Abstract

Images can be optically encrypted by random encoding in the phase, the polarization, or even the coherence of a light field. It is important for these optical encryption methods to undergo rigorous cryptanalysis. However, only phase-encoding-based encryption has been rigorously analyzed to date. In this manuscript, we demonstrate that the double random polarization encryption (DRPolE) is vulnerable to chosen-plaintext attack (CPA). We show that the keys can be retrieved if one can choose the polarization states of two plaintext images and collect the corresponding cyphertext images. Our study reveals a serious concern regarding the DRPolE that should be addressed in the design of polarization-based optical encryption methods.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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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]
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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]
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    [Crossref]
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    [Crossref]
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    [Crossref]

2019 (2)

2017 (3)

G. Li, W. Yang, D. Li, and G. Situ, “Cyphertext-only attack on the double random-phase encryption: experimental demonstration,” Opt. Express 25(8), 8690–8697 (2017).
[Crossref]

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

M. Liao, W. He, D. Lu, and X. Peng, “Ciphertext-only attack on optical cryptosystem with spatially incoherent illumination: from the view of imaging through scattering medium,” Sci. Rep. 7(1), 41789 (2017).
[Crossref]

2016 (3)

2015 (2)

2013 (1)

2012 (1)

M. Dubreuil, A. Alfalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using the Stokes-Mueller formalism,” J. Opt. 14(9), 094004 (2012).
[Crossref]

2010 (3)

2008 (2)

D. S. Monahgan, G. Situ, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Role of phase key in the double random phase encoding technique: an error analysis,” Appl. Opt. 47(21), 3808–3816 (2008).
[Crossref]

A. M. Youssef, “On the security of a cryptosystem based on multiple-parameters discrete fractional Fourier transform,” IEEE Signal Process. Lett. 15, 77–78 (2008).
[Crossref]

2007 (2)

2006 (3)

2005 (1)

2004 (5)

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

A. Zlotnik, Z. Zalevsky, and E. Marom, “Optical encryption by using a synthesized mutual intensity function,” Appl. Opt. 43(17), 3456–3465 (2004).
[Crossref]

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

C. J. Cheng and M. L. Chen, “Polarization encoding for optical encryption using twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 237(1-3), 45–52 (2004).
[Crossref]

2001 (2)

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

X Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40(14), 2310–2315 (2001).
[Crossref]

2000 (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]

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2035 (2000).
[Crossref]

T. Nomura and B. Javidi, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[Crossref]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000).
[Crossref]

1995 (1)

Abookasis, D.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

Alfalou, A.

M. Dubreuil, A. Alfalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using the Stokes-Mueller formalism,” J. Opt. 14(9), 094004 (2012).
[Crossref]

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

Andrade, A. H. R.

Arazi, O.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

Arcos, S.

Brosseau, C.

M. Dubreuil, A. Alfalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using the Stokes-Mueller formalism,” J. Opt. 14(9), 094004 (2012).
[Crossref]

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

Carnicer, A.

Castro, A.

Chen, K.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Chen, M. L.

C. J. Cheng and M. L. Chen, “Polarization encoding for optical encryption using twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 237(1-3), 45–52 (2004).
[Crossref]

Chen, W.

Chen, X.

Cheng, C. J.

C. J. Cheng and M. L. Chen, “Polarization encoding for optical encryption using twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 237(1-3), 45–52 (2004).
[Crossref]

Dubreuil, M.

M. Dubreuil, A. Alfalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using the Stokes-Mueller formalism,” J. Opt. 14(9), 094004 (2012).
[Crossref]

Falaggis, K.

Frauel, Y.

Gopinathan, U.

Guo, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Hai, H.

He, W.

Ide, M.

Javidi, B.

Y. Frauel, A. Castro, T. J. Nauqhton, and B. Javidi, “Resistance of the double random phase encryption against various attacks,” Opt. Express 15(16), 10253–10265 (2007).
[Crossref]

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

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2035 (2000).
[Crossref]

T. Nomura and B. Javidi, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[Crossref]

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]

Joseph, J.

