Abstract

Based on an optical gyrator transform system, an image encryption algorithm is designed and studied. An original secret image is regarded as the output intensity of the second gyrator transform. A coherent nonuniform optical beam is converted into the input of the first gyrator transform. A Gerchberg–Saxton phase retrieval algorithm is employed for obtaining the compensation phases in the first gyrator transform pair. The compensation phases are regarded as the encrypted image and key in this algorithm. The parameters of the laser beam and gyrator transform can serve as the additional key of encryption method. The decryption process of this encryption algorithm can be achieved with an optical system. Numerical simulations are performed to test the validity and capability of the encryption algorithm.

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
    [CrossRef] [PubMed]
  2. G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
    [CrossRef]
  3. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
    [CrossRef] [PubMed]
  4. X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31, 1044–1046 (2006).
    [CrossRef] [PubMed]
  5. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
    [CrossRef] [PubMed]
  6. W. Liu, G. Yang, and H. Xie, “A hybrid heuristic algorithm to improve known-plaintext attack on Fourier plane encryption,” Opt. Express 17, 13928–13938 (2009).
    [CrossRef] [PubMed]
  7. Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double-random phase encryption against various attacks,” Opt. Express 15, 10253–10265 (2007).
    [CrossRef] [PubMed]
  8. D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641–6647 (2007).
    [CrossRef] [PubMed]
  9. G. Situ, G. Pedrini, and W. Osten, “Strategy for cryptanalysis of optical encryption in the Fresnel domain,” Appl. Opt. 49, 457–462 (2010).
    [CrossRef] [PubMed]
  10. A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48, 5933–5947 (2009).
    [CrossRef] [PubMed]
  11. X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
    [CrossRef] [PubMed]
  12. S. Yuan, X. Zhou Xin, S. A. Mohammed, X. Lu, and X.-F. Li, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17, 3270–3284(2009).
    [CrossRef]
  13. R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007).
    [CrossRef] [PubMed]
  14. L. Chen and D. Zhao, “Optical image encryption with Hartley transforms,” Opt. Lett. 31, 3438–3440 (2006).
    [CrossRef] [PubMed]
  15. Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279, 285–290 (2007).
    [CrossRef]
  16. Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285 (2002).
    [CrossRef]
  17. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33, 2443–2445 (2008).
    [CrossRef] [PubMed]
  18. J. Rodrigo, T. Alieva, and M. L. Calvo, “Experimental implementation of the gyrator transform,” J. Opt. Soc. Am. A 24, 3135–3139 (2007).
    [CrossRef]
  19. J. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: Properties and applications,” Opt. Express 15, 2190–2203(2007).
    [CrossRef] [PubMed]
  20. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).
  21. H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749(2008).
    [CrossRef]
  22. Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
    [CrossRef]
  23. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
    [CrossRef] [PubMed]
  24. J. A. Rodrigo, T. Alieva, M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
    [CrossRef]
  25. H. Li and Y. Wang, “Information security system based on iterative multiple-phase retrieval in gyrator domain,” Opt. Laser Technol. 40, 962–966 (2008).
    [CrossRef]
  26. R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

2010

2009

2008

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

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

H. Li and Y. Wang, “Information security system based on iterative multiple-phase retrieval in gyrator domain,” Opt. Laser Technol. 40, 962–966 (2008).
[CrossRef]

2007

2006

2004

2002

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

2000

1995

1972

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Ahmad, M. A.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef] [PubMed]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279, 285–290 (2007).
[CrossRef]

Alfalou, A.

Alieva, T.

Cai, L. Z.

Calvo, M. L.

Castro, A.

Chen, H.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Chen, L.

Dai, J.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Dong, G. Y.

Frauel, Y.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Gopinathan, U.

Guo, Q.

Javidi, B.

Joseph, J.

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Li, H.

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

H. Li and Y. Wang, “Information security system based on iterative multiple-phase retrieval in gyrator domain,” Opt. Laser Technol. 40, 962–966 (2008).
[CrossRef]

Li, P.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Li, X.-F.

Liu, S.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef] [PubMed]

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279, 285–290 (2007).
[CrossRef]

Liu, T.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Liu, W.

Liu, Z.

Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18, 12033–12043 (2010).
[CrossRef] [PubMed]

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279, 285–290 (2007).
[CrossRef]

Lu, X.

Mansour, A.

Meng, X. F.

Mohammed, S. A.

Monaghan, D. S.

Naughton, T. J.

Osten, W.

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Pedrini, G.

Peng, X.

Refregier, P.

Rodrigo, J.

Rodrigo, J. A.

J. A. Rodrigo, T. Alieva, M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Shen, X. X.

Sheridan, J. T.

