Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding
Abstract
:1. Introduction
- (1)
- A new one-dimensional chaotic system model with large chaotic range and an unbroken periodic interval is proposed.
- (2)
- A fast image encryption algorithm based on parallel DNA encoding and decoding is proposed.
- (3)
- The complex chaotic characteristics of the chaotic system model are proved by a series of complexity criteria, and the security of the proposed image encryption scheme is verified by a large number of experiments and a security analysis.
2. The Newly Designed Chaotic Map
2.1. Boundedness Analysis
- (1)
- If xn = 0 or xn = 1, then f(xn) = 0. If 0 < xn < 1, then f(xn) > 0. The fact indicates that function f(xn) has a lower bound in the domain of xn ∊ (0, 1).
- (2)
- Since ≤ 1, therefore f(xn) ≤ μ × 1/(exn + μ) < 1. This indicates that the function f(xn) has an upper bound of μ/(exn + μ) < 1 when xn ∊ (0, 1).
2.2. Fixed Point and Its Stability Analysis
2.3. The Diagram of Bifurcation and Lyapunov Exponent
2.4. The Time-Series Diagram and Cobweb Graph
2.5. Correlation Analysis
2.6. Approximate Entropy Analysis
2.7. Correlation Dimension Analysis
2.8. NIST Test of the New Chaotic Map
3. The Proposed Image Encryption Scheme
3.1. Chaotic Secret Key Streams Generating
Algorithm 1. Generating a chaotic sequence with length of L. |
Input: Parameters {μ, e, x0} of chaotic system (1) and L. |
Output: The chaotic sequence X = {X(1), X(2),..., X(L)}. |
Step 1: X(1) ← x0. |
Step 2: For i = 2:L, do |
X(i) ← μ × sin(π × X(i − 1))/(e × X(i − 1) + μ); |
End for |
Step 3: Output the chaotic sequence X = {X(1), X(2),..., X(L)}. |
3.2. Parallel DNA Encoding
3.3. DNA Permutation
3.4. Parallel DNA Decoding
3.5. Parallel Pixel Encryption
4. Experimental Results and Security Analysis
4.1. Key Space Analysis
4.2. Histogram Analysis
4.3. Correlation Analysis
4.3.1. Correlations of Two Adjacent Pixels
4.3.2. Correlations between Original and Cipher-Images
4.4. MSE and Peak Signal-To-Noise Ratio Analysis
4.5. Information Entropy Analysis
4.6. Sensitivity Analysis
4.7. Robustness Analysis
4.8. Time Complexity Analysis
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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NNIST Statistical Test Item | p-Value | Pass Rate | Results |
---|---|---|---|
Frequency (monobit) | 0.867692 | 99/100 | Pass |
Block Frequency (m = 128) | 0.122325 | 99/100 | Pass |
Cumulative Sums (Forward) | 0.319084 | 100/100 | Pass |
Cumulative Sums (Reverse) | 0.224821 | 99/100 | Pass |
Runs | 0.181557 | 96/100 | Pass |
Longest Run of Ones | 0.419021 | 99/100 | Pass |
Rank | 0.637119 | 99/100 | Pass |
FFT | 0.289667 | 99/100 | Pass |
Non-Overlapping Templates (m = 9, B = 000000011) | 0.014550 | 97/100 | Pass |
Overlapping Templates (m = 9) | 0.030806 | 99/100 | Pass |
Universal | 0.437274 | 99/100 | Pass |
Approximate Entropy (m = 10) | 0.935716 | 100/100 | Pass |
Random-Excursions (X = 4) | 0.022503 | 65/66 | Pass |
Random-Excursions Variant (X = −9) | 0.671779 | 66/66 | Pass |
Serial Test 1 (m = 16) | 0.019188 | 98/100 | Pass |
Serial Test 2 (m = 16) | 0.719747 | 100/100 | Pass |
Linear Complexity | 0.334538 | 99/100 | Pass |
Digitals\Rules | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
---|---|---|---|---|---|---|---|---|
00 | A | A | G | C | G | C | T | T |
01 | G | C | A | A | T | T | G | C |
10 | C | G | T | T | A | A | C | G |
11 | T | T | C | G | C | G | A | A |
