## 4 Simulation Results and Security Evaluation
### Noise and Data Loss Analysis  
Our proposed encryption system has demonstrated superior performance when it comes to dealing with cipher data contamination that includes both data loss and noise. To test this, we applied 50%, 25%, 10%, 1.25%, and 0.5% noise to an encrypted Lena image and attempted to recover the original image. Our encryption system performed exceptionally well in the recovery process, highlighting its effectiveness in preserving the integrity and quality of encrypted data even in the presence of noise and data loss of less than 50%. We also tested our proposed encryption approach on encrypted images that experienced data loss, with the aim of recovering the original images. The test was conducted to measure data loss at different rates: 50%, 30%, 20%, and 10%.. Through our testing, we found that our encryption system was effective in reconstructing the original images from the cipher data, even when the encrypted images had experienced data loss of less than 30%. This underscores the robustness and reliability of our proposed encryption approach, which can be valuable in situations where the encrypted data are vulnerable to loss or corruption. See Figure 15 for noise effects and Figure 16 for data loss.  
The reason for conducting noise tests in our proposed encryption process lies in the inherent stability provided by the utilization of chaotic maps and fuzzy numbers. These elements contribute to a high level of randomness and complexity within the encryption process. Chaotic maps, being dynamic systems, exhibit a sensitive dependence on initial conditions, resulting in significant output variations from even minor changes in the initial conditions. Fuzzy numbers, on the other hand, introduce the capability to accommodate  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Encryption Time** \\ \hline Proposed & 0.030 \\ [PERSON] et al. [33] & 0.032 \\ [PERSON] and [PERSON] [37] & 0.19102 \\ [PERSON] et al. [38] & 0.606 \\ \hline \hline \end{tabular}
\end{table}
Table 10: Speed performance test.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** & **[PERSON]** \\ \hline Local entropy of plain image & 7.4710 & 6.9472 & 6.8835 & 7.4924 & 6.3964 \\ Local entropy of encrypted image & 7.8982 & 7.8988 & 7.8990 & 7.8973 & 7.8981 \\ \hline \hline \end{tabular}
\end{table}
Table 9: Local entropy of plain and encrypted images.  
uncertainty and imprecision. By incorporating both chaotic maps and fuzzy numbers, our encryption scheme generates a highly intricate and random sequence of numbers, thereby impeding attackers seeing to decipher the encrypted data. Additionally, the chaotic nature of the encryption process enhances its resistance to noise and other forms of interference. Even small alterations in the input data will lead to substantial changes in the encrypted output, reinforcing the scheme's resilience against noise. In conclusion, the integration of chaotic maps and fuzzy numbers in our proposed encryption scheme introduces an additional layer of randomness and complexity, making it more challenging for attackers to decrypt the data while bolstering resistance against noise and interference during the decryption process.