Abstract

We perform a numerical analysis on the double random phase encryption∕decryption technique. The key-space of an encryption technique is the set of possible keys that can be used to encode data using that technique. In the case of a strong encryption scheme, many keys must be tried in any brute-force attack on that technique. Traditionally, designers of optical image encryption systems demonstrate only how a small number of arbitrary keys cannot decrypt a chosen encrypted image in their system. However, this type of demonstration does not discuss the properties of the key-space nor refute the feasibility of an efficient brute-force attack. To clarify these issues we present a key-space analysis of the technique. For a range of problem instances we plot the distribution of decryption errors in the key-space indicating the lack of feasibility of a simple brute-force attack.

© 2007 Optical Society of America

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References

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2007 (1)

2006 (3)

2005 (3)

2004 (2)

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

M. Liebling, T. Blu, and M. Unser, "Complex-wave retrieval from a single off-axis hologram," J. Opt. Soc. Am. A 21, 367-377 (2004).
[CrossRef]

2003 (3)

B. H. Zhu, H. F. Zhao, and S. T. Liu, "Image encryption based on pure intensity random coding and digital holography technique," Optik 114, 95-99 (2003).
[CrossRef]

B. Hennelly and J. T. Sheridan, "Optical image encryption by random shifting in fractional Fourier domains," Opt. Lett. 28, 269-271 (2003).
[CrossRef] [PubMed]

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

2000 (3)

1998 (1)

1995 (1)

1994 (1)

1987 (1)

C. A. Deavours, Cryptology Yesterday, Today and Tomorrow (Artech House, 1987).

1982 (1)

1976 (1)

W. Diffie and M. E. Hellman, "New directions in cryptography," IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

1967 (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1939 (1)

G. F. Gaines, Cryptanalysis: a Study of Ciphers and Their Solution (Dover, 1939).

1931 (1)

H. O. Yardley, The American Black Chamber (U.S. Naval Institute Press, 1931).

Arcos, S.

Bell, K. M.

Blu, T.

Bollaro, F.

Brackenbury, L. E. M.

Carnicer, A.

Castro, A.

Deavours, C. A.

C. A. Deavours, Cryptology Yesterday, Today and Tomorrow (Artech House, 1987).

Diffie, W.

W. Diffie and M. E. Hellman, "New directions in cryptography," IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

Fienup, J. R.

Frauel, Y.

Gaines, G. F.

G. F. Gaines, Cryptanalysis: a Study of Ciphers and Their Solution (Dover, 1939).

Goodman, J. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Gopinathan, U.

Goudail, F.

Hellman, M. E.

W. Diffie and M. E. Hellman, "New directions in cryptography," IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

Hennelly, B.

Hennelly, B. M.

B. M. Hennelly and J. T. Sheridan, "Optical encryption and the space bandwidth product," Opt. Commun. 247, 291-305 (2005).
[CrossRef]

B. M. Hennelly and J. T. Sheridan, "Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms," J. Opt. Soc. Am. A 22, 917-927 (2005).
[CrossRef]

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

Javidi, B.

Joseph, J.

Juptner, W.

Juvells, I.

Lawrence, R. W.

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

Liebling, M.

Liu, S. T.

B. H. Zhu, H. F. Zhao, and S. T. Liu, "Image encryption based on pure intensity random coding and digital holography technique," Optik 114, 95-99 (2003).
[CrossRef]

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Peng, X.

Refregier, P.

Schnars, U.

Sheridan, J. T.

Singh, K.

Tajahuerce, E.

Unnikrishnan, G.

Unser, M.

Wei, H. Z.

Yardley, H. O.

H. O. Yardley, The American Black Chamber (U.S. Naval Institute Press, 1931).

Yu, B.

Zhang, P.

Zhao, H. F.

B. H. Zhu, H. F. Zhao, and S. T. Liu, "Image encryption based on pure intensity random coding and digital holography technique," Optik 114, 95-99 (2003).
[CrossRef]

Zhu, B. H.

