Abstract

A multiple-image encryption scheme based on cascaded fractional Fourier transform is proposed. In the scheme, images are successively coded into the amplitude and phase of the input by cascading stages, which ends up with an encrypted image and a series of keys. The scheme takes full advantage of multikeys and the cascaded relationships of all stages, and it not only realizes image encryption but also achieves higher safety and more diverse applications. So multiuser authentication and hierarchical encryption are achieved. Numerical simulation verifies the feasibility of the method and demonstrates the security of the scheme and decryption characteristics. Finally, flexibility and variability of the scheme in application are discussed, and the simple photoelectric mixed devices to realize the scheme are proposed.

© 2013 Optical Society of America

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References

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  1. 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]
  2. S. T. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multi-stage and multi- channel fractional Fourier domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
    [CrossRef]
  3. Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
    [CrossRef]
  4. Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
    [CrossRef]
  5. W. M. Jin and C. J. Yan, “Optical image encryption based on multi-channel fractional Fourier transform and double random phase encoding technique,” Optik 118, 38–41(2007).
    [CrossRef]
  6. C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
    [CrossRef]
  7. C. Lin and X. J. Shen, “Analysis and design of impulse attack free generalized joint transform correlator optical encryption scheme,” Opt. Laser Technol. 44, 2032–2036 (2012).
    [CrossRef]
  8. X. C. Cheng and L. Z. Cai, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33, 1575–1577 (2008).
    [CrossRef]
  9. R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple- parameter fractional Fourier transform,” Opt. Lett. 33, 581–583 (2008).
    [CrossRef]
  10. R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
    [CrossRef]
  11. D. M. Zhao, X. X. Li, and L. E. Chen, “Optical image encryption with redefined fractional Hartley transform,” Opt. Commun. 281, 5326–5329 (2008).
    [CrossRef]
  12. L. F. Chen and D. M. Zhao, “Color image encoding in dual fractional Fourier-wavelet domain with random phases,” Opt. Commun. 282, 3433–3438 (2009).
    [CrossRef]
  13. G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1307(2005).
    [CrossRef]
  14. D. Amaya, M. Tebaldi, R. Torroba, and N. Bolognini, “Wavelength multiplexing encryption using joint transform correlator architecture,” Appl. Opt. 48, 2099–2104(2009).
    [CrossRef]
  15. H. T. Chang, H.-E. Hwang, C. L. Lee, and M.-T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
    [CrossRef]
  16. L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
    [CrossRef]
  17. G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” Appl. Opt. 8, 391–397 (2006).
    [CrossRef]
  18. H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
    [CrossRef]
  19. Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
    [CrossRef]
  20. B. M. Hennelly, T. J. Naughton, and J. Mcdonald, “Spread-space spread-spectrum technique for secure multiplexing,” Opt. Lett. 32, 1060–1062 (2007).
    [CrossRef]
  21. M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
    [CrossRef]

2012 (2)

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

C. Lin and X. J. Shen, “Analysis and design of impulse attack free generalized joint transform correlator optical encryption scheme,” Opt. Laser Technol. 44, 2032–2036 (2012).
[CrossRef]

2011 (3)

H. T. Chang, H.-E. Hwang, C. L. Lee, and M.-T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
[CrossRef]

H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

2009 (3)

D. Amaya, M. Tebaldi, R. Torroba, and N. Bolognini, “Wavelength multiplexing encryption using joint transform correlator architecture,” Appl. Opt. 48, 2099–2104(2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
[CrossRef]

L. F. Chen and D. M. Zhao, “Color image encoding in dual fractional Fourier-wavelet domain with random phases,” Opt. Commun. 282, 3433–3438 (2009).
[CrossRef]

2008 (3)

2007 (2)

W. M. Jin and C. J. Yan, “Optical image encryption based on multi-channel fractional Fourier transform and double random phase encoding technique,” Optik 118, 38–41(2007).
[CrossRef]

B. M. Hennelly, T. J. Naughton, and J. Mcdonald, “Spread-space spread-spectrum technique for secure multiplexing,” Opt. Lett. 32, 1060–1062 (2007).
[CrossRef]

2006 (3)

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” Appl. Opt. 8, 391–397 (2006).
[CrossRef]

Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
[CrossRef]

2005 (3)

Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
[CrossRef]

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1307(2005).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

2001 (1)

2000 (1)

Ahmad, M. A.

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

Amaya, D.

Bolognini, N.

Cai, L. Z.

X. C. Cheng and L. Z. Cai, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33, 1575–1577 (2008).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Chang, H. T.

