Abstract

A novel image encryption method is proposed by utilizing random phase encoding in the fractional Fourier domain to encrypt two images into one encrypted image with stationary white distribution. By applying the correct keys which consist of the fractional orders, the random phase masks and the pixel scrambling operator, the two primary images can be recovered without cross-talk. The decryption process is robust against the loss of data. The phase-based image with a larger key space is more sensitive to keys and disturbances than the amplitude-based image. The pixel scrambling operation improves the quality of the decrypted image when noise perturbation occurs. The novel approach is verified by simulations.

© 2007 Optical Society of America

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References

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  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. B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
    [Crossref]
  3. N. Towghi, B. Javidi, and Z. Luo, “Fully phase encrypted image processor,” J. Opt. Soc. Am. A 16, 1915 (1999).
    [Crossref]
  4. 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]
  5. G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
    [Crossref]
  6. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
    [Crossref]
  7. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
    [Crossref]
  8. G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67, (2001).
    [Crossref]
  9. Y. Zhang, C.H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202, 277–285, (2002).
    [Crossref]
  10. B. Zhu and S. Liu, “Optical Image encryption based on the generalized fractional convolution operation,” Opt. Commun. 195, 371–381, (2001).
    [Crossref]
  11. B. Zhu and S. Liu, “Optical Image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244, (2001).
    [Crossref]
  12. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28, 269–271 (2003).
    [Crossref] [PubMed]
  13. N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
    [Crossref]
  14. J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
    [Crossref]
  15. A. Sinha and K. Singh, “Image encryption by using fractional Fourier transform and jigsaw transform in image bit planes,” Opt. Eng. 44, 057001 (2005).
    [Crossref]
  16. L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Exp.14, 8552–8560 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-19-8552.
    [Crossref]
  17. X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
    [Crossref]
  18. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24, 762–764 (1999).
    [Crossref]
  19. M. S. Millán, E. Pérez-Cabré, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
    [Crossref]
  20. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1308 (2005).
    [Crossref] [PubMed]
  21. G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A: Pure Appl. Opt. 8, 391 (2006).
    [Crossref]
  22. X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
    [Crossref]
  23. Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
    [Crossref]
  24. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing. (John Wiley & Sons, Chichester, 2001).
  25. A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” J. Opt. Soc. Am. A 10, 2181- (1993)
    [Crossref]
  26. B. Javidi, N. Towghi, N. Maghzi, and S. C. Verrall, “Error-reduction techniques and error analysis for fully phase- and amplitude-based encryption,” Appl. Opt. 39, 4117–4130 (2000).
    [Crossref]
  27. J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
    [Crossref]
  28. N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
    [Crossref]

2007 (1)

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

2006 (3)

M. S. Millán, E. Pérez-Cabré, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
[Crossref]

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A: Pure Appl. Opt. 8, 391 (2006).
[Crossref]

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

2005 (4)

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

A. Sinha and K. Singh, “Image encryption by using fractional Fourier transform and jigsaw transform in image bit planes,” Opt. Eng. 44, 057001 (2005).
[Crossref]

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

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

2004 (2)

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
[Crossref]

2003 (3)

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
[Crossref]

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

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[Crossref]

2002 (1)

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

2001 (3)

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

B. Zhu and S. Liu, “Optical Image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244, (2001).
[Crossref]

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67, (2001).
[Crossref]

2000 (3)

1999 (2)

1997 (2)

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

J. Hua, L. Liu, and G. Li, “Extended fractional Fourier transforms,” J. Opt. Soc. Am. A 14, 3316–3322 (1997).
[Crossref]

1995 (1)

1993 (1)

Cai, L. Z.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

Chen, L. F.

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Exp.14, 8552–8560 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-19-8552.
[Crossref]

Dong, G. Y.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

Guibert, L.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

He, M. Z.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

Hennelly, B.

Hua, J.

Javidi, B.

Jing, F.

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Joseph, J.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
[Crossref]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[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]

Kutay, M. A.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing. (John Wiley & Sons, Chichester, 2001).

Li, G.

Li, J.F.

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

Liu, L.

Liu, S.

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

B. Zhu and S. Liu, “Optical Image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244, (2001).
[Crossref]

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

Liu, Z.

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

Lohmann, A. W.

Lu, H.

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

Luo, Z.

Ma, Y.H.

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

Maghzi, N.

Matoba, O.

Meng, X. F.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

Millán, M. S.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
[Crossref]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[Crossref]

Ozaktas, H. M.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing. (John Wiley & Sons, Chichester, 2001).

