Abstract

We present a polarimetric-based optical encoder for image encryption and verification. A system for generating random polarized vector keys based on a Mach-Zehnder configuration combined with translucent liquid crystal displays in each path of the interferometer is developed. Polarization information of the encrypted signal is retrieved by taking advantage of the information provided by the Stokes parameters. Moreover, photon-counting model is used in the encryption process which provides data sparseness and nonlinear transformation to enhance security. An authorized user with access to the polarization keys and the optical design variables can retrieve and validate the photon-counting plain-text. Optical experimental results demonstrate the feasibility of the encryption method.

© 2015 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  10. 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]
  11. H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18(13), 13772–13781 (2010).
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    [Crossref] [PubMed]
  14. 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]
  15. 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).
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    [Crossref]
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    [Crossref] [PubMed]
  22. F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
    [PubMed]
  23. F. A. Sadjadi, “Passive three-dimensional imaging using polarimetric diversity,” Opt. Lett. 32(3), 229–231 (2007).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  25. Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
    [Crossref]
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    [Crossref]
  27. X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007).
    [Crossref] [PubMed]
  28. I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Express 19(5), 4583–4594 (2011).
    [Crossref] [PubMed]
  29. F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20(13), 14015–14029 (2012).
    [Crossref] [PubMed]
  30. W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21(18), 20692–20706 (2013).
    [Crossref] [PubMed]
  31. I. Moreno, J. A. Davis, D. M. Cottrell, and R. Donoso, “Encoding high-order cylindrically polarized light beams,” Appl. Opt. 53(24), 5493–5501 (2014).
    [Crossref] [PubMed]
  32. D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21(5), 5432–5439 (2013).
    [Crossref] [PubMed]
  33. D. Maluenda, R. Martínez-Herrero, I. Juvells, and A. Carnicer, “Synthesis of highly focused fields with circular polarization at any transverse plane,” Opt. Express 22(6), 6859–6867 (2014).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  37. B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28(12), 2358–2367 (1989).
    [Crossref] [PubMed]
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  40. E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (2000).
    [Crossref] [PubMed]

2014 (3)

2013 (4)

2012 (1)

2011 (3)

2010 (3)

2009 (4)

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]

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

2008 (1)

2007 (4)

2006 (1)

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

2005 (1)

2004 (1)

2001 (1)

2000 (3)

1999 (1)

1998 (1)

1996 (1)

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35(9), 2464–2469 (1996).
[Crossref]

1995 (1)

1994 (1)

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

1989 (1)

1984 (1)

Alfalou, A.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Arcos, S.

Barrera, J. F.

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
[PubMed]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Bernet, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Bollaro, F.

Bolognini, N.

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
[PubMed]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Brosseau, C.

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

Campos, J.

Carnicer, A.

Castro, A.

Chatwin, C.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35(9), 2464–2469 (1996).
[Crossref]

Chen, W.

Chen, X.

Cheng, W.

Cho, M.

Cottrell, D. M.

Dainty, C.

Davis, J. A.

Ding, J.

Donoso, R.

Dowling, T.

Frauel, Y.

Fürhapter, S.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Goudail, F.

Guo, C. S.

Han, W.

Henao, R.

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Hennelly, B. M.

Horner, J. L.

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Ide, M.

Iemmi, C.

Javidi, B.

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

A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2013).

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

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]

O. Matoba and B. Javidi, “Secure holographic memory by double-random polarization encryption,” Appl. Opt. 43(14), 2915–2919 (2004).
[Crossref] [PubMed]

E. Tajahuerce and B. Javidi, “Encrypting three-dimensional information with digital holography,” Appl. Opt. 39(35), 6595–6601 (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]

F. Goudail, F. Bollaro, B. Javidi, and P. Réfrégier, “Influence of a perturbation in a double phase-encoding system,” J. Opt. Soc. Am. A 15(10), 2629–2638 (1998).
[Crossref]

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

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

B. Javidi, “Nonlinear joint power spectrum based optical correlation,” Appl. Opt. 28(12), 2358–2367 (1989).
[Crossref] [PubMed]

Jesacher, A.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Joseph, J.

Juvells, I.

Kenny, F.

Kreske, K.

Kumar, A.

Kumar, P.

Kuroda, K.

Lara, D.

Li, Y.

Maluenda, D.

Markman, A.

A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2013).

Martínez-Herrero, R.

Matoba, O.

Maurer, C.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Millan, M. S.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Montes-Usategui, M.

Moreno, I.

Morris, G. M.

Mosso, F.

Nakano, K.

Naughton, T. J.

Ni, W. J.

Nomura, T.

