Abstract

Methods of image encryption based on fractional Fourier transform have an incipient flaw in security. We show that the schemes have the deficiency that one group of encryption keys has many groups of keys to decrypt the encrypted image correctly for several reasons. In some schemes, many factors result in the deficiencies, such as the encryption scheme based on multiple-parameter fractional Fourier transform [Opt. Lett. 33, 581 (2008) ]. A modified method is proposed to avoid all the deficiencies. Security and reliability are greatly improved without increasing the complexity of the encryption process. © 2009 Optical Society of America

© 2009 Optical Society of America

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Errata

Qiwen Ran, Haiying Zhang, Jin Zhang, Liying Tan, and Jing Ma, "Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform: erratum," Opt. Lett. 34, 2945-2945 (2009)
https://www.osapublishing.org/ol/abstract.cfm?uri=ol-34-19-2945

References

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    [PubMed]

2008 (1)

2007 (2)

R. Tao, B. Li, and Y. Wang, IEEE Trans. Signal Process. 55, 3541 (2007).

W. Jin and C. Yan, Optik (Jena) 118, 38 (2007).

2006 (1)

S.-C. Pei and W.-L. Hsuec, IEEE Signal Process. Lett. 13, 329 (2006).

2005 (1)

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

2003 (1)

B. Hennelly and J. T. Sheridan, Opt. Commun. 226, 61 (2003).

2002 (1)

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).

2000 (2)

B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

1995 (1)

1994 (1)

Cariolaro, G.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

Erseghe, T.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

Hennelly, B.

B. Hennelly and J. T. Sheridan, Opt. Commun. 226, 61 (2003).

Hsuec, W.-L.

S.-C. Pei and W.-L. Hsuec, IEEE Signal Process. Lett. 13, 329 (2006).

Jin, W.

W. Jin and C. Yan, Optik (Jena) 118, 38 (2007).

Kraniauskas, P.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

Lang, J.

Laurenti, N.

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

Li, B.

R. Tao, B. Li, and Y. Wang, IEEE Trans. Signal Process. 55, 3541 (2007).

Liu, S.

Lohmann, A. W.

Mendlovic, D.

Ozaktas, H. M.

Pei, S.-C.

S.-C. Pei and W.-L. Hsuec, IEEE Signal Process. Lett. 13, 329 (2006).

Ran, Q.

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

B. Zhu, S. Liu, and Q. Ran, Opt. Lett. 25, 1159 (2000).

Sheridan, J. T.

B. Hennelly and J. T. Sheridan, Opt. Commun. 226, 61 (2003).

Soffer, B. H.

Tanno, N.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).

Tao, R.

R. Tao, J. Lang, and Y. Wang, Opt. Lett. 33, 581 (2008).
[PubMed]

R. Tao, B. Li, and Y. Wang, IEEE Trans. Signal Process. 55, 3541 (2007).

Tsang, C. C.

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

Wang, Q.

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

Wang, Y.

R. Tao, J. Lang, and Y. Wang, Opt. Lett. 33, 581 (2008).
[PubMed]

R. Tao, B. Li, and Y. Wang, IEEE Trans. Signal Process. 55, 3541 (2007).

Yan, C.

W. Jin and C. Yan, Optik (Jena) 118, 38 (2007).

Yeung, D. S.

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

Zhang, Y.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).

Zheng, C. H.

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).

Zhu, B.

IEEE Signal Process. Lett. (1)

S.-C. Pei and W.-L. Hsuec, IEEE Signal Process. Lett. 13, 329 (2006).

IEEE Trans. Signal Process. (2)

R. Tao, B. Li, and Y. Wang, IEEE Trans. Signal Process. 55, 3541 (2007).

G. Cariolaro, T. Erseghe, P. Kraniauskas, and N. Laurenti, IEEE Trans. Signal Process. 48, 227 (2000).

IEEE Trans. Signal Processing (1)

Q. Ran, D. S. Yeung, C. C. Tsang, and Q. Wang, IEEE Trans. Signal Processing 53, 83 (2005).

J. Opt. Soc. Am. A (2)

Opt. Commun. (2)

Y. Zhang, C. H. Zheng, and N. Tanno, Opt. Commun. 202, 277 (2002).

B. Hennelly and J. T. Sheridan, Opt. Commun. 226, 61 (2003).

Opt. Lett. (2)

Optik (Jena) (1)

W. Jin and C. Yan, Optik (Jena) 118, 38 (2007).

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Figures (3)

Fig. 1
Fig. 1

Digital simulation of the encryption: (a) original image; (b) encryption with keys [ 4 , 2 / 4 , ( 7 , 3 , 9 , 5 ) , ( 3 , 2 , 9 , 11 ) ] ; (c) and (d) the decrypted image with different correct keys.

Fig. 2
Fig. 2

Deviation between Figs. 1c, 1d.

Fig. 3
Fig. 3

Digital simulation of the periodicity of MPFRFT: (a) encryption with keys ( M , α , M , N ) ; (b), (c), and (d) are the decrypted images with the keys ( M , α , M , N ) , ( M + 8 , α , M , N ) , and ( M + 24 , α , M , N ) , respectively.

Equations (10)

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

R M α = l = 0 M 1 H l ( α , M , N ) f l ( x ) ,
H l ( α , M , N ) = 1 M k = 0 M 1 exp { 2 π i M [ ( m k M + 1 ) α ( k + n k M ) l k ] } .
H l ( α + M , M , N ) = 1 M k = 0 M 1 exp ( 2 π l k i M ) exp [ 2 π i M ( m k M + 1 ) α ( k + n k M ) ] exp { 2 π i [ ( m k M + 1 ) ( k + n k M ) ] } .
H l ( α , M , N ) = 1 M k = 0 M 1 exp { 2 π i M [ α ( k + M n k ) l k ] } = H l ( α , N ) ,
n k = ( k m k + M m k n k + n k ) ,     l = 0 , 1 , , M 1 ,
N = ( n 0 , n 1 , , n M 1 ) Z M .
H l ( α , N ) = 1 M k = 0 M 1 exp { 2 π i M [ α ( k + n k M ) l k ] } .
R M α = l = 0 M 1 H l ( α , N ) f l ( x ) ,     N R M .
H l ( α , N ) = 1 M k = 0 M 1 exp { 2 π i M [ α ( k + r k M ) l k ] } ,
N = ( r 0 , r 1 , , r M 1 ) R M .

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