Abstract

A double-image encryption is proposed based on the discrete fractional random transform and logistic maps. First, an enlarged image is composited from two original images and scrambled in the confusion process which consists of a number of rounds. In each round, the pixel positions of the enlarged image are relocated by using cat maps which are generated based on two logistic maps. Then the scrambled enlarged image is decomposed into two components. Second, one of two components is directly separated into two phase masks and the other component is used to derive the ciphertext image with stationary white noise distribution by using the cascaded discrete fractional random transforms generated based on the logistic map. The cryptosystem is asymmetric and has high resistance against to the potential attacks such as chosen plaintext attack, in which the initial values of logistic maps and the fractional orders are considered as the encryption keys while two decryption keys are produced in the encryption process and directly related to the original images. Simulation results and security analysis verify the feasibility and effectiveness of the proposed encryption scheme.

© 2014 Optical Society of America

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L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[CrossRef]

2013 (7)

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. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51(3), 317–324 (2013).
[CrossRef]

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. 51(3), 230–239 (2013).
[CrossRef]

S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
[CrossRef] [PubMed]

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (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]

2012 (6)

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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]

X. Wang, D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012).
[CrossRef]

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

X. Deng, D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

W. Chen, X. Chen, C. J. R. Sheppard, “Optical color-image encryption and synthesis using coherent diffractive imaging in the Fresnel domain,” Opt. Express 20(4), 3853–3865 (2012).
[CrossRef] [PubMed]

X. Wang, D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012).
[CrossRef] [PubMed]

2011 (9)

X. Wang, D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011).
[CrossRef]

W. Chen, D. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284(16–17), 3913–3917 (2011).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011).
[CrossRef] [PubMed]

A. Alfalou, C. Brosseau, N. Abdallah, M. Jridi, “Simultaneous fusion, compression, and encryption of multiple images,” Opt. Express 19(24), 24023–24029 (2011).
[CrossRef] [PubMed]

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011).
[CrossRef]

X. Wang, D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011).
[CrossRef]

2010 (5)

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]

W. Chen, X. Chen, “Space-based optical image encryption,” Opt. Express 18(26), 27095–27104 (2010).
[CrossRef] [PubMed]

J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

N. Singh, A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010).
[CrossRef]

W. Qin, X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
[CrossRef] [PubMed]

2009 (6)

2008 (3)

H. Li, Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008).
[CrossRef]

N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008).
[CrossRef]

2006 (3)

2005 (3)

2004 (1)

2003 (1)

2000 (2)

1997 (1)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

1995 (1)

Abdallah, N.

Abuturab, M. R.

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51(3), 317–324 (2013).
[CrossRef]

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. 51(3), 230–239 (2013).
[CrossRef]

Ahmad, M. A.

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (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]

Ailing, T.

Alam, M. S.

Alfalou, A.

Arcos, S.

Brosseau, C.

Cao, L.

Carnicer, A.

Chang, H. T.

Chen, C. M.

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

Chen, C. Y.

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

Chen, D.

W. Chen, D. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284(16–17), 3913–3917 (2011).
[CrossRef]

Chen, L.

Chen, W.

Chen, X.

Dai, J.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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]

Deng, X.

X. Deng, D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

Dou, Y.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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]

Gong, M.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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, Q.

He, M.

He, Q.

Hennelly, B.

Huang, J. J.

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

Hwang, H. E.

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011).
[CrossRef] [PubMed]

H. T. Chang, H. E. Hwang, C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

H. E. Hwang, H. T. Chang, W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009).
[CrossRef] [PubMed]

Javidi, B.

Jin, G.

Joseph, J.

Jridi, M.

Juvells, I.

Lang, J.

J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

Lee, C. L.

H. T. Chang, H. E. Hwang, C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, M. T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50(5), 710–716 (2011).
[CrossRef] [PubMed]

Lee, M. T.

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]

H. Li, Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[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, 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, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Liansheng, S.

Lie, W. N.

Lin, S.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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, A. A. Muhammad, 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, S.

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (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, H. Zhao, S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005).
[CrossRef]

B. Zhu, S. Liu, Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
[CrossRef] [PubMed]

Liu, W.

