Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System
Abstract
:1. Introduction
2. New Three-Dimensional Chaotic System and Analysis
2.1. New Three-Dimensional Chaotic System
2.2. Complexity Analysis
- Suppose the original data is , and they are composed of D vectors in order.In which .
- The distance between and is
- Setting a threshold value , for each , we can obtain the statistics of .
- The mean of logarithm of is written as and can be calculated by
- Changing dimension and repeating step 1 to step 4, we can obtain the approximate entropy
3. Compressed Sensing and Scrambling
3.1. Compressed Sensing
3.2. Arnold Scrambling
4. Image Encryption and Decryption Schemes
- (1)
- The initial conditions of the newly designed three-dimensional chaotic system are determined as , and the parameters are defined as in order to iteratively generate the chaotic sequence.
- (2)
- The header data of the chaotic sequence generated by Equation (1) is discarded before the system enters the steady state. The steady state data is retained as the y sequence. It is then reorganized into a measurement matrix of size . The generated compressed sensing measurement matrix is then quantized.
- (3)
- The initial conditions and parameters of the new chaotic image are taken as key 1. That is, the parameters and initial values of the new three-dimensional discrete chaotic system are taken as key .
- (4)
- DWT is used to make the original image sparse in the wavelet domain, with a sparsity of . Then, two observations are performed on the original image according to the formula to obtain the I2 of , where I1 is a plaintext image and is the DWT transformation matrix.
- (5)
- Uniform quantization is performed on I2, so that the quantized value is an integer between 0 and 255.
- (6)
- To improve the effect of encryption, the image continues to undergo Arnold scrambling, as per Equation (12). At the same time, the ciphertext image is obtained, and the scrambling parameter and iteration number constitute key .
- (7)
- Decryption is the inverse process of encryption. Key 2 and key 1 are used sequentially to perform inverse Arnold scrambling and inverse DWT transform on the ciphertext image, and finally, compressed sensing and reconstruction using the CoSaMP algorithm is applied to obtain the original image.
5. Simulation Experiments and Performance Analysis
5.1. Simulation Conditions
5.2. Compression Ratio
5.3. NIST Test
5.4. Key Space Analysis
5.5. Analysis of Resistance to Statistical Attacks
5.5.1. Histogram Analysis
5.5.2. Analysis of Correlation between Adjacent Pixels
5.5.3. Information Entropy Analysis
5.6. Analysis of Resistance to Differential Attacks
6. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Chaotic System | Input Parameters | ApEn |
---|---|---|
Logistic | N = 2000, m = 2, r = 0.2SD | 0.4918 |
Henon | N = 2000, m = 2, r = 0.2SD | 0.4699 |
Lorenz | N = 2000, m = 2, r = 0.2SD | 0.3197 |
Ours | N = 2000, m = 2, r = 0.2SD | 0.6932 |
Statistical Test | p-Value | Result |
---|---|---|
Frequency | 0.843512 | Passed |
Block Frequency | 0.697188 | Passed |
Cumulative Sums | 0.593463 | Passed |
Runs | 0.689301 | Passed |
Longest Run | 0.314464 | Passed |
Rank | 0.894036 | Passed |
FFT | 0.421210 | Passed |
Non-Overlapping Templates | 0.904121 | Passed |
Overlapping Templates | 0.013027 | Passed |
Universal | 0.301746 | Passed |
Approximate Entropy | 0.693216 | Passed |
Random Excursions | 0.011393 | Passed |
Random Excursions Variant | 0.020299 | Passed |
Serial | 0.498839 | Passed |
Linear Complexity | 0.393688 | Passed |
Direction | Lena | Lake | Cameraman | Rice | ||||
---|---|---|---|---|---|---|---|---|
Plaintext | Ciphertext | Plaintext | Ciphertext | Plaintext | Ciphertext | Plaintext | Ciphertext | |
Horizontal | 0.9376 | 0.0 0 33 | 0.9526 | 0. 0028 | 0.9318 | 0.0021 | 0.9214 | 0.0056 |
Vertical | 0.9660 | 0.0027 | 0.89531 | 0.0112 | 0.9559 | 0.0098 | 0.9374 | 0.0031 |
Diagonal | 0.9753 | 0.0014 | 0.9206 | 0.0038 | 0.9076 | 0.0015 | 0.8934 | 0.0109 |
Image | Lena | Lake | Cameraman | Rice |
---|---|---|---|---|
Plaintext image | 7.5686 | 7.4644 | 7.0097 | 7.0115 |
Ciphertext mage | 7.9975 | 7.9973 | 7.9972 | 7.9976 |
Image | NPCR (ideal: 99.6093%) | UACI (ideal: 33.4635%) |
---|---|---|
Lena | 99.6154% | 33.3526% |
Lake | 99.5890% | 33.3848% |
Cameraman | 99.6017% | 33.3361% |
Rice | 99.6109% | 33.3746% |
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Xie, Y.; Yu, J.; Guo, S.; Ding, Q.; Wang, E. Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System. Entropy 2019, 21, 819. https://doi.org/10.3390/e21090819
Xie Y, Yu J, Guo S, Ding Q, Wang E. Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System. Entropy. 2019; 21(9):819. https://doi.org/10.3390/e21090819
Chicago/Turabian StyleXie, Yaqin, Jiayin Yu, Shiyu Guo, Qun Ding, and Erfu Wang. 2019. "Image Encryption Scheme with Compressed Sensing Based on New Three-Dimensional Chaotic System" Entropy 21, no. 9: 819. https://doi.org/10.3390/e21090819