Juvells, I.

Kuroda, K.

Li, D.

Li, G.

Li, T.

Liao, M.

Liu, J.

Liu, S.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Liu, X.

Liu, Z.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Lu, D.

H. Hai, S. Pan, M. Liao, D. Lu, W. He, and X. Peng, “Cryptanalysis of random-phase-encoding-based optical cryptosystem via deep learning,” Opt. Express 27(15), 21204–21213 (2019).
[Crossref]

M. Liao, W. He, D. Lu, and X. Peng, “Ciphertext-only attack on optical cryptosystem with spatially incoherent illumination: from the view of imaging through scattering medium,” Sci. Rep. 7(1), 41789 (2017).
[Crossref]

Luna, J. G. G.

Luo, Y.

Marom, E.

Matoba, O.

Monaghan, D. S.

Monahgan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Nauqhton, T. J.

Nishchal, N. K.

Nomura, T.

T. Nomura and B. Javidi, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[Crossref]

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2035 (2000).
[Crossref]

Ojeda, C. G.

Okada-Shudo, Y.

Osten, W.

Pan, S.

Pedrini, G.

Peng, X.

Pohit, M.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000).
[Crossref]

Porras-Aguilar, R.

Rajput, S. K.

Refregier, P.

Rosen, J.

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

Sheppard, C. J. R.

Sheridan, J. T.

Shi, Y. S.

Shimura, T.

Singh, K.

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000).
[Crossref]

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]

Situ, G.

L. Wang, G. Li, Q. Wu, and G. Situ, “Cyphertext-only attack on the joint-transform-correlator-based optical encryption: experimental demonstration,” Appl. Opt. 58(5), A197–A201 (2019).
[Crossref]

G. Li, W. Yang, D. Li, and G. Situ, “Cyphertext-only attack on the double random-phase encryption: experimental demonstration,” Opt. Express 25(8), 8690–8697 (2017).
[Crossref]

X. Xu, Q. Wu, J. Liu, and G. Situ, “Decryption with incomplete cyphertext and multiple-information encryption in phase space,” Opt. Express 24(2), 1734–1746 (2016).
[Crossref]

J. Liu, X. Xu, Q. Wu, J. T. Sheridan, and G. Situ, “Information encryption in phase space,” Opt. Lett. 40(6), 859–862 (2015).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Strategy for cryptanalysis of optical encryption in the Fresnel domain,” Appl. Opt. 49(3), 457–462 (2010).
[Crossref]

D. S. Monahgan, G. Situ, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Role of phase key in the double random phase encoding technique: an error analysis,” Appl. Opt. 47(21), 3808–3816 (2008).
[Crossref]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with signficant output images,” Appl. Opt. 46(22), 5257–5262 (2007).
[Crossref]

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

Tan, J.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Tan, X

Unnikrishnan, G.

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]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000).
[Crossref]

Wang, L.

Wei, C.

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Wei, H. Z.

Wu, J.

Wu, Q.

Xu, H. F.

Xu, W. H.

Xu, X.

Yang, W.

Youssef, A. M.

A. M. Youssef, “On the security of a cryptosystem based on multiple-parameters discrete fractional Fourier transform,” IEEE Signal Process. Lett. 15, 77–78 (2008).
[Crossref]

Yu, B.

Zalevsky, Z.

Zhang, C.

Zhang, J.

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

Zhang, P.

Zlotnik, A.