H. T. Chang, H.-E. Hwang, C. L. Lee, and M.-T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
[CrossRef]

H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

Chen, L. E.

D. M. Zhao, X. X. Li, and L. E. Chen, “Optical image encryption with redefined fractional Hartley transform,” Opt. Commun. 281, 5326–5329 (2008).
[CrossRef]

Chen, L. F.

L. F. Chen and D. M. Zhao, “Color image encoding in dual fractional Fourier-wavelet domain with random phases,” Opt. Commun. 282, 3433–3438 (2009).
[CrossRef]

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Cheng, X. C.

Guo, Q.

Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
[CrossRef]

He, M. Z.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Hennelly, B. M.

Hwang, H.-E.

H. T. Chang, H.-E. Hwang, C. L. Lee, and M.-T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
[CrossRef]

H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

Jin, W. M.

W. M. Jin and C. J. Yan, “Optical image encryption based on multi-channel fractional Fourier transform and double random phase encoding technique,” Optik 118, 38–41(2007).
[CrossRef]

Joseph, J.

Lang, J.

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple- parameter fractional Fourier transform,” Opt. Lett. 33, 581–583 (2008).
[CrossRef]

Lee, C. L.

H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

H. T. Chang, H.-E. Hwang, C. L. Lee, and M.-T. Lee, “Wavelength multiplexing multiple-image encryption using cascaded phase-only masks in the Fresnel transform domain,” Appl. Opt. 50, 710–716 (2011).
[CrossRef]

Lee, M.-T.

Li, X. X.

D. M. Zhao, X. X. Li, and L. E. Chen, “Optical image encryption with redefined fractional Hartley transform,” Opt. Commun. 281, 5326–5329 (2008).
[CrossRef]

Lin, C.

C. Lin and X. J. Shen, “Analysis and design of impulse attack free generalized joint transform correlator optical encryption scheme,” Opt. Laser Technol. 44, 2032–2036 (2012).
[CrossRef]

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

Liu, Q.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Liu, S. T.

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
[CrossRef]

Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
[CrossRef]

S. T. Liu, Q. Mi, and B. Zhu, “Optical image encryption with multi-stage and multi- channel fractional Fourier domain filtering,” Opt. Lett. 26, 1242–1244 (2001).
[CrossRef]

Liu, Z. J.

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
[CrossRef]

Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
[CrossRef]

Mcdonald, J.

Meng, X. F.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Mi, Q.

Naughton, T. J.

Shen, X. J.

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

C. Lin and X. J. Shen, “Analysis and design of impulse attack free generalized joint transform correlator optical encryption scheme,” Opt. Laser Technol. 44, 2032–2036 (2012).
[CrossRef]

Singh, K.

Situ, G.

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” Appl. Opt. 8, 391–397 (2006).
[CrossRef]

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1307(2005).
[CrossRef]

Tang, R.

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

Tao, R.

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple- parameter fractional Fourier transform,” Opt. Lett. 33, 581–583 (2008).
[CrossRef]

Tebaldi, M.

Torroba, R.

Unnikrishnan, G.

Wang, X. C.

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Wang, Y.

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple- parameter fractional Fourier transform,” Opt. Lett. 33, 581–583 (2008).
[CrossRef]

Yan, C. J.

W. M. Jin and C. J. Yan, “Optical image encryption based on multi-channel fractional Fourier transform and double random phase encoding technique,” Optik 118, 38–41(2007).
[CrossRef]

Zhang, J.

G. Situ and J. Zhang, “Position multiplexing for multiple-image encryption,” Appl. Opt. 8, 391–397 (2006).
[CrossRef]

G. Situ and J. Zhang, “Multiple-image encryption by wavelength multiplexing,” Opt. Lett. 30, 1306–1307(2005).
[CrossRef]

Zhang, Y.

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

Zhao, D. M.

L. F. Chen and D. M. Zhao, “Color image encoding in dual fractional Fourier-wavelet domain with random phases,” Opt. Commun. 282, 3433–3438 (2009).
[CrossRef]

D. M. Zhao, X. X. Li, and L. E. Chen, “Optical image encryption with redefined fractional Hartley transform,” Opt. Commun. 281, 5326–5329 (2008).
[CrossRef]

L. F. Chen and D. M. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Zhao, H. E.

Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
[CrossRef]

Zhao, H. F.

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

Zhu, B.

Zou, X.

