Abstract

In this paper, we propose a method using structured-illumination-based diffractive imaging with a laterally-translated phase grating for optical double-image cryptography. An optical cryptosystem is designed, and multiple random phase-only masks are placed in the optical path. When a phase grating is laterally translated just before the plaintexts, several diffraction intensity patterns (i.e., ciphertexts) can be correspondingly obtained. During image decryption, an iterative retrieval algorithm is developed to extract plaintexts from the ciphertexts. In addition, security and advantages of the proposed method are analyzed. Feasibility and effectiveness of the proposed method are demonstrated by numerical simulation results.

© 2011 Optical Society of America

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2011 (3)

2010 (5)

2009 (6)

W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95, 201103 (2009).
[CrossRef]

H. E. Hwang, H. T. Chang, and W. N. 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] [PubMed]

K. Matsushima and T. Shimobaba, “Band-limited angular spectrum method for numerical simulation of free-space propagation in far and near fields,” Opt. Express 17, 19662–19673 (2009).
[CrossRef] [PubMed]

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

Y. Nishino, Y. Takahashi, N. Imamoto, T. Ishikawa, and K. Maeshima, “Three-dimensional visualization of a human chromosome using coherent x-ray diffraction,” Phys. Rev. Lett. 102, 018101 (2009).
[CrossRef] [PubMed]

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

2008 (4)

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[CrossRef]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503(2008).
[CrossRef] [PubMed]

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in x-ray imaging based on using structured illumination,” Phys. Rev. A 78, 023817 (2008).
[CrossRef]

2007 (3)

V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104, 418–423(2007).
[CrossRef] [PubMed]

W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).
[CrossRef]

2006 (6)

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506(2006).
[CrossRef] [PubMed]

M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
[CrossRef] [PubMed]

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. R. Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97, 215503(2006).
[CrossRef] [PubMed]

X. F. Meng, L. Z. Cai, X. F. Xu, X. L. Yang, X. X. Shen, G. Y. Dong, and Y. R. Wang, “Two-step phase-shifting interferometry and its application in image encryption,” Opt. Lett. 31, 1414–1416 (2006).
[CrossRef] [PubMed]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31, 1044–1046 (2006).
[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, 3261–3263 (2006).
[CrossRef] [PubMed]

2004 (5)

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

H. Kim, D. H. Kim, and Y. H. Lee, “Encryption of digital hologram of 3-D object by virtual optics,” Opt. Express 12, 4912–4921 (2004).
[CrossRef] [PubMed]

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

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

2003 (3)

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1542 (2003).
[CrossRef] [PubMed]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

2002 (1)

2001 (1)

I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
[CrossRef] [PubMed]

2000 (2)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

1997 (1)

B. Javidi, “Securing information with optical technologies,” Phys. Today 50, 27–32 (1997).
[CrossRef]

1996 (1)

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

1995 (1)

1982 (1)

Abbey, B.

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[CrossRef]

Alieva, T.

Allen, L. J.

W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
[CrossRef] [PubMed]

Badizadegan, K.

W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503(2008).
[CrossRef] [PubMed]

Cabré, E. P.

O. Matoba, T. Nomura, E. P. Cabré, M. S. Millán, and B. Javidi, “Optical techniques for information security,” Proc. IEEE 97, 1128–1148 (2009).
[CrossRef]

Cai, L. Z.

Chang, H. T.

Chapman, H. N.

K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
[CrossRef] [PubMed]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometer-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[CrossRef]

Chatwin, C.

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

Chen, B.

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in x-ray imaging based on using structured illumination,” Phys. Rev. A 78, 023817 (2008).
[CrossRef]

Chen, C. C.

J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. R. Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97, 215503(2006).
[CrossRef] [PubMed]

Chen, W.

Chen, X.

Choi, W.

W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Clark, J. N.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[CrossRef]

Dao, L. V.

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

Dasari, R. R.

W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503(2008).
[CrossRef] [PubMed]

de Jonge, M.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. de Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4, 394–398 (2008).
[CrossRef]

de Jonge, M. D.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506(2006).
[CrossRef] [PubMed]

Dhal, B. B.

G. J. Williams, H. M. Quiney, B. B. Dhal, C. Q. Tran, K. A. Nugent, A. G. Peele, D. Paterson, and M. D. de Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97, 025506(2006).
[CrossRef] [PubMed]

Dierolf, M.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503(2008).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321, 379–382 (2008).
[CrossRef] [PubMed]

Dilanian, R. A.

