Robust Cryptosystem Analysis: Key Space Sensitivity and Differential Attack Resistance

Table of Links

Abstract & Introduction

Strategy Description

Encryption Speed Evaluation

Statistical Evaluation

Security Analysis

Parameter Setup

Comparison To Previous Works

Conclusions

Acknowledgments & References

5. Security Analysis

5.1. Key Space And Sensitivity

In addition, the cryptosystem should have high sensitivity to the change of key in encryption and decryption process. That is, even one bit of the key is changed, the attacker cannot obtain any information from the decrypted image. To analyze the key sensitivity of the proposed strategy, we randomly select key, use the deployed cryptosystems to ecnrypt an iamge, followed by decrypting the cipher image with the correct key. The images decrypted with PLCM and 2DLASM are plotted in Fig. 5 (a) and (e), respectively. Then we slight change the key by adding an increment δ = 0.000000001 on initial condition and control parameter, decrypted the cipher image with the slightly modified Key. The images decrypted by PLCM based cryptosystem using x0 + δ, p + δ, and x0 + δ, p + δ are shown in Fig. 5 (b), (c), and (d), respectively. The images decrypted by 2DLASM based cryptosystem using x0 + δ, µ + δ, and x0 + δ, µ + δ are shown in Fig. 5 (f), (g), and (h), respectively. We also calculate the correlation coefficients between the images decrypted with the correct key and the images decrypted with the slightly changed key, and list the results in Tab. 7.

5.2. Resistance To Differential Attacks

To resist differential attacks, the encryption algorithms should have high sensitivity to the plain image, that is, a minor change in the plain image will lead to a completely change in the cipher image [41]. To evaluate the ability of the proposed strategy to resist such attacks, Number of Pixels Change Rate (NPCR) and Unified Average Changing Intensity (UACI) are calculated [42]. NPCR can be calculated according to the following equation:

According to Refs. [43, 44], for an image of size 512 × 512, when the significance level is equal to 0.05, the expected values of NPCR and UACI are 99.5893% and [33.3730%, 33.5541%], respectively. An image encryption scheme passes the diffusion property test when NPCR is higher than the expected value, and UACI falls within the interval.

In the proposed strategy, despite all assistant threads perform diffusion operations independently, they take the last pixel of the next subframe as the diffusion seed to reconstruct the relationship between all subframes. This guarantees that one pixel changed in any subframe, will result in a completely different cipher frame. To fully evaluate the resistance of the proposed strategy to differential attacks, we generate byte sequences, encrypt a set of plain images, randomly select and change a pixel in the plain images, encrypt the modified images with the same byte sequences, calculate NPCR and UACI between the generated cipher images. For each plain image, we repeat above steps for 100 times, fetch the minimum, maximum, and average values, and list the results in Tab. 8.

5.3. Resistance To Noise And Data Loss

Images or video frames may be affected by noise while transmitting over the channel. When a cipher image has noise or losses part of data, the cyrptosystem needs to recover the original image with high visual quality [45]. To evaluate the robustness, we use the two deployed cryptosystems to encrypt a plain image with randomly selected keys, add 1%, 3%, and 5% salt-and-pepper noise [46] to the generated cipher images, use the same keys to decrypt the cipher images, and plot the plain and decrypted images in Fig. 6. Although the decrypted image are changed when there exists noise in the cipher image, the approximate information of the plain image is preserved, even when the slat-and-pepper noise as high as 5%.

In real application, image cropping is very common, which may lead to data loss [47]. To assess the capability of the proposed strategy to resist the data loss, similarly, we use the deployed cryptosystems to encrypt a plain image, randomly select and remove 64 × 64, 128 × 128, or 512 × 512 blocks from the cipher images, replace the removed block with white or black blocks, decrypt the modified cipher images with the same keys, and show the results in Fig. 7. Clearly, the contours of the plain images are preserved, even though 256 × 256 blocks are removed from the cipher images. That is, when there exist 25% data loss in the cipher image, the deployed cryptosystems can still recover the information of the plain image. The proposed strategy, therefore, can resist noise and data loss.