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Obi, T.

Ohyama, N.

Okada-Shudo, Y.

Pérez-Cabré, E.

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Réfrégier, P.

Ritsch-Marte, M.

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Rodríguez-Herrera, O. G.

Rosen, J.

Sadjadi, F. A.

Sheppard, C. J. R.

Shimura, T.

Singh, K.

Suzuki, H.

Tajahuerce, E.

Takeda, M.

Tan, X.

Tashima, H.

Tebaldi, M.

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
[PubMed]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Tehranipoor, M.

A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2013).

Torroba, R.

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
[PubMed]

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Unnikrishnan, G.

Wang, H. T.

Wang, R. K.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35(9), 2464–2469 (1996).
[Crossref]

Wang, X. L.

Watson, I. A.

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35(9), 2464–2469 (1996).
[Crossref]

Yamaguchi, M.

Yang, Y.

Yzuel, M. J.

Zhan, Q.

Adv. Opt. Photonics (3)

A. Alfalou and C. Brosseau, “Optical image compression and encryption methods,” Adv. Opt. Photonics 1(3), 589–636 (2009).
[Crossref]

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

Q. Zhan, “Cylindrical vector beams: from mathematical concepts to applications,” Adv. Opt. Photonics 1(1), 1–57 (2009).
[Crossref]

Appl. Opt. (8)

IEEE Photon. J. (1)

A. Markman, B. Javidi, and M. Tehranipoor, “Photon-counting security tagging and verification using optically encoded QR codes,” IEEE Photon. J. 6, 6800609 (2013).

J. Opt. Soc. Am. A (2)

New J. Phys. (1)

C. Maurer, A. Jesacher, S. Fürhapter, S. Bernet, and M. Ritsch-Marte, “Tailoring of arbitrary optical vector beams,” New J. Phys. 9(3), 78 (2007).
[Crossref]

Opt. Commun. (1)

J. F. Barrera, R. Henao, M. Tebaldi, R. Torroba, and N. Bolognini, “Multiplexing encrypted data by using polarized light,” Opt. Commun. 260(1), 109–112 (2006).
[Crossref]

Opt. Eng. (2)

R. K. Wang, I. A. Watson, and C. Chatwin, “Random phase encoding for optical security,” Opt. Eng. 35(9), 2464–2469 (1996).
[Crossref]

B. Javidi and J. L. Horner, “Optical pattern recognition for validation and security verification,” Opt. Eng. 33(6), 1752–1756 (1994).
[Crossref]

Opt. Express (9)

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]

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18(13), 13772–13781 (2010).
[Crossref] [PubMed]

I. Moreno, C. Iemmi, J. Campos, and M. J. Yzuel, “Jones matrix treatment for optical fourier processors with structured polarization,” Opt. Express 19(5), 4583–4594 (2011).
[Crossref] [PubMed]

F. Kenny, D. Lara, O. G. Rodríguez-Herrera, and C. Dainty, “Complete polarization and phase control for focus-shaping in high-NA microscopy,” Opt. Express 20(13), 14015–14029 (2012).
[Crossref] [PubMed]

W. Han, Y. Yang, W. Cheng, and Q. Zhan, “Vectorial optical field generator for the creation of arbitrarily complex fields,” Opt. Express 21(18), 20692–20706 (2013).
[Crossref] [PubMed]

W. Chen and X. Chen, “Space-based optical image encryption,” Opt. Express 18(26), 27095–27104 (2010).
[Crossref] [PubMed]

F. Mosso, J. F. Barrera, M. Tebaldi, N. Bolognini, and R. Torroba, “All-optical encrypted movie,” Opt. Express 19(6), 5706–5712 (2011).
[PubMed]

D. Maluenda, I. Juvells, R. Martínez-Herrero, and A. Carnicer, “Reconfigurable beams with arbitrary polarization and shape distributions at a given plane,” Opt. Express 21(5), 5432–5439 (2013).
[Crossref] [PubMed]

D. Maluenda, R. Martínez-Herrero, I. Juvells, and A. Carnicer, “Synthesis of highly focused fields with circular polarization at any transverse plane,” Opt. Express 22(6), 6859–6867 (2014).
[Crossref] [PubMed]

Opt. Lett. (9)

F. A. Sadjadi, “Passive three-dimensional imaging using polarimetric diversity,” Opt. Lett. 32(3), 229–231 (2007).
[Crossref] [PubMed]

E. Pérez-Cabré, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36(1), 22–24 (2011).
[Crossref] [PubMed]

W. Chen, X. Chen, and C. J. R. Sheppard, “Optical image encryption based on diffractive imaging,” Opt. Lett. 35(22), 3817–3819 (2010).
[Crossref] [PubMed]

X. L. Wang, J. Ding, W. J. Ni, C. S. Guo, and H. T. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32(24), 3549–3551 (2007).
[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. 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]

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]

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

Proc. IEEE (1)

O. Matoba, T. Nomura, E. Pérez-Cabré, M. S. Millan, and B. Javidi, “Optical Techniques for Information Security,” Proc. IEEE 97(6), 1128–1148 (2009).
[Crossref]

Other (3)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Cambridge University, 1999).

