Abstract

Several attacks are proposed against the double random phase encryption scheme. These attacks are demonstrated on computer-generated ciphered images. The scheme is shown to be resistant against brute force attacks but susceptible to chosen and known plaintext attacks. In particular, we describe a technique to recover the exact keys with only two known plain images. We compare this technique to other attacks proposed in the literature.

© 2007 Optical Society of America

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References

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2006 (5)

2005 (1)

2004 (2)

2003 (2)

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

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik 114, 95–99 (2003).
[Crossref]

2002 (3)

2001 (2)

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67 (2001).
[Crossref]

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 18757–63 (2001).
[Crossref]

2000 (4)

1999 (4)

1998 (1)

1997 (1)

S. R. Blackburn, S. Murphy, and K. G. Paterson, “Comments on ‘Theory and applications of cellular automata in cryptography’,” IEEE Trans. Comp. 46, 637–638 (1997).
[Crossref]

1996 (1)

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[Crossref]

1995 (1)

Ahouzi, E.

Arcos, S.

Beker, H.

H. Beker and F. Piper, Cipher systems (Van Nostrand, London, 1982).

Blackburn, S. R.

S. R. Blackburn, S. Murphy, and K. G. Paterson, “Comments on ‘Theory and applications of cellular automata in cryptography’,” IEEE Trans. Comp. 46, 637–638 (1997).
[Crossref]

Carnicer, A.

Castro, A.

Y. Frauel, A. Castro, T. J. Naughton, and B. JavidiE. M. Carapezza, “Security analysis of optical encryption,” inUnmanned/ Unattended Sensors and Sensor Networks II, ed., Proc. SPIE5986, 25–34 (2005).

Chang, H. T.

Chen, L.

X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. 260, 449–453 (2006).
[Crossref]

Cui, Z.

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212, 235–245 (2002).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992), Chap. 2.

Frauel, Y.

Y. Frauel, A. Castro, T. J. Naughton, and B. JavidiE. M. Carapezza, “Security analysis of optical encryption,” inUnmanned/ Unattended Sensors and Sensor Networks II, ed., Proc. SPIE5986, 25–34 (2005).

Fujimoto, A.

Gopinathan, U.

Han, J.-W.

J.-W. Han, C.-S. Park, D.-H. Ryu, and E.-S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Hennelly, B. M.

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

Horner, J. L.

J. L. Horner and B. Javidi, Opt. Eng.38, Special issue on Optical security, 1999.
[Crossref]

Javidi, B.

Joseph, J.

Juvells, I.

Kim, E.-S.

J.-W. Han, C.-S. Park, D.-H. Ryu, and E.-S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Kreske, K.

Kuo, C. L.

Lai, S.

S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178, 283–289 (2000).
[Crossref]

Li, Y.

Liu, S.

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik 114, 95–99 (2003).
[Crossref]

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 18757–63 (2001).
[Crossref]

Lu, W. C.

Luo, Z.

Matoba, O.

Monaghan, D. S.

Montes-Usategui, M.

Murphy, S.

S. R. Blackburn, S. Murphy, and K. G. Paterson, “Comments on ‘Theory and applications of cellular automata in cryptography’,” IEEE Trans. Comp. 46, 637–638 (1997).
[Crossref]

Naughton, T. J.

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]

Y. Frauel, A. Castro, T. J. Naughton, and B. JavidiE. M. Carapezza, “Security analysis of optical encryption,” inUnmanned/ Unattended Sensors and Sensor Networks II, ed., Proc. SPIE5986, 25–34 (2005).

Neifeld, M. A.

S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178, 283–289 (2000).
[Crossref]

Neto, L. G.

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[Crossref]

Obi, T.

Ohtsubo, J.

Ohyama, N.

Park, C.-S.

J.-W. Han, C.-S. Park, D.-H. Ryu, and E.-S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Paterson, K. G.

S. R. Blackburn, S. Murphy, and K. G. Paterson, “Comments on ‘Theory and applications of cellular automata in cryptography’,” IEEE Trans. Comp. 46, 637–638 (1997).
[Crossref]

Peng, X.

Piper, F.

H. Beker and F. Piper, Cipher systems (Van Nostrand, London, 1982).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992), Chap. 2.

Réfrégier, Ph.

Rosen, J.

Ryu, D.-H.

J.-W. Han, C.-S. Park, D.-H. Ryu, and E.-S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

Sheng, Y.

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[Crossref]

Sheridan, J. T.

Singh, K.

Situ, G.

Suzuki, H.

Tajahuerce, E.

Tan, T.

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212, 235–245 (2002).
[Crossref]

Tashima, H.

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992), Chap. 2.

Towghi, N.

Unnikrishnan, G.

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992), Chap. 2.

Wang, X.

X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. 260, 449–453 (2006).
[Crossref]

Wei, H.

Yachida, M.

Yamaguchi, M.

Yu, B.

Yu, L.

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 18757–63 (2001).
[Crossref]

Zhang, J.

Zhang, P.

Zhao, D.

X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. 260, 449–453 (2006).
[Crossref]

Zhao, H.

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik 114, 95–99 (2003).
[Crossref]

Zhu, B.

