Abstract

We propose an improvement method for an asymmetric cryptosystem based on spherical wave illumination. Compared with the phase-truncated Fourier transform-based cryptosystem and the reported improving methods, the encryption process uses a spherical wave to illuminate the encryption system, rather than a uniform plane wave. As a result, the proposed method can avoid various types of the currently existing attacks and maintain the asymmetric characteristic of the cryptosystem. Moreover, due to only changing the illuminating mode, the proposed method can be easily implemented in optics compared with the reported improving methods. Simulation results are presented to demonstrate the feasibility and the security performance of the proposed method.

© 2013 Optical Society of America

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References

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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]

2012

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and inter-modulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

2011

2010

2009

2008

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

2007

2006

2004

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

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (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

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

2000

1995

Cai, L. L.

Chang, H. T.

Chen, C. C.

Chen, C. T.

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (2004).
[CrossRef]

Chen, L. F.

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]

Deng, X.

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and inter-modulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

X. Deng and D. Zhao, “Single-channel color image encryption using a modified Gerchberg–Saxton algorithm and mutual encoding in the Fresnel domain,” Appl. Opt. 50, 6019–6025 (2011).
[CrossRef]

Han, P.

Hennelly, B.

Hwang, H. E.

Javidi, B.

Joseph, J.

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

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]

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. N.

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]

Lu, W. C.

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]

Peng, X.

Qin, W.

Refregier, P.

Sheridan, J. T.

Singh, 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]

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]

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.

Tao, R.

Unnikrishnan, G.

Wang, X.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

Wang, Y.

Xin, Y.

Yu, L. F.

Zhang, J.

Zhao, D.

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and inter-modulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

X. Wang and D. Zhao, “Security enhancement of a phase-truncation based image encryption algorithm,” Appl. Opt. 50, 6645–6651 (2011).
[CrossRef]

X. Deng and D. Zhao, “Single-channel color image encryption using a modified Gerchberg–Saxton algorithm and mutual encoding in the Fresnel domain,” Appl. Opt. 50, 6019–6025 (2011).
[CrossRef]

L. F. Chen and D. Zhao, “Optical color image encryption by wavelength multiplexing and lensless Fresnel transform holograms,” Opt. Express 14, 8552–8560 (2006).
[CrossRef]

Appl. Opt.

J. Opt. Soc. Am. A

Opt. Commun.

H. T. Chang and C. T. Chen, “Asymmetric-image verification for security optical system based on joint transform correlator,” Opt. Commun. 239, 43–54 (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]

X. Wang and D. Zhao, “A special attack on the asymmetric cryptosystem based on phase-truncated Fourier transforms,” Opt. Commun. 285, 1078–1081 (2012).
[CrossRef]

Opt. Eng.

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

Opt. Laser Technol.

X. Deng and D. Zhao, “Single-channel color image encryption based on asymmetric cryptosystem,” Opt. Laser Technol. 44, 136–140 (2012).
[CrossRef]

X. Deng and D. Zhao, “Multiple-image encryption using phase retrieve algorithm and inter-modulation in Fourier domain,” Opt. Laser Technol. 44, 374–377 (2012).
[CrossRef]

Opt. Lasers. Eng.

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

Opt. Lett.

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Figures (8)

Fig. 1.
Fig. 1.

Flowchart of (a) encryption process and (b) decryption process with the PTFT-based asymmetric cryptosystem.

Fig. 2.
Fig. 2.

Flowchart of encryption process with the PTFT-based asymmetric cryptosystem illuminated by spherical wave.

Fig. 3.
Fig. 3.

Optical setup for image (a) encryption and (b) decryption.

Fig. 4.
Fig. 4.

(a) Image to be encoded, (b) ciphertext, and (c) correctly decrypted image.

Fig. 5.
Fig. 5.

Decryption results with (a) no keys, (b) arbitrarily selected decryption keys, and (c) public keys.

Fig. 6.
Fig. 6.

Specific attack result (a) without knowledge of the spherical wave, (b) with the correct operation wavelength (λ=600nm) and the incorrect radius (z=4.1cm), and (c) with the incorrect operation wavelength (λ=601nm) and the correct radius (z=4cm).

Fig. 7.
Fig. 7.

Relation between iteration times and the MSE (a) in the first step without any knowledge of the spherical wave, (b) in the second step without any knowledge of the spherical wave, (c) in the first step with the correct operation wavelength (λ=600nm) and the incorrect radius (z=4.1cm), (d) in the second step with the correct operation wavelength (λ=600nm) and the incorrect radius (z=4.1cm), (e) in the first step with the incorrect operation wavelength (λ=601nm) and the correct radius (z=4cm), and (f) in the second step with the incorrect operation wavelength (λ=601nm) and the correct radius (z=4cm).

Fig. 8.
Fig. 8.

MSE variation versus (a) wavelength difference Δλ and (b) displacement Δz.

Equations (19)

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

g1(u,v)=PT{FT[f(x,y)·E1(x,y)]},
g(x,y)=PT{IFT[g1(u,v)·E2(u,v)]},
D1(x,y)=PR{IFT[g1(u,v)·E2(u,v)]},
D2(u,v)=PR{FT[f(x,y)·E1(x,y)]},
g1(u,v)=PT{FT[g(x,y)·D1(x,y)]},
f(x,y)=PT{IFT[g1(u,v)·D2(u,v)]}.
U1(x,y)=Ceik2z(x2+y2),
fM(x,y)=f(x,y)·U1(x,y)·E1(x,y).
g1(u,v)=PT{FT[fM(x,y)]}.
g(x,y)=PT{IFT[g1(u,v)·U2(u,v)·E2(u,v)]},
D1(x,y)=PR{IFT[g1(u,v)·U2(u,v)·E2(u,v)]},
D2(u,v)=PR{FT[fM(x,y)]}.
g1(u,v)D2(u,v)=FT[f(x,y)·U1(x,y)·E1(x,y)],
g(x,y)D1(x,y)=IFT[g1(u,v)·U2(u,v)·E2(u,v)].
FT[g(x,y)·D1(x,y)]=g1(u,v)·E2(u,v)·U2(u,v).
PT[g1(u,v)·E2(u,v)·U2(u,v)]=g1(u,v).
IFT[g1(u,v)·D2(u,v)]=f(x,y)·E1(x,y)·U1(x,y).
f(x,y)=PT[f(x,y)·E1(x,y)·U1(x,y)].
MSE=1Li=1L(fi|fi|)2,

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