Abstract

We propose a method to simultaneously transmit double random-phase encryption key and an encrypted image by making use of the fact that an acceptable decryption result can be obtained when only partial data of the encrypted image have been taken in the decryption process. First, the original image data are encoded as an encrypted image by a double random-phase encryption technique. Second, a double random-phase encryption key is encoded as an encoded key by the Rivest–Shamir–Adelman (RSA) public-key encryption algorithm. Then the amplitude of the encrypted image is modulated by the encoded key to form what we call an encoded image. Finally, the encoded image that carries both the encrypted image and the encoded key is delivered to the receiver. Based on such a method, the receiver can have an acceptable result and secure transmission can be guaranteed by the RSA cipher system.

© 2007 Optical Society of America

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References

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  1. D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).
  2. Y. Zhou and D. Feng, Public Key Cryptographic Algorithms and Its Fast Implementation (National Defence Industry Press, 2002), in Chinese.
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    [CrossRef]
  4. C.-C. Chang and S.-J. Hwang, "A simple approach for generating RSA keys," Inf. Process. Lett. 63, 19-21 (1997).
    [CrossRef]
  5. 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]
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    [CrossRef]
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    [CrossRef]
  8. B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1506 (2001).
    [CrossRef]
  9. J. A. Davis, D. E. McNamara, D. M. Cottrell, and T. Sonehara, "Two-dimensional polarization encoding with a phase-only liquid-crystal spatial light modulator," Appl. Opt. 39, 1549-1554 (2000).
    [CrossRef]
  10. D. Armitage and J. I. Thackara, "Liquid crystal differentiating spatial light modulators," Proc. SPIE 613, 165-170 (1986).
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    [CrossRef]
  12. T. Nomura and B. Javidi, "Optical encryption system with a binary key code," Appl. Opt. 39, 4783-4787 (2000).
    [CrossRef]
  13. B. Wang, C.-C. Sun, W.-C. Su, and A. E. T. Chiou, "Shift-tolerance property of an optical double-random phase-encoding encryption system," Appl. Opt. 39, 4788-4793 (2000).
    [CrossRef]
  14. 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]
  15. P. C. Mogensen and J. Glückstad, "Phase-only optical decryption of a fixed mask," Appl. Opt. 40, 1226-1235 (2001).
    [CrossRef]

2001

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1506 (2001).
[CrossRef]

P. C. Mogensen and J. Glückstad, "Phase-only optical decryption of a fixed mask," Appl. Opt. 40, 1226-1235 (2001).
[CrossRef]

2000

1998

B. Javidi, A. Sergent, and E. Ahouzi, "Performance of double phase encoding encryption technique using binarized encrypted images," Opt. Eng. 37, 565-569 (1998).
[CrossRef]

1997

B. Javidi, "Securing information with optical technologies," Phys. Today 50, 27-32 (1997).
[CrossRef]

C.-C. Chang and S.-J. Hwang, "A simple approach for generating RSA keys," Inf. Process. Lett. 63, 19-21 (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]

1996

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

1995

1986

D. Armitage and J. I. Thackara, "Liquid crystal differentiating spatial light modulators," Proc. SPIE 613, 165-170 (1986).

Ahouzi, E.

B. Javidi, A. Sergent, and E. Ahouzi, "Performance of double phase encoding encryption technique using binarized encrypted images," Opt. Eng. 37, 565-569 (1998).
[CrossRef]

Armitage, D.

D. Armitage and J. I. Thackara, "Liquid crystal differentiating spatial light modulators," Proc. SPIE 613, 165-170 (1986).

Chang, C.-C.

C.-C. Chang and S.-J. Hwang, "A simple approach for generating RSA keys," Inf. Process. Lett. 63, 19-21 (1997).
[CrossRef]

Chatwin, C.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Chiou, A. E. T.

Cottrell, D. M.

Davis, J. A.

Feng, D.

Y. Zhou and D. Feng, Public Key Cryptographic Algorithms and Its Fast Implementation (National Defence Industry Press, 2002), in Chinese.

Glückstad, J.

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]

Herson, D.

D. Herson, Chase Infosec Services, Cheltenham, UK, "The changing face of international cryptography policy: part 14--RSA and digital signatures," Comput. Fraud Secur. V 2000, 7-8 (2000).
[CrossRef]

Hwang, S.-J.

C.-C. Chang and S.-J. Hwang, "A simple approach for generating RSA keys," Inf. Process. Lett. 63, 19-21 (1997).
[CrossRef]

Javidi, B.

T. Nomura and B. Javidi, "Optical encryption system with a binary key code," Appl. Opt. 39, 4783-4787 (2000).
[CrossRef]

B. Javidi, A. Sergent, and E. Ahouzi, "Performance of double phase encoding encryption technique using binarized encrypted images," Opt. Eng. 37, 565-569 (1998).
[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]

B. Javidi, "Securing information with optical technologies," Phys. Today 50, 27-32 (1997).
[CrossRef]

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]

McNamara, D. E.

Mogensen, P. C.

Nomura, T.

Refregier, P.

Sergent, A.

B. Javidi, A. Sergent, and E. Ahouzi, "Performance of double phase encoding encryption technique using binarized encrypted images," Opt. Eng. 37, 565-569 (1998).
[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]

Sonehara, T.

Stinson, D. R.

D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).

Su, W.-C.

Sun, C.-C.

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1506 (2001).
[CrossRef]

B. Wang, C.-C. Sun, W.-C. Su, and A. E. T. Chiou, "Shift-tolerance property of an optical double-random phase-encoding encryption system," Appl. Opt. 39, 4788-4793 (2000).
[CrossRef]

Thackara, J. I.

D. Armitage and J. I. Thackara, "Liquid crystal differentiating spatial light modulators," Proc. SPIE 613, 165-170 (1986).

