## 3 Proposed Scheme
### Diffusion of Image Pixels  
The process involved summing up the pixel values from different columns and rows to determine a new value at a specific position. Additionally, a triangular membership function derived from fuzzy mathematics was employed to obtain a manipulation key number. To achieve image diffusion, a hybrid random chaos system was created using three different chaotic maps, namely, the logistic map, the sine map, and the Henon map. Step 4: In this step, the manipulation is performed separately on each block, \(\beta_{1},\beta_{2},\beta_{3},\text{ and }\beta_{4}\). If \(e_{i}\) represents any column to be selected from a square matrix \(\beta\), where \(i\in[1-128]\), then \(\beta(e_{i})\) denotes the selection of a specific column in the matrix \(\beta\). The summation of two-pixel values from different columns \(\beta(e_{i})\) and two adjacent pairwise rows is performed to derive a new value, as shown in Figure 4a-d, when Equation (6) is applied. For blocks \(\beta_{1}\)_and_\(\beta_{2}\), the value of \(\beta(e_{i})\) is 1, whereas for blocks \(\beta_{3}\) and \(\beta_{4}\), the value of \(\beta(e_{i})\) is 123. The process of summation is dependent on the number of iteration processes in each block, which can range from 1 to 4. However, for this study, only three iterations were conducted to increase performance. The top row remains unchanged, which simplifies the decryption process. The pseudo code algorithm of this process is shown in Algorithm 2. \[\left\{ \begin{array}{l}\beta\big{(}x_{i,j}\big{)}=\text{mod}\big{(}\ \big{(}\beta\big{(}x_{i,j-1}\big{)}+\beta\big{(}x_{i-1,j}\big{)}\big{)},256 \big{)}\\ x_{i,j}\leq 256\leq 256n,\ n=1,2,3,4,\ldots\ldots.\ n-1\end{array} \right. \tag{6}\]  
Step 5: Generation of fuzzy triangular membership for the \(k_{m}\) value. The functions described in Equation (7) refer to the membership values obtained from the image \(\beta^{xx}\), which takes values in the interval [0, 1]. To obtain values close to 0.5, the value \(\beta^{xx}(x_{i})\) is adjusted piecewise, as shown in Equation (8). This facilitates the determination numbers between 0 and 1 of the sums of all membership values of the plain image and its mean, as illustrated in Equation (9). Here, n represents the total number of pixels in the image \(\beta^{xx}\). Additionally, the \(k_{m}\) value is added to the initial keys of the fuzzy Henon map and the logistic sine map as illustrated in next step. \[\beta^{xx}(x_{i})=\left\{ \begin{array}{l}0\ \ \ \ \ \ \ \ \ \ \ ;\ \ x_{i}\leq a\ or\ x_{i}\geq c\\ \left(\frac{x_{i}-a}{b-a}\right)\times 255^{-1}\ ;\ a\leq x_{i}\leq b\\ \left(\frac{c-x_{i}}{c-b}\right)\times 255^{-1}\ ;\ b\leq x_{i}\leq c \end{array} \right. \tag{7}\]  
\[MK=\left\{ \begin{array}{l}2\times(\beta^{xx}(x_{i}))^{2};\ \beta^{xx}(x_{i})\leq 0.5\\ 1-(1-\beta^{xx}(x_{i}))^{2}\ ;\ \beta^{xx}(x_{i})>0.5\end{array} \right. \tag{8}\]  
\[k_{m}=\left(\sum_{i=1}^{n}MK_{i}+\sum_{i=1}^{n}\frac{MK_{i}}{n}\right)-floor \left(\sum_{i=1}^{n}MK_{i}+\sum_{i=1}^{n}\frac{MK_{i}}{n}\right) \tag{9}\]  
Step 6: Generate Cipher \(K_{1}\). The initial parameter is adjusted through the utilization of Equation (10). and subsequently integrated into the hybrid chaotic equation of the logistic sine map, as depicted in Equation (11). This incorporation facilitates the generation of a secret cipher key, \(K_{1}\), as illustrated in Equation (12). The values of \(x_{initial}\), \(y_{initial}\)\(\alpha,\text{a},\text{b},\text{c},\text{d},\text{e}\) are \(-1.2\), \(1.5\), \(2.1\), \(0.5\), \(-0.65\), and \(-0.7\), respectively.