Abstract

We propose a double image encryption by using random binary encoding and gyrator transform. Two secret images are first regarded as the real part and imaginary part of complex function. Chaotic map is used for obtaining random binary matrix. The real part and imaginary part of complex function are exchanged under the control of random binary data. An iterative structure composed of the random binary encoding method is designed and employed for enhancing the security of encryption algorithm. The parameters in chaotic map and gyrator transform serve as the keys of this encryption scheme. Some numerical simulations have been made, to demonstrate the performance this algorithm.

© 2010 OSA

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
  20. Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
    [CrossRef]
  21. Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
    [CrossRef]
  22. L. Chen and D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007).
    [CrossRef] [PubMed]
  23. M. Joshi, K. Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
    [CrossRef]
  24. Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
    [CrossRef]
  25. C. Jeffries and J. Perez, “Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26(4), 2117–2122 (1982).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]

2010 (3)

J. Wu, L. Zhang, and N. Zhou, “Image encryption based on the multiple-order discrete fractional cosine transform,” Opt. Commun. 283(9), 1720–1725 (2010).
[CrossRef]

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

2009 (6)

2008 (1)

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

2007 (7)

2006 (3)

2005 (1)

2004 (1)

2003 (1)

2002 (1)

Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

2000 (1)

1995 (1)

1982 (1)

C. Jeffries and J. Perez, “Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26(4), 2117–2122 (1982).
[CrossRef]

Ahmad, M. A.

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Alam, M. S.

Alieva, T.

Cai, L. Z.

Calvo, M. L.

Cao, L.

Castro, A.

Chandrashakher, K.

M. Joshi, K. Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Chen, H.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Chen, L.

Dai, J.

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Dong, G. Y.

Frauel, Y.

He, M.

He, Q.

Hennelly, B.

Javidi, B.

Jeffries, C.

C. Jeffries and J. Perez, “Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26(4), 2117–2122 (1982).
[CrossRef]

Jin, G.

Joseph, J.

Joshi, M.

M. Joshi, K. Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

Li, H.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

Li, P.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Li, Q.

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Liu, J.

Liu, S.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

Liu, T.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Liu, W.

Liu, Z.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

Meng, X. F.

Naughton, T. J.

Peng, X.

Perez, J.

C. Jeffries and J. Perez, “Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26(4), 2117–2122 (1982).
[CrossRef]

Refregier, P.

Rodrigo, J.

Rodrigo, J. A.

Shen, X. X.

Sheng, Y.

Sheridan, J. T.

Singh, K.

M. Joshi, K. Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25(12), 887–889 (2000).
[CrossRef]

Situ, G.

Sun, X.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Tan, Q.

Tanno, N.

Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

Tao, R.

Unnikrishnan, G.

Wang, Y.

Wang, Y. R.

Wei, H.

Wu, J.

J. Wu, L. Zhang, and N. Zhou, “Image encryption based on the multiple-order discrete fractional cosine transform,” Opt. Commun. 283(9), 1720–1725 (2010).
[CrossRef]

Xi, L.

Xiao-Feng, L.

Xie, H.

Xie, J.

Xin, Y.

Xin, Z.

Xu, X. F.

Yang, G.

Yang, X. L.

Yu, B.

Zhang, H.

Zhang, J.

Zhang, L.

J. Wu, L. Zhang, and N. Zhou, “Image encryption based on the multiple-order discrete fractional cosine transform,” Opt. Commun. 283(9), 1720–1725 (2010).
[CrossRef]

Zhang, P.

Zhang, Y.

Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

Zhao, D.

Zheng, C.-H.

Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

Zhou, N.

J. Wu, L. Zhang, and N. Zhou, “Image encryption based on the multiple-order discrete fractional cosine transform,” Opt. Commun. 283(9), 1720–1725 (2010).
[CrossRef]

Zhu, N.

J. Opt. (1)

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12(3), 035407 (2010).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Commun. (7)

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282(4), 518–522 (2009).
[CrossRef]

M. Joshi, K. Chandrashakher, and K. Singh, “Color image encryption and decryption using fractional Fourier transform,” Opt. Commun. 279(1), 35–42 (2007).
[CrossRef]

J. Wu, L. Zhang, and N. Zhou, “Image encryption based on the multiple-order discrete fractional cosine transform,” Opt. Commun. 283(9), 1720–1725 (2010).
[CrossRef]

Y. Zhang, C.-H. Zheng, and N. Tanno, “Optical encryption based on iterative fractional Fourier transform,” Opt. Commun. 202(4-6), 277–285 (2002).
[CrossRef]

Z. Liu, Q. Li, J. Dai, X. Sun, S. Liu, and M. A. Ahmad, “A new kind of double image encryption by using a cutting spectrum in the 1D fractional Fourier transform domains,” Opt. Commun. 282(8), 1536–1540 (2009).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275(2), 324–329 (2007).
[CrossRef]

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281(23), 5745–5749 (2008).
[CrossRef]

Opt. Express (8)

R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15(24), 16067–16079 (2007).
[CrossRef] [PubMed]

W. Liu, G. Yang, and H. Xie, “A hybrid heuristic algorithm to improve known-plaintext attack on Fourier plane encryption,” Opt. Express 17(16), 13928–13938 (2009).
[CrossRef] [PubMed]

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
[CrossRef] [PubMed]

