Abstract

We propose a novel optical information authentication scheme that uses Stokes polarimetry of vector beams. Phase-only functions of the plaintext are generated using a modified Gerchberg–Saxton algorithm. Partial information of these phase functions is used to tailor the phase of a vector beam. The Stokes parameters recording of the vector beam generates the ciphertext, which contains sparse information of the input image. In contrast to most authentication schemes, the proposed scheme can authenticate two images from a single real ciphertext. Computer simulation results prove the feasibility of the proposed scheme.

© 2018 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]
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    [Crossref]
  29. J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
    [Crossref]
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    [Crossref]
  32. A. Fatima and N. K. Nishchal, “Optical image security using Stokes polarimetry of spatially variant polarized beam,” Opt. Commun. 417, 30–36 (2018).
    [Crossref]
  33. X. Wang, J. Ding, W. Ni, C. Guo, and H. Wang, “Generation of arbitrary vector beams with a spatial light modulator and a common path interferometric arrangement,” Opt. Lett. 32, 3549–3551 (2007).
    [Crossref]
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    [Crossref]

2018 (1)

A. Fatima and N. K. Nishchal, “Optical image security using Stokes polarimetry of spatially variant polarized beam,” Opt. Commun. 417, 30–36 (2018).
[Crossref]

2017 (5)

F. Yi, Y. Jeoung, and I. Moon, “Three-dimensional image authentication scheme using sparse phase information in double random phase encoded integral imaging,” Appl. Opt. 56, 4381–4387 (2017).
[Crossref]

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

J. Cai, X. Shen, and M. Lei, “Optical asymmetric cryptography based on amplitude reconstruction of elliptically polarized light,” Opt. Commun. 403, 211–216 (2017).
[Crossref]

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

C. Lin and J. Ren, “Security validation based on orthogonal polarization multiplexing in three-dimensional space,” Appl. Opt. 56, 9583–9591 (2017).
[Crossref]

2016 (3)

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

A. Fatima, I. Mehra, and N. K. Nishchal, “Optical asymmetric cryptosystem using equal modulus decomposition and multiple diffractive imaging,” J. Opt. 18, 085701 (2016).
[Crossref]

E. A. Mohammed and H. L. Saadon, “Optical double-image encryption and authentication by sparse representation,” Appl. Opt. 55, 9939–9944 (2016).
[Crossref]

2015 (5)

X. Wang, W. Chen, and X. Chen, “Optical encryption and authentication based on phase retrieval and sparsity constraints,” IEEE Photon. J. 7, 7800310 (2015).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

A. Carnicer, A. Hassanfiroozi, P. Latorre-Carmona, Y. Huang, and B. Javidi, “Security authentication using phase-encoded nanoparticle structures and polarized light,” Opt. Lett. 40, 135–138 (2015).
[Crossref]

D. Maluenda, A. Carnicer, R. M. Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23, 655–666 (2015).
[Crossref]

C. Lin, X. Shen, B. Hua, and Z. Wang, “Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam,” Opt. Commun. 352, 25–32 (2015).
[Crossref]

2014 (5)

2013 (3)

2012 (1)

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification scheme using fractional nonconventional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

2011 (2)

2010 (2)

2009 (1)

2007 (1)

2006 (2)

2005 (2)

2000 (1)

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[Crossref]

1989 (1)

Alfalou, A.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

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

A. Alfalou, Q. Wang, and C. Brosseau, “Multiple-image authentication based on random sparse encoding of optical discrete cosine transform,” in Frontiers in Optics, OSA Technical Digest (Optical Society of America, 2017), paper JTu2A.96.

Arcos, S.

Barrera, J. F.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Biener, G.

Brosseau, C.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

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

A. Alfalou, Q. Wang, and C. Brosseau, “Multiple-image authentication based on random sparse encoding of optical discrete cosine transform,” in Frontiers in Optics, OSA Technical Digest (Optical Society of America, 2017), paper JTu2A.96.

Cai, J.

J. Cai, X. Shen, and M. Lei, “Optical asymmetric cryptography based on amplitude reconstruction of elliptically polarized light,” Opt. Commun. 403, 211–216 (2017).
[Crossref]

Carnicer, A.

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

A. Carnicer, A. Hassanfiroozi, P. Latorre-Carmona, Y. Huang, and B. Javidi, “Security authentication using phase-encoded nanoparticle structures and polarized light,” Opt. Lett. 40, 135–138 (2015).
[Crossref]

D. Maluenda, A. Carnicer, R. M. Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23, 655–666 (2015).
[Crossref]

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

Chang, H. T.

