Abstract

We present a novel image hiding method based on phase retrieval algorithm under the framework of nonlinear double random phase encoding in fractional Fourier domain. Two phase-only masks (POMs) are efficiently determined by using the phase retrieval algorithm, in which two cascaded phase-truncated fractional Fourier transforms (FrFTs) are involved. No undesired information disclosure, post-processing of the POMs or digital inverse computation appears in our proposed method. In order to achieve the reduction in key transmission, a modified image hiding method based on the modified phase retrieval algorithm and logistic map is further proposed in this paper, in which the fractional orders and the parameters with respect to the logistic map are regarded as encryption keys. Numerical results have demonstrated the feasibility and effectiveness of the proposed algorithms.

© 2014 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [Crossref] [PubMed]
  2. A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photon. 1(3), 589–636 (2009).
    [Crossref]
  3. W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6(2), 120–155 (2014).
    [Crossref]
  4. 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] [PubMed]
  5. O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
    [Crossref] [PubMed]
  6. G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
    [Crossref] [PubMed]
  7. Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. 32(13), 1914–1916 (2007).
    [Crossref] [PubMed]
  8. Z. Liu, Q. Guo, L. Xu, M. A. Ahmad, and S. Liu, “Double image encryption by using iterative random binary encoding in gyrator domains,” Opt. Express 18(11), 12033–12043 (2010).
    [Crossref] [PubMed]
  9. A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005).
    [Crossref] [PubMed]
  10. X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
    [Crossref] [PubMed]
  11. P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34(3), 331–333 (2009).
    [Crossref] [PubMed]
  12. A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35(13), 2185–2187 (2010).
    [Crossref] [PubMed]
  13. M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
    [Crossref] [PubMed]
  14. J. F. Barrera, A. Mira-Agudelo, and R. Torroba, “Experimental QR code optical encryption: noise-free data recovering,” Opt. Lett. 39(10), 3074–3077 (2014).
    [Crossref] [PubMed]
  15. W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
    [Crossref] [PubMed]
  16. I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
    [Crossref]
  17. W. Liu, Z. Liu, J. Wu, and S. Liu, “Asymmetric cryptosystem by using modular arithmetic operation based on double random phase encoding,” Opt. Commun. 301–302, 56–60 (2013).
    [Crossref]
  18. I. Mehra and N. K. Nishchal, “Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon,” Opt. Laser Technol. 60, 1–7 (2014).
    [Crossref]
  19. X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
    [Crossref]
  20. I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
    [Crossref]
  21. G. Situ and J. Zhang, “A lensless optical security system based on computer-generated phase only masks,” Opt. Commun. 232(1–6), 115–122 (2004).
    [Crossref]
  22. A. Alfalou and A. Mansour, “Double random phase encryption scheme to multiplex and simultaneous encode multiple images,” Appl. Opt. 48(31), 5933–5947 (2009).
    [Crossref] [PubMed]
  23. H. E. Hwang, H. T. Chang, and W. N. Lie, “Fast double-phase retrieval in Fresnel domain using modified Gerchberg-Saxton algorithm for lensless optical security systems,” Opt. Express 17(16), 13700–13710 (2009).
    [Crossref] [PubMed]
  24. 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] [PubMed]
  25. W. Liu, Z. Liu, and S. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
    [Crossref] [PubMed]
  26. Y. Y. Chen, J. H. Wang, C. C. Lin, and H. E. Hwang, “Lensless optical data hiding system based on phase encoding algorithm in the Fresnel domain,” Appl. Opt. 52(21), 5247–5255 (2013).
    [Crossref] [PubMed]
  27. W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
    [Crossref]
  28. X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
    [Crossref]
  29. Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
    [Crossref] [PubMed]
  30. Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding by information prechoosing,” Opt. Lett. 33(6), 542–544 (2008).
    [Crossref] [PubMed]
  31. H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Applications (Taylor and Francis, 1997).
  32. X. Wang, D. Zhao, and Y. Chen, “Double images encryption without information disclosure using phase-truncation Fourier transforms and a random amplitude mask,” Appl. Opt. 53(23), 5100–5108 (2014).
  33. Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
    [Crossref]
  34. P. Kumar, J. Joseph, and K. Singh, “Optical image encryption using a jigsaw transform for silhouette removal in interference-based methods and decryption with a single spatial light modulator,” Appl. Opt. 50(13), 1805–1811 (2011).
    [Crossref] [PubMed]
  35. Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
    [Crossref]
  36. X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
    [Crossref] [PubMed]
  37. Y. Qin and Q. Gong, “Interference-based multiple-image encryption with silhouette removal by position multiplexing,” Appl. Opt. 52(17), 3987–3992 (2013).
    [Crossref] [PubMed]
  38. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).
  39. K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2000).

