Abstract

The asymmetric cryptosystem, which is based on phase-truncated Fourier transforms (PTFTs), can break the linearity of conventional systems. However, it has been proven to be vulnerable to a specific attack based on iterative Fourier transforms when the two random phase masks are used as public keys to encrypt different plaintexts. An improvement from the asymmetric cryptosystem may be taken by relocating the amplitude values in the output plane. In this paper, two different methods are adopted to realize the amplitude modulation of the output image. The first one is to extend the PTFT-based asymmetrical cryptosystem into the anamorphic fractional Fourier transform domain directly, and the second is to add an amplitude mask in the Fourier plane of the encryption scheme. Some numerical simulations are presented to prove the good performance of the proposed cryptosystems.

© 2011 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  26. X. Wang and D. Zhao, “A specific attack on the asymmetric cryptosystem based on the phase-truncated Fourier transforms,” Opt. Commun. (to be published).
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    [CrossRef]

2011 (2)

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).

2010 (2)

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120(2010).
[CrossRef]

2009 (4)

2008 (2)

2007 (2)

2006 (4)

2005 (1)

2004 (2)

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

2003 (2)

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (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]

2002 (1)

2000 (2)

1995 (2)

Alfalou, A.

Arcos, S.

Barrera, J. F.

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

Bitran, Y.

Bolognini, N.

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

Brosseau, C.

Cai, L. Z.

Cao, L.

Carnicer, A.

Castro, A.

Chang, H.

Chang, H. T.

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

H. T. Chang, W. C. Lu, and C. J. Kuo, “Multiple-phase retrieval for optical security systems using random phase encoding,” Appl. Opt. 41, 4825–4834 (2002).
[CrossRef]

Chen, W.

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).

Chen, X.

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).

Cheng, X. C.

Chuang, C. H.

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Dong, G. Y.

Dorsch, R. G.

Ferreira, C.

Frauel, Y.

Garcia, J.

Gopinathan, U.

He, M.

He, Q.

Hennelly, B.

Hwang, H.

Javidi, B.

Jin, G.

Joseph, J.

Juvells, I.

Kumar, A.

Kumar, P.

Kuo, C. J.

Lai, W. N.

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Lie, W.

Lin, G. H.

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Liu, S.

Lu, W. C.

Mendlovic, D.

Meng, X. F.

Monaghan, D. S.

Montes-Usategui, M.

Naughton, T. J.

Nishchal, N. K.

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

Ozaktaz, H. M.

Peng, X.

Qin, W.

Ran, Q.

Refregier, P.

Shen, X. X.

Sheridan, J. T.

Singh, K.

Singh, N.

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Sinha, A.

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Situ, G.

Tan, Q.

Tao, R.

Tebaldi, M.

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

Torroba, R.

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

Unnikrishnan, G.

Vargas, C.

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

Wang, X.

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[CrossRef]

X. Wang and D. Zhao, “A specific attack on the asymmetric cryptosystem based on the phase-truncated Fourier transforms,” Opt. Commun. (to be published).

Wang, Y.

Wang, Y. R.

Wei, H.

Xin, Y.

Xu, X. F.

Yu, B.

Zhang, H.

Zhang, J.

Zhang, P.

Zhao, D.

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[CrossRef]

X. Wang and D. Zhao, “A specific attack on the asymmetric cryptosystem based on the phase-truncated Fourier transforms,” Opt. Commun. (to be published).

Zhu, B.

Adv. Opt. Photon. (1)

Appl. Opt. (2)

J. Opt. (1)

W. Chen and X. Chen, “Optical asymmetric cryptography using a three-dimensional space-based model,” J. Opt. 13, 075404 (2011).

Opt. Commun. (5)

X. Wang and D. Zhao, “A specific attack on the asymmetric cryptosystem based on the phase-truncated Fourier transforms,” Opt. Commun. (to be published).

N. K. Nishchal, J. Joseph, and K. Singh, “Securing information using fractional Fourier transform in digital holography,” Opt. Commun. 235, 253–259 (2004).
[CrossRef]

X. Wang and D. Zhao, “Image encryption based on anamorphic fractional Fourier transform and three-step phase-shifting interferometry,” Opt. Commun. 268, 240–244 (2006).
[CrossRef]

J. F. Barrera, C. Vargas, M. Tebaldi, N. Bolognini, and R. Torroba, “Chosen-plaintext attack on a joint transform correlator encrypting system,” Opt. Commun. 283, 3917–3921 (2010).
[CrossRef]

X. Wang and D. Zhao, “Multiple-image encryption based on nonlinear amplitude-truncation and phase-truncation in Fourier domain,” Opt. Commun. 284, 148–152 (2011).
[CrossRef]

Opt. Eng. (1)

G. H. Lin, H. T. Chang, W. N. Lai, and C. H. Chuang, “Public-key-based optical image cryptosystem with data embedding techniques,” Opt. Eng. 42, 2331–2339 (2003).
[CrossRef]

Opt. Express (4)

Opt. Lasers Eng. (1)

N. Singh and A. Sinha, “Optical image encryption using fractional Fourier transform and chaos,” Opt. Lasers Eng. 46, 117–123 (2008).
[CrossRef]

Opt. Lett. (12)

G. Unnikrishnan, J. Joseph, and K. Singh, “Optical encryption by double-random phase encoding in the fractional Fourier domain,” Opt. Lett. 25, 887–889 (2000).
[CrossRef]

P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20, 767–769 (1995).
[CrossRef]