Singh, K.

Situ, G.

Sun, X.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Tanno, N.

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

Tao, R.

Unnikrishnan, G.

Wang, B.

Wang, Y.

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

H. Li and Y. Wang, “Information security system based on iterative multiple-phase retrieval in gyrator domain,” Opt. Laser Technol. 40, 962–966 (2008).
[CrossRef]

R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007).
[CrossRef] [PubMed]

Wang, Y. R.

Wei, H.

Xie, H.

Xin, X. Zhou

Xin, Y.

Xu, L.

Xu, X. F.

Yang, G.

Yang, X. L.

Yu, B.

Yuan, S.

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Zhang, J.

Zhang, P.

Zhang, Y.

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

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

Zhao, D.

Zheng, C.-H.

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

Appl. Opt.

J. Opt.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279, 285–290 (2007).
[CrossRef]

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

J. A. Rodrigo, T. Alieva, M. L. Calvo, “Applications of gyrator transform for image processing,” Opt. Commun. 278, 279–284 (2007).
[CrossRef]

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

Opt. Express

Opt. Laser Technol.

H. Li and Y. Wang, “Information security system based on iterative multiple-phase retrieval in gyrator domain,” Opt. Laser Technol. 40, 962–966 (2008).
[CrossRef]

Opt. Lett.

Optik (Jena)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik (Jena) 35, 237–246 (1972).

Other

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2000).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Algorithm illustration of the proposed image decryption.

Fig. 2
Fig. 2

Gerchberg–Saxton phase retrieval algorithm between the optical beam and amplitude function A ( u , v ) .

Fig. 3
Fig. 3

Result of the phase retrieval algorithm: (a) the original secret image, (b) the pattern of laser beam HG 1 , 3 , and (c) the NMSE curve. Here, the original image has 256 × 256 pixels.

Fig. 4
Fig. 4

Distribution of compensation phases: (a) Δ φ B ( x , y ) and (b) Δ φ A ( u , v ) .

Fig. 5
Fig. 5

Decrypted image obtained with all the correct keys. Here, NMSE = 0.012 .

Fig. 6
Fig. 6

Result decrypted by wrong beam: (a) plane wave, (b) Gaussian beam, (c) Laguerre–Gaussian beam LG 3 , 0 , (d) HG 0 , 2 , (e) HG 0 , 3 , and (f) HG 1 , 2 .

Fig. 7
Fig. 7

NMSE curve calculated with various values of angle α.

Fig. 8
Fig. 8

Decrypted result calculated with half of the known data of the phases Δ φ A and Δ φ B : (a) the phase Δ φ A , (b) the phase Δ φ B , (c) the recovered image with Δ φ A , and (d) the recovered image with Δ φ B .

Equations (11)

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

Q ( u , v ) = G α [ q ( x , y ) ] ( u , v ) = q ( x , y ) | sin α | exp [ i 2 π ( x y + u v ) cos α ( x v + y u ) sin α ] d x d y ,
E ( x , y ) = B ( x , y ) exp [ i ϕ B ( x , y ) ] ,
A ( u , v ) exp [ i ϕ A ( u , v ) ] = G α [ C 0 h ( x , y ) ] , C 0 2 = A 2 ( u , v ) d u d v h ( x , y ) d x d y ,
A ( u , v ) exp [ i ϕ A ( u , v ) ] = G α { B ( x , y ) exp [ i ϕ B ( x , y ) ] } ,
A ( u , v ) exp [ i φ A ( u , v ) ] = G α { B ( x , y ) exp [ i φ B ( x , y ) ] } .
Δ φ B ( x , y ) = φ B ( x , y ) ϕ B ( x , y ) Δ φ A ( u , v ) = ϕ A ( u , v ) φ A ( u , v ) .
h ( x , y ) = 1 C 0 2 | G α { G α [ E exp ( i Δ φ B ) ] exp ( i Δ φ A ) } | 2 .
E ( x , y ) = HG m , n ( x , y ) = H m ( 2 x ω 0 ) H n ( 2 y ω 0 ) exp ( x 2 + y 2 ω 0 2 ) ,
NMSE = nmse ( I s , I t ) = m , n [ I s ( m , n ) I t ( m , n ) ] 2 m , n I s 2 ( m , n ) ,
G ( x , y ) = c g exp ( x 2 + y 2 ω 0 2 ) ,
LG p , s ( x , y ) = c l [ 2 ( x 2 + y 2 ) ω 0 ] p exp ( x 2 + y 2 ω 0 2 ) × L s p ( 2 x 2 + y 2 ω 0 2 ) cos [ p · arg ( x + i y ) ] ,

Metrics