Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation
Abstract
:1. Introduction
2. N-Dimensional Modular Chaotic Model
2.1. Construction of ND-CMCM
2.2. Examples of the Proposed Chaotic Model
2.2.1. 2D Logistic–PWLCM Coupled Modular Map
2.2.2. 3D Cubic–Fraction–IST Coupled Modular Map
2.2.3. 4D ICMIC–ICMIC–1DSP–1DCP Coupled Modular Map
3. Performance Analysis of New Systems
3.1. Bifurcation Diagram and Phase Diagram
3.2. Lyapunov Exponent
3.3. Sample Entropy
4. Proposed Encryption Algorithm
4.1. Overlapping Block
4.2. Newly Defined Operations Based on Latin Squares and Lookup Table
Algorithm 1 Pseudo-code for generating Latin squares | |
Input: chaotic sequences Q1 and Q2 of length n Output: nth order Latin square L | |
1: | [~,I1] = sort(Q1); |
2: | [~,I2] = sort(Q2); |
3: | I1 = I1 − 1; |
4: | for i = 0 to n − 1 |
5: | L(i + 1,:) = circshift(I1,I2(i + 1)); |
6: | end for |
Algorithm 2 Pseudo-code for generating reverse Latin squares | |
Input: nth Latin square L Output:nth reverse Latin square L’ | |
1: | for i = 0 to n − 1 |
2: | for j = 0 to n − 1 |
3: | L’(L(i + 1,j + 1) + 1,j + 1) = i; |
4: | end for |
5: | end for |
4.3. Encryption Process
4.3.1. Generation of Secret Key and Chaotic Sequences
4.3.2. Overlapping Block Permutation–Diffusion
Algorithm 3 Pseudo-code for index processing | |
Input: E, A Output: E1, A1 | |
1: | Temp = [E A]; |
2: | Temp = unique(Temp,’stable’); |
3: | Stemp = length(Temp); |
4: | if mod(Stemp,2) = 1 |
5: | Temp(end) = []; |
6: | Stemp = Stemp − 1; |
7: | end if |
8: | E1 = Temp(1: Stemp/2); |
9: | A1 =Temp(Stemp/2 + 1:end); |
Algorithm 4 Pseudo-code for forward permutation–diffusion | |
Input: P, Blen, Bmin, S1, CS3, CS4, CS5, CS9, CS10, CS13, Output: P | |
1: | Ct = 1; Plen = Blen; |
2: | for i = 1 to Bn |
3: | if Plen < S1(i) |
4: | S = Plen; |
5: | end if |
6: | L1 = mod(ceil(CS3(i) × 1014),Lnum) + 1; L2 = mod(ceil(CS4(i) × 1014),Lnum) + 1; |
7: | L3 = mod(ceil(CS5(i) × 1014),Lnum) + 1; E = mod(ceil(CS9(Ct:Ct + S − 1) × 1014),Plen) + 1; |
8: | D = mod(ceil(CS10(Ct:Ct + S − 1) × 1014),256); Ct = Ct + S; |
9: | A = 1 to S; |
10: | [E1,A1] = Algorithm 2(E,A); E1 =E1 + (i − 1) × Bmin; A1 =A1 + (i − 1) × Bmin; |
11: | P(E1)⇔P(A1) |
12: | if i > 1 |
13: | if Plen > Bmin |
14: | P((i − 1) × Bmin + 1:i × Bmin) = LL(P((i − 1) × Bmin + 1:i × Bmin),P((i − 2) × Bmin + 1:(i − 1) × Bmin),L{L1}); |
15: | else |
16: | P((i − 1) × Bmin + 1:end)= LL(P((i − 1) × Bmin + 1:end),P((i − 2) × Bmin + 1:(i − 1) × Bmin),L{L1}); |
17: | end if |
18: | for k = 1 to S |
19: | P((i − 1) × Bmin + k) = LL(P((i − 1) × Bmin + k),P((i − 1) × Bmin + k − 1),L{L2}); |
20: | end for |
21: | else |
22: | P(1:Blen − ((Bn − 1) × Bmin)) = LL(P(1:Blen − ((Bn − 1) × Bmin)),P((Bn − 1) × Bmin + 1:end), L{L1}); |
23: | P(1) = LL(P(1),P(Bn),L{L2}); |
24: | for k = 2 to S |
25: | P(k) = LL(P(k),P(k − 1),L{L2}); |
26: | end for |
27: | end if |
28: | P((i − 1) × Bmin + 1:(i − 1) × Bmin + S) = LL(P((i − 1) × Bmin + 1:(i − 1) × Bmin + S),D,L{L3}); |
29: | Plen = Plen − Bmin; |
30: | end for |
4.4. Decryption Process
5. Experimental Results and Security Analysis
5.1. Key Space Analysis
5.2. Key Sensitivity Analysis
5.3. Histogram Analysis
5.4. Chosen Plaintext Attack Analysis
