Abstract

In this paper, we propose a novel secure image sharing scheme based on Shamir’s three-pass protocol and the multiple-parameter fractional Fourier transform (MPFRFT), which can safely exchange information with no advance distribution of either secret keys or public keys between users. The image is encrypted directly by the MPFRFT spectrum without the use of phase keys, and information can be shared by transmitting the encrypted image (or message) three times between users. Numerical simulation results are given to verify the performance of the proposed algorithm.

© 2012 OSA

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References

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  1. P. Refregier and B. Javidi, “Optical image encryption based on input plane and Fourier plane random encoding,” Opt. Lett. 20(7), 767–769 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
    [CrossRef]
  4. B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
    [CrossRef] [PubMed]
  5. B. Hennelly and J. T. Sheridan, “Optical image encryption by random shifting in fractional Fourier domains,” Opt. Lett. 28(4), 269–271 (2003).
    [CrossRef] [PubMed]
  6. J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
    [CrossRef]
  7. X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
    [CrossRef]
  8. Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
    [CrossRef]
  9. R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
    [CrossRef] [PubMed]
  10. Q. Ran, H. Zhang, J. Zhang, L. Tan, and J. Ma, “Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform,” Opt. Lett. 34(11), 1729–1731 (2009).
    [CrossRef] [PubMed]
  11. R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
    [CrossRef]
  12. Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
    [CrossRef]
  13. Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
    [CrossRef]
  14. C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010).
    [CrossRef]
  15. N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
    [CrossRef]
  16. J. Massey, “An introduction to contemporary cryptology,” Proc. IEEE 76(5), 533–549 (1988).
    [CrossRef]
  17. L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
    [CrossRef]
  18. H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, Chichester, 2001).
  19. J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
    [CrossRef]
  20. J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
    [CrossRef]

2011 (2)

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

2010 (4)

Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
[CrossRef]

C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
[CrossRef]

2009 (2)

Q. Ran, H. Zhang, J. Zhang, L. Tan, and J. Ma, “Deficiencies of the cryptography based on multiple-parameter fractional Fourier transform,” Opt. Lett. 34(11), 1729–1731 (2009).
[CrossRef] [PubMed]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
[CrossRef]

2008 (3)

Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
[CrossRef] [PubMed]

J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
[CrossRef]

2007 (1)

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
[CrossRef]

2003 (1)

2002 (1)

L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
[CrossRef]

2001 (1)

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
[CrossRef]

2000 (2)

1995 (1)

1988 (1)

J. Massey, “An introduction to contemporary cryptology,” Proc. IEEE 76(5), 533–549 (1988).
[CrossRef]

Ahmad, M. A.

Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
[CrossRef]

Brouzet, R.

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

Hayat, K.

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

Hennelly, B.

Huang, S. M.

C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010).
[CrossRef]

Islam, N.

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

Javidi, B.

Joseph, J.

Lang, J.

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
[CrossRef] [PubMed]

J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
[CrossRef]

Ling, A. W.

L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
[CrossRef]

Liu, S.

Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
[CrossRef]

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
[CrossRef] [PubMed]

Liu, S. H.

L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
[CrossRef]

Liu, Z.

Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
[CrossRef]

Ma, J.

Massey, J.

J. Massey, “An introduction to contemporary cryptology,” Proc. IEEE 76(5), 533–549 (1988).
[CrossRef]

Puech, W.

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

Ran, Q.

Refregier, P.

Sheridan, J. T.

Singh, K.

Tan, L.

Tao, R.

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
[CrossRef] [PubMed]

J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
[CrossRef]

Unnikrishnan, G.

Wang, X.

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

Wang, Y.

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “Optical image encryption based on the multiple-parameter fractional Fourier transform,” Opt. Lett. 33(6), 581–583 (2008).
[CrossRef] [PubMed]

J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
[CrossRef]

Yang, C. N.

C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010).
[CrossRef]

Yang, L.

L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
[CrossRef]

Yu, L.

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
[CrossRef]

Zhang, H.

Zhang, J.

Zhao, D.

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

Zhu, B.

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
[CrossRef]

B. Zhu, S. Liu, and Q. Ran, “Optical image encryption based on multifractional Fourier transforms,” Opt. Lett. 25(16), 1159–1161 (2000).
[CrossRef] [PubMed]

Opt. Commun. (8)

S. Liu, L. Yu, and B. Zhu, “Optical image encryption by cascaded fractional Fourier transforms with random phase filtering,” Opt. Commun. 187(1-3), 57–63 (2001).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “Image encryption based on the multiple-parameter discrete fractional Fourier transform and chaos function,” Opt. Commun. 283(10), 2092–2096 (2010).
[CrossRef]

X. Wang and D. Zhao, “Double-image self-encoding and hiding based on phase-truncated Fourier transforms and phase retrieval,” Opt. Commun. 284(19), 4441–4445 (2011).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image encryption scheme based on the commutation and anti-commutation rules,” Opt. Commun. 279(2), 285–290 (2007).
[CrossRef]

R. Tao, J. Lang, and Y. Wang, “The multiple-parameter discrete fractional Hadamard transform,” Opt. Commun. 282(8), 1531–1535 (2009).
[CrossRef]

Z. Liu, M. A. Ahmad, and S. Liu, “Image sharing scheme based on combination theory,” Opt. Commun. 281(21), 5322–5325 (2008).
[CrossRef]

C. N. Yang and S. M. Huang, “Constructions and properties of k out of n scalable secret image sharing,” Opt. Commun. 283(9), 1750–1762 (2010).
[CrossRef]

