \(x_{0}=0.5\) and \(r=3.87\). Subsequently, the extraction process is repeated six times with different values of \(x_{0}\) and a constant value of \(r\). The secret and cover images employed for this analysis are named rainbow and lenna, respectively.  
The six extraction results presented in Table 4 exhibit distinct images, even with variances at the precision level of \(10^{-30}\). Such pronounced differences at this decimal precision indicate the system's high sensitivity to the key. This sensitivity is further enhanced by the decimal library, which permits adjustments in the precision level used in calculations. This theoretically ensures the system's continual sensitivity to the key, provided that memory is consistently sufficient.  
In addition to visual inspection, sensitivity can be quantified through Mean Squared Error (MSE) and Peak Signal-to-Noise Ratio (PSNR). To demonstrate this, we conduct a brute force experiment with a precision level of five digits, using the secret image rainbow and the cover image lenna. Embedding is performed with the values \(x_{0}=0.5\) and \(r=3.7\), while hacking attempts cover the range \(0.45\leq x_{0}\leq 0.56\) and \(3.65\leq r\leq 3.75\). The increment in values for \(x_{0}\) and \(r\) for each brute force attempt is \(0.01\) or \(10^{-2}\), consistent with the precision level used. Verification of a successful hack involves comparing the PSNR value of the extracted secret image with the original; if the PSNR value is infinite  
\begin{table}
\begin{tabular}{c c c} \hline \hline \(x_{0}=0.5+10^{-15}\) & \(x_{0}=0.5+10^{-18}\) & \(x_{0}=0.5+10^{-21}\) \\ \hline \hline \(x_{0}=0.5+10^{-24}\) & \(x_{0}=0.5+10^{-27}\) & \(x_{0}=0.5+10^{-30}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Extraction results for six different \(x_{0}\) values.  
dB, then the extracted secret image from the brute force attempt is considered the target secret image.  
Figure 8 illustrates the successful hacking of the image for the specific combination \(x_{0}=0.5\) and \(r=3.7\), matching the configuration employed during the embedding process. For this particular combination, the MSE value is 0, and the PSNR value is infinite. In contrast, the MSE values for other combinations exceed 90, and the PSNR values for those combinations fall below 30 dB. It is evident that, except at the correct \(x_{0}\) and \(r\) values, both MSE and PSNR values consistently provide similar error for other combinations. Consequently, this steganography method exhibits notable resistance to brute force attacks, demonstrating no indication that proximity in the \(x_{0}\) and \(r\) values results in smaller MSE or higher PSNR.