5.5. Differential Attack Analysis
5.6. The Correlation between Adjacent Pixels Analysis
5.7. Information Entropy Analysis
5.8. Robustness Analysis
5.9. Speed Analysis
5.10. Comparison with Other Schemes
6. Conclusions
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
References
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Key Type | Image | Size | Level 1 | Level 2 | Level 3 | |||
---|---|---|---|---|---|---|---|---|
NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | NPCR (%) | UACI (%) | |||
Key generated by SHA-512 | 5.1.11 | 256 × 256 × 1 | 99.6184 | 33.4516 | 99.6120 | 33.4797 | 99.6445 | 33.5498 |
1.1.01 | 512 × 512 × 1 | 99.6088 | 33.4600 | 99.6132 | 33.4604 | 99.6235 | 33.5580 | |
1.3.01 | 1024 × 1024 × 1 | 99.6108 | 33.4651 | 99.6078 | 33.4424 | 99.6058 | 33.4144 | |
4.1.01 | 256 × 256 × 3 | 99.6049 | 33.4838 | 99.6119 | 33.4793 | 99.5900 | 33.4514 | |
4.2.01 | 512 × 512 × 3 | 99.6146 | 33.4744 | 99.6131 | 33.4682 | 99.6260 | 33.4593 | |
2.2.01 | 1024 × 1024 × 3 | 99.6087 | 33.4596 | 99.6102 | 33.4637 | 99.6088 | 33.4732 | |
Pure black | 512 × 512 × 3 | 99.6084 | 33.4465 | 99.6111 | 33.4456 | 99.6241 | 33.4705 | |
Mean | - | 99.6107 | 33.4630 | 99.6113 | 33.4627 | 99.6175 | 33.4824 | |
Customized key | 5.1.12 | 256 × 256 × 1 | 99.5911 | 33.2539 | 99.6030 | 33.5674 | 99.6124 | 33.5947 |
5.2.10 | 512 × 512 × 1 | 99.5975 | 33.3703 | 99.6088 | 33.4564 | 99.6029 | 33.4332 | |
5.3.01 | 1024 × 1024 × 1 | 99.6031 | 33.4669 | 99.6107 | 33.4592 | 99.6011 | 33.4326 | |
4.1.05 | 256 × 256 × 3 | 99.5967 | 33.4399 | 99.6014 | 33.4807 | 99.6282 | 33.4856 | |
4.2.05 | 512 × 512 × 3 | 99.6015 | 33.4614 | 99.6084 | 33.4686 | 99.6160 | 33.4847 | |
6.2.02 | 1024 × 1024 × 3 | 99.6111 | 33.4849 | 99.6099 | 33.4630 | 99.6044 | 33.4669 | |
Pure white | 512 × 512 × 3 | 99.6168 | 33.4971 | 99.6032 | 33.4590 | 99.6170 | 33.4838 | |
Mean | - | 99.6025 | 33.4249 | 99.6065 | 33.4792 | 99.6117 | 33.4831 |
Image | Size | Direction | Plaintext Image | Ciphertext Image | ||
---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | ||||
4.1.01 | 256 × 256 × 3 | H | 0.9679 | −0.0032 | 0.0040 | 0.0019 |
V | 0.9588 | 0.0065 | 0.0068 | 0.0067 | ||
D | 0.9447 | 0.0019 | −0.0010 | 0.0028 | ||
5.2.10 | 512 × 512 × 1 | H | 0.9317 | 0.0199 | −0.0054 | −0.0022 |
V | 0.9435 | −0.0029 | 0.0000 | 0.0063 | ||
D | 0.9036 | −0.0075 | 0.0003 | 0.0009 | ||
4.2.05 | 512 × 512 × 3 | H | 0.9604 | 0.0022 | −0.0044 | −0.0015 |
V | 0.9543 | 0.0066 | 0.0036 | 0.0009 | ||
D | 0.9251 | 0.0027 | −0.0011 | −0.0014 | ||
5.3.01 | 1024 × 1024 × 1 | H | 0.9818 | −0.0062 | 0.0051 | 0.0001 |
V | 0.9769 | 0.0074 | 0.0005 | −0.0018 | ||
D | 0.9665 | 0.0087 | 0.0047 | 0.0001 | ||
Pure black | 512 × 512 × 3 | H | - | −0.0045 | −0.0020 | −0.0014 |
V | - | 0.0082 | 0.0005 | −0.0065 | ||
D | - | 0.0066 | 0.0040 | −0.0013 |
Image | Size | PI | CI | Image | Size | PI | CI | ||||
---|---|---|---|---|---|---|---|---|---|---|---|
Level 1 | Level 2 | Level 3 | Level 1 | Level 2 | Level 3 | ||||||
5.1.11 | 256 × 256 × 1 | 6.4523 | 7.9971 | 7.9975 | 7.9978 | 5.1.12 | 256 × 256 × 1 | 6.8981 | 7.9977 | 7.9981 | 7.9979 |
1.1.01 | 512 × 512 × 1 | 6.7057 | 7.9993 | 7.9992 | 7.9994 | 5.2.10 | 512 × 512 × 1 | 7.0686 | 7.9993 | 7.9992 | 7.9993 |
