Abstract

Although initial research shows that double-random phase encoding (DRPE) is vulnerable to known-plaintext attacks that use phase retrieval algorithms, subsequent research has shown that phase-only DRPE, in which the Fourier amplitude component of an image encrypted with classical DRPE remains constant, is resistant to attacks that apply phase retrieval algorithms. Herein, we numerically analyze the key-space of DRPE and investigate the distribution property of decryption keys for classical and phase-only DRPE. We determine the difference in the distribution property of successful decryption keys for these DRPE techniques from the numerical analysis results and then discuss the security offered by them.

© 2013 Optical Society of America

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References

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2011 (2)

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

E. Perez-Cabre, M. Cho, and B. Javidi, “Information authentication using photon-counting double-random-phase encrypted images,” Opt. Lett. 36, 22–24 (2011).
[CrossRef]

2010 (2)

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

2009 (4)

2008 (2)

2007 (3)

2006 (4)

2005 (1)

2004 (2)

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

Y. Hayasaki, Y. Matsuba, A. Nagaoka, H. Yamamoto, and N. Nishida, “Hiding an image with a light scattering medium and use of a contrast-discrimination method for readout,” Appl. Opt. 43, 1552–1558 (2004).
[CrossRef]

2003 (2)

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

S. Kishk and B. Javidi, “3D object watermarking by a 3D hidden object,” Opt. Express 11, 874–888 (2003).
[CrossRef]

2002 (2)

2001 (1)

1999 (2)

1995 (1)

Alam, M. S.

Arcos, S.

Cao, L.

Carnicer, A.

Chen, H.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Cho, M.

Dai, J.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
[CrossRef]

Ge, Q.

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

Gopinathan, U.

Hayasaki, Y.

He, M.

He, Q.

Hennelly, B. M.

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

Javidi, B.

Jin, G.

Juvells, I.

Kishk, S.

Li, H.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[CrossRef]

Li, P.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Li, Z.

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

Liu, S.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

Liu, T.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Liu, Z.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
[CrossRef]

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

Lu, W.

Matoba, O.

Matsuba, Y.

Mifune, Y.

Monaghan, D. S.

Montes-Usategui, M.

Nagaoka, A.

Naughton, T. J.

Nishida, N.

Obi, T.

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Ohyama, N.

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Peng, X.

Perez-Cabre, E.

Refregier, P.

Rosen, J.

Sheng, Y.

Sheridan, J. T.

Situ, G.

Sun, X.

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
[CrossRef]

Suzuki, H.

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Takai, N.

Takeda, M.

Tan, Q.

Tao, R.

Tashima, H.

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Wang, X.

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

Wang, Y.

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[CrossRef]

R. Tao, Y. Xin, and Y. Wang, “Double image encryption based on random phase encoding in the fractional Fourier domain,” Opt. Express 15, 16067–16079 (2007).
[CrossRef]

Wei, H.

Xi, L.

Xiao-feng, L.

Xie, H.

Xin, Y.

Xin, Z.

Yachida, M.

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Yamaguchi, M.

H. Tashima, M. Takeda, H. Suzuki, T. Obi, M. Yamaguchi, and N. Ohyama, “Known plaintext attack on double random phase encoding using fingerprint as key and a method for avoiding the attack,” Opt. Express 18, 13772–13781 (2010).
[CrossRef]

H. Suzuki, M. Yamaguchi, M. Yachida, N. Ohyama, H. Tashima, and T. Obi, “Experimental evaluation of fingerprint verification system based on double random phase encoding,” Opt. Express 14, 1755–1766 (2006).
[CrossRef]

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Yamamoto, H.

Yamaya, T.

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

Yang, G.

Yu, B.

Zhai, H.

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

Zhang, P.

Appl. Opt. (7)

J. Opt. (1)

Z. Liu, H. Chen, T. Liu, P. Li, J. Dai, X. Sun, and S. Liu, “Double-image encryption based on the affine transform and the gyrator transform,” J. Opt. 12, 035407 (2010).
[CrossRef]

Jpn. J. Opt. (1)

H. Suzuki, T. Yamaya, T. Obi, M. Yamaguchi, and N. Ohyama, “Fingerprint verification for smart card holders identification based on optical image encryption,” Jpn. J. Opt. 33, 37–44 (2004) (in Japanese).

