Abstract

A novel information hiding method based on double random-phase encoding (DRPE) and Rivest-Shamir-Adleman (RSA) public-key cryptosystem is proposed. In the proposed technique, the inherent diffusion property of DRPE is cleverly utilized to make up the diffusion insufficiency of RSA public-key cryptography, while the RSA cryptosystem is utilized for simultaneous transmission of the cipher text and the two phase-masks, which is not possible under the DRPE technique. This technique combines the complementary advantages of the DPRE and RSA encryption techniques and brings security and convenience for efficient information transmission. Extensive numerical simulation results are presented to verify the performance of the proposed technique.

© 2009 Optical Society of America

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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. 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. O. Matoba and B. Javidi, "Encrypted optical memory system using three-dimensional keys in the Fresnel domain," Opt. Lett. 24, 762-764 (1999).
    [CrossRef]
  4. 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]
  5. B. Javidi and T. Nomura, "Securing information by use of digital holography," Opt. Lett. 25, 28-30 (2000).
    [CrossRef]
  6. X. Zhou, S. Yuan, S. W. Wang, and J. Xie, "Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform," Appl. Opt. 45, 8434-8439 (2006).
    [CrossRef]
  7. S. Kishk and B. Javidi, "Information hiding technique with double phase encoding," Appl. Opt. 41, 5462-5470 (2002).
    [CrossRef] [PubMed]
  8. J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).
    [CrossRef]
  9. Y. S. Shi, G. H. Situ, and J. J. Zhang, "Multiple-image hiding in the Fresnel domain," Opt. Lett. 32, 1914-1916 (2007).
    [CrossRef] [PubMed]
  10. K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
    [CrossRef]
  11. J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
    [CrossRef]
  12. X. Zhou and J. G. Chen, "Information hiding based on double random phase encoding technology," J. Mod. Optics. 53, 1777-1783 (2006).
    [CrossRef]
  13. H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
    [CrossRef]
  14. D. R. Stinson, Cryptography: Theory and Practice (CRC Press 2002).
  15. B. Yang, Modern Cryptography (Tsinghua University Press 2007) (in Chinese).
  16. S. Yuan, X. Zhou, D. H. Li, and D. F. Zhou, "Simultaneous transmission for an encrypted image and a double random-phase encryption key," Appl. Opt. 46, 3747-3753 (2007).
    [CrossRef] [PubMed]
  17. Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).
  18. J. R. Fienup, "Phase retrieval algorithms: a comparison," Appl. Opt. 21, 2758-2769 (1982).
    [CrossRef] [PubMed]
  19. Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003).
    [CrossRef]
  20. Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

2007 (3)

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

S. Yuan, X. Zhou, D. H. Li, and D. F. Zhou, "Simultaneous transmission for an encrypted image and a double random-phase encryption key," Appl. Opt. 46, 3747-3753 (2007).
[CrossRef] [PubMed]

Y. S. Shi, G. H. Situ, and J. J. Zhang, "Multiple-image hiding in the Fresnel domain," Opt. Lett. 32, 1914-1916 (2007).
[CrossRef] [PubMed]

2006 (2)

X. Zhou, S. Yuan, S. W. Wang, and J. Xie, "Affine cryptosystem of double-random-phase encryption based on the fractional Fourier transform," Appl. Opt. 45, 8434-8439 (2006).
[CrossRef]

X. Zhou and J. G. Chen, "Information hiding based on double random phase encoding technology," J. Mod. Optics. 53, 1777-1783 (2006).
[CrossRef]

2004 (2)

J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

2003 (1)

Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003).
[CrossRef]

2002 (1)

2001 (2)

J. Rosen and B. Javidi, "Hidden images in halftone pictures," Appl. Opt. 40, 3346-3353 (2001).
[CrossRef]

K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
[CrossRef]

2000 (2)

1999 (1)

1997 (1)

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 (1)

1982 (1)

1972 (1)

Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).

Cai, L. Z.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Chen, J. G.

X. Zhou and J. G. Chen, "Information hiding based on double random phase encoding technology," J. Mod. Optics. 53, 1777-1783 (2006).
[CrossRef]

Choi, J. H.

J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Dong, G. Y.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Fienup, J. R.

Gerchberg, R. W.

Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).

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]

Javidi, B.

Joseph, J.

Kim, E. S.

J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
[CrossRef]

Kim, J. H.

K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
[CrossRef]

Kim, J. J.

J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Kim, K. T.

K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
[CrossRef]

Kishk, S.

Li, D. H.