2014 (6)

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6(2), 120–155 (2014).
[Crossref]

J. F. Barrera, A. Mira-Agudelo, and R. Torroba, “Experimental QR code optical encryption: noise-free data recovering,” Opt. Lett. 39(10), 3074–3077 (2014).
[Crossref] [PubMed]

I. Mehra and N. K. Nishchal, “Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon,” Opt. Laser Technol. 60, 1–7 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
[Crossref]

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

X. Wang, D. Zhao, and Y. Chen, “Double images encryption without information disclosure using phase-truncation Fourier transforms and a random amplitude mask,” Appl. Opt. 53(23), 5100–5108 (2014).

2013 (7)

Y. Qin and Q. Gong, “Interference-based multiple-image encryption with silhouette removal by position multiplexing,” Appl. Opt. 52(17), 3987–3992 (2013).
[Crossref] [PubMed]

W. Liu, Z. Liu, and S. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[Crossref] [PubMed]

Y. Y. Chen, J. H. Wang, C. C. Lin, and H. E. Hwang, “Lensless optical data hiding system based on phase encoding algorithm in the Fresnel domain,” Appl. Opt. 52(21), 5247–5255 (2013).
[Crossref] [PubMed]

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
[Crossref]

W. Liu, Z. Liu, J. Wu, and S. Liu, “Asymmetric cryptosystem by using modular arithmetic operation based on double random phase encoding,” Opt. Commun. 301–302, 56–60 (2013).
[Crossref]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
[Crossref] [PubMed]

2012 (3)

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (3)

2009 (6)

2008 (2)

2007 (1)

2006 (1)

2005 (1)

2004 (2)

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
[Crossref] [PubMed]

G. Situ and J. Zhang, “A lensless optical security system based on computer-generated phase only masks,” Opt. Commun. 232(1–6), 115–122 (2004).
[Crossref]

2000 (1)

1999 (1)

1995 (1)

Ahmad, M. A.

Alfalou, A.

Arcos, S.

Barrera, J. F.

Brosseau, C.

Cao, L.

Carnicer, A.

Chang, H. T.

Chen, J.

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

Chen, W.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6(2), 120–155 (2014).
[Crossref]

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
[Crossref]

Chen, X.

W. Chen, B. Javidi, and X. Chen, “Advances in optical security systems,” Adv. Opt. Photon. 6(2), 120–155 (2014).
[Crossref]

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
[Crossref]

Chen, Y.

Chen, Y. Y.

Cho, M.

Dai, C.

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

Dong, Z.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Gong, Q.

Guo, Q.

He, M.

He, Q.

Hwang, H. E.

Javidi, B.

Jin, G.

Joseph, J.

Juvells, I.

Kumar, A.

Kumar, P.

Lie, W. N.

Lin, C. C.

Liu, S.

Liu, W.

W. Liu, Z. Liu, and S. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[Crossref] [PubMed]

W. Liu, Z. Liu, J. Wu, and S. Liu, “Asymmetric cryptosystem by using modular arithmetic operation based on double random phase encoding,” Opt. Commun. 301–302, 56–60 (2013).
[Crossref]

Liu, Z.

Mansour, A.

Matoba, O.

Mehra, I.

