## 3 Proposed Approach
### Network Architecture
#### 3.2.2 Image Permutation  
During information hiding, we found that the presence of a large, solid colour area in the chosen cover image or complex texture in the secret image can lead to the error image, i.e., the residual image between the stego image and the cover image, containing discernible texture information from the secret image. This poses a significant security risk if the set of cover images used for training is inadvertently disclosed. To address this concern, we integrated image permutation with image steganography to minimize the distinction between the stego image and the cover image. This ensures that even if the image set is exposed, attackers cannot extract valuable information from the error image. Leveraging the sensitivity of chaotic systems to initial values and their ability to generate variable and intricate pseudo-random sequences [35], we proposed an image permutation algorithm based on Logistic chaotic mapping. The permutation process, depicted in Figure 3, involves four primary steps, and its reversal process serves as the inverse of the scrambling process. Figure 3: The first image represents the original image, the second image is generated by traversing the original image according to a specific rule, and subsequent permutation produces the third image. Without the correct key and the inverse permutation algorithm, an incorrectly restored image (fourth image) is obtained. Figure 2: In the first row, the first column of images is the original RGB image, and the second, third, and fourth columns are, in order, images with \(R\), \(G\), and \(B\) channels. And in the second row, the first column image is the greyscale image of the original RGB image, and the second, third, and fourth columns are the images of \(Y\), \(Cr\), and \(Cb\) channels in turn. Step 1: Pixel Position Choosing  
Iterate over image rows and columns, selecting pixel values alternatively at odd and even positions for each row and column. This process creates four sub-images with pixel positions denoted as (odd, odd), (odd, even), (even, odd), and (even, even). The dimensions of these sub-images are one-quarter of the original image. Step 2: Chaotic Sequence Generation  
The Logistic chaotic mapping is expressed in Equation (3), where the control parameter \(\mu\) ranges between (0, 4), but when the value range is within the interval of (3.5699456, 4), it enters the chaotic state [36]. We utilized the logistic chaotic map twice with different parameters to generate random sequences. As we can see in Figure 4, initially, using the Logistic chaotic mapping (F_Logistic) to generate a chaotic sequence (F_Sequence) of the same size as the sub-image for reordering, the position index of each element in this ordered sequence is calculated based on the original chaotic sequence. Subsequently, we used Logistic chaotic mapping (S_Logistic) again with distinct control parameters and initial values to generate four copies of the sequence (S_Sequence) for reordering, each having a quarter of the length of F_Sequence. Calculations are performed to obtain the position index of each element in the ordered sequence, guiding the reordering of F_Sequence sequentially to intensity its disorder. Finally, Guide_Index is acquired and utilized as guidance to determine the pixel positions of the sub-image to be scrambled. \[X_{n+1}=\mu\times X_{n}\times(1-X_{n}),\ \ \ n=0,1,2\ldots \tag{3}\]  
Step 3: Iterative Scrambling of Sub-Images  
Continuously repeat Steps 2 and 3 until all four sub-images have undergone scrambling. Step 4: Random Exchange of Sub-Images  
Perform a random exchange of positions for the scrambled sub-images. To maintain consistency in size with the cover image, concatenate these sub-images back into the original image based on their row and column positions. Output the final scrambled image after this exchange. This algorithm iteratively scrambles sub-images in a chaotic manner, guiding the disordering of pixel positions.