Images | Original Image | This Work | Ref. [30] | Ref. [31] |
---|---|---|---|---|
Lena(256 × 256) | 3.0666 × 104 | 223.2891 | 230.1484 | 217.8984 |
Cameraman(256 × 256) | 1.1097 × 105 | 239.4844 | 234.3047 | 219.4609 |
Lena(512 × 512) | 1.5802 × 105 | 248.8750 | 239.7539 | 249.7266 |
Cameraman(512 × 512) | 4.1853 × 105 | 271.9883 | 278.0410 | 261.8965 |
Barbara(512 × 512) | 1.4410 × 105 | 239.5566 | 253.9297 | 227.0996 |
Mandrill(512 × 512) | 1.8760 × 105 | 244.7559 | 245.0137 | 241.0781 |
Algorithm | Image Name | Horizontal | Vertical | Diagonal |
---|---|---|---|---|
This work | 5.1.10 | 0.0090266 | −0.0059255 | 0.0055227 |
Ref. [32] | 5.1.10 | −0.002971 | −0.000897 | 0.003682 |
Ref. [23] | 5.1.10 | −0.007100 | 0.008500 | 0.000200 |
This work | 5.1.11 | 0.0042465 | −0.0074077 | −0.0045656 |
Ref. [32] | 5.1.11 | 0.001757 | −0.010444 | 0.001124 |
Ref. [23] | 5.1.11 | −0.004800 | −0.001700 | 0.006800 |
This work | 5.1.12 | −0.0041533 | −0.0011927 | 0.0044498 |
Ref. [32] | 5.1.12 | 0.009575 | −0.002502 | −0.000582 |
Ref. [23] | 5.1.12 | 0.005500 | −0.004900 | 0.000100 |
This work | 5.1.13 | 0.00088813 | 0.00060857 | 0.004348 |
Ref. [32] | 5.1.13 | 0.000347 | 0.004691 | −0.009999 |
Ref. [23] | 5.1.13 | 0.003800 | 0.002500 | 0.003200 |
This work | 5.1.14 | −0.0029507 | −0.00010549 | −0.0016555 |
Ref. [32] | 5.1.14 | 0.008773 | −0.011971 | 0.000220 |
Ref. [23] | 5.1.14 | 0.000400 | 0.000400 | 0.001200 |
This work | 5.2.08 | 0.0011946 | 0.0013694 | 0.0010658 |
Ref. [32] | 5.2.08 | −0.002389 | −0.003528 | −0.003059 |
Ref. [23] | 5.2.08 | 0.004100 | 0.001400 | 0.000054 |
This work | 5.2.09 | −0.0019694 | 0.00042113 | 0.0015425 |
Ref. [32] | 5.2.09 | 0.000783 | −0.003316 | −0.000207 |
Ref. [23] | 5.2.09 | −0.001700 | −0.001800 | −0.001900 |
This work | 5.2.10 | −0.001072 | −0.0014797 | 0.0051372 |
Ref. [32] | 5.2.10 | −0.006168 | −0.007614 | 0.000369 |
Ref. [23] | 5.2.10 | 0.000007 | 0.002100 | 0.001200 |
This work | 7.1.01 | 0.0024507 | −0.0021272 | −0.0019601 |
Ref. [32] | 7.1.01 | −0.002843 | 0.000667 | 0.004116 |
Ref. [23] | 7.1.01 | −0.000100 | 0.001300 | −0.001300 |
This work | 7.1.02 | −0.001142 | −6.4625 × 10−5 | 0.0026359 |
Ref. [32] | 7.1.02 | −0.003666 | −0.001386 | −0.001295 |
Ref. [23] | 7.1.02 | 0.000900 | 0.001600 | 0.005700 |
This work | 7.1.03 | 0.0015743 | 0.0024114 | −4.7521 × 10−5 |
Ref. [32] | 7.1.03 | −0.002931 | −0.004124 | 0.003147 |
Ref. [23] | 7.1.03 | 0.000100 | 0.000200 | 0.003100 |
This work | 7.1.04 | −0.001451 | −4.903 × 10−5 | 0.0035602 |
Ref. [32] | 7.1.04 | −0.004028 | −0.001065 | −0.000901 |
Ref. [23] | 7.1.04 | −0.001400 | 0.000811 | −0.003100 |
This work | 7.1.05 | −0.005103 | 0.0019053 | 0.0020445 |
Ref. [32] | 7.1.05 | 0.001735 | −0.003046 | −0.002081 |
Ref. [23] | 7.1.05 | −0.002400 | −0.000700 | 0.003400 |
This work | 7.1.06 | −0.00022585 | 0.0040692 | 0.003082 |
Ref. [32] | 7.1.06 | −0.001395 | −0.003363 | −0.001516 |
Ref. [23] | 7.1.06 | 0.000832 | 0.001700 | 0.001800 |
This work | 7.1.07 | −0.0012256 | −0.0032455 | −0.0044288 |
Ref. [32] | 7.1.07 | −0.000608 | 0.000682 | −0.000090 |
Ref. [23] | 7.1.07 | 0.003900 | 0.002100 | 0.002500 |
This work | 5.3.01 | −6.562 × 10−5 | −3.2479 × 10−5 | −0.0013003 |
Ref. [32] | 5.3.01 | 0.000606 | 0.000090 | 0.002417 |
Ref. [23] | 5.3.01 | 0.000400 | 0.002600 | 0.001200 |
This work | 5.3.02 | 0.0011262 | −0.00055271 | −0.00089431 |
Ref. [32] | 5.3.02 | 0.000502 | 0.001669 | −0.000435 |