Pérez-Cabré, E.

Refregier, P.

Sergent, A.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

Shen, X. X.

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

Sheridan, J. T.

Singh, K.

A. Sinha and K. Singh, “Image encryption by using fractional Fourier transform and jigsaw transform in image bit planes,” Opt. Eng. 44, 057001 (2005).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
[Crossref]

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[Crossref]

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67, (2001).
[Crossref]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[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]

Sinha, A.

A. Sinha and K. Singh, “Image encryption by using fractional Fourier transform and jigsaw transform in image bit planes,” Opt. Eng. 44, 057001 (2005).
[Crossref]

Situ, G.

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A: Pure Appl. Opt. 8, 391 (2006).
[Crossref]

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

Song, X.S.

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[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]

Towghi, N.

Unnikrishnan, G.

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[Crossref]

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67, (2001).
[Crossref]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[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]

Verrall, S. C.

Wang, X.

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Wei, X.

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Zalevsky, Z.

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing. (John Wiley & Sons, Chichester, 2001).

Zhang, G.

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

Zhang, J.

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A: Pure Appl. Opt. 8, 391 (2006).
[Crossref]

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

Zhang, Y.

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.

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Zhao, D. M.

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Exp.14, 8552–8560 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-19-8552.
[Crossref]

Zhao, J.

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

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]

Zhu, B.

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

B. Zhu and S. Liu, “Optical Image encryption with multistage and multichannel fractional Fourier-domain filtering,” Opt. Lett. 26, 1242–1244, (2001).
[Crossref]

Appl. Opt. (1)

J. Opt. A: Pure Appl. Opt. (2)

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” J. Opt. A: Pure Appl. Opt. 8, 391 (2006).
[Crossref]

X. F. Meng, L. Z. Cai, M. Z. He, G. Y. Dong, and X. X. Shen, “Cross-talk-free double-image encryption and watermarking with amplitude-phase separate modulations,” J. Opt. A: Pure Appl. Opt. 7, 624 (2005).
[Crossref]

J. Opt. Soc. Am. A (3)

Opt. Commun. (5)

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67, (2001).
[Crossref]

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

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

J. Zhao, H. Lu, X.S. Song, J.F. Li, and Y.H. Ma, “Optical image encryption based on multistage fractional Fourier transforms and pixel scrambling technique,” Opt. Commun. 249, 493–499, (2005).
[Crossref]

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

Opt. Eng. (6)

N. K. Nishchal, G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption using a localized fractional Fourier transform,” Opt. Eng. 42, 3566–3571, (2003).
[Crossref]

A. Sinha and K. Singh, “Image encryption by using fractional Fourier transform and jigsaw transform in image bit planes,” Opt. Eng. 44, 057001 (2005).
[Crossref]

G. Unnikrishnan and K. Singh, “Double random fractional Fourier-domain encoding for optical security,” Opt. Eng. 39, 2853–2859 (2000).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase encryption using fractional Fourier transform,” Opt. Eng. 42, 1583–1588 (2003).
[Crossref]

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-based encryption using fractional order Fourier domain random phase encoding: Error analysis,” Opt. Eng. 43, 2266–2273 (2004).
[Crossref]

B. Javidi, A. Sergent, G. Zhang, and L. Guibert, “Fault tolerance properties of a double phase encoding encryption technique,” Opt. Eng. 36, 992–998 (1997).
[Crossref]

Opt. Exp. (1)

X. Wang, D. Zhao, F. Jing, and X. Wei, “Information synthesis (complex amplitude addition and subtraction) and encryption with digital holography and virtual optics” Opt. Exp. 14, 1476–1486 (2006).
[Crossref]

Opt. Lasers Eng. (1)

N. K. Nishchal, J. Joseph, and K. Singh, “Fully phase-encrypted memory using cascaded extended fractional Fourier transform,” Opt. Lasers Eng. 42, 141–151 (2004).
[Crossref]

Opt. Lett. (7)

Other (2)

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Exp.14, 8552–8560 (2006), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-14-19-8552.
[Crossref]

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The fractional Fourier transform with Applications in Optics and Signal Processing. (John Wiley & Sons, Chichester, 2001).

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Figures (15)

Fig. 1.
Fig. 1.

Schematic of encryption

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Fig. 2.
Fig. 2.

Schematic of decryption

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Fig. 3.
Fig. 3.

Optical implementation of the encryption

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Fig. 4.
Fig. 4.

(a). Primary grayscale image of Lena; (b) Primary binary text; (c) The encrypted result.