Matoba, O.

Meng, X. F.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Nomura, T.

Refregier, P.

Rosen, J.

Saxton, W. O.

Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).

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.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Shi, Y. S.

Singh, K.

Situ, G. H.

Y. S. Shi, G. H. Situ, and J. J. Zhang, "Multiple-image hiding in the Fresnel domain," Opt. Lett. 32, 1914-1916 (2007).
[CrossRef] [PubMed]

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003).
[CrossRef]

Unnikrishnan, G.

Wang, S. W.

Xie, J.

Xu, X. F.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Yang, X. L.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Yuan, S.

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, H.

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Zhang, J. J.

Y. S. Shi, G. H. Situ, and J. J. Zhang, "Multiple-image hiding in the Fresnel domain," Opt. Lett. 32, 1914-1916 (2007).
[CrossRef] [PubMed]

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003).
[CrossRef]

Zhang, Y.

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

Zhao, Z. S.

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

Zhou, D. F.

Zhou, X.

Appl. Opt. (5)

J. Mod. Optics. (1)

X. Zhou and J. G. Chen, "Information hiding based on double random phase encoding technology," J. Mod. Optics. 53, 1777-1783 (2006).
[CrossRef]

J. Optoelectron. Laser. (1)

Q3. G. H. Situ, J. J. Zhang, Y. Zhang, and Z. S. Zhao, "A cascaded-phase retrieval algorithm for optical image encryption," J. Optoelectron. Laser. 15, 341-343 (2004) (in Chinese).

Opt. Commun. (1)

H. Zhang, L. Z. Cai, X. F. Meng, X. F. Xu, X. L. Yang, X. X. Shen, and G. Y. Dong, "Image watermarking based on an iterative phase retrieval algorithm and sine-cosine modulation in the discrete-cosine-transform domain," Opt. Commun. 278, 257-263 (2007).
[CrossRef]

Opt. Eng. (3)

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]

K. T. Kim and J. H. Kim, and E. S. Kim, "Multiple information hiding technique using random sequence and Hadamard matrix," Opt. Eng. 40, 2489-2494, (2001).
[CrossRef]

J. J. Kim, J. H. Choi, and E. S. Kim, "Optodigital implementation of multiple information hiding and extraction system," Opt. Eng. 43, 113-125 (2004).
[CrossRef]

Opt. Lett. (5)

Optik. (2)

Q1. R. W. Gerchberg and W. O. Saxton, "A practical algorithm for the determination of phase from image and diffraction plane pictures," Optik. 35, 237-246 (1972).

Q2. G. H. Situ and J. J. Zhang, "A cascaded iterative Fourier transform algorithm for optical security applications," Optik. 114, 473-477 (2003).
[CrossRef]

Other (2)

D. R. Stinson, Cryptography: Theory and Practice (CRC Press 2002).

B. Yang, Modern Cryptography (Tsinghua University Press 2007) (in Chinese).

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Figures (12)

Fig. 1.
Fig. 1.

The double random-phase encoding system

Fig. 2.
Fig. 2.

Illustration of the information hiding and extraction processes

Fig. 3.
Fig. 3.

The computer simulation result for the method in this paper: (a) is the image to be hidden, (b) the enlarged host image, (c) the combined image, and (d) the decrypted image.

Fig. 4.
Fig. 4.

PSNR for the combined image (circles) and for the decrypted image (points) for different α

Fig. 5.
Fig. 5.

(a) The original binary image, and (b) the encrypted image by RSA

Fig. 6.
Fig. 6.

Effect of occluding 25% image pixels on the encrypted image by RSA and the combined image: (a) 25% of the image pixels are occluded after it is directly encrypted by the RSA, (b) the decrypted image from (a), (c) 25% of the combined image pixels are occluded, (d) the recovered image from (c), (e) the recovered image when only a single phase-mask φ is used and 25% of the combined image pixels are occluded.

Fig. 7.
Fig. 7.

Effect of adding Gaussian white noise to the combined image: (a) combined image with added Gaussian white noise, and (b) the recovered image.

Fig. 8.
Fig. 8.

Combined and recovered images when JPEG compression is applied to the combined image: (a) 90% quality JPEG compressed combined image, (b) recovered image corresponding to image (a), (c) 85% quality JPEG compressed transmitted image, (d) recovered image corresponding to image (c).

Fig. 9.
Fig. 9.

The effect of filtering the combined image in the frequency domain: (a) filtered combined image via Gaussian low-pass filter, (b) the recovered image from (a) with PSNR of 12.9 dB.