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Liu, Z.

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (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, H. Zhao, S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005).
[CrossRef]

Lu, H.

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[CrossRef]

L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

Luo, X.

J. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

Mansour, A.

Meiting, X.

Montes-Usategui, M.

Muhammad, A. A.

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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]

Ning, X.

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[CrossRef]

Peng, X.

Qin, W.

Ran, Q.

Refregier, P.

Sheng, Y.

Sheppard, C. J. R.

Sheridan, J. T.

Singh, K.

Singh, N.

N. Singh, A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010).
[CrossRef]

N. Singh, A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008).
[CrossRef]

Sinha, A.

N. Singh, A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010).
[CrossRef]

N. Singh, A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008).
[CrossRef]

Situ, G.

Sui, L.

L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[CrossRef]

Sun, Q.

L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

Tan, Q.

Tao, R.

J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

Unnikrishnan, G.

Wang, X.

X. Wang, D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012).
[CrossRef]

X. Wang, D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012).
[CrossRef] [PubMed]

X. Wang, D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

X. Wang, D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011).
[CrossRef]

Wang, Y.

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[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, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011).
[CrossRef]

N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011).
[CrossRef]

J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008).
[CrossRef]

Wang, Z.

L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

Wei, H.

Wu, J.

J. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011).
[CrossRef]

Xi, L.

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]

Xiao-Feng, L.

Xin, Z.

Xu, L.

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]

Zhang, J.

Zhang, P.

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, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011).
[CrossRef]

Zhao, D.

X. Wang, D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012).
[CrossRef]

X. Deng, D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

X. Wang, D. Zhao, “Double images encryption method with resistance against the specific attack based on an asymmetric algorithm,” Opt. Express 20(11), 11994–12003 (2012).
[CrossRef] [PubMed]

X. Wang, D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

X. Wang, D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011).
[CrossRef]

L. Chen, D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14(19), 8552–8560 (2006).
[CrossRef] [PubMed]

Zhao, H.

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011).
[CrossRef]

Z. Liu, H. Zhao, S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005).
[CrossRef]

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]

Zhou, N.

J. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011).
[CrossRef]

Zhu, B.

Adv. Opt. Photonics (1)

A. Alfalou, C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[CrossRef]

Appl. Opt. (2)

J. Mod. Opt. (1)

H. Li, Y. Wang, “Double-image encryption by iterative phase retrieval algorithm in fractional Fourier domain,” J. Mod. Opt. 55(21), 3601–3609 (2008).
[CrossRef]

J. Opt. A, Pure Appl. Opt. (1)

G. Situ, J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A, Pure Appl. Opt. 8(5), 391–397 (2006).
[CrossRef]

Opt. Commun. (9)

N. Zhou, Y. Wang, J. Wu, “Image encryption algorithm based on the multi-order discrete fractional Mellin transform,” Opt. Commun. 284(24), 5588–5597 (2011).
[CrossRef]

W. Chen, D. Chen, “Optical color image encryption based on an asymmetric cryptosystem in the Fresnel domain,” Opt. Commun. 284(16–17), 3913–3917 (2011).
[CrossRef]

H. T. Chang, H. E. Hwang, C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284(18), 4146–4151 (2011).
[CrossRef]

H. Li, Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

X. Wang, D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

X. Wang, D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284(1), 148–152 (2011).
[CrossRef]

X. Wang, D. Zhao, “Fully phase multiple-image encryption based on superposition principle and the digital holographic technique,” Opt. Commun. 285(21–22), 4280–4284 (2012).
[CrossRef]

J. Lang, R. Tao, Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

Z. Liu, H. Zhao, S. Liu, “A discrete fractional random transform,” Opt. Commun. 255(4-6), 357–365 (2005).
[CrossRef]

Opt. Eng. (1)

L. Sui, H. Lu, X. Ning, Y. Wang, “Asymmetric double-image encryption method by using iterative phase retrieve algorithm in fractional Fourier transform domain,” Opt. Eng. 53(2), 026108 (2014).
[CrossRef]

Opt. Express (8)

Opt. Laser Technol. (5)

