Abstract

A single-channel color image encryption is proposed based on the modified Gerchberg–Saxton algorithm (MGSA) and mutual encoding in the Fresnel domain. Similar to the double random phase encoding (DRPE), this encryption scheme also employs a pair of phase-only functions (POFs) as encryption keys. But the two POFs are generated by the use of the MGSA rather than a random function generator. In the encryption process, only one color component is needed to be encrypted when these POFs are mutually served as the second encryption keys. As a result, a more compact and simple color encryption system based on one-time-pad, enabling only one gray cipheretext to be recorded and transmitted when holographic recording is used, is obtained. Moreover, the optical setup is lensless, thus easy to be implemented and the system parameters and wavelength can be served as additional keys to further enhance the security of the system. The feasibility and effectiveness of the proposed method are demonstrated by numerical results.

© 2011 Optical Society of America

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References

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2010 (2)

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

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, 5632–5637 (2010).
[CrossRef] [PubMed]

2009 (3)

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

M. Joshi, C. Shakher, and K. Singh, “Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach,” Opt. Lasers Eng. 47, 721–727(2009).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

2007 (1)

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

2006 (2)

2004 (2)

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef] [PubMed]

G. Situ and J. Zhang, “A lensless optical security system based on computer- generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

2002 (1)

2000 (5)

1999 (1)

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

1997 (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

1995 (1)

Cai, L.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

X. Peng, L. Yu, and L. Cai, “Double-lock for image encryption with virtual optical wavelength,” Opt. Express 10, 41–45 (2002).
[PubMed]

Chang, H. T.

Chen, L. F.

Chen, W.

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Cheng, X.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Dong, G.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Gluckstad, J.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Guo, Q.

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

Hwang, H. E.

Iemmi, C.

Javidi, B.

Joseph, J.

Joshi, M.

M. Joshi, C. Shakher, and K. Singh, “Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach,” Opt. Lasers Eng. 47, 721–727(2009).
[CrossRef]

Karim, M. A.

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

Konforti, N.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Kreske, K.

Li, Y.

Lie, W. N.

Lin, C.

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, 5632–5637 (2010).
[CrossRef] [PubMed]

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

Liu, S.

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

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, 5632–5637 (2010).
[CrossRef] [PubMed]

Liu, Z.

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, 5632–5637 (2010).
[CrossRef] [PubMed]

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

Matoba, O.

Mela, C. L.

Mendlovic, D.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Meng, X.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Mogensen, P. C.

Nomura, T.

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (2000).
[CrossRef]

Peng, X.

Quan, C.

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Refregier, P.

Rosen, J.

Shakher, C.

M. Joshi, C. Shakher, and K. Singh, “Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach,” Opt. Lasers Eng. 47, 721–727(2009).
[CrossRef]

Shen, X.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Singh, K.

M. Joshi, C. Shakher, and K. Singh, “Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach,” Opt. Lasers Eng. 47, 721–727(2009).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
[CrossRef]

Situ, G.

G. Situ and J. Zhang, “A lensless optical security system based on computer- generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef] [PubMed]

Tajahuerce, E.

Tay, C. J.

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Unnikrishnan, G.

Verrall, S. C.

Wang, Y.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Xu, L.

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

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, 5632–5637 (2010).
[CrossRef] [PubMed]

Xu, X.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Yang, X.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Yu, L.

Zalevsky, Z.

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

Zhang, H.

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Zhang, J.

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef] [PubMed]

G. Situ and J. Zhang, “A lensless optical security system based on computer- generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

Zhang, S. Q.

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

Zhao, D. M.