I. Mehra and N. K. Nishchal, “Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon,” Opt. Laser Technol. 60, 1–7 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
[Crossref]

Mira-Agudelo, A.

Montes-Usategui, M.

Nishchal, N. K.

I. Mehra and N. K. Nishchal, “Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon,” Opt. Laser Technol. 60, 1–7 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
[Crossref]

Peng, X.

Qin, W.

Qin, Y.

Rajput, S. K.

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
[Crossref]

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
[Crossref]

Refregier, P.

Shi, Y.

Singh, K.

Situ, G.

Stern, A.

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
[Crossref]

Tan, Q.

Torroba, R.

Unnikrishnan, G.

Wang, B.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Wang, J. H.

Wang, Q.

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

Wang, X.

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

X. Wang, D. Zhao, and Y. Chen, “Double images encryption without information disclosure using phase-truncation Fourier transforms and a random amplitude mask,” Appl. Opt. 53(23), 5100–5108 (2014).

X. Wang and D. Zhao, “Optical image hiding with silhouette removal based on the optical interference principle,” Appl. Opt. 51(6), 686–691 (2012).
[Crossref] [PubMed]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

Wei, H.

Wu, J.

W. Liu, Z. Liu, J. Wu, and S. Liu, “Asymmetric cryptosystem by using modular arithmetic operation based on double random phase encoding,” Opt. Commun. 301–302, 56–60 (2013).
[Crossref]

Xu, L.

Zhang, J.

Zhang, P.

Zhang, Y.

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Zhao, D.

Adv. Opt. Photon. (2)

Appl. Opt. (6)

IEEE Photon. J. (1)

W. Chen, X. Chen, A. Stern, and B. Javidi, “Phase-modulated optical system with sparse representation for information encoding and authentication,” IEEE Photon. J. 5(2), 6900113 (2013).
[Crossref]

J. Opt. A, Pure Appl. Opt. (1)

Y. Zhang, B. Wang, and Z. Dong, “Enhancement of image hiding by exchanging two phase masks,” J. Opt. A, Pure Appl. Opt. 11(12), 125406 (2009).
[Crossref]

Opt. Commun. (5)

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285(6), 1078–1081 (2012).
[Crossref]

X. Wang, C. Dai, and J. Chen, “Optical image encryption via reverse engineering of a modified amplitude-phase retrieval-based attack,” Opt. Commun. 328, 67–72 (2014).
[Crossref]

W. Liu, Z. Liu, J. Wu, and S. Liu, “Asymmetric cryptosystem by using modular arithmetic operation based on double random phase encoding,” Opt. Commun. 301–302, 56–60 (2013).
[Crossref]

Q. Wang, “Optical image encryption with silhouette removal based on interference and phase blend processing,” Opt. Commun. 285(21–22), 4294–4301 (2012).
[Crossref]

G. Situ and J. Zhang, “A lensless optical security system based on computer-generated phase only masks,” Opt. Commun. 232(1–6), 115–122 (2004).
[Crossref]

Opt. Eng. (1)

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Collision in Fresnel domain asymmetric cryptosystem using phase truncation and authentication verification,” Opt. Eng. 52(2), 028202 (2013).
[Crossref]

Opt. Express (3)

Opt. Laser Technol. (1)

I. Mehra and N. K. Nishchal, “Asymmetric cryptosystem for securing multiple images using two beam interference phenomenon,” Opt. Laser Technol. 60, 1–7 (2014).
[Crossref]

Opt. Lasers Eng. (1)

I. Mehra, S. K. Rajput, and N. K. Nishchal, “Cryptanalysis of an image encryption scheme based on joint transform correlator with amplitude- and phase- truncation approach,” Opt. Lasers Eng. 52, 167–173 (2014).
[Crossref]

Opt. Lett. (15)

W. Liu, Z. Liu, and S. Liu, “Asymmetric cryptosystem using random binary phase modulation based on mixture retrieval type of Yang-Gu algorithm,” Opt. Lett. 38(10), 1651–1653 (2013).
[Crossref] [PubMed]