Chen, H.

Chen, J.

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Chen, W.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

X. Wang, W. Chen, and X. Chen, “Optical encryption and authentication based on phase retrieval and sparsity constraints,” IEEE Photon. J. 7, 7800310 (2015).
[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, 6900113 (2013).
[Crossref]

Chen, X.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

X. Wang, W. Chen, and X. Chen, “Optical encryption and authentication based on phase retrieval and sparsity constraints,” IEEE Photon. J. 7, 7800310 (2015).
[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, 6900113 (2013).
[Crossref]

Chen, Z.

Cho, M.

Ding, J.

Fatima, A.

A. Fatima and N. K. Nishchal, “Optical image security using Stokes polarimetry of spatially variant polarized beam,” Opt. Commun. 417, 30–36 (2018).
[Crossref]

A. Fatima, I. Mehra, and N. K. Nishchal, “Optical asymmetric cryptosystem using equal modulus decomposition and multiple diffractive imaging,” J. Opt. 18, 085701 (2016).
[Crossref]

Fu, C.

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Gong, Q.

Guo, C.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

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

Hao, J.

Hasman, E.

Hassanfiroozi, A.

He, W.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Herrero, R. M.

Hua, B.

C. Lin, X. Shen, B. Hua, and Z. Wang, “Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam,” Opt. Commun. 352, 25–32 (2015).
[Crossref]

Huang, Y.

Hwang, H.

Javidi, B.

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

A. Carnicer, A. Hassanfiroozi, P. Latorre-Carmona, Y. Huang, and B. Javidi, “Security authentication using phase-encoded nanoparticle structures and polarized light,” Opt. Lett. 40, 135–138 (2015).
[Crossref]

D. Maluenda, A. Carnicer, R. M. Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23, 655–666 (2015).
[Crossref]

A. Markman and B. Javidi, “Full-phase photon-counting double random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (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, 6900113 (2013).
[Crossref]

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

M. S. Millan, E. Perez-Cabre, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
[Crossref]

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

Jeoung, Y.

Juvells, I.

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

D. Maluenda, A. Carnicer, R. M. Herrero, I. Juvells, and B. Javidi, “Optical encryption using photon-counting polarimetric imaging,” Opt. Express 23, 655–666 (2015).
[Crossref]

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

Kleiner, V.

Kumar, D.

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16, 125406 (2014).
[Crossref]

Lancis, J.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Latorre-Carmona, P.

Lei, M.

J. Cai, X. Shen, and M. Lei, “Optical asymmetric cryptography based on amplitude reconstruction of elliptically polarized light,” Opt. Commun. 403, 211–216 (2017).
[Crossref]

Li, G.

Lie, W.

Lin, C.

Liu, X.

Maluenda, D.

Markman, A.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

A. Markman and B. Javidi, “Full-phase photon-counting double random-phase encryption,” J. Opt. Soc. Am. A 31, 394–403 (2014).
[Crossref]

Martinez-Herrero, R.

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

Matsumoto, T.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Mehra, I.

A. Fatima, I. Mehra, and N. K. Nishchal, “Optical asymmetric cryptosystem using equal modulus decomposition and multiple diffractive imaging,” J. Opt. 18, 085701 (2016).
[Crossref]

Millan, M. S.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

M. S. Millan, E. Perez-Cabre, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
[Crossref]

Mohammed, E. A.

E. A. Mohammed and H. L. Saadon, “Optical double-image encryption and authentication by sparse representation,” Appl. Opt. 55, 9939–9944 (2016).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

Moon, I.

Mosk, A. P.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Naruse, M.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Naughton, T. J.

Ni, W.

Nishchal, N. K.

A. Fatima and N. K. Nishchal, “Optical image security using Stokes polarimetry of spatially variant polarized beam,” Opt. Commun. 417, 30–36 (2018).
[Crossref]

A. Fatima, I. Mehra, and N. K. Nishchal, “Optical asymmetric cryptosystem using equal modulus decomposition and multiple diffractive imaging,” J. Opt. 18, 085701 (2016).
[Crossref]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16, 125406 (2014).
[Crossref]

S. K. Rajput and N. K. Nishchal, “An optical encryption and authentication scheme using asymmetric keys,” J. Opt. Soc. Am. A 31, 1233–1238 (2014).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in Fresnel domain,” Appl. Opt. 52, 4343–4352 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification scheme using fractional nonconventional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

Niv, A.

Nomura, T.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Peng, X.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

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

Perez-Cabre, E.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

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

M. S. Millan, E. Perez-Cabre, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
[Crossref]

Pinkse, P. W. H.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Pohit, M.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[Crossref]

Qin, W.

