Abstract

A novel asymmetric color information cryptosystem based on an optical coherent superposition method and phase-truncated gyrator transform (GT) is proposed. In this proposal, an original color image is converted into three independent channels, i.e., red, green, and blue. Each channel is separated into a random phase masks (RPM) and a key phase mask (KPM) using a coherent superposition method. The KPM is a modulation of the RPM by the color channel and used as decryption key. The same RPM, which is independent of plaintext, can be chosen for different images of the same size; however, KPMs, which are related to the original color images, are different. The RPM and the KPM are independently gyrator transformed. Then two gyrator spectra are, respectively, phase truncated to obtain two encoded images and amplitude truncated to generate two asymmetric phase keys. The KPM and two phase keys provide asymmetric keys. The transformation angles of the GT give additional keys for each channel and thus offer a high-level robustness against existing attacks. The proposed optical design is free from axial movement. Numerical simulations are demonstrated to verify the flexibility and effectiveness of the proposed method.

© 2012 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2013

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

2012

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

M. R. Abuturab, “Securing color information using Arnold transform in gyrator transform domain,” Opt. Lasers Eng. 50, 772–779 (2012).
[CrossRef]

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).

M. R. Abuturab, “Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding,” Opt. Lasers Eng. 50, 1383–1390 (2012).
[CrossRef]

M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51, 3006–3016 (2012).
[CrossRef]

W. Chen and X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51, 6076–6083 (2012).
[CrossRef]

2011

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[CrossRef]

P. W. M. Tsang, T.-C. Poon, and K. W. K. Cheung, “Fast numerical generation and encryption of computer-generated Fresnel holograms,” Appl. Opt. 50, B46–B52 (2011).
[CrossRef]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

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

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

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

2010

2009

2007

2006

1999

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

1995

Abuturab, M. R.

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).

M. R. Abuturab, “Securing color information using Arnold transform in gyrator transform domain,” Opt. Lasers Eng. 50, 772–779 (2012).
[CrossRef]

M. R. Abuturab, “Color image security system using double random-structured phase encoding in gyrator transform domain,” Appl. Opt. 51, 3006–3016 (2012).
[CrossRef]

M. R. Abuturab, “Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding,” Opt. Lasers Eng. 50, 1383–1390 (2012).
[CrossRef]

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. (to be published).

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. (to be published).

Alfalou, A.

Alieva, T.

Brosseau, C.

Calvo, M. L.

Chang, H. T.

Chen, C.

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Chen, H.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Chen, L.

Chen, W.

W. Chen and X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51, 6076–6083 (2012).
[CrossRef]

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

Chen, X.

W. Chen and X. Chen, “Interference-based optical image encryption using three-dimensional phase retrieval,” Appl. Opt. 51, 6076–6083 (2012).
[CrossRef]

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

Cheung, K. W. K.

Chin, C.

Cho, M.

Clemente, P.

Dai, J.

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Deng, X.

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

Durán, V.

Gao, B.

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

Gong, M.

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Hwang, H.-E.

Javidi, B.

Joshi, M.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Karim, M. A.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Lancis, J.

Li, P.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Li, S.

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Lie, W.-N.

Lin, C.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Liu, S.

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Z. Liu, L. Xu, C. Chin, and S. Liu, “Image encryption by encoding with a non-uniform optical beam in gyrator transform domains,” Appl. Opt. 49, 5632–5637 (2010).
[CrossRef]

Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
[CrossRef]

Liu, T.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Liu, W.

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Liu, Z.

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Z. Liu, L. Xu, C. Chin, and S. Liu, “Image encryption by encoding with a non-uniform optical beam in gyrator transform domains,” Appl. Opt. 49, 5632–5637 (2010).
[CrossRef]

Z. Liu and S. Liu, “Random fractional Fourier transform,” Opt. Lett. 32, 2088–2090 (2007).
[CrossRef]

Meng, X. F.

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

Peng, X.

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

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

Pérez-Cabré, E.

