A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains |
Optics Express, Vol. 22, Issue 6, pp. 7349-7361 (2014)
http://dx.doi.org/10.1364/OE.22.007349
Acrobat PDF (8200 KB)
Abstract
Recently, a number of double-image cryptosystems have been developed. However, there are notable security performance differences between the two encryption channels in these algorithms. This weakness downgrades the security level and practicability of these cryptosystems, as the cryptosystems cannot guarantee all the input images be transmitted in the channel with higher security level. In this paper, we propose a novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains. The two input images are firstly shuffled by the proposed cross-image pixel scrambling approach, which can well balance the pixel distribution across the input images. The two scrambled images will be encoded into the real and imaginary parts of a complex function, and then converted into gyrator domains. An iterative architecture is designed to enhance the security level of the cryptosystem, and the cross-image pixel scrambling operation is performed to the real and imaginary parts of the generated complex encrypted data in each round. Numerical simulation results prove that a satisfactory and balanced security performance can be achieved in both channels.
© 2014 Optical Society of America
1. Introduction
1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed]
2. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed]
3. B. Zhu and S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001). [CrossRef]
9. Z. Liu, S. Liu, W. Liu, Y. Wang, and S. Liu, “Image encryption algorithm by using fractional Fourier transform and pixel scrambling operation based on double random phase encoding,” Opt. Lasers Eng. 51(1), 8–14 (2013). [CrossRef]
10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed]
13. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed]
14. Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, and S. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011). [CrossRef]
15. L. Chen and D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007). [CrossRef] [PubMed]
17. L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43(15), 3078–3084 (2004). [CrossRef] [PubMed]
18. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef] [PubMed]
20. W. Chen and X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51(25), 6076–6083 (2012). [CrossRef] [PubMed]
21. W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef] [PubMed]
22. Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008). [CrossRef]
23. C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010). [CrossRef]
24. X. F. Meng, L. Z. Cai, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, H. Zhang, and Y. R. Wang, “Digital color image watermarking based on phase-shifting interferometry and neighboring pixel value subtraction algorithm in the discrete-cosine-transform domain,” Appl. Opt. 46(21), 4694–4701 (2007). [CrossRef] [PubMed]
25. M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009). [CrossRef] [PubMed]
26. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed]
27. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]
32. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007). [CrossRef]
27. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]
32. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007). [CrossRef]
27. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]
27. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]
33. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24(10), 3135–3139 (2007). [CrossRef] [PubMed]
2. The proposed double-image encryption scheme
2.1 Cross-image pixel scrambling strategy
2.2 Gyrator transform
10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed]
33. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24(10), 3135–3139 (2007). [CrossRef] [PubMed]
10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed]
14. Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, and S. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011). [CrossRef]
2.3 The proposed double-image encryption scheme
10. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed]
33. J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24(10), 3135–3139 (2007). [CrossRef] [PubMed]
34. S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014). [CrossRef]
3. Numerical simulation and discussion
31. H. Li, Y. Wang, H. Yan, L. Li, Q. Li, and X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef]
3.1 Robustness to noise perturbation
3.2 Tolerance against occlusion attacks
3.3 Sensitivity of the rotation angle
3.4 Sensitivity of the Chebyshev map
36. C. Fu, J. J. Chen, H. Zou, W. H. Meng, Y. F. Zhan, and Y. W. Yu, “A chaos-based digital image encryption scheme with an improved diffusion strategy,” Opt. Express 20(3), 2363–2378 (2012). [CrossRef] [PubMed]
3.5 Further discussion
4. Conclusions
Acknowledgments
References and links
1. | P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed] |
2. | G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed] |
3. | B. Zhu and S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001). [CrossRef] |