Qin, Y.

Rajput, S. K.

S. K. Rajput and N. K. Nishchal, “An optical encryption and authentication scheme using asymmetric keys,” J. Opt. Soc. Am. A 31, 1233–1238 (2014).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16, 125406 (2014).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in Fresnel domain,” Appl. Opt. 52, 4343–4352 (2013).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification scheme using fractional nonconventional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

Ren, J.

Rivenson, Y.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Saadon, H. L.

E. A. Mohammed and H. L. Saadon, “Optical double-image encryption and authentication by sparse representation,” Appl. Opt. 55, 9939–9944 (2016).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

Shen, X.

J. Cai, X. Shen, and M. Lei, “Optical asymmetric cryptography based on amplitude reconstruction of elliptically polarized light,” Opt. Commun. 403, 211–216 (2017).
[Crossref]

C. Lin, X. Shen, B. Hua, and Z. Wang, “Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam,” Opt. Commun. 352, 25–32 (2015).
[Crossref]

C. Lin, X. Shen, Z. Wang, and C. Zhao, “Optical asymmetric cryptography based on elliptical polarized light linear truncation and a numerical reconstruction technique,” Appl. Opt. 53, 3920–3928 (2014).
[Crossref]

Sheridan, J. T.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45, 5693–5700 (2006).
[Crossref]

Singh, K.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[Crossref]

Situ, G.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Stern, A.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[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, 6900113 (2013).
[Crossref]

Tajahuerce, E.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Torroba, R.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Unnikrishnan, G.

G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45, 5693–5700 (2006).
[Crossref]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[Crossref]

Usategui, M. M.

Wang, H.

Wang, Q.

A. Alfalou, Q. Wang, and C. Brosseau, “Multiple-image authentication based on random sparse encoding of optical discrete cosine transform,” in Frontiers in Optics, OSA Technical Digest (Optical Society of America, 2017), paper JTu2A.96.

Wang, X.

X. Wang, W. Chen, and X. Chen, “Optical encryption and authentication based on phase retrieval and sparsity constraints,” IEEE Photon. J. 7, 7800310 (2015).
[Crossref]

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

Wang, Z.

C. Lin, X. Shen, B. Hua, and Z. Wang, “Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam,” Opt. Commun. 352, 25–32 (2015).
[Crossref]

C. Lin, X. Shen, Z. Wang, and C. Zhao, “Optical asymmetric cryptography based on elliptical polarized light linear truncation and a numerical reconstruction technique,” Appl. Opt. 53, 3920–3928 (2014).
[Crossref]

Xu, J.

Yamaguchi, M.

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

Yi, F.

Yu, Z.

Zhang, B.

Zhang, L.

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Zhang, Y.

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Zhao, C.

Zhu, Z.

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Appl. Opt. (9)

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

G. Unnikrishnan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45, 5693–5700 (2006).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in Fresnel domain,” Appl. Opt. 52, 4343–4352 (2013).
[Crossref]

Q. Gong, X. Liu, G. Li, and Y. Qin, “Multiple-image encryption and authentication with sparse representation by space multiplexing,” Appl. Opt. 52, 7486–7493 (2013).
[Crossref]

J. Hao, Z. Yu, H. Chen, Z. Chen, H. Wang, and J. Ding, “Light field shaping by tailoring both phase and polarization,” Appl. Opt. 53, 785–791 (2014).
[Crossref]

C. Lin, X. Shen, Z. Wang, and C. Zhao, “Optical asymmetric cryptography based on elliptical polarized light linear truncation and a numerical reconstruction technique,” Appl. Opt. 53, 3920–3928 (2014).
[Crossref]

E. A. Mohammed and H. L. Saadon, “Optical double-image encryption and authentication by sparse representation,” Appl. Opt. 55, 9939–9944 (2016).
[Crossref]

F. Yi, Y. Jeoung, and I. Moon, “Three-dimensional image authentication scheme using sparse phase information in double random phase encoded integral imaging,” Appl. Opt. 56, 4381–4387 (2017).
[Crossref]

C. Lin and J. Ren, “Security validation based on orthogonal polarization multiplexing in three-dimensional space,” Appl. Opt. 56, 9583–9591 (2017).
[Crossref]

IEEE Photon. J. (2)

X. Wang, W. Chen, and X. Chen, “Optical encryption and authentication based on phase retrieval and sparsity constraints,” IEEE Photon. J. 7, 7800310 (2015).
[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, 6900113 (2013).
[Crossref]

