## 4 Experimental Results  
### Implementation and Setup Details  
**Datasets and settings. ** During model training, we utilized the DIV2K training set [41], comprising 800 high-quality 2 K resolution images, to train our CHASE. The model'sperformance was assessed using the DIV2K test set (100 2 K images) [41] and a validation set consisting of 120 images randomly selected from the COCO test set [42] and the ImageNet test set [43], respectively. Data augmentation during training involved random horizontal and vertical flipping, as well as random cropping of images. For the test set, images underwent centre cropping exclusively, and the resolution size for both training and test images was maintained at 256 \(\times\) 256. The forward hiding network and the backward reconstructing network underwent training and validation with 16 invertible blocks, respectively, optimizing their parameters using the Adam [44] optimizer with \(\beta_{1}=0.9\) and \(\beta_{2}=0.999\). In stage 1, the total number of epochs was set to 3 K, with an initial learning rate of \(1\times 10^{-4}\), halved every 1 K iterations. Hyperparameters in Equation (11) are configured as \(\lambda_{h}=8,\ \lambda_{r}=1,\ \lambda_{l}=1\). In stage 2, the total number of epochs was 1.5 K, the initial learning rate was \(1\times 10^{-4}\), and the discriminator underwent pre-training for 500 epochs. The learning rate was halved every 200 epochs, followed by joint training of the discriminator and CHASE for 1 K epoch, with the learning rate halved every 500 epochs. The hyperparameters in Equation (13) were set as \(\lambda_{h}=16,\lambda_{r}=1,\ \lambda_{gan}=1\). The hiding loss employed an \(L2\) regularized loss function, while the reconstruction loss utilized an \(L1\) regularized loss function. **Benchmarks. ** To assess the efficacy of our steganography method, we conducted a comparative analysis against various state-of-the-art image steganography methods, including a version of our proposed steganography method without image permutation algorithm named CHASE_WO, a spatial-domain image steganography method known as 4 bit-LSB, [PERSON]'s steganography [22] based on convolutional neural network, and DeepMIH founded on the invertible neural network proposed by [PERSON] [34]. To ensure fairness in the comparison, we acquired the relevant open-source code for each method and adhered to the parameter configurations specified in their respective papers. The retraining process was executed using the aforementioned dataset. **Evaluation metrics. ** To gauge the quality of the cover-stego image pairs and the secret-reconstructed secret image, we employed two widely used evaluation metrics in image steganography, namely the peak signal-to-noise ratio (PSNR) and Structural Similarity Index (SSIM) [45]. The PSNR serves as a standard image quality metric to assess the impact of a steganography method on an image. It calculates the peak signal-to-noise ratio between the original image and the steganographic image, providing a quantitative measure of image quality. Higher PSNR values indicate superior image quality, with results typically expressed in decibels (dB). The mathematical expressions for PSNR, given an original image \(X\) and a generated image \(Y\) with an input size of \(m\times n\), are as follows:  
\[\text{MSE}=\frac{1}{mn}\sum_{i=0}^{m-1}\sum_{j=0}^{n-1}\left[X(i,j)-Y(i,j) \right]^{2} \tag{14}\]  
\[\text{PSNR}=10\cdot log_{10}\left(\frac{MAX_{X}^{2}}{MSE}\right) \tag{15}\]  
In the PSNR formula, \(MAX_{X}\) represents the maximum pixel value of the image, typically 255 in the case of an 8-bit image. It is important to note that the PSNR value used for comparison is the average of the PSNR values calculated for the \(R\), \(G\), and \(B\) channels of the two images. On the other hand, SSIM is a metric that considers not only brightness (luminance contrast) differences but also structural similarity. It assesses the structural similarity between the original image and the steganographic image, taking into account luminance and contrast similarities.