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

Dong, G. Y.

Duadi, H.

Elser, V.

V. Elser, I. Rankenburg, and P. Thibault, “Searching with iterated maps,” Proc. Natl. Acad. Sci. U.S.A. 104, 418–423(2007).
[CrossRef] [PubMed]

V. Elser, “Phase retrieval by iterated projections,” J. Opt. Soc. Am. A 20, 40–55 (2003).
[CrossRef]

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93, 023903 (2004).
[CrossRef] [PubMed]

Feld, M. S.

W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
[CrossRef] [PubMed]

Fienup, J. R.

Gao, M.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1542 (2003).
[CrossRef] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Gray, J.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503(2008).
[CrossRef] [PubMed]

Hannaford, P.

B. Chen, R. A. Dilanian, S. Teichmann, B. Abbey, A. G. Peele, G. J. Williams, P. Hannaford, L. V. Dao, H. M. Quiney, and K. A. Nugent, “Multiple wavelength diffractive imaging,” Phys. Rev. A 79, 023809 (2009).
[CrossRef]

Harder, R.

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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. R. Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97, 215503(2006).
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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. R. Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97, 215503(2006).
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J. Miao, C. C. Chen, C. Song, Y. Nishino, Y. Kohmura, T. Ishikawa, D. R. Johnson, T. K. Lee, and S. H. Risbud, “Three-dimensional GaN-Ga2O3 core shell structure revealed by x-ray diffraction microscopy,” Phys. Rev. Lett. 97, 215503(2006).
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K. A. Nugent, A. G. Peele, H. N. Chapman, and A. P. Mancuso, “Unique phase recovery for nonperiodic objects,” Phys. Rev. Lett. 91, 203902 (2003).
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W. McBride, N. L. O’Leary, and L. J. Allen, “Retrieval of a complex-valued object from its diffraction pattern,” Phys. Rev. Lett. 93, 233902 (2004).
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W. Choi, C. F. Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717–719 (2007).
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R. P. Yu and D. M. Paganin, “Blind phase retrieval for aberrated linear shift-invariant imaging systems,” New J. Phys. 12, 073040 (2010).
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Pfeifer, M. A.

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I. K. Robinson, I. A. Vartanyants, G. J. Williams, M. A. Pfeifer, and J. A. Pitney, “Reconstruction of the shapes of gold nanocrystals using coherent x-ray diffraction,” Phys. Rev. Lett. 87, 195505 (2001).
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M. A. Pfeifer, G. J. Williams, I. A. Vartanyants, R. Harder, and I. K. Robinson, “Three-dimensional mapping of a deformation field inside a nanocrystal,” Nature 442, 63–66 (2006).
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W. Chen, C. Quan, and C. J. Tay, “Extended depth of focus in a particle field measurement using a single-shot digital hologram,” Appl. Phys. Lett. 95, 201103 (2009).
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Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in x-ray imaging based on using structured illumination,” Phys. Rev. A 78, 023817 (2008).
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G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
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Zhang, R.

J. M. Zou, I. Vartanyants, M. Gao, R. Zhang, and L. A. Nagahara, “Atomic resolution imaging of a carbon nanotube from diffraction intensities,” Science 300, 1542 (2003).
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Y. J. Liu, B. Chen, E. R. Li, J. Y. Wang, A. Marcelli, S. W. Wilkins, H. Ming, Y. C. Tian, K. A. Nugent, P. P. Zhu, and Z. Y. Wu, “Phase retrieval in x-ray imaging based on using structured illumination,” Phys. Rev. A 78, 023817 (2008).
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Appl. Opt. (5)

Appl. Phys. Lett. (1)

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Original plaintexts (Lena and Peppers): http://sipi.usc.edu/database.

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Figures (9)

Fig. 1
Fig. 1

Schematic experimental arrangement for the proposed structured-illumination-based diffractive imaging: G, phase grating; O, object; CCD, charge-coupled device. Phase grating could be embedded in or displayed by a spatial light modulator (SLM). The distances of pairs 1-1, 2-2, and 3-3 are 512 pixels. During lateral translation in the x - y plane, the displayed area (i.e., pair 1-1) will be fully replaced by information in the pair 2-2 or 3-3 region.

