Three-Dimensional and Multiple Image Encryption Algorithm Using a Fractional-Order Chaotic System
Abstract
:1. Introduction
2. Fractional-Based Hybrid Chaotic System
2.1. 4D Fractional Chaotic System
2.2. Hybrid Chaotic System
- Step 1. Consider and as input values.
- Step 2. Define and run the following algorithm.
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- Step 3. Consider .
- Step 4. If , stop, orelse go to Step 2.
3. 3D and Multiple Image Encryption Algorithm
3.1. The Key of Encryption Algorithm
- Step 1. In this step, in order to reduce the volume of calculations, we reduce the size of the image used in the calculations by wavelet transformation. For , derive as the low–low pass filter of the Daubechies wavelet transform at level 5 (db5) for .
- Step 2. Obtain the sequence by using the following formulae.
- Step 3. Define and .
- Step 4. Calculate the and using the following algorithm.
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3.2. Proposed Shift and Rearrange
Algorithm 1 PermutationMatrix |
Input: A permutation Output: A permutation matrix
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Algorithm 2 Shift |
Input: Matrices and permutations Output: Matrices
|
Algorithm 3 RowCol |
Input: Natural umbers Output: Natural numbers
|
Algorithm 4 Rearrange |
Input: A matrix and a permutation Output: A matrix
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Algorithm 5 Rearrange3D |
Input: A multilayer matrix and a permutation Output: A matrix
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3.3. Proposed Encryption Algorithm
- Step 1. By using the input keys and hybrid chaotic system, obtain the vector with as follows:
- Step 2. Obtain the sequence as
- Step 3.
- (a) By using input keys and obtain with as
- (b) Calculate as follows:
- (c) Consider and as input values for shift Algorithm 2 and obtain as output of this function.
- (a) Obtain as
- (b) Define as
- (c) Define as
- (d) Define a multilayer matrix , and then consider and A as input values for Rearrange3D Algorithm 5 and obtain final encrypted images .
4. Simulation Results and Security Analysis
4.1. Simulation Infrastructure
4.2. Analysis of the Proposed Systems
- .
- .
- .
- , , .
- , , , , , , , .
- , , .
4.3. Key Space Analysis
- Aerial 5.1.10(256 × 256), Airplane 5.1.11(256 × 256), Chemical plant 5.1.14(256 × 256), Clock 5.1.12(256 × 256)}.
- Female 4.1.02(256 × 256 × 3), Female 4.1.03(256 × 256 × 3), Jelly beans 4.1.08(256 × 256 × 3), House 4.1.05(256 × 256 × 3)}.
- Airplane 5.1.11(256 × 256), Clock 5.1.12(256 × 256), Jelly Jellybeans 4.1.08(256 × 256 × 3), House 4.1.05(256 × 256 × 3)}.
- Watch(512 × 512), Lena(512 × 512), Peppers(512 × 512), Fishing Boat(512 × 512)}.
- Case i.
- Without changing the input data.
- Case ii.
- The following changes have been applied to images and : and .
- Case iii.
- has been changed to .
- Case iv.
- The value of r has been updated to .
- Case v.
- The new value assigned to h is .