1.3.01 | 1024 × 1024 × 1 | 7.6702 | 7.9998 | 7.9998 | 7.9998 | 5.3.01 | 1024 × 1024 × 1 | 7.2428 | 7.9999 | 7.9998 | 7.9998 |
4.1.01 | 256 × 256 × 3 | 5.7056 | 7.9990 | 7.9992 | 7.9992 | 4.1.05 | 256 × 256 × 3 | 6.6639 | 7.9998 | 7.9998 | 7.9998 |
4.2.01 | 512 × 512 × 3 | 7.4404 | 7.9997 | 7.9998 | 7.9998 | 4.2.05 | 512 × 512 × 3 | 7.6133 | 7.9999 | 7.9999 | 7.9999 |
2.2.01 | 1024 × 1024 × 3 | 7.5237 | 7.9999 | 7.9999 | 7.9999 | 6.2.02 | 1024 × 1024 × 3 | 6.5786 | 7.9999 | 7.9999 | 7.9999 |
Pure black | 512 × 512 × 3 | 0 | 7.9997 | 7.9998 | 7.9998 | Pure white | 512 × 512 × 3 | 0 | 7.9998 | 7.9998 | 7.9998 |
Size | Level 1 | Level 2 | Level 3 | |||
---|---|---|---|---|---|---|
Time (s) | Time (s/Unit) | Time (s) | Time (s/Unit) | Time (s) | Time (s/Unit) | |
256 × 256 × 1 | 0.064 | 0.064 | 0.121 | 0.121 | 0.229 | 0.229 |
512 × 512 × 1 | 0.226 | 0.057 | 0.337 | 0.084 | 0.749 | 0.187 |
1024 × 1024 × 1 | 0.807 | 0.050 | 1.161 | 0.073 | 2.502 | 0.156 |
256 × 256 × 3 | 0.171 | 0.057 | 0.264 | 0.088 | 0.579 | 0.193 |
512 × 512 × 3 | 0.624 | 0.052 | 0.886 | 0.074 | 1.944 | 0.162 |
1024 × 1024 × 3 | 2.221 | 0.046 | 3.153 | 0.066 | 6.626 | 0.138 |
Mean | - | 0.054 | - | 0.084 | - | 0.178 |
Scheme | NPCR (%) | UACI (%) | Correlation | IE | Time (s/Unit) | ||
---|---|---|---|---|---|---|---|
Horizontal | Vertical | Diagonal | |||||
Ref. [18] | 99.6229 | 33.4809 | 0.0019 | 0.0012 | 0.0020 | 7.9994 | - |
Ref. [43] | 99.6060 | 33.5126 | −0.0003 | 0.0011 | 0.0013 | 7.9998 | 0.107 |
Ref. [44] | - | - | 0.0044 | −0.0050 | −0.0002 | - | 0.515 |
Ref. [45] | 99.6167 | 33.4772 | −0.0036 | −0.0049 | −0.0023 | 7.9993 | - |
Ref. [46] | 99.6200 | 33.4600 | 0.0013 | 0.0009 | −0.0018 | 7.9999 | 0.540 |
Ref. [47] | 99.6085 | 33.4634 | 0.0011 | 0.0008 | 0.0015 | 7.9993 | 0.320 |
Ref. [48] | 99.6012 | 33.4418 | 0.0015 | −0.0002 | −0.0004 | 7.9996 | 0.503 |
Ref. [49] | 99.6077 | 33.4399 | −0.0016 | 0.0057 | −0.0189 | 7.9996 | 1.711 |
Ref. [50] | 99.6367 | 33.3733 | 0.0116 | 0.0057 | 0.0039 | 7.9994 | - |
Ref. [51] | 99.6174 | 33.4657 | −0.0164 | 0.0056 | 0.0289 | 7.9993 | 0.123 |
Level 1 | 99.6107 | 33.4630 | 0.0022 | 0.0066 | 0.0027 | 7.9993 | 0.054 |
Level 2 | 99.6113 | 33.4627 | −0.0044 | 0.0036 | −0.0011 | 7.9994 | 0.084 |
Level 3 | 99.6175 | 33.4824 | −0.0015 | 0.0009 | −0.0014 | 7.9994 | 0.178 |
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Zhou, Z.; Xu, X.; Jiang, Z.; Sun, K. Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation. Mathematics 2023, 11, 3373. https://doi.org/10.3390/math11153373
Zhou Z, Xu X, Jiang Z, Sun K. Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation. Mathematics. 2023; 11(15):3373. https://doi.org/10.3390/math11153373
Chicago/Turabian StyleZhou, Ziqi, Xuemei Xu, Zhaohui Jiang, and Kehui Sun. 2023. "Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation" Mathematics 11, no. 15: 3373. https://doi.org/10.3390/math11153373
APA StyleZhou, Z., Xu, X., Jiang, Z., & Sun, K. (2023). Multiple-Image Encryption Scheme Based on an N-Dimensional Chaotic Modular Model and Overlapping Block Permutation–Diffusion Using Newly Defined Operation. Mathematics, 11(15), 3373. https://doi.org/10.3390/math11153373