Abstract

Traditionally, cryptanalysis of optical security systems attempts to find original keys. Usually, by use of this kind of method, one can find a set of keys located close to the original keys in the key space. We call such a set the region of original key (ROK). For an optical encryption system in the Fresnel domain, such a strategy is ineffective since it needs to perform an exhaustive search to determine the system geometry or to solve an extremely large set of system equations. We propose to employ an alternative search strategy: to find a region of possible key (RPK). Since there is only one ROK for a cypher system but there are many RPKs, the probability to find a key in the RPK would be higher than in the ROK. Our analysis reveals that even a Fresnel-based encryption system has larger key space, but there are also serious security problems to be resolved.

© 2010 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  6. G. Situ, U. Gopinathan, D. S. Monaghan, and J. T. Sheridan, “Cryptanalysis of optical security systems with significant output images,” Appl. Opt. 46, 5257-5262 (2007).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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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]
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    [CrossRef] [PubMed]
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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] [PubMed]
  29. P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388-1390 (1994).
    [CrossRef] [PubMed]
  30. B. Javidi, G. Zhang, and J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506-2512 (1996).
    [CrossRef]
  31. D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
    [CrossRef]

2009

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. A: Pure Appl. Opt. 11, 075402 (2009).
[CrossRef]

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[CrossRef]

2008

C. Cuadrado-Laborde, R. Duchowicz, R. Torroba, and E. E. Sicre, “Dual random phase encoding: a temporal approach for fiber optic applications,” Appl. Opt. 47, 1940-1946 (2008).
[CrossRef] [PubMed]

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]

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

D. S. Monaghan, 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, 3808-3816 (2008).
[CrossRef] [PubMed]

G. Situ, D. S. Monaghan, T. J. Naughton, J. T. Sheridan, G. Pedrini and W. Osten, “Collision in double random phase encoding,” Opt. Commun. 281, 5122-5125 (2008).
[CrossRef]

2007

2006

2005

2004

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

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

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

2003

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

2000

1999

1996

Z. Zalevsky and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

B. Javidi, G. Zhang, and J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506-2512 (1996).
[CrossRef]

1995

1994

1984

Arcos, S.

Carnicer, A.

Castro, A.

Cheng, C. -J

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

Chiang, J. -S

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

Chiou, A. E. T.

Chou, J. -W

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

Cuadrado-Laborde, C.

Dorsch, R. G.

Duchowicz, R.

Frauel, Y.

Goodman, J. W.

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

Gopinathan, U.

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

Javidi, B.

Juvells, I.

Kelly, D. P.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[CrossRef]

Levi, A.

Li, J.

B. Javidi, G. Zhang, and J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506-2512 (1996).
[CrossRef]

Matoba, O.

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Osten, W.

G. Situ, D. S. Monaghan, T. J. Naughton, J. T. Sheridan, G. Pedrini and W. Osten, “Collision in double random phase encoding,” Opt. Commun. 281, 5122-5125 (2008).
[CrossRef]

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]

Pedrini, G.

G. Situ, D. S. Monaghan, T. J. Naughton, J. T. Sheridan, G. Pedrini and W. Osten, “Collision in double random phase encoding,” Opt. Commun. 281, 5122-5125 (2008).
[CrossRef]

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]

Pellat-Finet, P.

Peng, X.

Qin, W.

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. A: Pure Appl. Opt. 11, 075402 (2009).
[CrossRef]

Réfrégier, P.

Schneier, B.

B. Schneier, Applied Cryptography: Protocols, Algorthms, and Source Code in C, 2nd ed. (Wiley, 1996).

Sheridan, J. T.

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[CrossRef]

G. Situ, D. S. Monaghan, T. J. Naughton, J. T. Sheridan, G. Pedrini and W. Osten, “Collision in double random phase encoding,” Opt. Commun. 281, 5122-5125 (2008).
[CrossRef]

D. S. Monaghan, 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, 3808-3816 (2008).
[CrossRef] [PubMed]

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

D. S. Monaghan, U. Gopinathan, T. J. Naughton, and J. T. Sheridan, “Key-space analysis of double random phase encryption technique,” Appl. Opt. 46, 6641-6647 (2007).
[CrossRef] [PubMed]

U. Gopinathan, D. S. Monaghan, T. J. Naughton, and J. T. Sheridan, “A known-plaintext heuristic attack on the Fourier plane encryption algorithm,” Opt. Express 14, 3181-3186 (2006).
[CrossRef] [PubMed]

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

Sicre, E. E.

Situ, G.

Stark, H.

Su, W. -C

Sun, C. -C

Torroba, R.

Tu, H. -Y

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

Wang, B.

Wei, H.

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, F.

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]

Zhang, G.

B. Javidi, G. Zhang, and J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506-2512 (1996).
[CrossRef]

Zhang, J.

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

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

Zhang, P.

Appl. Opt.

IEEE Signal Process. Lett.

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. A: Pure Appl. Opt.