Fig. 2
Fig. 2

Schematic experimental setup for the proposed optical cryptography: P, plaintexts; M1–M4, random phase-only masks. Note that two plaintexts can be separately displayed by using two SLMs.

Fig. 3
Fig. 3

Plaintexts embedded in (a) real amplitude and (b) phase parts of O ( x , y ) \ ; (c) the phase grating with a pitch of 32 pixels; phase gratings translated with (d) 12 pixels and (e) 24 pixels; (f) a typical random phase-only mask (i.e., M1).

Fig. 4
Fig. 4

Diffraction patterns (i.e., ciphertexts) obtained when phase grating information within the region of pair (a) 1-1, (b) 2-2, and (c) 3-3 is used.

Fig. 5
Fig. 5

Correct security keys: relationships between the number of iterations and CC values for the plaintexts embedded at (a) real amplitude and (b) phase parts; decrypted images at (c) real amplitude and (d) phase parts.

Fig. 6
Fig. 6

Decrypted images at (a) real amplitude and (b) phase parts using a wrong phase-only mask M2; decrypted images at (c) real amplitude and (d) phase parts using a wrong wavelength.

Fig. 7
Fig. 7

Decrypted images at (a) real amplitude and (b) phase parts when phase grating pitch is wrong; relationships between the number of iterations and CC values for the plaintexts embedded at (c) real amplitude and (d) phase parts.

Fig. 8
Fig. 8

Decrypted images at (a) real amplitude and (b) phase parts when ciphertexts are contaminated by noise; relationships between the number of iterations, and CC values for the plaintexts embedded at (c) real amplitude and (d) phase parts.

Fig. 9
Fig. 9

Decrypted images at (a) real amplitude and (b) phase parts when ciphertexts are contaminated by occlusions; relationships between the number of iterations and CC values for the plaintexts embedded at (c) real amplitude and (d) phase parts.

Equations (10)

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O ( μ , ν ) = j λ + + O ( x , y ) exp ( j k Ω ) Ω d x d y ,
O ( μ , ν ) = j λ d exp ( j 2 π λ d ) exp [ j π λ d ( μ 2 + ν 2 ) ] × + + O ( x , y ) exp [ j π λ d ( x 2 + y 2 ) ] × exp [ j 2 π λ d ( x μ + y ν ) ] d x d y .
I h ( μ , ν ) = | W P d 4 [ ( W P d 3 { [ W P d 2 ( { W P d 1 [ W h ( x , y ) O ( x , y ) M 1 ( x , y ) ] } × M 2 ( ξ 1 , η 1 ) ) ] M 3 ( ξ 2 , η 2 ) } ) M 4 ( ξ 3 , η 3 ) ] | 2 ,
O n 1 ( μ , v ) = W P d 4 [ ( W P d 3 { [ W P d 2 ( { W P d 1 [ T n ( x , y ) M 1 ( x , y ) ] } × M 2 ( ξ 1 , η 1 ) ) ] M 3 ( ξ 2 , η 2 ) } ) M 4 ( ξ 3 , η 3 ) ] ;
T n 1 ( x , y ) ¯ = ( W P d 1 { ( W P d 2 { [ W P d 3 ( { W P d 4 [ O n 1 ( μ , ν ) ¯ ] } M 4 * ( ξ 3 , η 3 ) ) ] × [ M 3 * ( ξ 2 , η 2 ) ] } ) [ M 2 * ( ξ 1 , η 1 ) ] } ) { M 1 * ( x , y ) [ W 1 ( x , y ) ] * } ,
Error = x , y [ | T n 3 ( x , y ) ¯ | | T 1 ( x , y ) | ] 2 ( n = 1 ) ; Error = x , y [ | T n 3 ( x , y ) ¯ | | T n 1 3 ( x , y ) ¯ | ] 2 ( n > 1 ) .
O r ( x , y ) = abs [ T n 3 ( x , y ) ¯ ] ,
O p ( x , y ) = angle [ T n 3 ( x , y ) ¯ ] ,
CC _ R = x y [ abs ( O ) abs ( O ) ¯ ] ( O r O r ¯ ) { x y [ abs ( O ) abs ( O ) ¯ ] 2 } [ x y ( O r O r ¯ ) 2 ] ,
CC _ P = x y [ angle ( O ) angle ( O ) ¯ ] ( O p O p ¯ ) { x y [ angle ( O ) angle ( O ) ¯ ] 2 } [ x y ( O p O p ¯ ) 2 ] ,

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