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

2005

2004

2003

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]

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

2002

2001

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193,51-67 (2001).
[CrossRef]

S. Liu, L. Yu, B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering, " Opt. Commun. 18757-63 (2001).
[CrossRef]

2000

1999

1998

1997

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

L. G. Neto, and Y. Sheng, "Optical implementation of image encryption using random phase encoding," Opt. Eng. 35,2459-2463 (1996).
[CrossRef]

1995

Appl. Opt.

IEEE Trans. Comp.

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

Opt. Commun.

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

S. Lai and M. A. Neifeld, "Digital wavefront reconstruction and its application to image encryption," Opt. Commun. 178,283-289 (2000).
[CrossRef]

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

G. Unnikrishnan and K. Singh, "Optical encryption using quadratic phase systems," Opt. Commun. 193,51-67 (2001).
[CrossRef]

S. Liu, L. Yu, B. Zhu, "Optical image encryption by cascaded fractional Fourier transforms with random phase filtering, " Opt. Commun. 18757-63 (2001).
[CrossRef]

Opt. Eng.

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]

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

Opt. Lett.

Optik

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]

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

Other

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

Y. Frauel, A. Castro, T. J. Naughton, and B. Javidi, "Security analysis of optical encryption," in Unmanned/Unattended Sensors and Sensor Networks II, E. M. Carapezza, ed., Proc. SPIE 5986, 25-34 (2005).

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

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

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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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