4.4. Analysis of Program Execution Time
4.5. Histogram and Correlation Coefficients Analysis
4.6. NPCR, UACI and Information Entropy Analysis
4.7. Data Loss and Noise Attacks
5. Conclusions
Author Contributions
Funding
Data Availability Statement
Conflicts of Interest
References
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Images | Key | Case i | Case ii | Case iii | Case iv | Case v |
---|---|---|---|---|---|---|
0.3699 | 0.3607 | 0.0053 | 0.2009 | 0.1936 | ||
0.3302 | 0.3437 | 0.2468 | 0.2260 | 0.3965 | ||
0.3310 | 0.1241 | 0.5127 | 0.3521 | 0.2145 | ||
0.3041 | 0.3462 | 0.4062 | 0.1765 | 0.1814 | ||
0.4680 | 0.1866 | 0.1718 | 0.1439 | 0.1860 | ||
0.3314 | 0.3966 | 0.1107 | 0.3376 | 0.3431 | ||
0.3057 | 0.4664 | 0.2168 | 0.1822 | 0.3739 | ||
0.2933 | 0.1508 | 0.4643 | 0.1746 | 0.3616 | ||
0.2307 | 0.1978 | 0.4691 | 0.4069 | 0.2901 | ||
0.1547 | 0.1771 | 0.1116 | 0.3248 | 0.3889 | ||
0.4242 | 0.2972 | 0.3734 | 0.3846 | 0.2588 | ||
0.4377 | 0.5214 | 0.2964 | 0.3770 | 0.3176 | ||
0.2633 | 0.2028 | 0.2784 | 0.2385 | 0.3297 | ||
0.0873 | 0.3831 | 0.3515 | 0.1748 | 0.0777 | ||
0.1848 | 0.3284 | 0.5258 | 0.2342 | 0.2752 | ||
0.2583 | 0.3127 | 0.4522 | 0.2607 | 0.2202 | ||
Engine | 0.5381 | 0.3147 | 0.3814 | 0.2705 | 0.0153 | |
0.2818 | 0.5031 | 0.3812 | 0.2064 | 0.1799 | ||
0.2737 | 0.1674 | 0.2143 | 0.0793 | 0.2559 | ||
0.1766 | 0.4007 | 0.0623 | 0.0133 | 0.5334 |
Algorithm | Proposed | Ref. [25] | Ref. [26] | Ref. [27] | Ref. [28] |
Key size | |||||
Algorithm | Ref. [29] | Ref. [30] | Ref. [31] | Ref. [32] | Ref. [33] |
Key size |
Images | Key Generation | Encryption | Decryption |
---|---|---|---|
0.2084 | 3.7334 | 3.8196 | |
0.2677 | 6.6504 | 6.4169 | |
0.2648 | 5.1215 | 5.0604 | |
0.3203 | 14.1428 | 14.1293 |
Input images | Proposed | Ref. [34] | |
Horizontal | |||
Vertical | |||
Diagonal | |||
Ref. [35] | Ref. [36] | Ref. [37] | |
Horizontal | |||
Vertical | |||
Diagonal |
Input images | Proposed | Ref. [38] | |
Horizontal | |||
Vertical | |||
Diagonal | |||
Ref. [39] | Ref. [40] | Ref. [41] | |
Horizontal | |||
Vertical | |||
Diagonal |
UACI Critical Values [42] | NPCR Critical Values [42] | |||||||
---|---|---|---|---|---|---|---|---|
= 33.2824 | = 33.2255 | = 33.1594 | ||||||
Image | UACI | = 33.6447 | = 33.7016 | = 33.7677 | NPCR | 99.5693 | 99.5527 | 99.5341 |
33.3277 | Pass | Pass | Pass | 99.6063 | Pass | Pass | Pass | |
33.4920 | Pass | Pass | Pass | 99.6190 | Pass | Pass | Pass | |
33.4685 | Pass | Pass | Pass | 99.6002 | Pass | Pass | Pass | |
Engine | 33.4090 | Pass | Pass | Pass | 99.6107 | Pass | Pass | Pass |
Proposed | Ref. [34] | Ref. [37] | Ref. [43] | |
---|---|---|---|---|
NPCR | ||||
UACI | ||||
Information entropy |
Engine | ||||
---|---|---|---|---|
Information entropy |
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Share and Cite
Ghasemi, G.; Parvaz, R.; Khedmati Yengejeh, Y. Three-Dimensional and Multiple Image Encryption Algorithm Using a Fractional-Order Chaotic System. Computation 2025, 13, 115. https://doi.org/10.3390/computation13050115
Ghasemi G, Parvaz R, Khedmati Yengejeh Y. Three-Dimensional and Multiple Image Encryption Algorithm Using a Fractional-Order Chaotic System. Computation. 2025; 13(5):115. https://doi.org/10.3390/computation13050115
Chicago/Turabian StyleGhasemi, Ghader, Reza Parvaz, and Yavar Khedmati Yengejeh. 2025. "Three-Dimensional and Multiple Image Encryption Algorithm Using a Fractional-Order Chaotic System" Computation 13, no. 5: 115. https://doi.org/10.3390/computation13050115
APA StyleGhasemi, G., Parvaz, R., & Khedmati Yengejeh, Y. (2025). Three-Dimensional and Multiple Image Encryption Algorithm Using a Fractional-Order Chaotic System. Computation, 13(5), 115. https://doi.org/10.3390/computation13050115