N. Islam, W. Puech, K. Hayat, and R. Brouzet, “Application of homomorphism to secure image sharing,” Opt. Commun. 284(19), 4412–4429 (2011).
[CrossRef]

Opt. Lett. (6)

Optik (Stuttg.) (1)

Z. Liu, S. Liu, and M. A. Ahmad, “Image sharing scheme based on discrete fractional random transform,” Optik (Stuttg.) 121(6), 495–499 (2010).
[CrossRef]

Proc. IEEE (1)

J. Massey, “An introduction to contemporary cryptology,” Proc. IEEE 76(5), 533–549 (1988).
[CrossRef]

Proc. SPIE (1)

L. Yang, A. W. Ling, and S. H. Liu, “Quantum three-pass cryptography protocol,” Proc. SPIE 4917, 106–111 (2002).
[CrossRef]

Sci. China Ser. F, Inf. Sci. (2)

J. Lang, R. Tao, Q. Ran, and Y. Wang, “The multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 51(8), 1010–1024 (2008).
[CrossRef]

J. Lang, R. Tao, and Y. Wang, “The discrete multiple-parameter fractional Fourier transform,” Sci. China Ser. F, Inf. Sci. 53(11), 2287–2299 (2010).
[CrossRef]

Other (1)

H. M. Ozaktas, Z. Zalevsky, and M. A. Kutay, The Fractional Fourier Transform with Applications in Optics and Signal Processing (John Wiley & Sons, Chichester, 2001).

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Figures (8)

Fig. 1
Fig. 1

Illustration of Shamir’s three-pass protocol for secrecy sharing.

Fig. 2
Fig. 2

Results of the proposed secure image sharing scheme based MPFRFT and Shamir’s three-pass protocol. (a) Original image. (b) Cryptogram C 1 of Alice with her private secret keys. (c) Cryptogram C 2 of Bob with his private secret keys. (d) Cryptogram C 3 of Alice with her private secret keys. (e) Cryptogram C 4 of Bob with his private secret keys.

Fig. 3
Fig. 3

Security analysis of the proposed secure image sharing scheme based MPFRFT and Shamir’s three-pass protocol. (a) Bob received message if Alice decrypts C 2 with wrong fractional order ( α L , α R )=( -14.098,-23.598 ) . (b) Bob received message if Alice decrypts C 2 with wrong periodicity ( M L , M R )=( 22,30 ) . (c) Bob received message if Alice decrypts C 2 with wrong vector parameter ( M L , M R ) . (d) Bob received message if Bob decrypts C 3 with wrong vector parameter ( N L , N R ) .

Fig. 4
Fig. 4

Mean Square Error plotted as a function of error in the decryption keys. (a) The MSE for deviation of fractional order α. (b) The MSE for deviation of periodicity M. (c) Derivation of the MSE versus error numbers occurred in vector parameters N .

Fig. 5
Fig. 5

Autocorrelation and histogram of the original and encrypted images. (a) Histogram of the original image. (b) Histogram of the encrypted image. (c) Autocorrelation of the original image. (d) Autocorrelation of the encrypted image.

Fig. 6
Fig. 6

Robustness performance against data loss. (a) When 25% pixels of C 1 (Fig. 2b) are occluded. (b) Recovered message (PSNR = 16.34 dB). (c) When 50% pixels of C 1 (Fig. 2b) are occluded. (d) Recovered message (PSNR = 10.97 dB). (e) When 75% pixels of C 1 (Fig. 2b) are occluded. (f) Recovered message (PSNR = 9.38 dB).

Fig. 7
Fig. 7

Decrypted images with different σ: (a) σ=0.05 (PSNR = 37.76 dB), (b) σ=0.1 (PSNR = 31.66 dB), (c) σ=0.2 (PSNR = 25.73 dB), (d) σ=0.5 (PSNR = 18.52 dB), (e) σ=0.8 (PSNR = 16.57 dB), and (f) σ=1 (PSNR = 10.69 dB).

Fig. 8
Fig. 8

Robustness performance of the proposed method against noise perturbation with different σ

Equations (13)

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

E k 1 ( E k 2 ( X ) )= E k 2 ( E k 1 ( X ) )
α β = α+β = β α
α [ f( x ) ]= + f( x ) K α ( x, x α )dx
K α ( x, x α )={ A ϕ exp[ iπ( x 2 cotϕ2x x a cscϕ+ x a 2 cotϕ ) ] δ( x x α ) δ( x+ x α ) , ifα2n ifα=4n ifα=4n±2
A ϕ = | sinϕ | 1 /2 exp[ iπsgn( ϕ ) 4 + iϕ 2 ], ϕ= απ 2
M α ( N )[ f( x ) ]= l=0 M1 l ( α,N ) f l ( x )
l ( α,N )= 1 M k=0 ( M1 ) exp{ ( 2πi /M )[ α( k+ n k M )lk ] } ,N=( n 0 , n 1 ,, n ( M1 ) ) M
M α ( N ) N β ( M )= l=0 M1 n=0 N1 l ( α,N ) n ( β,M ) 4l /M 4n /N = n=0 N1 n ( β,M ) 4n /N l=0 M1 l ( α,N ) 4l /M = N β ( M ) M α ( N )
( M L , M R ) ( α L , α R ) ( N L ; N R )[ f( x,y ) ]= M L α L ( x ; N L )f( x,y ) M R α R ( y ; N R )
MSE= XD =( 1 N×M n=1 N m=1 M | x n,m d n,m | 2 )
α i = α i +δ, β i = β i +δ, N i = N i +δ, M i = M i +δ
A C 3 B= C 2
E =E( 1+σG )

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