Appl. Opt. (9)

X Tan, O. Matoba, Y. Okada-Shudo, M. Ide, T. Shimura, and K. Kuroda, “Secure optical memory system with polarization encryption,” Appl. Opt. 40(14), 2310–2315 (2001).
[Crossref]

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

A. Zlotnik, Z. Zalevsky, and E. Marom, “Optical encryption by using a synthesized mutual intensity function,” Appl. Opt. 43(17), 3456–3465 (2004).
[Crossref]

U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Polarization encoding and multiplexing of two-dimensional signals: application to image encryption,” Appl. Opt. 45(22), 5693–5700 (2006).
[Crossref]

D. S. Monahgan, G. Situ, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Role of phase key in the double random phase encoding technique: an error analysis,” Appl. Opt. 47(21), 3808–3816 (2008).
[Crossref]

G. Situ, G. Pedrini, and W. Osten, “Strategy for cryptanalysis of optical encryption in the Fresnel domain,” Appl. Opt. 49(3), 457–462 (2010).
[Crossref]

G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with signficant output images,” Appl. Opt. 46(22), 5257–5262 (2007).
[Crossref]

S. K. Rajput and N. K. Nishchal, “Image encryption using polarized light encoding and amplitude and phase truncation in the Fresnel domain,” Appl. Opt. 52(18), 4343–4352 (2013).
[Crossref]

L. Wang, G. Li, Q. Wu, and G. Situ, “Cyphertext-only attack on the joint-transform-correlator-based optical encryption: experimental demonstration,” Appl. Opt. 58(5), A197–A201 (2019).
[Crossref]

IEEE Signal Process. Lett. (1)

A. M. Youssef, “On the security of a cryptosystem based on multiple-parameters discrete fractional Fourier transform,” IEEE Signal Process. Lett. 15, 77–78 (2008).
[Crossref]

J. Opt. (1)

M. Dubreuil, A. Alfalou, and C. Brosseau, “Robustness against attacks of dual polarization encryption using the Stokes-Mueller formalism,” J. Opt. 14(9), 094004 (2012).
[Crossref]

Opt. Commun. (3)

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]

G. Unnikrishnan, M. Pohit, and K. Singh, “A polarization encoded optical encryption system using ferroelectric spatial light modulator,” Opt. Commun. 185(1-3), 25–31 (2000).
[Crossref]

C. J. Cheng and M. L. Chen, “Polarization encoding for optical encryption using twisted nematic liquid crystal spatial light modulators,” Opt. Commun. 237(1-3), 45–52 (2004).
[Crossref]

Opt. Eng. (3)

T. Nomura and B. Javidi, “Optical encryption using a joint transform correlator architecture,” Opt. Eng. 39(8), 2031–2035 (2000).
[Crossref]

D. Abookasis, O. Arazi, J. Rosen, and B. Javidi, “Security optical systems based on a joint transform correlator with significant output images,” Opt. Eng. 40(8), 1584–1589 (2001).
[Crossref]

T. Nomura and B. Javidi, “Polarization encoding for optical security systems,” Opt. Eng. 39(9), 2439–2443 (2000).
[Crossref]

Opt. Express (6)

Opt. Laser Eng. (1)

C. Guo, C. Wei, J. Tan, K. Chen, S. Liu, Q. Wu, and Z. Liu, “A review of iterative phase retrieval for measurement and encryption,” Opt. Laser Eng. 89, 2–12 (2017).
[Crossref]

Opt. Lett. (10)

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]

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]

X. Peng, H. Z. 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]

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

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]

X. Peng, P. Zhang, H. Z. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31(8), 1044–1046 (2006).
[Crossref]

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[Crossref]

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[Crossref]

K. Falaggis, A. H. R. Andrade, J. G. G. Luna, C. G. Ojeda, and R. Porras-Aguilar, “Optical encryption with protection against Dirac delta and plain signal attacks,” Opt. Lett. 41(20), 4787–4790 (2016).
[Crossref]

Sci. Rep. (1)

M. Liao, W. He, D. Lu, and X. Peng, “Ciphertext-only attack on optical cryptosystem with spatially incoherent illumination: from the view of imaging through scattering medium,” Sci. Rep. 7(1), 41789 (2017).
[Crossref]

