## 3 Proposed Scheme
### Diffusion of Image Pixels
\[x_{initial}=(x_{0}+k_{m})-floor(x_{0}+k_{m}) \tag{10}\]  
\[w_{new}=\alpha\sin(\pi b(y_{initial}+a)x_{initial}(1-x_{initial}))+d \tag{11}\]  
\[v_{new}=\alpha\sin(\pi w_{new}(w_{new}+c)y_{initial}(1-y_{initial}))+e\]  
\[w_{values}(n+1)=w_{new}\]  
\[K_{1}=w_{values}-floor(w_{values}/y)\times y \tag{12}\]  
Given that \(y=1\) to ensures that the result of the operation is always between 0 and y and \(w_{values}\) is the array of \(w\) values, \(n\) is the index of the last element in the array and \(w_{new}\)is the new \(w\) value to be added to the end of the array. Algorithm 3 shows the algorithm for generating secret keys \(k_{1}\). Step 7: Generate Cipher \(K_{2}\). The initial value of the parameter \(t=0.0018901\) is adjusted using Equation (13) and incorporated into a hybrid of the Henon map and the triangular membership function to be used as the initial parameter, \(x\), in Equation (14). The values from \(\beta^{xx}\) are converted to membership values, as in Equation (16) with the left edge \(a_{l}\), peak \(c_{p}\) and right edge \(b_{r}\) of the triangle as parameters, and used to find the centroid value C as in Equation (15). Then, the minimum and maximum possible values of \(C\) with respect to \(\beta^{xx}\) are found and normalize the value of \(C\) to the range [0, 0.5] using Equation (16). The resulting value of \(C\) represents the normalized crisp value that best represents the input values according to the triangular membership function, and it falls within the desired range. Based on Equation (17), the value of \(C\) is then applied to the fuzzy Henon map. The secret key \(k_{2}\) is obtained by finding the range of numbers between 0 and \(y\), as shown in Equation (18). The initial value for \(c=0.0124\), \(a_{1}=10\), \(c_{p}=100\), \(b_{r}=200\), \(a=1.4\), and \(b=0.2\), respectively. \[x(1)=(t_{0}+k_{m})-floor(t_{0}+k_{m}) \tag{13}\]  
\[mem=trim\big{(}\beta^{xx},\ \big{[}a_{l},\ \mathrm{c}_{p},\ \mathrm{b}_{r}\big{]} \big{)}; \tag{14}\]  
\[C=\frac{\sum_{i=1}^{m}\sum_{j=1}^{n}mem_{i,j}\times\beta^{xx}_{i,j}}{\sum_{i=1 }^{m}\sum_{j=1}^{n}mem_{i,j}} \tag{15}\]  
\[C_{min}=min(min(double(\beta^{xx})));\]  
\[x_{initial}=(x_{0}+k_{m})-floor(x_{0}+k_{m}) \tag{10}\]0  
\[x_{initial}=(x_{0}+k_{m})-floor(x_{0}+k_{m}) \tag{10}\]1  
\[\left\{ \begin{aligned} x_{i}=1-(a\times x_{i-1}^{2})+y_{i-1}\\ y_{i}=(b\times x_{i})+c\end{aligned} \right. \tag{17}\]  
\[K_{2}=y-floor(y) \tag{18}\]  
Algorithm 4 shows the algorithm for generating the randomly sequenced set of secret keys from fuzzy numbers and the Henon map. Step 8: Cipher image \(C\) Based on Equation (19), the final step involves conducting an exclusive operation between \(\beta^{\ast}\), \(\mathrm{K}_{1}\) and \(\mathrm{K}_{2}\) to obtain the encrypted image, \(C\). Exclusive operation is used in this step because it transforms intermediary ciphertext into final ciphertext by combining it with a secret key, which makes it difficult for unauthorized users to decrypt the data. The resulting final Ciphertext \((C)\) is the exclusive operation of the intermediary ciphertext and the secret keys from the logistic sine map and the fuzzy Henon map. \[C=\beta^{xx}\oplus K_{1}\oplus K_{2} \tag{19}\]  
The flowchart of how the encryption process is shown Figure 5. The decryption process of this scheme is the inverse of the encryption process from step 8 to step 1 above. Figure 4: Value manipulation. **Algorithm 2**. The image pixel manipulation  
```
1. Start
2. Input scrambled image \(\beta\) of size \(M\times N\)
3. Select any Column \(\beta(e)\) from each block of size \(\frac{N}{2}\); \(N\in(1-256)\)
4. Adding two diagonal pixel values in pairwise rows
4.1\(for\ i=1:M\times N\)
4.2\(for\ j=1:M\times N\)
4.3\(\beta^{x}\gets mod(\beta^{x}(i,j-1)+\beta^{x}(i-1,j),256)\)
4.4\(end\)
5.\(n\leftarrow\) length\((\beta^{x})\)
6. Repeat Step 4, four times
7. Combine all the blocks
7.1\(\beta^{xx}\leftarrow[\beta_{\omega},\beta_{3},\beta_{2},\beta_{1}]\)
8. Display \(\beta^{\cdot}\)
9. Stop
```  
**Algorithm 3** Generate cipher key \(K_{1}\)  
```
1.