W. Qin and X. Peng, “Vulnerability to known-plaintext attack of optical encryption schemes based on two fractional Fourier transform order keys and double random phase keys,” J. Opt. A: Pure Appl. Opt. 11, 075402 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Jpn. J. Appl. Phys.

H. -Y Tu, J. -S Chiang, J. -W Chou, and C. -J Cheng, “Full phase encoding for digital holographic encryption using liquid crystal spatial light modulators,” Jpn. J. Appl. Phys. 47, 8838-8843 (2008).
[CrossRef]

Opt. Commun.

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

G. Situ, D. S. Monaghan, T. J. Naughton, J. T. Sheridan, G. Pedrini and W. Osten, “Collision in double random phase encoding,” Opt. Commun. 281, 5122-5125 (2008).
[CrossRef]

Opt. Eng.

B. M. Hennelly and J. T. Sheridan, “Random phase and jigsaw encryption in the Fresnel domain,” Opt. Eng. 43, 2239-2249 (2004).
[CrossRef]

B. Javidi, G. Zhang, and J. Li, “Experimental demonstration of the random phase encoding technique for image encryption and security verification,” Opt. Eng. 35, 2506-2512 (1996).
[CrossRef]

D. S. Monaghan, U. Gopinathan, D. P. Kelly, T. J. Naughton, and J. T. Sheridan, “Systematic errors of an optical encryption system due to the discrete values of a spatial light modulator,” Opt. Eng. 48, 027001 (2009).
[CrossRef]

Opt. Express

Opt. Lett.

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]

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, 1644-1646 (2005).
[CrossRef] [PubMed]

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

C. Cuadrado-Laborde, “Time-variant signal encryption by lensless dual random phase encoding applied to fiber optic links,” Opt. Lett. 32, 2867-2869 (2007).
[CrossRef] [PubMed]

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

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

P. Pellat-Finet, “Fresnel diffraction and the fractional-order Fourier transform,” Opt. Lett. 19, 1388-1390 (1994).
[CrossRef] [PubMed]

Z. Zalevsky and R. G. Dorsch, “Gerchberg-Saxton algorithm applied in the fractional Fourier or the Fresnel domain,” Opt. Lett. 21, 842-844 (1996).
[CrossRef] [PubMed]

Optik (Jena)

B. M. Hennelly and J. T. Sheridan, “Image encryption and the fractional Fourier transform,” Optik (Jena) 114, 251-265(2003).
[CrossRef]

Phys. Rev. A

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]

Other

http://sipi.usc.edu/database/

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

B. Schneier, Applied Cryptography: Protocols, Algorthms, and Source Code in C, 2nd ed. (Wiley, 1996).

B. Javidi, ed. Optical and Digital Techniques for Information Security (Springer Verlag, 2005).
[CrossRef]

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Figures (7)

Fig. 1
Fig. 1

Schematic setup of the double-phase random encoding in the Fresnel domain.

Fig. 2
Fig. 2

Schematic illustration of the key space mapped into a two-dimensional key plane.

Fig. 3
Fig. 3

Situation of an attack. The cypher system is regarded as a black box to the attacker. The available information includes the f k g k pairs and the cyphertext of interest c ( x , y ) .

Fig. 4
Fig. 4

Flow chart of the proposed algorithm.

Fig. 5
Fig. 5

Revealed secret image with a key in the RPK.

Fig. 6
Fig. 6

Differences between the generated and the original phase keys: (a)  Δ ϕ 1 = ψ 1 ϕ 1 and (b)  Δ ϕ 2 = ψ 2 ϕ 2 .

Fig. 7
Fig. 7

Decryption output with the original geometric keys.

Equations (7)

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

g ( x , y ) = P λ , z 2 { P λ , z 1 { f ( x 1 , y 1 ) exp [ j ϕ 1 ( x 1 , y 1 ) ] } exp [ j ϕ 2 ( x 2 , y 2 ) ] } ,
P λ , z i { f ( x 1 , y 1 ) } = F 1 { F { f ( x 1 , y 1 ) } H λ , z i ( ξ x , ξ y ) } ,
H λ , z i ( ξ x , ξ y ) = exp ( j 2 π z i λ ) exp [ j π λ z i ( ξ x 2 + ξ y 2 ) ] ,
f ( x 1 , y 1 ) = P λ , z 1 1 { P λ , z 2 1 { g ( x , y ) } exp [ j ϕ 2 ( x 2 , y 2 ) ] } exp [ j ϕ 1 ( x 1 , y 1 ) ] ,
u k ( x 2 , y 2 ) = P Λ , Z 2 1 { g k ( x , y ) } ,
| u k ( x 2 , y 2 ) | = | P Λ , Z 1 { f k ( x 1 , y 1 ) exp [ j ψ 1 ( x 1 , y 1 ) ] } | .
ψ 2 ( x 2 , y 2 ) = arg { u k ( x 2 , y 2 ) } arg { P Λ , Z 1 { f k ( x 1 , y 1 ) exp [ j ψ 1 ( x 1 , y 1 ) ] } .

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