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Double image encryption based on random phase encoding in the fractional Fourier domain

Ran Tao, Yi Xin, and Yue Wang

Open Access Open Access

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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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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