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

Appl. Opt. (3)

Opt. Commun. (8)

H. T. Chang, H.-E. Hwang, and C. L. Lee, “Position multiplexing multiple-image encryption using cascaded phase-only masks in Fresnel transform domain,” Opt. Commun. 284, 4146–4151 (2011).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter fractional Hadamard transform,” Opt. Commun. 282, 1531–1535 (2009).
[CrossRef]

D. M. Zhao, X. X. Li, and L. E. Chen, “Optical image encryption with redefined fractional Hartley transform,” Opt. Commun. 281, 5326–5329 (2008).
[CrossRef]

L. F. Chen and D. M. Zhao, “Color image encoding in dual fractional Fourier-wavelet domain with random phases,” Opt. Commun. 282, 3433–3438 (2009).
[CrossRef]

Z. J. Liu, H. E. Zhao, and S. T. Liu, “Discrete fractional random transform,” Opt. Commun. 255, 357–365 (2005).
[CrossRef]

Z. J. Liu, Q. Guo, and S. T. Liu, “The discrete fractional random cosine and sine transforms,” Opt. Commun. 265, 100–105 (2006).
[CrossRef]

C. Lin, X. J. Shen, R. Tang, and X. Zou, “Multiple images encryption based on Fourier transform hologram,” Opt. Commun. 285, 1023–1028 (2012).
[CrossRef]

M. Z. He, L. Z. Cai, Q. Liu, X. C. Wang, and X. F. Meng, “Multiple image encryption an water marking by random phase matching,” Opt. Commun. 247, 29–37 (2005).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (1)

C. Lin and X. J. Shen, “Analysis and design of impulse attack free generalized joint transform correlator optical encryption scheme,” Opt. Laser Technol. 44, 2032–2036 (2012).
[CrossRef]

Opt. Lett. (6)

Optik (2)

Z. J. Liu, Y. Zhang, H. F. Zhao, M. A. Ahmad, and S. T. Liu, “Optical multi-image encryption based on frequency shift,” Optik 122, 1010–1013 (2011).
[CrossRef]

W. M. Jin and C. J. Yan, “Optical image encryption based on multi-channel fractional Fourier transform and double random phase encoding technique,” Optik 118, 38–41(2007).
[CrossRef]

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Figures (7)

Fig. 1.
Fig. 1.

Flow chart of the scheme. (a) Encryption process and (b) decryption process.

Fig. 2.
Fig. 2.

Photoelectric mixed devices for the scheme. d denotes the distance between the lens and the plane and SLM denotes the spatial light modulator.

Fig. 3.
Fig. 3.

Original image. (a) Lena, (b) Barbara, (c) Baboon, and (d) Elaine.

Fig. 4.
Fig. 4.

Encrypted image and part of keys. (a) Final encrypted image, (b) the key C2, and (c) the key ε2.

Fig. 5.
Fig. 5.

Simulation results of decryption. (a) Simulation results of Lena, (b) simulation results of Barbara, (c) simulation results of Baboon, and (d) simulation results of Elaine.

Fig. 6.
Fig. 6.

Original images, corresponding decrypted images, and MSEs with P changing. (a), (e), (i), (m) Original images. (b), (f), (j), (n) Corresponding decrypted images when p=1.205. (c), (g), (k), (o) Corresponding decrypted images when p=1.21. (d), (h), (l), (p) MSEs when p is changing.

Fig. 7.
Fig. 7.

Simulation results of decryption with wrong keys. (a)–(e) are the decrypted image when only one of C2, α2, β2, ε2, and A3, respectively, is unknown.

Equations (8)

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

Bn={{Cn+αn[(i=2n+12mAi)+βn(A2n1+A2n)]}×exp{εn+αn[(i=2n+12mθi)+βnθ2n]}(1<n<m)[Cn+αnβn(A2n1+A2n)]×exp(εn+αnβnθ2n)(n=m).
θi=j2πAi/256.
B1={C1+α1[(i=32mAi)+β1(A1+A2)]}×exp{ε1+α1[(i=32mθi)+β1θ2]}.
F(Bn)=Cn+1×exp(εn+1).
Bm+1=(Cm+1)×exp(εm+1+θ2m+1),
A2n1={[(|Bn|Cn)/αni=2n+12mAi]/βnA2n(0<n<m)(|Bn|Cn)/(αnβn)A2n(n=m),
A2n=Arg(θ2n)={Arg({[h(Bn)εn]/αni=2n+12mθi}/βn)(0<n<m)Arg{[h(Bn)εn]/(αnβn)}(n=m).
MSE=1N2i=0N1j=0N1I(i,j)K(i,j)2,

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