X. Peng, H. Wei, and P. Zhang, “Chosen-plaintext attack on lensless double-random phase encoding in the Fresnel domain,” Opt. Lett. 31, 3261–3263 (2006).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multi-fractional Fourier transforms,” Opt. Lett. 25, 1159–1161 (2000).
[CrossRef]

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

G. Situ and J. Zhang, “Double random-phase encoding in the Fresnel domain,” Opt. Lett. 29, 1584–1586 (2004).
[CrossRef]

A. Carnicer, M. Montes-Usategui, S. Arcos, and I. Juvells, “Vulnerability to chosen-cyphertext attacks of optical encryption schemes based on double random phase keys,” Opt. Lett. 30, 1644–1646 (2005).
[CrossRef]

X. Peng, P. Zhang, H. Wei, and B. Yu, “Known-plaintext attack on optical encryption based on double random phase keys,” Opt. Lett. 31, 1044–1046 (2006).
[CrossRef]

X. C. Cheng, L. Z. Cai, Y. R. Wang, X. F. Meng, H. Zhang, X. F. Xu, X. X. Shen, and G. Y. Dong, “Security enhancement of double-random phase encryption by amplitude modulation,” Opt. Lett. 33, 1575–1577 (2008).
[CrossRef]

P. Kumar, A. Kumar, J. Joseph, and K. Singh, “Impulse attack free double-random-phase encryption scheme with randomized lens-phase functions,” Opt. Lett. 34, 331–333 (2009).
[CrossRef]

H. Hwang, H. Chang, and W. Lie, “Multiple-image encryption and multiplexing using a modified Gerchberg-Saxton algorithm and phase modulation in Fresnel-transform domain,” Opt. Lett. 34, 3917–3919 (2009).
[CrossRef]

W. Qin and X. Peng, “Asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Lett. 35, 118–120(2010).
[CrossRef]

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Figures (9)

Fig. 1.
Fig. 1.

Flowchart of (a) encryption process, (b) decryption process based on phase-truncated anamorphic FrFT.

Fig. 2.
Fig. 2.

Flowchart of (a) encryption process and (b) decryption process with amplitude modulation.

Fig. 3.
Fig. 3.

(a) Original image and (b) encrypted result with the fractional orders α1=0.3, α2=0.5, α3=0.4 and α4=0.6.

Fig. 4.
Fig. 4.

Decryption results by using (a) no keys, (b) arbitrarily selected phase keys and fractional orders, (c) arbitrarily selected phase keys but correct fractional orders, (d) arbitrarily fractional orders (α1=0.31, α2=0.51, α3=0.41, and α4=0.61) but correct phase keys, (e) correct fractional orders but fake decryption keys generated from a fake plaintext, (f) arbitrarily selected fractional orders but fake decryption keys, (g) the specific attack (m=300, n=2), and (h) all the correct keys. (i) The fake plaintext.

Fig. 5.
Fig. 5.

Relation between iteration times and the MSE (a) in the first step, (b) in the second step.

Fig. 6.
Fig. 6.

(a) AM A1 with k=0.03 and p=0.1005, (b) ciphertext obtained with the AM, and (c) the decryption result with all the correct keys.

Fig. 7.
Fig. 7.

Decryption results by using (a) no keys, (b) arbitrarily selected phase keys, (c) the correct phase keys but with a different amplitude modulator (a reverse distribution of the AM with k=0.02 and p=0.2011), (d) fake decryption keys generated from Fig. 4(h) but with correct amplitude modulator, (e) the specific attack (m=300, n=2).

Fig. 8.
Fig. 8.

Relation between MSE and the parameter (a) k (p=1.02), (b) p(k=0.01) of the AM.

Fig. 9.
Fig. 9.

Relation between iteration times and the MSE (a) in the first step, (b) in the second step.

Equations (17)

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

g1(u,υ)=PT{FT[f(x,y)R1(x,y)]},
g(x,y)=PT{IFT[g1(u,υ)R2(u,υ)]},
K1(u,υ)=PR{FT[f(x,y)R1(x,y)]},
K2(x,y)=PR{IFT[g1(u,υ)R2(u,υ)]},
Fα1[f(x)](u)=+Kα1(x,u)f(x)dx,
Kα1(x,u)={A1exp[iπ(x2cotϕ12xucscϕ1+u2cotϕ1)],ϕ1nπ,δ(xu),ϕ1=2nπ,δ(x+u),ϕ1=(2n+1)π,,
Fα1,α2[f(x,y)](u,υ)=+Kα1,α2(x,y;u,υ)f(x,y)dxdy,
Kα1,α2(x,y;u,υ)=Kα1(x,u)Kα2(y,υ),
P1(u,υ)=PR{Fα1,α2[f(x,y)R1(x,y)]},
P2(x,y)=PR{Fα3,α4[g1(u,υ)R2(u,υ)]}.
g1(u,υ)=PT{Fα3,α4[E(x,y)P2(x,y)]},
f(x,y)=PT{Fα1,α2[g1(u,υ)P1(u,υ)]},
E(x,y)=PT{IFT[g1(u,υ)·A1(u,υ)·R2(u,υ)]},
P2(x,y)=PR{IFT[g1(u,υ)·A1(u,υ)·R2(u,υ)]},
PT{FT[E(x,y)P2(x,y)]·A2(u,υ)=k·g1(u,υ),
PT{IFT[k·g1(u,υ)P1(u,υ)]}=k·f(x,y).
MSE(f,f)=1Li=1L|fifi|2,

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