A deep learning based known-plaintext attack method for chaotic cryptosystem

As a classic encoder-decoder network in the field of medical image processing, Unet [17] is usually conducted for image segmentation, image compression, image denoising, etc. It can also be used to decrypt encrypted information and restore images. A straightforward modification is made to Unet’ output channels since the datasets contain vast small-size images, and its network structure is shown in Fig. 1.

In addition to the last convolution layer with kernel size 1x1, each convolution consists of three consecutive operations: convolution (Conv) with 3x3 kernel size, batch normalization (BN), and rectified linear unit (ReLU). The down-sampling phase adopts the max-pooling of stride 2x2. The up-sampling layers are concatenated with the output of the corresponding down-sampling layer and deconvolved with a 2x2 filter.

In this paper, we propose an encoder-decoder network for decryption, named Multi-Stage Encoder Decoder Network. The first five layers of the left column of the network are encoding layers, and each layer is composed of convolution (Conv) with filter size 3x3, batch normalization (BN), rectified linear unit (ReLU), and average pooling layer with stride 2x2. Through feature extraction, higher-precision pixel-by-pixel estimation can be achieved while maintaining the per-pixel position information on the feature map.

In the decoding stage, we integrate the feature map of multiple encoding stages to fully fuse the multi-scale context information and conduct pixel-by-pixel regression considering that simple interpolation operation cannot restore the information lost in the previous pooling layer, even if skip connections are employed. The network structure is presented in Fig. 2.

This part briefly introduces three types of classic one-dimensional chaotic maps: Logistic map, Sine map, and Chebyshev map [18]. In view of the space limitation and the similar nature of the experiment, we only employ Logistic map as the experimental scheme in one-dimensional chaotic map.

This section will introduce the above-mentioned one-dimensional hybrid chaotic map. In [6], one novel image encryption scheme was proposed to combine two different chaotic maps and derive a new encryption formula, which was used to encrypt the image and reduce the correlation between the three channels of image R, G and B. Based on this idea, three different one-dimensional chaotic maps are adopted in this paper to perform chaotic processing on R, G and B channels of the color image. Compared with using only one-dimensional Logistic map to generate a sequence of length W x H x C (W, H, C denotes width, height, channels of the image), the hybrid chaotic map is more secure and can further reflect the ability of convolution neural network to learn the encryption mechanism of chaotic systems. In Section 5, we observe that the hybrid chaotic map is better than the one-dimensional Logistic map in terms of the chaotic effect through the evaluation metrics in Fig. 3 and Fig. 5. The mathematical formula is as follows:

where Eq.(4) indicates that the color image $Y$ is divided into three single-channel images $Y_{i}$ along channel; in Eq.(5), Eq.(6), Eq.(7), $Y_{i,n+1}$ represents the chaotic processing of the R, G, and B channels of the color image $Y$ with Logistic map, Sine map, and Chebyshev map respectively.

In this paper, we generate the “plaintext-ciphertext” pairs required for the experiment on MNIST [19] and CIFAR-10 [20]. Since MNIST are all single-channel images, we employ an one-dimensional Logistic map to encrypt the datasets, while utilize hybrid chaotic map to encrypt the three-channel images of cifar-10. These “plaintext-ciphertext” pairs are randomly divided, 90$\%$ of which are used as training sets of neural networks, while the rest are employed as testing sets to detect the decryption ability of model.

To minimize the information loss between the decrypted images and the plaintext images, the absolute value loss function (L1loss) is used to supervise the training of all the networks in this paper. The equation is defined as follows:

where $N$ is the number of the training images batch, ${O({X_{i}};\theta)}$ is the decrypted image output by the network with parameters ${\theta}$, ${P({X_{i}})}$ is the supervision label (plaintext) of the ciphertext ${X_{i}}$.

All experiments are implemented in the environment of CPU-Python3.5-Pytorch1.0, employing Adam optimizer with L2 regularization weight decay rate of 1e-4. The initial learning rate is 1e-5 and it drops by 10% every 20 epochs when training with a batch size of 8. In addition, a dropout layer with ratio of 0.5 is added to the partial convolution layer to prevent the network from over-fitting.

In the experiment, Pearson correlation coefficient [21] was taken as the evaluation metrics for two aspects: (1) to estimate the correlation between plaintext images and ciphertext images in different chaotic encryption systems in Section3; (2) to evaluate the correlation between the output (decryption image) and the label (plaintext image) of the known-plaintext attack method based on deep learning. It can be defined by the following equation:

where $W$, $H$ represent the width and height of the images respectively, $O(i,j)$ is the output of the network (decrypted image), $P(i,j)$ is the label of the network (plaintext image). This Eq.(11) is applicable to two encryption schemes and is regarded as training/testing accuracy.

Table. 1 shows the experimental results of two convolution neural networks to crack the Logistic map on the MNIST dataset, including training Loss, training average accuracy, testing average accuracy, epoch and the running time of each epoch.

Fig. 3 gives the visualization results of one-dimensional Logistic map on the MNIST dataset, where the first column shows the plaintext image; the second column displays the corresponding ciphertext image; the third-fourth columns list the decrypted images of MSEDNet, and Unet severally. At the same time, the correlation coefficients of the ciphertext image and the plaintext image, as well as the decrypted images and the plaintext image are given separately. From the results we can observe that the testing ciphertext images can be well reconstructed by the neural network, and the correlation with the plaintext gradually tends to 1. However, the reconstruction effect of diverse networks is different, among which the decryption effect is Unet ¿ MSEDNet.

The curve changes of training Loss and testing evaluation metrics of two networks on the MNIST dataset are plotted in Fig. 4 (a) and Fig. 4 (b). We reach the conclusion from the figure that the selections of neural network for the same encryption scheme greatly affect the training time and accuracy. The training efficiency and accuracy of Unet [17] are better than the MSEDNet.

Table. 2 presents the experimental results of different convolution neural networks cracking the hybrid chaotic map (Cifar-10), similar to Table. 1.

Fig. 5 shows the visualization results of the hybrid chaotic map on the Cifar-10 dataset, which is greatly similar to MNIST, i.e., after more epochs training, the ciphertext image in testing set can be reconstructed pretty well, but the two reconstruction effects have slight differences, among which the decryption effect is Unet ¿ MSEDNet. Fig. 6 displays the curve changes of training Loss and testing evaluation metrics of two networks on the Cifar-10 dataset. Similar to Fig. 4, it can be observed that the training efficiency and accuracy of Unet are significantly higher than MSEDNet.

This paper aims to investigate the cryptanalysis of performing known-plaintext attacks with multiple convolution neural networks against different chaotic encryption systems. The following conclusions can be drawn from the above experimental data in the figure and table: (1) For the same encryption scheme, different neural networks can be used as the decryption model. However, the reason for different ciphertext reconstruction effects may be related to the number of participants and learning ability of the neural network, which will be one of our next research works. (2) The same convolution neural network can be used to attack different encryption systems, but the ciphertext reconstruction effect is also diverse, which may be related to the complexity of the encryption mechanism. This will be our other future research works.

Compared with traditional known-plaintext attack methods specific to a certain chaotic cryptosystem, our scheme deep learning-based is more flexible and extensible. Moreover, this method can be transferred to other chaotic cryptosystems, even to the field of non-chaotic encryption.