4. | Y. Zhang, C. H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002). [CrossRef] |
5. | X. Wang and D. Zhao, “Simultaneous nonlinear encryption of grayscale and color images based on phase-truncated fractional Fourier transform and optical superposition principle,” Appl. Opt. 52(25), 6170–6178 (2013). [CrossRef] [PubMed] |
6. | N. Zhou, T. Dong, and J. Wu, “Novel image encryption algorithm based on multiple-parameter discrete fractional random transform,” Opt. Commun. 283(15), 3037–3042 (2010). [CrossRef] |
7. | J. Lang, “Image encryption based on the reality-preserving multiple-parameter fractional Fourier transform and chaos permutation,” Opt. Lasers Eng. 50(7), 929–937 (2012). [CrossRef] |
8. | Z. Liu, S. Li, M. Yang, W. Liu, and S. Liu, “Image encryption based on the random rotation operation in the fractional Fourier transform domains,” Opt. Lasers Eng. 50(10), 1352–1358 (2012). [CrossRef] |
9. | Z. Liu, S. Liu, W. Liu, Y. Wang, and S. Liu, “Image encryption algorithm by using fractional Fourier transform and pixel scrambling operation based on double random phase encoding,” Opt. Lasers Eng. 51(1), 8–14 (2013). [CrossRef] |
10. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed] |
11. | Q. Wang, Q. Guo, L. Lei, and J. Zhou, “Linear exchanging operation and random phase encoding in gyrator transform domain for double image encryption,” Optik (Stuttg.) 124(24), 6707–6712 (2013). [CrossRef] |
12. | Z. Liu, L. Xu, C. Lin, and S. Liu, “Image encryption by encoding with a nonuniform optical beam in gyrator transform domains,” Appl. Opt. 49(29), 5632–5637 (2010). [CrossRef] [PubMed] |
13. | Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed] |
14. | Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, and S. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011). [CrossRef] |
15. | L. Chen and D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007). [CrossRef] [PubMed] |
16. | E. Tajahuerce, O. Matoba, S. C. Verrall, and B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39(14), 2313–2320 (2000). [CrossRef] [PubMed] |
17. | L. Z. Cai, M. Z. He, Q. Liu, and X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43(15), 3078–3084 (2004). [CrossRef] [PubMed] |
18. | Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef] [PubMed] |
19. | X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012). [CrossRef] [PubMed] |
20. | W. Chen and X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51(25), 6076–6083 (2012). [CrossRef] [PubMed] |
21. | W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef] [PubMed] |
22. | Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008). [CrossRef] |
23. | C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010). [CrossRef] |
24. | X. F. Meng, L. Z. Cai, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, H. Zhang, and Y. R. Wang, “Digital color image watermarking based on phase-shifting interferometry and neighboring pixel value subtraction algorithm in the discrete-cosine-transform domain,” Appl. Opt. 46(21), 4694–4701 (2007). [CrossRef] [PubMed] |
25. | M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009). [CrossRef] [PubMed] |
26. | G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed] |
27. | R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed] |
28. | Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010). [CrossRef] |
29. | Y. Zhang and D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef] |
30. | Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, M. A. Ahmad, J. Dai, and S. Liu, “Double image encryption by using Arnold transform and discrete fractional angular transform,” Opt. Lasers Eng. 50(2), 248–255 (2012). [CrossRef] |
31. | H. Li, Y. Wang, H. Yan, L. Li, Q. Li, and X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef] |
32. | Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007). [CrossRef] |
33. | J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24(10), 3135–3139 (2007). [CrossRef] [PubMed] |
34. | S. Liu, C. Guo, and J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014). [CrossRef] |
35. | IEEE Computer Society, “IEEE standard for binary floating-point arithmetic,” ANSI/IEEE std. 754–1985 (1985). |
36. | C. Fu, J. J. Chen, H. Zou, W. H. Meng, Y. F. Zhan, and Y. W. Yu, “A chaos-based digital image encryption scheme with an improved diffusion strategy,” Opt. Express 20(3), 2363–2378 (2012). [CrossRef] [PubMed] |
OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(070.4560) Fourier optics and signal processing : Data processing by optical means
(100.0100) Image processing : Image processing
(100.2960) Image processing : Image analysis
ToC Category:
Image Processing
History
Original Manuscript: January 24, 2014
Revised Manuscript: March 12, 2014
Manuscript Accepted: March 13, 2014
Published: March 21, 2014
Citation
Jun-xin Chen, Zhi-liang Zhu, Zhengjun Liu, Chong Fu, Li-bo Zhang, and Hai Yu, "A novel double-image encryption scheme based on cross-image pixel scrambling in gyrator domains," Opt. Express 22, 7349-7361 (2014)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-6-7349
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References