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik 114, 95–99 (2003).
[Crossref]

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 18757–63 (2001).
[Crossref]

Appl. Opt. (7)

IEEE Trans. Comp. (1)

S. R. Blackburn, S. Murphy, and K. G. Paterson, “Comments on ‘Theory and applications of cellular automata in cryptography’,” IEEE Trans. Comp. 46, 637–638 (1997).
[Crossref]

J. Opt. Soc. Am. A (1)

Opt. Commun. (5)

G. Unnikrishnan and K. Singh, “Optical encryption using quadratic phase systems,” Opt. Commun. 193, 51–67 (2001).
[Crossref]

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 18757–63 (2001).
[Crossref]

X. Peng, Z. Cui, and T. Tan, “Information encryption with virtual-optics imaging system,” Opt. Commun. 212, 235–245 (2002).
[Crossref]

S. Lai and M. A. Neifeld, “Digital wavefront reconstruction and its application to image encryption,” Opt. Commun. 178, 283–289 (2000).
[Crossref]

X. Wang, D. Zhao, and L. Chen, “Image encryption based on extended fractional Fourier transform and digital holography technique,” Opt. Commun. 260, 449–453 (2006).
[Crossref]

Opt. Eng. (2)

J.-W. Han, C.-S. Park, D.-H. Ryu, and E.-S. Kim, “Optical image encryption based on XOR operations,” Opt. Eng. 38, 47–54 (1999).
[Crossref]

L. G. Neto and Y. Sheng, “Optical implementation of image encryption using random phase encoding,” Opt. Eng. 35, 2459–2463 (1996).
[Crossref]

Opt. Express (2)

Opt. Lett. (7)

Optik (2)

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

B. Zhu, H. Zhao, and S. Liu, “Image encryption based on pure intensity random coding and digital holography technique,” Optik 114, 95–99 (2003).
[Crossref]

Other (5)

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipes in C (Cambridge University Press, Cambridge, 1992), Chap. 2.

Y. Frauel, A. Castro, T. J. Naughton, and B. JavidiE. M. Carapezza, “Security analysis of optical encryption,” inUnmanned/ Unattended Sensors and Sensor Networks II, ed., Proc. SPIE5986, 25–34 (2005).

H. Beker and F. Piper, Cipher systems (Van Nostrand, London, 1982).

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

J. L. Horner and B. Javidi, Opt. Eng.38, Special issue on Optical security, 1999.
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1.

Principle of the double random phase encryption scheme.

Fig. 2.
Fig. 2.

Decryption of images encrypted with 16-level phase keys. (a)–(b) 16-level decryption key, (c)–(d) 4-level decryption key, and (e)–(f) 2-level decryption key.

Fig. 3.
Fig. 3.

Decryption using partial windows of the original 100×100 key. (a)–(b) 50×50 window, (c)–(d) 40×40 window, and (e)–(f) 30×30 window.

Fig. 4.
Fig. 4.

Decryption using partial windows of the original 100×100 key and reduction to three phase levels. (a)–(b) 50×50 window, (c)–(d) 40×40 window, and (e)–(f) 30×30 window.

Fig. 5.
Fig. 5.

Decryption using Attack 9. (a) and (b) are the two known plain images. (c) Ciphered image corresponding to an unknown plain image. (d) Unknown image decrypted using the keys retrieved by Attack 9.

Fig. 6.
Fig. 6.

Decryption of noisy ciphered images using an adaptation of Attack 9. (a) Plain images. (b) Decrypted images using the keys retrieved by simple system solving. (c) Decrypted images using the keys retrieved by least-square solving.

Equations (26)

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

C = Y · 𝓕 ( P · X ) ,
P · X = 𝓕 1 ( C ÷ Y )
P = P · X = 𝓕 1 ( C ÷ Y ) ,
P 3 · X = 𝓕 1 ( C 3 ÷ Y ) ,
X 𝓕 1 ( C 3 ÷ Y )
C = T P ,
C = T P ,
T = C P 1 ,
P = T 1 C ,
c i = y i j = 1 N F ij x j p j with 1 i N ,
c i = y i j = 1 N F ij x j p j with 1 i N .
c i j = 1 N F ij x j p j = c i j = 1 N F ij x j p j with 1 i N ,
j = 1 N F ij x j ( c i p j c i p j ) = 0 with 1 i N .
j = 1 N S ij x j = 0 with 1 i N ,
j = 1 N 1 S ij x j = S iN with 1 i N 1 .
y i = c i j = 1 N F ij x j p j with 1 i N .
j = 1 N S ij αβ x j = 0 with 1 i N ,
S αβ X = 0 ,
[ S 12 S 13 S 14 S 23 S 24 S 34 0 0 1 ] x = [ 0 0 1 ] ,
x ̂ i = x i x i with 1 i N .
λ i α y i = c i α with 1 i N ,
λ i α = j = 1 N F ij x ̂ j p j α .
Λ α Y = C α ,
Λ α = [ λ 1 α λ N α ] .
[ Λ 1 Λ 2 Λ 3 Λ 4 ] Y = [ C 1 C 2 C 3 C 4 ] .
y ̂ i = y i y i with 1 i N .

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