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Fig. 5.
Fig. 5.

When decrypting with wrong mask M 1: (a) The recovered amplitude-based image; (b) The recovered phase-based image.

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Fig. 6.
Fig. 6.

When decrypting with wrong mask M 2: (a) The recovered amplitude-based image; (b) The recovered phase-based image.

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Fig. 7.
Fig. 7.

When decrypting with correct keys: (a) The recovered amplitude-based image; (b) The recovered phase-based image.

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Fig. 8.
Fig. 8.

The recovered phase-based image with wrong inverse pixel scrambling operator

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Fig. 9.
Fig. 9.

MSE between the original image and the decrypted image when errors are introduced in fractional order parameters

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Fig. 10.
Fig. 10.

(a) When 25% pixels of Fig. 4c are occluded; (b) The recovered amplitude-based image; (c) The recovered phase-based image

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Fig. 11.
Fig. 11.

(a) When 50% pixels of Fig. 4c are occluded; (b) The recovered amplitude-based image; (c) The recovered phase-based image

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Fig. 12.
Fig. 12.

Decrypting from the phase information of the encrypted image: (a) The recovered amplitude-based image; (b) The recovered phase-based image.

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Fig. 13.
Fig. 13.

Performance of noise perturbation

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Fig. 14.
Fig. 14.

The recovered two images obtained from the noisy encrypted image when the forward and inverse pixel scrambling is not applied. (a) The recovered amplitude-based image; (b) The recovered phase-based image

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Fig. 15.
Fig. 15.

The recovered two images obtained from the noisy encrypted image when the forward and inverse pixel scrambling is applied. (a) The recovered amplitude-based image; (b) The recovered phase-based image

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Equations (14)

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F a [ f ( x 0 ) ] = ∫ − ∞ + ∞ f ( x 0 ) K a ( x 0 , x a ) d x 0 ,
K a ( x 0 , x a ) = { A ϕ exp [ i π ( x 0 2 cot ⁡ ϕ − 2 x 0 x a csc ϕ + x a 2 cot ⁡ ϕ ) ] if a ≠ 2 n δ ( x 0 − x a ) if a = 4 n δ ( x 0 + x a ) if a = 4 n ± 2 ,
A ϕ = ∣ sin ⁡ ϕ ∣ − 1 ⁄ 2 exp [ − i π sgn ( ϕ ) 4 + i ϕ 2 ] , ϕ = a π 2 .
C ( x 0 ) = f ( x 0 ) exp [ i π J [ g ( x 0 ) ] ] .
ψ ( x b ) = F b − a { F a [ C ( x 0 ) • M 1 ( x 0 ) ] • M 2 ( x a ) }
= F b − a { F a { f ( x 0 ) exp [ i π J [ g ( x 0 ) ] ] • M 1 ( x 0 ) } • M 2 ( x a ) } .
C ( x 0 ) = F − a { F a − b [ ψ ( x b ) ] • M 2 * ( x a ) } • M 1 * ( x 0 ) = f ( x 0 ) exp { i π J [ g ( x 0 ) ] } .
ψ ′ ( x b ) = ψ ( x b ) + n ( x b ) ,
C ′ ( x 0 ) = f ′ ( x 0 ) exp [ i π g ˜ ( x 0 ) ] = f ( x 0 ) exp [ i π J [ g ( x 0 ) ] ] + n ′ ( x 0 ) ,
n ′ ( x 0 ) = F − a { F − ( b − a ) [ n ( x b ) ] • M 2 * ( x a ) } M 1 * ( x 0 ) ,
f ′ ( x 0 ) = ∣ C ′ ( x 0 ) ∣ , g ′ ( x 0 ) = J − 1 [ g ˜ ( x 0 ) ] = J − 1 [ ∣ arg { C ′ ( x 0 ) } ⁄ π ∣ ] ,
∣ f ′ ( x 0 ) ∣ 2 = ∣ f ( x 0 ) exp { i π J [ g ( x 0 ) ] } + n ′ ( x 0 ) ∣ 2
= ∣ f ( x 0 ) ∣ 2 + ∣ n ′ ( x 0 ) ∣ 2 + f ( x 0 ) exp { − i π J [ g ( x 0 ) ] } n ′ ( x 0 ) + f ( x 0 ) exp { i π J [ g ( x 0 ) ] } n ′ * ( x 0 ) .
MSE = 1 MN ∑ i = 1 M ∑ j = 1 N ( I ( i , j ) − H ( i , j ) ) 2 ,

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