Poon, T.-C.

Qin, W.

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

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

Refregier, P.

Rodrigo, J. A.

Shakher, C.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Singh, K.

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Stallings, W.

W. Stallings, Cryptography and Network Security: Principles and Practices, 5th ed. (Prentice Hall, 2011).

Tajahuerce, E.

Torres-Company, V.

Tsang, P. W. M.

Wang, X.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

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

Xu, L.

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

Z. Liu, L. Xu, C. Chin, and S. Liu, “Image encryption by encoding with a non-uniform optical beam in gyrator transform domains,” Appl. Opt. 49, 5632–5637 (2010).
[CrossRef]

Yang, M.

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

Zhang, S. Q.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Zhao, D.

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

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

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

Appl. Opt.

J. Opt. Soc. Am. A

Microwave Opt. Technol. Lett.

S. Q. Zhang and M. A. Karim, “Color image encryption using double random phase encoding,” Microwave Opt. Technol. Lett. 21, 318–323 (1999).
[CrossRef]

Opt. Commun.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

M. Joshi, C. Shakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279, 35–42 (2007).
[CrossRef]

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Opt. Commun. 284, 123–128 (2011).
[CrossRef]

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

Z. Liu, M. Yang, W. Liu, S. Li, M. Gong, W. Liu, and S. Liu, “Image encryption algorithm based on the random local phase encoding in gyrator transform domains,” Opt. Commun. 285, 3921–3925 (2012).
[CrossRef]

X. Wang and D. Zhao, “Image encoding based on coherent superposition and basic vector operations,” Opt. Commun. 284, 945–951 (2011).
[CrossRef]

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

Opt. Eng.

W. Qin, X. Peng, X. F. Meng, and B. Gao, “Universal and special keys based on phase-truncated Fourier transform,” Opt. Eng. 50, 080501 (2011).
[CrossRef]

Opt. Express

Opt. Laser Technol.

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

M. R. Abuturab, “Color information security system using Arnold transform and double structured phase encoding in gyrator transform domain,” Opt. Laser Technol. 45, 525–532 (2013).
[CrossRef]

Z. Liu, S. Li, W. Liu, W. Liu, and S. Liu, “Image hiding scheme by use of rotating squared sub-image in the gyrator transform domains,” Opt. Laser Technol. 45, 198–203 (2013).
[CrossRef]

Opt. Lasers Eng.

M. R. Abuturab, “Securing color information using Arnold transform in gyrator transform domain,” Opt. Lasers Eng. 50, 772–779 (2012).
[CrossRef]

M. R. Abuturab, “Color information security system using discrete cosine transform in gyrator transform domain radial-Hilbert phase encoding,” Opt. Lasers Eng. 50, 1209–1216 (2012).

M. R. Abuturab, “Securing color image using discrete cosine transform in gyrator transform domain structured-phase encoding,” Opt. Lasers Eng. 50, 1383–1390 (2012).
[CrossRef]

Z. Liu, L. Xu, C. Chen, J. Dai, and S. Liu, “Image encryption scheme by using iterative random phase encoding in gyrator transform domains,” Opt. Lasers Eng. 49, 542–546 (2011).
[CrossRef]

Opt. Lett.

Other

M. R. Abuturab, “Color image security system based on discrete Hartley transform in gyrator transform domain,” Opt. Lasers Eng. (to be published).

M. R. Abuturab, “Noise-free recovery of color information using a joint-extended gyrator transform correlator,” Opt. Lasers Eng. (to be published).

W. Stallings, Cryptography and Network Security: Principles and Practices, 5th ed. (Prentice Hall, 2011).

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Figures (8)

Fig. 1.
Fig. 1.

(a) Flow diagram for proposed color image encryption and (b) decryption algorithms.

Fig. 2.
Fig. 2.

Optical setup for coherent superposition-based color image decryption system.

Fig. 3.
Fig. 3.

