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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2007

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 275, 324-329 (2007).
[CrossRef]

2006

2005

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

X. F. Meng, L. Z. Cai, M. Z. He, and 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. 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]

2004

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]

2003

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]

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]

2002

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

2001

B. Zhu and S. Liu, "Optical Image encryption based on the generalized fractional convolution operation," Opt. Commun. 195, 371-381 (2001).
[CrossRef]

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (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]

2000

1999

1997

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

1995

Appl. Opt.

J. Opt. A: Pure Appl. Opt.

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, and 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

Opt. Commun.

Z. Liu and S. Liu, "Double image encryption based on iterative fractional Fourier transform," Opt. Commun. 275, 324-329 (2007).
[CrossRef]

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193, 51-67 (2001).
[CrossRef]

Y. Zhang, C. H. Zheng, 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]

Opt. Eng.

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]

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]

Opt. Express

Opt. Lasers Eng.

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.

Other

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).

A. W. Lohmann, "Image rotation, Wigner rotation, and the fractional Fourier transform," J. Opt. Soc. Am. A 10, 2181- (1993).
[CrossRef]

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Figures (15)

Fig. 1.
Fig. 1.

Schematic of encryption

Fig. 2.
Fig. 2.

Schematic of decryption

Fig. 3.
Fig. 3.

Optical implementation of the encryption

Fig. 4.
Fig. 4.

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

Fig. 5.
Fig. 5.

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

Fig. 6.
Fig. 6.

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

Fig. 7.
Fig. 7.

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

Fig. 8.
Fig. 8.

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

Fig. 9.
Fig. 9.

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

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

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

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.

Fig. 13.
Fig. 13.

Performance of noise perturbation

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

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

Equations (14)

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

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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