- P. Refregier, B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995). [CrossRef] [PubMed]
- G. Unnikrishnan, J. Joseph, K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000). [CrossRef] [PubMed]
- B. Zhu, S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001). [CrossRef]
- Y. Zhang, C. H. Zheng, N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002). [CrossRef]
- X. Wang, D. Zhao, “Simultaneous nonlinear encryption of grayscale and color images based on phase-truncated fractional Fourier transform and optical superposition principle,” Appl. Opt. 52(25), 6170–6178 (2013). [CrossRef] [PubMed]
- N. Zhou, T. Dong, J. Wu, “Novel image encryption algorithm based on multiple-parameter discrete fractional random transform,” Opt. Commun. 283(15), 3037–3042 (2010). [CrossRef]
- J. Lang, “Image encryption based on the reality-preserving multiple-parameter fractional Fourier transform and chaos permutation,” Opt. Lasers Eng. 50(7), 929–937 (2012). [CrossRef]
- Z. Liu, S. Li, M. Yang, W. Liu, S. Liu, “Image encryption based on the random rotation operation in the fractional Fourier transform domains,” Opt. Lasers Eng. 50(10), 1352–1358 (2012). [CrossRef]
- Z. Liu, S. Liu, W. Liu, Y. Wang, S. Liu, “Image encryption algorithm by using fractional Fourier transform and pixel scrambling operation based on double random phase encoding,” Opt. Lasers Eng. 51(1), 8–14 (2013). [CrossRef]
- J. A. Rodrigo, T. Alieva, M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007). [CrossRef] [PubMed]
- Q. Wang, Q. Guo, L. Lei, J. Zhou, “Linear exchanging operation and random phase encoding in gyrator transform domain for double image encryption,” Optik (Stuttg.) 124(24), 6707–6712 (2013). [CrossRef]
- Z. Liu, L. Xu, C. Lin, S. Liu, “Image encryption by encoding with a nonuniform optical beam in gyrator transform domains,” Appl. Opt. 49(29), 5632–5637 (2010). [CrossRef] [PubMed]
- Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010). [CrossRef] [PubMed]
- Z. Liu, D. Chen, J. Ma, S. Wei, Y. Zhang, J. Dai, S. Liu, “Fast algorithm of discrete gyrator transform based on convolution operation,” Optik (Stuttg.) 122(10), 864–867 (2011). [CrossRef]
- L. Chen, D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007). [CrossRef] [PubMed]
- E. Tajahuerce, O. Matoba, S. C. Verrall, B. Javidi, “Optoelectronic information encryption with phase-shifting interferometry,” Appl. Opt. 39(14), 2313–2320 (2000). [CrossRef] [PubMed]
- L. Z. Cai, M. Z. He, Q. Liu, X. L. Yang, “Digital image encryption and watermarking by phase-shifting interferometry,” Appl. Opt. 43(15), 3078–3084 (2004). [CrossRef] [PubMed]
- Y. Zhang, B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008). [CrossRef] [PubMed]
- X. Wang, D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012). [CrossRef] [PubMed]
- W. Chen, X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51(25), 6076–6083 (2012). [CrossRef] [PubMed]
- W. Chen, X. Chen, C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010). [CrossRef] [PubMed]
- Z. Liu, M. A. Ahmad, S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008). [CrossRef]
- C. N. Yang, S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010). [CrossRef]
- X. F. Meng, L. Z. Cai, X. L. Yang, X. F. Xu, G. Y. Dong, X. X. Shen, H. Zhang, Y. R. Wang, “Digital color image watermarking based on phase-shifting interferometry and neighboring pixel value subtraction algorithm in the discrete-cosine-transform domain,” Appl. Opt. 46(21), 4694–4701 (2007). [CrossRef] [PubMed]
- M. He, Q. Tan, L. Cao, Q. He, G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009). [CrossRef] [PubMed]
- G. Situ, J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005). [CrossRef] [PubMed]
- R. Tao, Y. Xin, Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007). [CrossRef] [PubMed]
- Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010). [CrossRef]
- Y. Zhang, D. Xiao, “Double optical image encryption using discrete Chirikov standard map and chaos-based fractional random transform,” Opt. Lasers Eng. 51(4), 472–480 (2013). [CrossRef]
- Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, M. A. Ahmad, J. Dai, S. Liu, “Double image encryption by using Arnold transform and discrete fractional angular transform,” Opt. Lasers Eng. 50(2), 248–255 (2012). [CrossRef]
- H. Li, Y. Wang, H. Yan, L. Li, Q. Li, X. Zhao, “Double-image encryption by using chaos-based local pixel scrambling technique and gyrator transform,” Opt. Lasers Eng. 51(12), 1327–1331 (2013). [CrossRef]
- Z. Liu, S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007). [CrossRef]
- J. A. Rodrigo, T. Alieva, M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24(10), 3135–3139 (2007). [CrossRef] [PubMed]
- S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014). [CrossRef]
- IEEE Computer Society, “IEEE standard for binary floating-point arithmetic,” ANSI/IEEE std. 754–1985 (1985).
- C. Fu, J. J. Chen, H. Zou, W. H. Meng, Y. F. Zhan, Y. W. Yu, “A chaos-based digital image encryption scheme with an improved diffusion strategy,” Opt. Express 20(3), 2363–2378 (2012). [CrossRef] [PubMed]
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