Z. Liu, Y. Zhang, S. Li, W. Liu, W. Liu, Y. Wang, S. Liu, “Double image encryption scheme by using random phase encoding and pixel exchanging in the gyrator transform domains,” Opt. Laser Technol. 47, 152–158 (2013).
[CrossRef]

J. J. Huang, H. E. Hwang, C. Y. Chen, C. M. Chen, “Optical multiple-image encryption based on phase encoding algorithm in the Fresnel transform domain,” Opt. Laser Technol. 44(7), 2238–2244 (2012).
[CrossRef]

X. Deng, D. Zhao, “Multiple-image encryption using phase retrieve algorithm and intermodulation in Fourier domain,” Opt. Laser Technol. 44(2), 374–377 (2012).
[CrossRef]

N. Singh, A. Sinha, “Chaos based multiple image encryption using multiple canonical transforms,” Opt. Laser Technol. 42(5), 724–731 (2010).
[CrossRef]

J. Wu, X. Luo, N. Zhou, “Four-image encryption method based on spectrum truncation, chaos and the MODFrFT,” Opt. Laser Technol. 45, 571–577 (2013).
[CrossRef]

Opt. Lasers Eng. (9)

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]

N. Singh, A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46(2), 117–123 (2008).
[CrossRef]

N. Singh, A. Sinha, “Gyrator transform-based optical image encryption, using chaos,” Opt. Lasers Eng. 47(5), 539–546 (2009).
[CrossRef]

Z. Liu, M. Gong, Y. Dou, F. Liu, S. Lin, A. A. Muhammad, 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]

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]

H. Li, Y. Wang, “Double-image encryption based on discrete fractional random transform and chaotic maps,” Opt. Lasers Eng. 49(7), 753–757 (2011).
[CrossRef]

L. Sui, H. Lu, Z. Wang, Q. Sun, “Double-image encryption using discrete fractional random transform and logistic maps,” Opt. Lasers Eng. 56, 1–12 (2014).
[CrossRef]

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. 51(3), 317–324 (2013).
[CrossRef]

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. 51(3), 230–239 (2013).
[CrossRef]

Opt. Lett. (11)

G. Situ, J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30(11), 1306–1308 (2005).
[CrossRef] [PubMed]

G. Situ, J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
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P. Refregier, B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
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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, Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
[CrossRef] [PubMed]

B. Hennelly, J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
[CrossRef] [PubMed]

S. Liansheng, X. Meiting, T. Ailing, “Multiple-image encryption based on phase mask multiplexing in fractional Fourier transform domain,” Opt. Lett. 38(11), 1996–1998 (2013).
[CrossRef] [PubMed]

H. E. Hwang, H. T. Chang, W. N. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34(24), 3917–3919 (2009).
[CrossRef] [PubMed]

A. Carnicer, M. Montes-Usategui, S. Arcos, I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005).
[CrossRef] [PubMed]

X. Peng, H. Wei, P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
[CrossRef] [PubMed]

W. Qin, X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
[CrossRef] [PubMed]

Optik (Stuttg.) (1)

Z. Liu, Y. Zhang, H. Zhao, M. A. Ahmad, S. Liu, “Optical multi-image encryption based on frequency shift,” Optik (Stuttg.) 122(11), 1010–1013 (2011).
[CrossRef]

Phys. Today (1)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50(3), 27–32 (1997).
[CrossRef]

Other (1)

Original images: http://sipi.use.edu/database/database.php .

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Figures (16)

Fig. 1
Fig. 1

Flowchart of encryption process.

Fig. 2
Fig. 2

(a) Plaintext images “Lena” and “Baboon” and (b) two new components.

Fig. 3
Fig. 3

Flowchart of decryption process.

Fig. 4
Fig. 4

Optical implementation of DFrRT.

Fig. 5
Fig. 5

(a) Ciphertext image, (b) decrypted “Lena” and (c) decrypted “Baboon”.

Fig. 6
Fig. 6

Decrypted “Lena” with (a) incorrect x 01 , (b) incorrect x 02 and (c) incorrect x 03 .