Ref. [23] | 5.3.02 | −0.000377 | −0.000474 | −0.000301 |
Images | This Paper | Ref. [5] | Ref. [1] | Ref. [33] |
---|---|---|---|---|
Lena | −0.0035 | 0.002851 | −0.0033 | 0.001744 |
Baboon | −7.3104 × 10−5 | −0.006632 | 0.0020 | 0.005929 |
Cameraman | 0.0026 | −0.003558 | −0.0025 | - |
Pepper | −0.0046 | −0.001650 | 0.0043 | −0.001332 |
Images | MSE | PSNR | ||||||
---|---|---|---|---|---|---|---|---|
Ours | Ref. [5] | Ref. [1] | Ref. [33] | Ours | Ref. [5] | Ref. [1] | Ref. [33] | |
lena | 9072.7 | 7760.0 | 9048.1 | 8692.3 | 8.5534 | 9.2322 | 8.5652 | 8.1636 |
baboon | 7188.3 | 7218.1 | 7200.6 | 8325.0 | 9.5645 | 9.5466 | 9.5571 | 8.9400 |
cameraman | 9400.4 | 9466.7 | 9460.6 | - | 8.3993 | 8.3688 | 8.3716 | - |
peppers | 8199.6 | 8202.4 | 8093.0 | 9647.4 | 8.9929 | 8.9914 | 9.0497 | 7.5889 |
Image Name | Image Size | This Work | Ref. [18] | Ref. [23] | Ref. [32] |
---|---|---|---|---|---|
5.1.10 | 256 × 256 | 7.9976948 | 7.99720 | 7.99680 | 7.99717 |
5.1.11 | 256 × 256 | 7.9970808 | 7.99730 | 7.99710 | 7.96999 |
5.1.12 | 256 × 256 | 7.9975342 | 7.99540 | 7.99730 | 7.99757 |
5.1.13 | 256 × 256 | 7.9971601 | 7.99630 | 7.99680 | 7.99735 |
5.1.14 | 256 × 256 | 7.9972009 | 7.99730 | 7.99690 | 7.99674 |
5.2.08 | 512 × 512 | 7.9993163 | 7.99920 | 7.99920 | 7.99934 |
5.2.09 | 512 × 512 | 7.9993178 | 7.99900 | 7.99940 | 7.99930 |
5.2.10 | 512 × 512 | 7.9992856 | 7.99870 | 7.99930 | 7.99926 |
7.1.01 | 512 × 512 | 7.9992931 | 7.99800 | 7.99930 | 7.99929 |
7.1.02 | 512 × 512 | 7.9992812 | 7.99490 | 7.99930 | 7.99931 |
7.1.03 | 512 × 512 | 7.9992667 | 7.99830 | 7.99940 | 7.99925 |
7.1.04 | 512 × 512 | 7.9992436 | 7.99850 | 7.99940 | 7.99923 |
7.1.05 | 512 × 512 | 7.9993051 | 7.99880 | 7.99930 | 7.99929 |
7.1.06 | 512 × 512 | 7.9993503 | 7.99900 | 7.99930 | 7.99933 |
7.1.07 | 512 × 512 | 7.9993485 | 7.99870 | 7.99910 | 7.99931 |
7.1.08 | 512 × 512 | 7.9993392 | 7.99880 | 7.99920 | 7.99923 |
7.1.09 | 512 × 512 | 7.9993272 | 7.99850 | 7.99920 | 7.99219 |
elaine.512 | 512 × 512 | 7.9992569 | 7.99930 | 7.99930 | 7.99922 |
5.3.01 | 1024 × 1024 | 7.9998174 | 7.99930 | 7.99980 | 7.99983 |
5.3.02 | 1024 × 1024 | 7.9998495 | 7.99920 | 7.99990 | 7.99981 |
testpat.1k | 1024 × 1024 | 7.9997892 | 7.98470 | 7.99980 | 7.99982 |
The Proposed Method in this Paper | The Method Proposed in Ref. [5] | ||||
---|---|---|---|---|---|
Changes of Key | NPCR | UACI | Changes of Key | NPCR | UACI |
Δμ = 10−15 | 99.6124 | 33.3809 | Δx0 = 10−10 | 99.6300 | 33.5100 |
Δx01 = 10−15 | 99.6201 | 33.6414 | Δy0 = 10−10 | 99.6000 | 33.5100 |
Δy01 = 10−15 | 99.6262 | 33.5600 | Δz0 = 10−10 | 99.6000 | 33.5000 |
Δs01 = 10−15 | 99.6552 | 33.5363 | Δw0 = 10−10 | 99.6200 | 33.5100 |
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Zhu, S.; Deng, X.; Zhang, W.; Zhu, C. Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding. Mathematics 2023, 11, 231. https://doi.org/10.3390/math11010231
Zhu S, Deng X, Zhang W, Zhu C. Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding. Mathematics. 2023; 11(1):231. https://doi.org/10.3390/math11010231
Chicago/Turabian StyleZhu, Shenli, Xiaoheng Deng, Wendong Zhang, and Congxu Zhu. 2023. "Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding" Mathematics 11, no. 1: 231. https://doi.org/10.3390/math11010231
APA StyleZhu, S., Deng, X., Zhang, W., & Zhu, C. (2023). Image Encryption Scheme Based on Newly Designed Chaotic Map and Parallel DNA Coding. Mathematics, 11(1), 231. https://doi.org/10.3390/math11010231