N. Zhu, Y. Wang, J. Liu, J. Xie, and H. Zhang, “Optical image encryption based on interference of polarized light,” Opt. Express 17(16), 13418–13424 (2009).
[CrossRef] [PubMed]

Y. Sheng, Z. Xin, M. S. Alam, L. Xi, and L. Xiao-Feng, “Information hiding based on double random-phase encoding and public-key cryptography,” Opt. Express 17(5), 3270–3284 (2009).
[CrossRef] [PubMed]

M. He, Q. Tan, L. Cao, Q. He, and G. Jin, “Security enhanced optical encryption system by random phase key and permutation key,” Opt. Express 17(25), 22462–22473 (2009).
[CrossRef]

L. Chen and D. Zhao, “Color information processing (coding and synthesis) with fractional Fourier transforms and digital holography,” Opt. Express 15(24), 16080–16089 (2007).
[CrossRef] [PubMed]

J. A. Rodrigo, T. Alieva, and M. L. Calvo, “Gyrator transform: properties and applications,” Opt. Express 15(5), 2190–2203 (2007).
[CrossRef] [PubMed]

Opt. Lasers Eng. (1)

Z. Liu, J. Dai, X. Sun, and S. Liu, “Color image encryption by using the rotation of color vector in Hartley transform domains,” Opt. Lasers Eng. 48(7-8), 800–805 (2010).
[CrossRef]

Opt. Lett. (8)

Phys. Rev. A (1)

C. Jeffries and J. Perez, “Observation of a Pomeau-Manneville intermittent route to chaos in a nonlinear oscillator,” Phys. Rev. A 26(4), 2117–2122 (1982).
[CrossRef]

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Figures (9)

Fig. 1
Fig. 1

An example of chaotic map: (a) logistic map, (b) the map expressed in Eq. (2)

Fig. 2
Fig. 2

The process of double image encryption.

Fig. 3
Fig. 3

The result of double image encryption: (a) Elaine regarded as I 1 , (b) leopard regarded as I 2 , (c) the amplitude of encrypted data, (d) the phase of encrypted data, (e) the decrypted image for Elaine, (f) the decrypted image for leopard.

Fig. 4
Fig. 4

The decrypted results with various values of the angle α : (a) the NMSE curves, (b) the decrypted image for Elaine, (c) the decrypted image for leopard. Here the two recovered images are obtained by using the angle α = 0.543.

Fig. 5
Fig. 5

The decrypted result by using a wrong series n : (a) for Elaine and (b) for leopard with the series n ( k ) 1 , (c) for Elaine and (d) for leopard with the series n ( k ) + 1 , (e) and (f) are the decrypted images under the case n ( 1 ) = 2 and other parameters are correct.

Fig. 6
Fig. 6

the decryption results by use of a half of correct data in random matrix s 0: (a) a half of data in the key s 0 is known by attacker, (b) the retrieved image for DRPE, (c) the recovered image for Elaine, (d) the recovered image for leopard.

Fig. 7
Fig. 7

The flowchart of (a) the designed DRPE and (b) the phase retrieval algorithm for the ciphertext-only attack on this encryption algorithm.

Fig. 8
Fig. 8

The result of the ciphertext-only attack: (a) NMSE curve, (b) recovered image for Elaine, (c) recovered image for leopard. α = 0.55.

Fig. 9
Fig. 9

The robustness test of noise attack: (a) NMSE curves, (b) Elaine, (c) Leopard

Equations (14)

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

s n + 1 = r s n ( 1 s n ) ,
s n + 1 = ( 3.57 + s n / 4 ) s n ( 1 s n ) .
R ( x , y ) = round ( C [ s 0 ( x , y ) , n ] ) ,
F ( u , v ) = G α [ f ( x , y ) ] ( u , v ) = 1 | sin α | f ( x , y ) exp [ i2π ( x y + u v ) cos α ( x v + y u ) sin α ] d x d y ,
I 1 + i I 2 = B ( x , y ) exp [ i ϕ ( x , y ) ] ,
A 1 ( r ) = A 1 ( r ) [ 1 R k ( r ) ] + A 2 ( r ) R k ( r ) A 2 ( r ) = A 1 ( r ) R k ( r ) + A 2 ( r ) [ 1 R k ( r ) ] ,
n ( k ) = 1 , 1 , 2 , 3 , 5 , 8 , ...
NMSE = nmse ( I r , I o ) = m , n | I r ( m , n ) I o ( m , n ) | 2 m , n | I o ( m , n ) | 2 .
G α ( E n ) = B 2 ( u , v ) exp [ i ϕ 2 ( u , v ) ] ,
B exp ( i ϕ ) = G α [ B 2 exp ( i ϕ 2 ) ] .
γ : B ( x , y ) = { B ( x , y ) , if 0< B ( x , y ) cos [ ϕ ( x , y ) ] , B ( x , y ) sin [ ϕ ( x , y ) ] I m I m W ( x , y ) , otherwise .
B 2 exp ( i ϕ 2 ) = G α [ B exp ( i ϕ ) ] ,
I 1 , r = B ( x , y ) cos [ ϕ ( x , y ) ] I 2 , r = B ( x , y ) sin [ ϕ ( x , y ) ] .
E n = E n ( 1 + d Q 0 , 1 ) ,

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