Fig. 7
Fig. 7

(a) Decrypted “Lena” with incorrect α , (b) decrypted “Baboon” with incorrect α , (c) decrypted “Lena” with incorrect β , (d) decrypted “Baboon” with incorrect β .

Fig. 8
Fig. 8

(a) Decrypted “Lena” with incorrect ξ 1 , (b) decrypted “Baboon” with incorrect ξ 1 , (c) decrypted “Lena” with incorrect ξ 2 , (d) decrypted “Baboon” with incorrect ξ 2 .

Fig. 9
Fig. 9

Average MSE versus (a) the deviation of fractional order α and (b) the deviation of fractional order β .

Fig. 10
Fig. 10

(a) Top one-third of ξ 1 is unknown, (b) decrypted “Lena” with ξ 1 , (c) decrypted “Baboon” with ξ 1 , (d) top half of ξ 2 is unknown, (e) decrypted “Lena” with ξ 2 , (f) decrypted “Baboon” with ξ 2 .

Fig. 11
Fig. 11

(a) Histograms of the ciphertext of “Lena” and “Baboon”, (b) histograms of the decryption key ξ 1 and (c) histogram of the decryption key ξ 2 .

Fig. 12
Fig. 12

(a) Plaintext image “Aerial” and (b) plaintext image “Peppers”.

Fig. 13
Fig. 13

(a) Histograms of the ciphertext of “Aerial” and “Peppers”, (b) histograms of the decryption key ξ 1 and (c) histogram of the decryption key ξ 2 .

Fig. 14
Fig. 14

Decrypted images with coefficient k : (a) k = 0.4, (b) k = 0.6, (c) k = 0.8 and (d) k = 1.0.

Fig. 15
Fig. 15

(a) Ciphertext with 50% occlusion from the left side, (b) decrypted “Lena” and (c) decrypted “Baboon”.

Fig. 16
Fig. 16

(a) Decrypted “Lena” and (b) decrypted “Baboon”.

Tables (1)

Tables Icon

Table 1 Correlation Results in the Plaintext Images, Ciphertext and Phase Distribution

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

f( x )= 22cos( πα( x ) ) .
α( x )=πarccos 2 f 2 ( x ) 2 .
exp( i ϕ 2 ( x ) )=exp( i( ϕ 1 ( x )+α( x ) ) ).
f( x )=| exp( i ϕ 1 ( x ) )+exp( i ϕ 2 ( x ) ) |.
f(x)=px(1x).
x n+1 =p x n (1 x n ).
F α ( x )= R α x,
R α =V D α V t .
D α =diag{ 1,exp(i 2πα T ),,exp(i 2(N1)πα T ) },
S= Q+ Q t 2 .
F (α,α) (f)= R 1 α f R 2 α ,
[ x y ]=[ 1 p i q i p i q i +1 ][ x y ],
x = x mod2N,
y = y modN.
p i =( s 1 (i)× 10 9 )mod2N,
q i =( s 2 (i)× 10 9 )mod2N,
hexp( i ξ 1 )= F (α,α) ( f 2 exp( i ϕ 1 ) )= R 1 α ( f 2 exp( i ϕ 1 ) ) R 2 α ,
gexp( i ξ 2 )= F (β,β) ( hexp( i ϕ 2 ) )= R 1 β ( hexp( i ϕ 2 ) ) R 2 β ,
h 2 exp(i ϕ 2 )= F (β,β) (gexp(i ξ 2 ))= R 1 β (gexp(i ξ 2 )) R 2 β .
h 1 exp(i ϕ 1 )= F (α,α) ( h 2 exp(i ξ 1 ))= R 1 α ( h 2 exp(i ξ 1 )) R 2 α .
f 1 =| exp(i ϕ 1 )+exp(i ϕ 2 ) |.
MSE= 1 N 2 i=1 N j=1 N | g(i,j) g (i,j) | 2 ,
MS E ave =(MS E first +MS E second )/2,
Cor= i=1 N ( x i x ¯ )( y i y ¯ ) ( i=1 N ( x i x ¯ ) 2 )( i=1 N ( y i y ¯ ) 2 ) ,
C =C(1+kG),

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