Y. Zhang and B. Wang, “Optical image encryption based on interference,” Opt. Lett. 33(21), 2443–2445 (2008).
[Crossref] [PubMed]

Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding by information prechoosing,” Opt. Lett. 33(6), 542–544 (2008).
[Crossref] [PubMed]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
[Crossref] [PubMed]

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] [PubMed]

O. Matoba and B. Javidi, “Encrypted optical memory system using three-dimensional keys in the Fresnel domain,” Opt. Lett. 24(11), 762–764 (1999).
[Crossref] [PubMed]

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29(14), 1584–1586 (2004).
[Crossref] [PubMed]

Y. Shi, G. Situ, and J. Zhang, “Multiple-image hiding in the Fresnel domain,” Opt. Lett. 32(13), 1914–1916 (2007).
[Crossref] [PubMed]

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30(13), 1644–1646 (2005).
[Crossref] [PubMed]

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31(22), 3261–3263 (2006).
[Crossref] [PubMed]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34(3), 331–333 (2009).
[Crossref] [PubMed]

A. Alfalou and C. Brosseau, “Dual encryption scheme of images using polarized light,” Opt. Lett. 35(13), 2185–2187 (2010).
[Crossref] [PubMed]

M. Cho and B. Javidi, “Three-dimensional photon counting double-random-phase encryption,” Opt. Lett. 38(17), 3198–3201 (2013).
[Crossref] [PubMed]

J. F. Barrera, A. Mira-Agudelo, and R. Torroba, “Experimental QR code optical encryption: noise-free data recovering,” Opt. Lett. 39(10), 3074–3077 (2014).
[Crossref] [PubMed]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35(2), 118–120 (2010).
[Crossref] [PubMed]

Other (3)

H. P. Herzig, ed., Micro-Optics: Elements, Systems, and Applications (Taylor and Francis, 1997).

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (Wiley, 2001).

K. T. Alligood, T. D. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (18)

Fig. 1
Fig. 1

Flowchart of (a) encryption process and (b) decryption of the nonlinear DRPE based on FrFT.

Fig. 2
Fig. 2

Flowchart of the kth loop of the iterative process.

Fig. 3
Fig. 3

Decryption process with linear DRPE in fractional Fourier domain.

Fig. 4
Fig. 4

Flowchart of the modified amplitude-phase retrieval algorithm.

Fig. 5
Fig. 5

Proposed optical setup of the decryption process. The POM E ( u , ν ) is the cyphertext and the CRPM C ( x , y ) is the key for decryption.

Fig. 6
Fig. 6

(a) Secret image; POMs generated after 50 iterations using the first algorithm: (b) K 1 ( u , ν ) , (c) K 2 ( x , y ) .

Fig. 7
Fig. 7

(a) MSE and (b) correlation coefficient versus number of iterations.

Fig. 8
Fig. 8

Reconstructed results with respect to different numbers of iterations: (a) 3; (b) 5; (c) 10; (d) 50.

Fig. 9
Fig. 9

Images reconstructed from (a) K 1 ( u , ν ) , (b) K 2 ( x , y ) .

Fig. 10
Fig. 10

Decrypted images by using (a) K 1 ( u , ν ) and a RPM (b) K 2 ( x , y ) and a RPM.

Fig. 11
Fig. 11

(a) MSE versus the fractional orders, decrypted images using a wrong fractional order (b) α = 0.59 ( Δ α = 0.01 ) , (c) α = 0.58 ( Δ α = 0.02 ) .

Fig. 12
Fig. 12

(a) MSE versus number of iterations in the attack; (b) the final decrypted image.

Fig. 13
Fig. 13

Outcomes of the second method: (a) C ( x , y ) (decryption key) and (b) E ( u , ν ) (encrypted result).

Fig. 14
Fig. 14

(a) MSE versus number of iterations, (b) Correlation coefficient versus number of iterations.

