## 2 Proposed Scheme
### Image Encryption  
We introduce a dual encryption technique involving Fisher-Yates shuffle and chaotic pixel diffusion, combining these two encryption methods to enhance the disorderliness of the image, thereby further improving its security and confidentiality. We first represent the original grayscale image sized \(W\times H\) as a matrix \(OI\). Each pixel in the original image, \(OI\), is shuffled at the 8-bit level within itself. Then, convert the bit-level shuffled image into a one-dimensional vector denoted as \(S\_OI\) sized as \(W\times H\). Next, the Fisher-Yates shuffle algorithm is deployed to shuffle the pixels in a completely random order within the \(S\_OI\). Subsequently, the one-dimensional vector \(S\_OI\) is converted into a two-dimensional matrix \(OI^{\prime}\). We employ the logistic mapping defined in Equation (1) to generate a one-dimensional chaotic encryption matrix by calculating the next value \(x^{\prime}\) of \(x\) as  
\[x^{\prime}=\mu\times x\times(1-x), \tag{1}\]  
where \(x\) is the initial value for the logistic map, which is used to generate chaotic values and \(\mu\) is the chaotic parameter. The value range of \(x\) must be between 0 and 1, as logistic mapping operates within this domain to maintain its behavior. The parameter \(\mu\) must be maintained within the range of \(3.5699456\leq\mu\leq 4\) to preserve the chaotic state.  
Therefore, a one-dimensional chaotic matrix \(CH\) of size \(W\times H\) can be generated as  
\[CH(i+1)=\mu\times CH(i)\times(1-CH(i)), \tag{2}\]  
where \(W\) and \(H\) were defined as the width and height of the original image and \(i\) is an indicator representing the index in the one-dimensional chaotic matrix \(CH\). It is used to calculate the next value in the sequence using the logistic map formula. It is noted that the value range of \(i\) depends on the size of the chaotic matrix, which matches the total number of pixels in the image (i.e., \(W\times H\)). In other words, \(i\) typically ranges from 1 to \(W\times H\) to iterate through all indices in chaotic matrix \(CH\). \(x^{\prime}\) is assigned to \(CH(1)\). Subsequently, a one-dimensional chaotic matrix \(CH\) is converted to the unit8 type as \(CH_{8}=unit8(255*CH)\), and then a two-dimensional chaotic encryption matrix \(D_{CH8}\) is generated as \(D_{CH8}=\text{double(reshape(}CH_{8},W,H)\text{)}\). Finally, a bitwise XOR operation is conducted between matrix \(O^{\prime}\) and chaotic encryption matrix \(D_{CH8}\) to obtain the dual encryption matrix \(E\) as  
\[E=OI^{\prime}\ \oplus\ D_{CH8} \tag{3}\]  
Here, \(E\) also represents the encrypted image. It is noted that the pixel values of grayscale images range from 0 to 255, they are converted to their 8-bit binary representation before performing the XOR operation. For example, a pixel value of 120 in \(OI^{\prime}\) (binary:  
Figure 1: The flowchart of the proposed scheme.  
01111000) corresponds to a value of 45 in \(D_{CH8}\) (binary: 00101101), and the XOR operation produces a result of 85 (binary: 01010101). It is noted that the same concept can also be extended to color images. Moreover, the proposed framework is highly flexible, allowing the image encryption method to be replaced with other well-known encryption algorithms, such as AES, to meet the user's requirements. Despite these substitutions, the framework can still work seamlessly with the proposed image interpolation, data hiding, data extraction, and image recovery processes.