Fig. A1.
Fig. A1.

The decrypted secret images recovered from part of the input image pixels: (a), (b), and (c) 1/2, 3/4, and 7/8 of the input image pixels are occluded; (d), (e), and (f) are the decrypted images corresponding to (a), (b), and (c), respectively.

Fig. A2.
Fig. A2.

(a) The random binary image; (b) The image with 1 gray value for all pixels; (c) Decrypted image.

Fig. A3.
Fig. A3.

(a) Input image whose gray values are uniformly distributed from 0 to 7; (b) Decrypted image obtained by a random 8 gray scales image which is selected arbitrarily; (c)–(j) Decrypted images respectively obtained by the images with 0, 1, 2, 3, 4, 5, 6, or 7 gray value for all pixels.

Equations (30)

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c m e mod n .
m c d mod n .
θ ( x , y ) = exp [ i 2 π θ 0 ( x , y ) ] ,
φ ( u , v ) = exp [ i 2 π φ 0 ( u , v ) ] ,
g ( x , y ) = F T 1 { FT { f ( x , y ) · θ ( x , y ) } · φ ( u , v ) } ,
MSE ( k ) = 1 M × N m = 1 M n = 1 N [ g ( m , n ) g k ( m , n ) ] 2
R ( k ) = m = 1 M n = 1 N { g ( m , n ) E [ g ] } { g k ( m , n ) E [ g k ] } { { m = 1 M n = 1 N [ g ( m , n ) E [ g ] ] 2 } { m = 1 M n = 1 N [ g k ( m , n ) E [ g k ] ] 2 } } ½ ,
H ( 2 m 1 , 2 n 1 ) = h ' ( m , n ) , H ( 2 m 1,2 n ) = h ' ( m , n ) ,
H ( 2 m , 2 n 1 ) = h ' ( m , n ) , H ( 2 m , 2 n ) = h ' ( m , n ) .
m = 1,2,3 , , M n = 1,2,3 , , N
H ' ( 2 m 1,2 n 1 ) = H ( 2 m 1,2 n 1 ) + α cos ( 2 π θ 0 ( m , n ) ) ,
H ' ( 2 m 1,2 n ) = H ( 2 m 1,2 n ) + α sin ( 2 π θ 0 ( m , n ) ) ,
H ' ( 2 m , 2 n 1 ) = H ( 2 m , 2 n 1 ) + α φ 0 ( m , n ) ,
H ' ( 2 m , 2 n ) = H ( 2 m , 2 n ) ,
α cos ( 2 π θ 0 ( m , n ) ) = H ' ( 2 m 1,2 n 1 ) H ' ( 2 m , 2 n ) ,
α sin ( 2 π θ 0 ( m , n ) ) = H ' ( 2 m , 2 n 1 ) H ' ( 2 m , 2 n ) ,
α φ 0 ( m , n ) ) = H ' ( 2 m , 2 n 1 ) H ' ( 2 m , 2 n ) .
A ( m , n ) = [ H ' ( 2 m 1,2 n 1 ) H ' ( 2 m , 2 n ) ] + i [ H ' ( 2 m , 2 n 1 ) H ' ( 2 m , 2 n ) ]
= α cos ( 2 π θ 0 ( m , n ) ) + sin ( 2 π θ 0 ( m , n ) )
= α exp ( i 2 π θ 0 ( m , n ) ) .
θ 0 ( m , n ) = angle ( A ( m , n ) ) 2 π ,
α = abs ( A ) ,
φ 0 ( m , n ) = H ' ( 2 m , 2 n 1 ) H ' ( 2 m , 2 n ) abs ( A ) .
h ' ( m , n ) = H ' ( 2 m , 2 n ) .
PSNR = 10 * lg { 1 ( 2 k 1 ) 2 M × N m = 1 M n = 1 N [ g ' ( m , n ) g ( m , n ) ] 2 }
P = n = 0 3 4 M × M 7 n C M × M n 8 M × M .
P = n = 0 49152 7 n C 65536 n 8 65536 .
a ( n ) a ( n 1 ) = 7 n C 65536 n 7 n 1 C 65536 n 1 = 7 ( 65536 n + 1 ) n > 1 ,
a ( n ) > a ( n 1 ) .
P = n = 0 49152 7 n C 65536 n 8 65536 < 49153 * a ( 49152 ) 8 65536 = 49153 * ( 7 ¾ 8 ) 65536 C 65536 49152 = 5.5 × 10 1640

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