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Figures (3)

Fig. 1.
Fig. 1. Schematic setup of the double random polarization encryption system.
Fig. 2.
Fig. 2. Simulation results for $N$ being even. (a) a binary plaintext image “USTS”, (b) its cyphertext, (c) the reconstructed plaintext image “USTS” with the proposed method, (d) a gray-tone plaintext image, (e) its cyphertext, and (f) the reconstructed plaintext image with the proposed method.
Fig. 3.
Fig. 3. Simulation results for $N$ being odd. (a) a binary plaintext image “USTS”, (b) its cyphertext, (c) the reconstructed plaintext image “USTS” with the proposed method, (d) a gray-tone plaintext image, (e) its cyphertext, and (f) the reconstructed plaintext image with the proposed method.

Equations (18)

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e ( j , k ) = F { Q ( l , m ) F { P ( j , k ) f ( j , k ) } } ,
P ( j , k ) = [ cos [ θ ( j , k ) ] sin [ θ ( j , k ) ] sin [ θ ( j , k ) ] cos [ θ ( j , k ) ] ] [ e i Δ ( j , k ) 2 0 0 e i Δ ( j , k ) 2 ] [ cos [ θ ( j , k ) ] sin [ θ ( j , k ) ] sin [ θ ( j , k ) ] cos [ θ ( j , k ) ] ] = [ cos [ Δ ( j , k ) 2 ] i sin [ Δ ( j , k ) 2 ] cos [ 2 θ ( j , k ) ] i sin [ Δ ( j , k ) 2 ] sin [ 2 θ ( j , k ) ] i sin [ Δ ( j , k ) 2 ] sin [ 2 θ ( j , k ) ] cos [ Δ ( j , k ) 2 ] + i sin [ Δ ( j , k ) 2 ] cos [ 2 θ ( j , k ) ] ] ,
Q ( l , m ) = [ cos [ θ ( l , m ) ] sin [ θ ( l , m ) ] sin [ θ ( l , m ) ] cos [ θ ( l , m ) ] ] [ e i Δ ( l , m ) 2 0 0 e i Δ ( l , m ) 2 ] [ cos [ θ ( l , m ) ] sin [ θ ( l , m ) ] sin [ θ ( l , m ) ] cos [ θ ( l , m ) ] ] = [ cos [ Δ ( l , m ) 2 ] i sin [ Δ ( l , m ) 2 ] cos [ 2 θ ( l , m ) ] i sin [ Δ ( l , m ) 2 ] sin [ 2 θ ( l , m ) ] i sin [ Δ ( l , m ) 2 ] sin [ 2 θ ( l , m ) ] cos [ Δ ( l , m ) 2 ] + i sin [ Δ ( l , m ) 2 ] cos [ 2 θ ( l , m ) ] ] = [ a ( l , m ) b ( l , m ) i 1 a ( l , m ) 2 b ( l , m ) 2 i 1 a ( l , m ) 2 b ( l , m ) 2 i a ( l , m ) + b ( l , m ) i ]
r ( j , k ) = P 1 ( j , k ) F { Q 1 ( l , m ) F { e ( j , k ) } } = f ( j , k ) .
p 1 ( j , k ) = P ( j , k ) f 1 ( j , k ) = [ cos [ Δ ( j , k ) 2 ] i sin [ Δ ( j , k ) 2 ] cos [ 2 θ ( j , k ) ] i sin [ Δ ( l , m ) 2 ] sin [ 2 θ ( l , m ) ] ] p 2 ( j , k ) = P ( j , k ) f 2 ( j , k ) = [ i sin [ Δ ( l , m ) 2 ] sin [ 2 θ ( l , m ) ] cos [ Δ ( j , k ) 2 ] + i sin [ Δ ( j , k ) 2 ] cos [ 2 θ ( j , k ) ] ]