J. W. Goodman, Statistical Optics (John Wiley & Sons, 1985).

A. Carnicer, A. Hassanfiroozi, P. Latorre-Carmona, Y. P. Huang, and B. Javidi, “Security Authentication using Phase-Encoded Nanoparticle Structures and Polarized Light,” Opt. Lett. doc. ID 225126 (posted 26 November 2014, in press).

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Figures (11)

Fig. 1
Fig. 1 sketch of the optical setup.
Fig. 2
Fig. 2 (a) plain-text image t. (b) false-class plain text image f.
Fig. 3
Fig. 3 Vector cypher-text (T): (a) Stokes parameter S0 = |(T)|2. Green and red ellipses indicate right-handed left-handed polarization respectively. (b) Histogram of |(T)|2.
Fig. 4
Fig. 4 Stokes parameters (a) S1, (b) S2, (c) S3 displayed using the jet colormap.
Fig. 5
Fig. 5 (a) Nonlinear indexes estimation: Peak to correlation energy as a function of k and l. (b) true image (k = 0.1 and l = 0.7); (c) false image (k = 0.1 and l = 0.7).
Fig. 6
Fig. 6 SLM modulation polar plot: (a) SLM1: high-contrast configuration; (b) SLM2: phase-mostly configuration.
Fig. 7
Fig. 7 Alignment procedure: (a) test pattern; (b) Misaligned patterns; (c) aligned patterns.
Fig. 8
Fig. 8 Stokes parameters (a) S0 = |(T)|2, (b) S1, (c) S2, (d) S3.
Fig. 9
Fig. 9 Photon counting encrypted signals N = 3200 (about 1% of the total encrypted signal): (a) true class tph; (b) false class fph.
Fig. 10
Fig. 10 Photon counting encrypted nonlinear cross-correlation. Indexes used k = 0.1 and l = 0.7: (a) true class: tpht (b) false class: f pht.
Fig. 11
Fig. 11 Phase-encoded photon counting encrypted nonlinear cross-correlation. Indexes used k = 0.1 and l = 0.7: (a) true class: tpht (b) false class: f pht.

Tables (1)

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Table 1 Values of the variables used in the optical implementation

Equations (10)

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T= T x e x + T y e y =Pr[ M 1 Pr[ t M 0 , d 1 ], d 2 ] e x +Pr[ M 2 , d 2 ] e y M 0 =exp( i ϕ 0 ( x,y ) ) M 2 =Cexp( i ϕ 2 ( x,y ) ),
S 0 =I(0°, 0) + I(90°, 0) S 1 =I(0°, 0)I(90°, 0) S 2 = I(45°, 0) I(135°, 0) S 3 = I(45°, π/2) I(135°, π/2).
S 0 = | T x | 2 + | T y | 2 S 1 = | T x | 2 | T y | 2 S 2 =2| T x || T y |cos( τ y τ x ) S 3 =2| T x || T y |sin( τ y τ x )
| T x |= S 0 + S 1 2 | T y |= S 0 S 1 2 tan( τ y τ x )= S 3 S 2 .
τ x = τ y arctan S 3 S 2 =arg[ Pr[ M 2 , d 2 ] ]arctan S 3 S 2 t=| Pr[ Pr[ | T x |exp( i τ x ), d 2 ] M 1 1 , - d 1 ] |.
P( 0;( x i , y i ) )=exp( n p ( x i , y i ) ),
n p ( x i , y i )= N p | T x ( x i , y i ) | 2 i,j=1 N,M | T x ( x i , y i ) | 2 .
| T x | ph ( x i , y i )={ 0, if rand( x i , y i )P( 0;( x i , y i ) ) 1, otherwise
t ph =| Pr[ Pr[ | T x | ph exp( i( arg[ Pr[ M 2 , d 2 ] ]arctan S 3 S 2 ) ), d 2 ] M 1 1 , - d 1 ] |.
t ph t= FT 1 [ FT[ t ph ]FT [ t ] * | FT[ t ph ] | 1k | FT[ t ] | 1l ].

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