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

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2014 (1)

S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[CrossRef]

2013 (5)

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]

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]

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]

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]

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]

2012 (6)

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, 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]

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]

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]

2011 (1)

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]

2010 (6)

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]

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]

W. Chen, X. Chen, C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
[CrossRef] [PubMed]

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]

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]

2009 (1)

2008 (2)

Z. Liu, M. A. Ahmad, S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Y. Zhang, B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[CrossRef] [PubMed]

2007 (6)

2005 (1)

2004 (1)

2002 (1)

Y. Zhang, C. H. Zheng, N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002).
[CrossRef]

2001 (1)

B. Zhu, S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001).
[CrossRef]

2000 (2)

1995 (1)

Ahmad, M. A.

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]

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, M. A. Ahmad, S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Alieva, T.

Cai, L. Z.

Calvo, M. L.

Cao, L.

Chen, D.

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]

Chen, H.

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]

Chen, J. J.

Chen, L.

Chen, W.

Chen, X.

Dai, J.

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]

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]

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]

Dong, G. Y.

Dong, T.

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]

Dou, Y.

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]

Fu, C.

Gong, M.

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]

Guo, C.

S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[CrossRef]

Guo, Q.

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, 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]

He, M.

He, M. Z.

He, Q.

Huang, S. M.

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]

Javidi, B.

Jin, G.

Joseph, J.

Lang, J.

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]

Lei, L.

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]

Li, H.

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]

Li, L.

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]

Li, P.

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]

Li, Q.

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]

Li, S.

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]

Lin, C.

Lin, S.

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]

Liu, F.

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]

Liu, Q.

Liu, S.

S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[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]

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]

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, 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]

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]

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, 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, 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]

Z. Liu, M. A. Ahmad, S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Z. Liu, S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

B. Zhu, S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001).
[CrossRef]

Liu, T.

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]

Liu, W.

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]

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]

Liu, Z.

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]

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, 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]

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]

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, 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, 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]

Z. Liu, M. A. Ahmad, S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Z. Liu, S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

Ma, J.

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]

Matoba, O.

Meng, W. H.

Meng, X. F.

Refregier, P.

Rodrigo, J. A.

Shen, X. X.

Sheppard, C. J. R.

Sheridan, J. T.

S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[CrossRef]

Singh, K.

Situ, G.

Sun, X.

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]

Tajahuerce, E.

Tan, Q.

Tanno, N.

Y. Zhang, C. H. Zheng, N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002).
[CrossRef]

Tao, R.

Unnikrishnan, G.

Verrall, S. C.

Wang, B.

Wang, Q.

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]

Wang, X.

Wang, Y.

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]

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]

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]

Wang, Y. R.

Wei, S.

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]

Wu, J.

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]

Xiao, D.

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]

Xin, Y.

Xu, L.

Xu, X. F.

Yan, H.

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]

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[CrossRef]

Yang, M.

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]

Yang, X. L.

Yu, Y. W.

Zhan, Y. F.

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Zhang, J.

Zhang, Y.

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, 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]

Y. Zhang, B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[CrossRef] [PubMed]

Y. Zhang, C. H. Zheng, N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002).
[CrossRef]

Zhao, D.

Zhao, X.

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]

Zheng, C. H.

Y. Zhang, C. H. Zheng, N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4–6), 277–285 (2002).
[CrossRef]

Zhou, J.

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]

Zhou, N.

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).
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B. Zhu, S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195(5–6), 371–381 (2001).
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Appl. Opt. (7)

J. Opt. (1)

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]

J. Opt. Soc. Am. A (1)

Opt. Commun. (6)

Z. Liu, S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

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]

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]

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]

Opt. Express (6)

Opt. Laser Technol. (1)

S. Liu, C. Guo, J. T. Sheridan, “A review of optical image encryption techniques,” Opt. Laser Technol. 57, 327–342 (2014).
[CrossRef]

Opt. Lasers Eng. (6)

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]

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]

Opt. Lett. (5)

Optik (Stuttg.) (2)

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, 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]

Other (1)

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Figures (17)

Fig. 1
Fig. 1

The block diagram of a traditional double-image cryptosystem.

Fig. 2
Fig. 2

The block diagram of double-image cryptosystem using cross-image pixel scrambling approach.

Fig. 3
Fig. 3

The testing images used for cross-image pixel scrambling operation. (a) Barb image; (b) Boats image and (c) the combined image.

Fig. 4
Fig. 4

The resultant images using cross-image pixel scrambling operation. (a) The scrambled version of the combined image; (b) Scrambled image of Barb and (c) Scrambled image of Boats.

Fig. 5
Fig. 5

The histograms of the original and scrambled images. (a) The histogram of Barb; (b) the histogram of Boats; (c) the histogram of the scrambled Barb; and (d) the histogram of the scrambled Boats.

Fig. 6
Fig. 6

The flowchart of the proposed double-image encryption algorithm.

Fig. 7
Fig. 7

Optoelectronic implementation of the encryption.

Fig. 8
Fig. 8

The simulation results of the proposed scheme. (a) Barb image regarded as img1; (b) Boats image served as img2; (c) amplitude of the encrypted image; (d) the decrypted image of Barb; and (e) the decrypted image of Boats.

Fig. 9
Fig. 9

Robustness to noise attack. (a) Barb image regarded as img1; (b) Boats image as img2; (c) decrypted Barb image with Gaussian noise of standard deviation 0.01; (d) decrypted Boats image with Gaussian noise of standard deviation 0.01; (e) decrypted Barb image with Gaussian noise of standard deviation 0.05; and (f) decrypted Boats image with Gaussian noise of standard deviation 0.05.

Fig. 10
Fig. 10

The MSE curves with various values of standard deviations of the Gaussian noise.

Fig. 11
Fig. 11

Tolerance to occlusion attack I. (a) Encrypted image with 25% occlusion; (b) decrypted Barb image; and (c) decrypted Boats image.

Fig. 12
Fig. 12

Tolerance to occlusion attack II. (a) Encrypted image with 50% occlusion; (b) decrypted Barb image; and (c) decrypted Boats image.

Fig. 13
Fig. 13

The MSE curves with various values of the rotation angle.

Fig. 14
Fig. 14

The decryption images with α = 0.52. (a) Decrypted Barb image; and (b) decrypted Boats image.

Fig. 15
Fig. 15

The MSE curves with various values of the rotation angle when using Boats as img1 and Barb as img2.

Fig. 16
Fig. 16

The decrypted images using incorrect initial value of Chebyshev map. (a) The decrypted Barb image; and (b) the decrypted Boats image.

Fig. 17
Fig. 17

The decrypted images using incorrect control parameter of Chebyshev map. (a) The decrypted Barb image; and (b) the decrypted Boats image.

Equations (6)

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r m =m+mod(floor(state_variable× 10 1 5 ), Mm+1),
x n+1 =cos(k cos 1 x n ), x n [1, 1],
F(u,v)= G α [f(x,y)](u,v)= 1 |sinα| f(x,y)exp[ i2π (xy+uv)cosα(xv+yu) sinα ]dxdy .
C= I 1 +i* I 2 .
C'= A 1 +i* A 2 .
MSE= 1 M×N m=1 M n=1 N | I o (m,n) I d (m,n) | 2 .

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