g 1 ( l , m ) = F { p 1 ( j , k ) } = j = 0 N 1 k = 0 N 1 p 1 ( j , k ) exp [ i 2 π N ( l j + m k ) ] = j = 0 N 1 k = 0 N 1 [ [ cos Δ ( j , k ) 2 i sin Δ ( j , k ) 2 cos 2 θ ( j , k ) ] exp [ i 2 π N ( l j + m k ) ] i sin ( Δ ( l , m ) 2 ) sin 2 θ ( l , m ) exp [ i 2 π N ( l j + m k ) ] ]
g 2 ( l , m ) = F { p 2 ( j , k ) } = j = 0 N 1 k = 0 N 1 p 2 ( j , k ) exp [ i 2 π N ( l j + m k ) ] = j = 0 N 1 k = 0 N 1 [ i sin Δ ( l , m ) 2 sin 2 θ ( l , m ) exp [ i 2 π N ( l j + m k ) ] [ cos Δ ( j , k ) 2 + i sin Δ ( j , k ) 2 cos 2 θ ( j , k ) ] exp [ i 2 π N ( l j + m k ) ] ]
G 1 ( l , m ) = Q ( l , m ) g 1 ( l , m ) = [ a ( l , m ) i b ( l , m ) i 1 a ( l , m ) 2 b ( l , m ) 2 i 1 a ( l , m ) 2 b ( l , m ) 2 a ( l , m ) + i b ( l , m ) ] [ c 1 ( l , m ) + i d 1 ( l , m ) c 2 ( l , m ) + i d 2 ( l , m ) ]
G 2 ( l , m ) = Q ( l , m ) g 2 ( l , m ) = [ a ( l , m ) i b ( l , m ) i 1 a ( l , m ) 2 b ( l , m ) 2 i 1 a ( l , m ) 2 b ( l , m ) 2 a ( l , m ) + i b ( l , m ) ] [ c 2 ( l , m ) + i d 2 ( l , m ) c 3 ( l , m ) + i d 3 ( l , m ) ]
G 1 ( l , m ) = [ h 1 ( l , m ) + i t 1 ( l , m ) h 2 ( l , m ) + i t 2 ( l , m ) ]
G 2 ( l , m ) = [ h 3 ( l , m ) + i t 3 ( l , m ) h 4 ( l , m ) + i t 4 ( l , m ) ] .
{ a ( l , m ) c 1 ( l , m ) + b ( l , m ) d 1 ( l , m ) + d 2 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = h 1 ( l , m ) a ( l , m ) d 1 ( l , m ) b ( l , m ) c 1 ( l , m ) c 2 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = t 1 ( l , m ) a ( l , m ) c 2 ( l , m ) b ( l , m ) d 2 ( l , m ) + d 1 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = h 2 ( l , m ) a ( l , m ) d 2 ( l , m ) + b ( l , m ) c 2 ( l , m ) c 1 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = t 2 ( l , m ) a ( l , m ) c 2 ( l , m ) + b ( l , m ) d 2 ( l , m ) + d 3 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = h 3 ( l , m ) a ( l , m ) d 2 ( l , m ) b ( l , m ) c 2 ( l , m ) c 3 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = t 3 ( l , m ) a ( l , m ) c 3 ( l , m ) b ( l , m ) d 3 ( l , m ) + d 2 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = h 4 ( l , m ) a ( l , m ) d 3 ( l , m ) + b ( l , m ) c 3 ( l , m ) c 2 ( l , m ) 1 a ( l , m ) 2 b ( l , m ) 2 = t 4 ( l , m )