Opt. Commun. (3)

Z. Liu and S. Liu, “Double image encryption based on iterative fractional Fourier transform,” Opt. Commun. 275, 324–329 (2007).
[CrossRef]

H. Li and Y. Wang, “Double-image encryption based on iterative gyrator transform,” Opt. Commun. 281, 5745–5749 (2008).
[CrossRef]

Z. Liu, J. Dai, X. Sun, and S. Liu, “Triple image encryption scheme in fractional Fourier transform domains,” Opt. Commun. 282, 518–522 (2009).
[CrossRef]

Opt. Express (8)

Opt. Lett. (6)

Optik (1)

X. Wang, H. Zhai, Z. Li, and Q. Ge, “Double random phase encryption based on discrete quaternion Fourier transforms,” Optik 122, 1856–1859 (2011).
[CrossRef]

Optik (Jena) (1)

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

Other (3)

H. Suzuki, H. Tashima, M. Yamaguchi, T. Obi, M. Yachida, and N. Ohyama, “File encryption software using fingerprint keys based on double random encoding,” in Frontiers in Optics, OSA Technical Digest Series (Optical Society of America, 2005), paper JWA50. http://www.opticsinfobase.org/abstract.cfm?URI=FiO-2005-JWA50 .

Data Encryption Standard (DES) http://csrc.nist.gov/publications/fips/fips46-3/fips46-3.pdf .

Advanced Encryption Standard (AES) http://csrc.nist.gov/archive/aes/index.html .

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Figures (12)

Fig. 1.
Fig. 1.

Optical system of classical DRPE.

Fig. 2.
Fig. 2.

Encryption and decryption procedures.

Fig. 3.
Fig. 3.

Method for key-space analysis.

Fig. 4.
Fig. 4.

(a) Plaintext image and (b) encryption key (phase mask at the Fourier plane).

Fig. 5.
Fig. 5.

Distance between the discrete point that is one of the possible keys (circles) and the sets of the keys (line A B ) that are equivalent to an encryption key (a square) (two pixels and four phase quantization levels).

Fig. 6.
Fig. 6.

Schematic illustrations of key-space analysis (two pixels and four phase quantization levels). (a) NRMS error of the plaintext image that appears in each decrypted image using the corresponding decryption key in two-dimensional key-space. (b) Distance (double-headed arrows) between all decryption keys ( 4 1 × 2 discrete points) and the sets of the keys that are equivalent to the encryption key (straight lines).

Fig. 7.
Fig. 7.

Histograms of NRMS error (a) in classical DRPE and (b) in phase-only DRPE. (c) Close-up plot of (a) in the range of 0 < NRMS error < 0.3 and (d) close-up plot of (b) in the range of 0 < NRMS error < 0.3 .

Fig. 8.
Fig. 8.

Distribution of the distance between an encryption key and a decryption key (a) in classical DRPE and (b) in phase-only DRPE.

Fig. 9.
Fig. 9.

Intensity distribution of the reconstructed image in phase-only DRPE (a) small plaintext image shown in Fig. 4(a) ( 3 × 3 pixels ), (b) large plaintext image ( 256 × 256 pixels ), (c) reconstructed image of small plaintext image (a), in phase-only DRPE, and (d) reconstructed image of small plaintext image (b), in phase-only DRPE.

Fig. 10.
Fig. 10.

Distribution of the distance between an encryption key and a decryption key in classical DRPE (circles) and in phase-only DRPE (triangles) for 0 < NRMS error < 0.3 .

Fig. 11.
Fig. 11.

Number of keys that can correctly decrypt the image versus the variance of the Fourier amplitude spectrum in classical DRPE (circles) and in phase-only DRPE (triangles).

Fig. 12.
Fig. 12.

Examples of the Fourier amplitude spectrum in classical DRPE.

Tables (1)

Tables Icon

Table 1. Comparison of the Size of the Key Space and the Number of Successful Decryption Keys between Classical and Phase-only DRPE

Equations (10)

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

f m ( x , y ) = f ( x , y ) · exp { j r ( x , y ) } ,
c ( x , y ) = IF [ F m ( μ , ν ) · exp { j R E ( μ , ν ) } ] ,
F [ c ( x , y ) ] · exp { j R D ( μ , ν ) } = F m ( μ , ν ) · exp [ j { R E ( μ , ν ) + R D ( μ , ν ) } ] ,
f r ( x , y ) = IF [ F m ( x , y ) · N ( μ , ν ) ] = f m ( x , y ) * n ( x , y ) ,
N ( μ , ν ) = exp [ j { R E ( μ , ν ) + R D ( μ , ν ) } ] ,
n ( x , y ) = IF [ N ( μ , ν ) ] .
f ^ PO ( x , y ) = IF [ F m ( μ , ν ) | F m ( μ , ν ) | N ( μ , ν ) ] = f PO ( x , y ) * n ( x , y ) ,
f PO ( x , y ) = IF [ F m ( μ , ν ) | F m ( μ , ν ) | ] .
NRMS error = i = 1 N j = 1 N | I d ( i , j ) I ( i , j ) | 2 / i = 1 N j = 1 N | I ( i , j ) | 2 ,
bit error rate = N error N total ,

Metrics