Wang, B.

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1506 (2001).
[CrossRef]

B. Wang, C.-C. Sun, W.-C. Su, and A. E. T. Chiou, "Shift-tolerance property of an optical double-random phase-encoding encryption system," Appl. Opt. 39, 4788-4793 (2000).
[CrossRef]

Wang, R. K.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

Watson, I. A.

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[CrossRef]

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]

Zhou, Y.

Y. Zhou and D. Feng, Public Key Cryptographic Algorithms and Its Fast Implementation (National Defence Industry Press, 2002), in Chinese.

Appl. Opt.

Comput. Fraud Secur. V

D. Herson, Chase Infosec Services, Cheltenham, UK, "The changing face of international cryptography policy: part 14--RSA and digital signatures," Comput. Fraud Secur. V 2000, 7-8 (2000).
[CrossRef]

Inf. Process. Lett.

C.-C. Chang and S.-J. Hwang, "A simple approach for generating RSA keys," Inf. Process. Lett. 63, 19-21 (1997).
[CrossRef]

Opt. Eng.

B. Javidi, A. Sergent, and E. Ahouzi, "Performance of double phase encoding encryption technique using binarized encrypted images," Opt. Eng. 37, 565-569 (1998).
[CrossRef]

B. Wang and C.-C. Sun, "Enhancement of signal-to-noise ratio of a double random phase encoding encryption system," Opt. Eng. 40, 1502-1506 (2001).
[CrossRef]

R. K. Wang, I. A. Watson, and C. Chatwin, "Random phase encoding for optical security," Opt. Eng. 35, 2464-2469 (1996).
[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. Lett.

Phys. Today

B. Javidi, "Securing information with optical technologies," Phys. Today 50, 27-32 (1997).
[CrossRef]

Proc. SPIE

D. Armitage and J. I. Thackara, "Liquid crystal differentiating spatial light modulators," Proc. SPIE 613, 165-170 (1986).

Other

D. R. Stinson, Cryptography: Theory and Practice, 2nd ed. (CRC Press, 2002).

Y. Zhou and D. Feng, Public Key Cryptographic Algorithms and Its Fast Implementation (National Defence Industry Press, 2002), in Chinese.

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Figures (6)

Fig. 1
Fig. 1

Numerical simulation of (a) the original image, (b) an encrypted image (real part), (c) a decrypted image.

Fig. 2
Fig. 2

Decrypted image obtained with a gray-scale image by use of (a) only the real part and (b) only the imaginary part. Decrypted image obtained with a binary image by use of (c) only the real part and (d) only the imaginary part.

Fig. 3
Fig. 3

Decrypted image obtained by use of only (a), (c) the amplitude and (b), (d) the phase.

Fig. 4
Fig. 4

Illustration of the optical setup to achieve transmission of the key with the RSA public-key cryptosystem during double random-phase encryption.

Fig. 5
Fig. 5

Distribution images of the double random-phase encryption key (a) before encryption and (b) after encryption with the RSA public key.

Fig. 6
Fig. 6

Numerical simulation of (a) the original image, (b) the decrypted image when all the keys were extracted correctly, (c) the decrypted image when some of the keys were extracted incorrectly.

Equations (103)

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

θ 0 ( x )
φ 0 ( u )
θ ( x ) = exp [ i 2 π θ 0 ( x ) ] ,
φ ( u ) = exp [ i 2 π φ 0 ( u ) ]
f ( x )
g ( x )
g ( x ) = FT 1 { FT [ f ( x ) θ ( x ) ] φ ( u ) } ,
f ( x ) = FT 1 { FT [ g ( x ) ] φ * ( u ) } θ * ( x ) ,
FT 1
256 × 256
g ( x )
g ( x ) = g R ( x ) + j g I ( x ) ,
g R
g I
Δ g ( x )
g ( x )
Δ g ( x ) = j g I ( x )
Δ g ( x ) = g R ( x )
0.02
0.01   dB
g ( x )
g ( x ) = g A ( x ) g Ψ ( x ) ,
g A ( x )
g Ψ ( x )
g A ( x ) = | g ( x ) | , g Ψ ( x ) = g ( x ) g A ( x ) .
Δ g ( x ) = | g ( x ) | g ( x ) ;
Δ g ( x ) = [ 1 | g ( x ) | 1 ] g ( x ) .
46.41
42.94
5.65   dB
n = p q
Φ ( n ) = ( p 1 ) ( q 1 )
Φ ( n )
Φ ( n )
Φ ( n )
e d 1 [ mod Φ ( n ) ] ,
Φ ( n )
m i ( i = 1 , 2 , 3 )
m i
n 1
c i m i e ( mod n ) ,
c i
c i
m i
m i e
m i
m i c i d ( mod n ) ,
c i d
c i
f ( x )
g ( x )
g ( x )
g A ( x )
g ( x )
g ( x ) = A g Ψ ( x ) ,
g Ψ ( x )
θ 0 ( x )
φ 0 ( u )
θ 0 ( x )
φ 0 ( u )
m i
c i
g ( x )
c i
c i
m i
m i
g ( x )
N × N
f ( x )
g ( x )
g ( x )
8 × N × N
Φ ( n )
m i
c i
8 × N × N
c i
N × N
c i
c i
g ( x )
c i
m i
m i
1001101101001110001111111010001110010000010010100111110000010011.
p = 17
q = 31
n = 527
Φ ( n ) = 480
e = 7
d = 343
m i
m i < n
2 9 = 512
100110110,100111000,111111101,000111001,000001001,010011111,000001001,1.
011011001,011111100,001010100,010010110,110111100,110000100,110111100,1.
10 6
308 × 10 9
4.9 × 10 15
2 100
5.66 d B
5.66 d B

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