B. H. Zhu, H. F. Zhao, and S. T. Liu, "Image encryption based on pure intensity random coding and digital holography technique," Optik 114, 95-99 (2003).
[CrossRef]

Appl. Opt. (5)

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

IEEE Trans. Inf. Theory (1)

W. Diffie and M. E. Hellman, "New directions in cryptography," IEEE Trans. Inf. Theory 22, 644-654 (1976).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Commun. (1)

B. M. Hennelly and J. T. Sheridan, "Optical encryption and the space bandwidth product," Opt. Commun. 247, 291-305 (2005).
[CrossRef]

Opt. Eng. (1)

T. J. Naughton and B. Javidi, "Compression of encrypted three-dimensional objects using digital holography," Opt. Eng. 43, 2233-2238 (2004).
[CrossRef]

Opt. Express (2)

Opt. Lett. (5)

Optik (2)

B. H. Zhu, H. F. Zhao, and S. T. 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 (4)

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

C. A. Deavours, Cryptology Yesterday, Today and Tomorrow (Artech House, 1987).

G. F. Gaines, Cryptanalysis: a Study of Ciphers and Their Solution (Dover, 1939).

H. O. Yardley, The American Black Chamber (U.S. Naval Institute Press, 1931).

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Figures (11)

Fig. 1
Fig. 1

Double random phase encoding (DRPE).

Fig. 2
Fig. 2

Schematic of one possible optical implementation of DRPE.

Fig. 3
Fig. 3

(a) Original image is a binary real image and (b) R2 phase values (multiply by 2 π ). The numbers in parentheses correspond to those in Fig. 11.

Fig. 4
Fig. 4

(Color online) Error produced by each phase-key when used to decrypt the system. There are four exact, NRMS = 0, encircled solutions, and 56 cases with NRMS < 0.2 . The four exact solutions are labeled as in Fig. 11.

Fig. 5
Fig. 5

(Color online) (a) Histogram of the NRMS error associated with every phase-key in key-space, which shows the number of phase-keys that decrypt to a certain error. (b) A zoomed-in plot of the section of (a) near the origin, showing there are four phase-keys that achieve exact decryption and 24 with 0.04 < NRMS < 0.07 (also see Fig. 11).

Fig. 6
Fig. 6

Decrypted images with NMRS errors of: (a) 0, (b) 0.2, (c) 0.4, (d) 0.6, (e) 0.8, and (f) 1. The application of a threshold at 0.5, which sets each pixel to either 1 or 0, will mean output (a) and (b) give the correct solution and output (c) has only one incorrect pixel.

Fig. 7
Fig. 7

(Color online) Distribution of 1 × 106 keys randomly generated in a 256 × 256 system with Q = 8. Note that the y-axis is a logarithmic scale. For comparison see Fig. 5(a).

Fig. 8
Fig. 8

(Color online) The exact phase-key is taken and a constant phase is added to an increasing number of pixels to see how it affects the decryption. There are 65,536 pixels in the phase-key and the maximum number of pixels changed is 3,500, which is 5.3% of the total number of pixels.

Fig. 9
Fig. 9

(Color online) The exact phase-key R2 is taken and seven equally likely phase values are chosen randomly and added to an increasing number of pixels to see how it affects the decryption (A, solid curve). The values from Fig. 8 are averaged and plotted (B, dotted curve). The random selection of phases produces a slightly higher NRMS error. NRMS < 0.2 when 1.5%–1.9% of the total number of pixels are in error.

Fig. 10
Fig. 10

(Color online) Visualization aid of the key-space for a system with two pixels and Q = 8.

Fig. 11
Fig. 11

(Color online) Visualization aid for mapping out the phase-keys for the cases examined in Figs. 4 and 5. The thick solid line is R2 (i). The dashed lines (ii), (iii), and (iv) are the other correct keys. The labels (a)–(f) correspond to keys giving low NRMS error decryption.

Tables (1)

Tables Icon

Table 1 For a 3 × 3 Pixel System With Q = 2, 3, and 4 There Is a Comparison of the Number of Keys in the Key-Space to the Fraction of Keys That Produce an Output an Exact Solution and the Fraction of Keys That Produce an Output With NRMS < 0.02. The Increase in Exact Solutions and Solutions With NRMS < 0.2 Is Much Less Than the Increase in Key-Space

Equations (1)

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NRMS = ( i = 1 N j = 1 N | I d ( i , j ) I ( i , j ) | 2 ) / ( i = 1 N j = 1 N | I ( i , j ) | 2 ) ,

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