Simulation results of the proposed method. (a) Original color image with 512×512 pixels and 24 bits to be encoded, (b) real part of the random phase mask, (c) real part of the key phase mask for decryption, (d) real part of the phase key of the random phase mask for decryption, (e) real part of the phase key of the key phase mask for decryption, (f) encrypted random phase mask, (g) encrypted key phase mask, (h) encrypted image obtained from the superposition of (f) and (g), (i) decrypted random phase mask with the transformation angle for each channel changed by 0.02° but all the other parameters are correct, (j) decrypted key phase mask with the transformation angle for each channel changed by 0.02° but all the other parameters are correct, (k) decrypted image without phase key of the random phase mask with other right decryption keys, (l) decrypted image without phase key of the key phase mask with other correct decryption keys, (m) decrypted image without key phase masks with other right decryption keys, and (n) decrypted image with all the correct decryption keys.

Fig. 4.
Fig. 4.

(a) Relation between the MSE for the transformation angle errors of random phase mask of original red, green, and blue channels and their corresponding decrypted images and (b) relationship between the MSE for the transformation angle errors of key phase mask of original red, green, and blue channels, and their corresponding decrypted images.

Fig. 5.
Fig. 5.

(a) Fake plaintext, (b) real part of the fake random phase mask of the fake plaintext, (c) real part of the fake key phase mask of the fake plaintext, (d) decrypted image using the fake phase key generated by the fake random phase mask of the fake plaintext, (e) decrypted image using the fake phase key generated by the fake key phase mask of fake plaintext, and (f) decrypted image using the fake key phase mask of the fake plaintext.

Fig. 6.
Fig. 6.

Robustness test of the proposed method against occlusion attack. (a) Encrypted random phase mask with 50% occlusion, (b) encrypted key phase mask with 50% occlusion, (c) corresponding recovered image with all the correct keys from the superposition of (a) and (b), (d) encrypted random phase mask with 70% occlusion, (e) encrypted key phase mask with 70% occlusion, and (f) corresponding reconstructed image with all the right keys from the superposition of (d) and (e).

Fig. 7.
Fig. 7.

Robustness test of the proposed method against Gaussian noise attack. (a) Gaussian-noised encrypted random phase mask with a variance 0.1, (b) Gaussian-noised encrypted key phase mask with a variance 0.1, (c) encrypted image obtained from the superposition of (a) and (b), and (d) retrieved image with all the correct keys from superposition of (a) and (b).

Fig. 8.
Fig. 8.

(a) Relationship between the MSE of the occlusion part on the encrypted red, green, and blue channels and their corresponding recovered images and (b) relationship between the MSE of the Gaussian noise attack on the encrypted red, green, and blue channels, and their corresponding decrypted images.

Equations (15)

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fo(xo,yo)=Gα[fi(xi,yi)](xo,yo)=1|sinα|+fi(xi,yi)exp(i2π(xoyo+xiyi)cosα(xiyo+xoyi)sinα)dxidyi,
fr(xi,yi)=|Rr(xi,yi)+Kr(xi,yi)|,
fr(xi,yi)=22cos(πθr(xi,yi)),
θr(xi,yi)=πarccos(1|fr(xi,yi)|22).
Kr(xi,yi)=Rr(xi,yi)+exp[iθr(xi,yi)],
Er1R(x,y)=Gαr1[Rr(xi,yi)],
Er1K(x,y)=Gαr2[Kr(xi,yi)].
Er2R(x,y)=PT[Er1R(x,y)],
Er2K(x,y)=PT[Er1K(x,y)].
PrR(x,y)=AT[Er1R(x,y)],
PrK(x,y)=AT[Er1K(x,y)].
DrR(xi,yi)=Gαr1[Er2R(x,y)PrR(x,y)],
DrK(xi,yi)=Gαr2[Er2K(x,y)PrK(x,y)],
fr(xi,yi)=|DrR(xi,yi)+DrK(xi,yi)|.
MSE=1M×Ni=1Mj=1N|Io(m,n)Id(m,n)|2,

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