g 1 ( 0 , N 2 ) = j = 0 N 1 k = 0 N 1 p 1 ( j , k ) exp [ i 2 π N ( N 2 ( k ) ) ] = j = 0 N 1 k = 0 N 1 p 1 ( j , k ) ( 1 ) k = j = 0 N 1 k = 0 N 1 [ (   cos ( Δ ( j , k ) 2 ) i sin ( Δ ( j , k ) 2 ) cos ( 2 θ ( j , k ) ) ) ( 1 ) k i sin ( Δ ( l , m ) 2 ) sin ( 2 θ ( l , m ) ) ( 1 ) k ]
g 2 ( 0 , N 2 ) = j = 0 N 1 k = 0 N 1 p 2 ( j , k ) exp [ i 2 π N ( N 2 ( k ) ) ] = j = 0 N 1 k = 0 N 1 p 2 ( j , k ) ( 1 ) k = j = 0 N 1 k = 0 N 1 [ i sin ( Δ ( l , m ) 2 ) sin ( 2 θ ( l , m ) ) ( 1 ) k ( cos ( Δ ( j , k ) 2 ) + i sin ( Δ ( j , k ) 2 ) cos ( 2 θ ( j , k ) ) ) ( 1 ) k ]
{ a ( 0 , N 2 ) c 1 ( 0 , N 2 ) + b ( 0 , N 2 ) d 1 ( 0 , N 2 ) + d 2 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 1 ( 0 , N 2 ) a ( 0 , N 2 ) d 1 ( 0 , N 2 ) b ( 0 , N 2 ) c 1 ( 0 , N 2 ) = t 1 ( 0 , N 2 ) b ( 0 , N 2 ) d 2 ( 0 , N 2 ) + d 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 2 ( 0 , N 2 ) a ( 0 , N 2 ) d 2 ( 0 , N 2 ) c 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = t 2 ( 0 , N 2 ) b ( 0 , N 2 ) d 2 ( 0 , N 2 ) d 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 3 ( 0 , N 2 ) a ( 0 , N 2 ) d 2 ( 0 , N 2 ) c 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = t 3 ( 0 , N 2 ) a ( 0 , N 2 ) c 1 ( 0 , N 2 ) + b ( 0 , N 2 ) d 1 ( 0 , N 2 ) + d 2 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 4 ( 0 , N 2 ) a ( 0 , N 2 ) d 1 ( 0 , N 2 ) + b ( 0 , N 2 ) c 1 ( 0 , N 2 ) = t 4 ( 0 , N 2 )
{ a ( 0 , N 2 ) c 1 ( 0 , N 2 ) + b ( 0 , N 2 ) d 1 ( 0 , N 2 ) + d 2 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 1 ( 0 , N 2 ) a ( 0 , N 2 ) d 1 ( 0 , N 2 ) b ( 0 , N 2 ) c 1 ( 0 , N 2 ) = t 1 ( 0 , N 2 ) b ( 0 , N 2 ) d 2 ( 0 , N 2 ) + d 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = h 2 ( 0 , N 2 ) a ( 0 , N 2 ) d 2 ( 0 , N 2 ) c 1 ( 0 , N 2 ) 1 a ( 0 , N 2 ) 2 b ( 0 , N 2 ) 2 = t 2 ( 0 , N 2 )
g 1 ( 0 , 0 ) = j = 0 N 1 k = 0 N 1 p 1 ( j , k ) exp [ 0 ] = j = 0 N 1 k = 0 N 1 p 1 ( j , k ) = j = 0 N 1 k = 0 N 1 [ cos ( Δ ( j , k ) 2 ) i sin ( Δ ( j , k ) 2 ) cos ( 2 θ ( j , k ) ) i sin ( Δ ( l , m ) 2 ) sin ( 2 θ ( l , m ) ) ]
g 2 ( 0 , 0 ) = j = 0 N 1 k = 0 N 1 p 2 ( j , k ) exp [ 0 ] = j = 0 N 1 k = 0 N 1 p 2 ( j , k ) = j = 0 N 1 k = 0 N 1 [ i sin ( Δ ( l , m ) 2 ) sin ( 2 θ ( l , m ) ) cos ( Δ ( j , k ) 2 ) + i sin ( Δ ( j , k ) 2 ) cos ( 2 θ ( j , k ) ) ]