J. Opt. (4)

A. Fatima, I. Mehra, and N. K. Nishchal, “Optical asymmetric cryptosystem using equal modulus decomposition and multiple diffractive imaging,” J. Opt. 18, 085701 (2016).
[Crossref]

B. Javidi, A. Carnicer, M. Yamaguchi, T. Nomura, E. Perez-Cabre, M. S. Millan, N. K. Nishchal, R. Torroba, J. F. Barrera, W. He, X. Peng, A. Stern, Y. Rivenson, A. Alfalou, C. Brosseau, C. Guo, J. T. Sheridan, G. Situ, M. Naruse, T. Matsumoto, I. Juvells, E. Tajahuerce, J. Lancis, W. Chen, X. Chen, P. W. H. Pinkse, A. P. Mosk, and A. Markman, “Roadmap to optical security,” J. Opt. 18, 083001 (2016).
[Crossref]

S. K. Rajput, D. Kumar, and N. K. Nishchal, “Photon counting imaging and polarized light encoding for secure image verification and hologram watermarking,” J. Opt. 16, 125406 (2014).
[Crossref]

E. Perez-Cabre, E. A. Mohammed, M. S. Millan, and H. L. Saadon, “Photon-counting multifactor optical encryption and authentication,” J. Opt. 17, 025706 (2015).
[Crossref]

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

Opt. Commun. (4)

A. Fatima and N. K. Nishchal, “Optical image security using Stokes polarimetry of spatially variant polarized beam,” Opt. Commun. 417, 30–36 (2018).
[Crossref]

C. Lin, X. Shen, B. Hua, and Z. Wang, “Three-dimensional polarization marked multiple-QR code encryption by optimizing a single vectorial beam,” Opt. Commun. 352, 25–32 (2015).
[Crossref]

J. Cai, X. Shen, and M. Lei, “Optical asymmetric cryptography based on amplitude reconstruction of elliptically polarized light,” Opt. Commun. 403, 211–216 (2017).
[Crossref]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185, 25–31 (2000).
[Crossref]

Opt. Express (1)

Opt. Lasers Eng. (2)

A. Carnicer, I. Juvells, B. Javidi, and R. Martinez-Herrero, “Optical encryption in the axial domain using beams with arbitrary polarization,” Opt. Lasers Eng. 89, 145–149 (2017).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption and authentication verification scheme using fractional nonconventional joint transform correlator,” Opt. Lasers Eng. 50, 1474–1483 (2012).
[Crossref]

Opt. Lett. (10)

G. Biener, A. Niv, V. Kleiner, and E. Hasman, “Geometrical phase image encryption obtained with space-variant subwavelength gratings,” Opt. Lett. 30, 1096–1098 (2005).
[Crossref]

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

M. S. Millan, E. Perez-Cabre, and B. Javidi, “Multifactor authentication reinforces optical security,” Opt. Lett. 31, 721–723 (2006).
[Crossref]

A. Carnicer, A. Hassanfiroozi, P. Latorre-Carmona, Y. Huang, and B. Javidi, “Security authentication using phase-encoded nanoparticle structures and polarized light,” Opt. Lett. 40, 135–138 (2015).
[Crossref]

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

H. Hwang, H. T. Chang, and W. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
[Crossref]

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

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

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

H. Chen, J. Hao, B. Zhang, J. Xu, J. Ding, and H. Wang, “Generation of vector beam with space-variant distribution of both polarization and phase,” Opt. Lett. 36, 3179–3181 (2011).
[Crossref]

Optik (1)

J. Chen, Z. Zhu, C. Fu, L. Zhang, and Y. Zhang, “Information authentication using sparse representation of double random phase encoding in fractional Fourier transform domain,” Optik 136, 1–7 (2017).
[Crossref]

Other (1)

A. Alfalou, Q. Wang, and C. Brosseau, “Multiple-image authentication based on random sparse encoding of optical discrete cosine transform,” in Frontiers in Optics, OSA Technical Digest (Optical Society of America, 2017), paper JTu2A.96.