Fig. 15
Fig. 15

Reconstructed results with respect to different numbers of iterations: (a) 10; (b) 50; (c) 150; (d) 500.

Fig. 16
Fig. 16

Images reconstructed from (a) E ( u , ν ) , (b) C ( x , y ) .

Fig. 17
Fig. 17

MSE versus the fractional order.

Fig. 18
Fig. 18

Decrypted images using incorrect CRPM produced by (a) μ = 3.93 ( Δ μ = 0.01 ) , x 0 = 0.12 , (b) μ = 3.94 , x 0 = 0.11 ( Δ x 0 = 0.01 ) .

Equations (29)

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

FrFT α [ f ( x ) ] ( u ) = K α ( x , u ) f ( x ) d x
K α ( x , u ) = { A exp [ i π ( x 2 cot φ 2 x u csc φ + u 2 cot φ ) ] φ n π δ ( x u ) φ = 2 n π δ ( x + u ) φ = ( 2 n + 1 ) π
FrFT α 1 , α 2 [ f ( x , y ) ] ( u , ν ) = + K α 1 , α 2 ( x , y ; u , ν ) f ( x , y ) d x d y
K α 1 , α 2 ( x , y ; u , ν ) = K α 1 ( x , u ) K α 2 ( y , ν )
g ( u , ν ) = PT { FrFT α [ f ( x , y ) R ( x , y ) ] }
E ( x , y ) = PT { FrFT β [ g ( u , ν ) R ( x , y ) ] }
P ( u , ν ) = PR { FrFT α [ f ( x , y ) R ( x , y ) ] }
P ( x , y ) = PR { FrFT β [ g ( u , ν ) R ( u , ν ) ] }
g ( u , ν ) = PT { FrFT β [ E ( x , y ) p ( x , y ) ] }
f ( x , y ) = PT { FrFT α [ g ( u , ν ) p ( u , ν ) ] }
g k ( u , ν ) = PT { FrFT α [ f ( x , y ) R k ( x , y ) ] }
P k ( u , ν ) = PR { FrFT α [ f ( x , y ) R k ( x , y ) ] }
P k ( x , y ) = PR { FrFT β [ g k ( u , ν ) R k ( u , ν ) ] }
g k ( u , ν ) = PT { FrFT β [ P k ( x , y ) ] }
R k + 1 ( u , ν ) = PR { FrFT β [ P k ( x , y ) ] }
f k ( x , y ) = PT { FrFT α [ g k ( u , ν ) P k ( u , ν ) ] }
R k + 1 ( x , y ) = PR { FrFT α [ g k ( u , ν ) P k ( u , ν ) ] }
f ( x , y ) = f m ( x , y )
K 1 ( u , ν ) = R m + 1 ( u , ν ) P m ( u , ν )
K 2 ( x , y ) = P m ( x , y )
f ( x , y ) = f m ( x , y ) = PT { FrFT - α [ g m ( u , ν ) P m ( u , ν ) ] } = PT { FrFT - α [ FrFT β [ P m ( x , y ) ] R m + 1 ( u , ν ) P m ( u , ν ) ] } = PT { FrFT - α [ FrFT β [ K 2 ( x , y ) ] K 1 ( u , ν ) ] }
x n + 1 = μ x n ( 1 x n )
R k + 1 ( u , ν ) = PR { FrFT β [ C ( x , y ) ] }
g ( u , ν ) = PT { FrFT β [ C ( x , y ) ] }
R k + 1 ( x , y ) = PR { FrFT α [ g ( u , ν ) P k ( u , ν ) ] }
f k ( x , y ) = PT { FrFT α [ g ( u , ν ) P k ( u , ν ) ] }
f ( x , y ) = PT { FrFT α [ FrFT β [ C ( x , y ) ] E ( u , ν ) ] }
MSE ( f , f ) = 1 M | f ( x , y ) f ( x , y ) | 2
CC = E { [ f E [ f ] ] [ f E [ f ] ] } E { [ f E [ f ] ] 2 } E { [ f E [ f ] ] 2 }

Metrics