Appl. Opt. (3)

J. Mod. Opt. (1)

D. Mendlovic, Z. Zalevsky, and N. Konforti, “Computation considerations and fast algorithms for calculating the diffraction integral,” J. Mod. Opt. 44, 407–414 (1997).
[CrossRef]

J. Opt. A (1)

X. Meng, L. Cai, Y. Wang, X. Yang, X. Xu, G. Dong, X. Shen, H. Zhang, and X. Cheng, “Hierarchical image encryption based on cascaded iterative phase retrieval algorithm in the Fresnel domain,” J. Opt. A 9, 1070–1075(2007).
[CrossRef]

Microw. Opt. Technol. Lett. (1)

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

Opt. Commun. (3)

G. Situ and J. Zhang, “A lensless optical security system based on computer- generated phase only masks,” Opt. Commun. 232, 115–122 (2004).
[CrossRef]

Z. Liu, L. Xu, Q. Guo, C. Lin, and S. Liu, “Image watermarking by using phase retrieval algorithm in gyrator transform domain,” Opt. Commun. 283, 4923–4927(2010).
[CrossRef]

W. Chen, C. Quan, and C. J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Opt. Commun. 282, 3680–3685 (2009).
[CrossRef]

Opt. Eng. (1)

B. Javidi and T. Nomura, “Polarization encoding for optical security systems,” Opt. Eng. 39, 2439–2443 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lasers Eng. (1)

M. Joshi, C. Shakher, and K. Singh, “Logarithms-based RGB image encryption in the fractional Fourier domain: a non-linear approach,” Opt. Lasers Eng. 47, 721–727(2009).
[CrossRef]

Opt. Lett. (6)

Other (2)

H.P.Herzig, ed., Micro-Optics (Taylor & Francis, 1996).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

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Figures (6)

Fig. 1
Fig. 1

Flow chart of the MGSA based on the Fresnel domain.

Fig. 2
Fig. 2

Flow charts of the (a) encryption process and (b) decryption process.

Fig. 3
Fig. 3

Optical setup of this proposed method.

Fig. 4
Fig. 4

(a) Original color image. (b)–(d) The embedded red, green, and blue components. (e) The CC variation versus iteration number. (f) The encrypted image.

Fig. 5
Fig. 5

Real part of the main decryption keys respectively corresponding to the (a) red, (b) green, and (c) blue components. The decrypted image obtained when (d) all the correct decryption keys are correct, (e) the main decryption keys are not used and the other keys are correct, (f) random phase masks are used as the main decryption keys and the other keys are correct, (g) the main decryption keys of the red is incorrect and the other keys are correct, (h) the main decryption keys of the green is incorrect and the other keys are correct, (i) the main decryption keys of the blue is incorrect and the other keys are correct, (j)  λ = 610 nm and the other keys are correct, (k)  z = 21 mm and the other keys are correct, (l)  z 1 = 31 mm and the other keys are correct.

Fig. 6
Fig. 6

CC variation versus (a) wavelength difference Δ λ , and (b) displacement Δ z .

Equations (11)

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FrT λ ; z { h ^ ( x , y ) exp [ j θ ( x , y ) ] } = exp [ j φ ( u , v ) ] ,
FrT λ ; z { f ^ R ( x , y ) exp [ j θ R ( x , y ) ] } = exp [ j φ R ( u , v ) ] ,
FrT λ ; z { f ^ G ( x , y ) exp [ j θ G ( x , y ) ] } = exp [ j φ G ( u , v ) ] ,
FrT λ ; z { f ^ B ( x , y ) exp [ j θ B ( x , y ) ] } = exp [ j φ B ( u , v ) ] .
exp [ j φ R ( u , v ) ] = FrT λ ; z [ f ^ R exp ( j θ R ( x , y ) ] .
g ( x 0 , y 0 ) = FrT λ ; z 1 { exp [ j φ R ( u , v ) + j φ G ( u , v ) + j φ B ( u , v ) ] } ,
| f ^ R ( x , y ) | = | FrT λ ; z { FrT λ ; z 1 [ g * ( x , y ) ] DK R ( u , v ) } | ,
DK R ( u , v ) = exp { j [ φ G ( u , v ) + φ B ( u , v ) ] } .
DK G ( u , v ) = exp { j [ φ R ( u , v ) + φ B ( u , v ) ] } ,
DK B ( u , v ) = exp { j [ φ R ( u , v ) + φ G ( u , v ) ] } .
CC = E { [ h E [ h ] ] [ | h ^ | E [ | h ^ | ] ] } { E { [ h E [ h ] ] 2 } E { [ | h ^ | E [ | h ^ | ] ] 2 } } 1 2 ,

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