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Figures (14)

Fig. 1.
Fig. 1. Flowchart showing MGSA to evaluate POFs. n subscript represents the iteration number of the process.
Fig. 2.
Fig. 2. Schematic for the proposed optical setup for the image authentication system. HG, holographic grating; L1, L2, lenses; f, focal length; SF, spatial filter; QW, quarter wave plate; RG, Ronchi grating.
Fig. 3.
Fig. 3. Flowchart for the proposed image authentication scheme.
Fig. 4.
Fig. 4. (a) True class object. (b) False class object.
Fig. 5.
Fig. 5. (a) First POF distribution ϕ1. (b) Second POF distribution ϕ2 for true class image. (c) First POF distribution ϕ1f. (d) Second POF distribution ϕ2f for false class image. (e) First sparse POF distribution ϕ1. (f) Second sparse POF distribution ϕ2 for true class image. (g) First POF distribution ϕ1. (h) Second POF distribution ϕ2 for false class image, with sparsity percentage as 4% (for all the above four cases).
Fig. 6.
Fig. 6. Sparse encrypted image: (a) true class and (b) false class. (c) Spatially variant polarization distribution of the encoded vector beam. Decrypted image from sparse encryption: (d) true class and (e) false class.
Fig. 7.
Fig. 7. (a)–(c) PCE results of decrypted images for true class and false class binary images used in single image authentication. (d) DR values for different nonlinearity index and sparsity percentage.
Fig. 8.
Fig. 8. Sparse phase (4%) single image encrypted nonlinear cross correlation. Index used k=0.3: (a) true class object and (b) false class object.
Fig. 9.
Fig. 9. True class binary image encryption: (a) input image T1, (b) input Image T2, (c) sparse POF α1, (d) sparse POF α2, (e) first decrypted image Td1, and (f) second decrypted image Td2. False class binary image encryption: (g) input image F1, (h) input image F2, (i) sparse POF α1f, (j) sparse POF α2f, (k) first decrypted image Fd1, and (l) second decrypted image Fd2. Sparsity percentage is 4%.
Fig. 10.
Fig. 10. Sparse phase (4%) double binary images encrypted nonlinear cross correlation. Index used k=0.3: (a) true class, Td1T1; (b) false class, Fd1T1; (c) true class, Td2T2; and (d) false class, Fd2T2. Symbol represents nonlinear correlation here.
Fig. 11.
Fig. 11. True class image encryption: (a) input image Tg1, (b) input Image Tg2, (c) sparse POF ϕ1, (d) sparse POF ϕ2, (e) first decrypted image Tgd1, and (f) second decrypted image Tgd2. False class image encryption: (g) input image Fg1, (h) input image Fg2, (i) sparse POF ϕ1f, (j) sparse POF ϕ2f, (k) first decrypted image Fgd1, and (l) second decrypted image Fgd2. Sparsity percentage is 14%.
Fig. 12.
Fig. 12. Sparse phase (14%) double images encrypted nonlinear cross correlation. Index used k=0.3: (a) true class, Tgd1Tg1; (b) false class, Fgd1Tg1; (c) true class, Tgd2Tg2; and (d) false class, Fgd2Tg2. Symbol represents nonlinear correlation here.
Fig. 13.
Fig. 13. (a), (b) PCE results of decrypted images for true class and false class gray-scale images used in double image authentication. (c) DR values for different nonlinearity index and sparsity percentage.
Fig. 14.
Fig. 14. Gaussian noise contaminated ciphertext (a) corresponding to Fig. 4(a) of binary true class image and (b) corresponding to Fig. 4(b) of binary false class image. (c) Autocorrelation peak between decrypted image from true class ciphertext of (a) and true class image of Fig. 4(a); (d) cross-correlation peak between decrypted image from false class ciphertext of (b) and true class image of Fig. 4(a). Gaussian noise contaminated ciphertext (e) corresponding to gray true class image of Fig. 11(a) and (f) corresponding to gray false class image Fig. 11(g). (g) Autocorrelation between decrypted image from true class ciphertext of (e) and true class image of Fig. 11(a); (h) cross-correlation peak between decrypted image from false class ciphertext of (f) and true class image of Fig. 11(a). Nonlinearity index k=0.3, and sparsity percentage is 4% and 14% for binary and gray images, respectively.

Equations (11)

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t(x,y)=0.5+m[cos(2πf0x+ϕ1(x,y))+cos(2πf0y+ϕ2(x,y))]/4.
E(u,v)=exp(iϕ1)(1i)+exp(iϕ2)(1i).
S1=2Im(ErEl*)=2Sin(ϕ2ϕ1),
S2=2Re(ErEl*)=2Cos(ϕ2ϕ1).
H=phase(exp[i(ϕ2ϕ1)]).
P=H+ϕ1.
J=tan1S1S2.
ϕ2=PJ.
I=|IFT[exp(i(PJ))]|.
C(x,y)=IFT[|T(u,v)R(u,v)|kexp{i(ϕT(u,v)ϕR(u,v))}].
PCE=max[|C(x,y)|2]x=1My=1N|C(x,y)|2,

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