Abstract

Chaos-based image cipher has been widely investigated over the last decade or so to meet the increasing demand for real-time secure image transmission over public networks. In this paper, an improved diffusion strategy is proposed to promote the efficiency of the most widely investigated permutation-diffusion type image cipher. By using the novel bidirectional diffusion strategy, the spreading process is significantly accelerated and hence the same level of security can be achieved with fewer overall encryption rounds. Moreover, to further enhance the security of the cryptosystem, a plain-text related chaotic orbit turbulence mechanism is introduced in diffusion procedure by perturbing the control parameter of the employed chaotic system according to the cipher-pixel. Extensive cryptanalysis has been performed on the proposed scheme using differential analysis, key space analysis, various statistical analyses and key sensitivity analysis. Results of our analyses indicate that the new scheme has a satisfactory security level with a low computational complexity, which renders it a good candidate for real-time secure image transmission applications.

© 2012 OSA

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    [CrossRef]
  4. F. Belkhouche, I. Gokcen, and U. Qidwai, “Chaotic gray-level image transformation,” J. Electron. Imaging 14(4), 043001 (2005).
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  5. N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
    [CrossRef]
  6. H. S. Kwok and W. K. S. Tang, “A fast image encryption system based on chaotic maps with finite precision representation,” Chaos Solitons Fractals 32(4), 1518–1529 (2007).
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  7. S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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  22. K. W. Wong, B. S. H. Kwok, and W. S. Law, “A fast image encryption scheme based on chaotic standard map,” Phys. Lett. A 372(15), 2645–2652 (2008).
    [CrossRef]
  23. K. W. Wong, B. S. H. Kwok, and C. H. Yuen, “An efficient diffusion approach for chaos-based image encryption,” Chaos Solitons Fractals 41(5), 2652–2663 (2009).
    [CrossRef]
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    [CrossRef]

2011 (1)

X. Ma, C. Fu, W. M. Lei, and S. Li, “A novel chaos-based image encryption scheme with an improved permutation process,” Int. J. Adv. Comput. Technol. 3(5), 223–233 (2011).
[CrossRef]

2009 (9)

K. W. Wong, B. S. H. Kwok, and C. H. Yuen, “An efficient diffusion approach for chaos-based image encryption,” Chaos Solitons Fractals 41(5), 2652–2663 (2009).
[CrossRef]

X. J. Tong and M. G. Cui, “Image encryption scheme based on 3D baker with dynamical compound chaotic sequence cipher generator,” Signal Process. 89(4), 480–491 (2009).
[CrossRef]

V. Patidar, N. K. Pareek, and K. K. Sud, “A new substitution-diffusion based image cipher using chaotic standard and logistic maps,” Commun. Nonlinear Sci. Numer. Simul. 14(7), 3056–3075 (2009).
[CrossRef]

R. Rhouma, S. Meherzi, and S. Belghith, “OCML-based colour image encryption,” Chaos Solitons Fractals 40(1), 309–318 (2009).
[CrossRef]

C. K. Huang and H. H. Nien, “Multi chaotic systems based pixel shuffle for image encryption,” Opt. Commun. 282(11), 2123–2127 (2009).
[CrossRef]

S. Mazloom and A. M. Eftekhari-Moghadam, “Color image encryption based on coupled nonlinear chaotic map,” Chaos Solitons Fractals 42(3), 1745–1754 (2009).
[CrossRef]

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
[CrossRef]

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

S. E. Borujeni and M. Eshghi, “Chaotic image encryption design using Tompkins-Paige algorithm,” Math. Probl. Eng. 2009, 762652 (2009).

2008 (4)

F. Y. Sun, S. T. Liu, Z. Q. Li, and Z. W. Lü, “A novel image encryption scheme based on spatial chaos map,” Chaos Solitons Fractals 38(3), 631–640 (2008).
[CrossRef]

S. Behnia, A. Akhshani, H. Mahmodi, and A. Akhavan, “A novel algorithm for image encryption based on mixture of chaotic maps,” Chaos Solitons Fractals 35(2), 408–419 (2008).
[CrossRef]

T. G. Gao and Z. Q. Chen, “A new image encryption algorithm based on hyper-chaos,” Phys. Lett. A 372(4), 394–400 (2008).
[CrossRef]

K. W. Wong, B. S. H. Kwok, and W. S. Law, “A fast image encryption scheme based on chaotic standard map,” Phys. Lett. A 372(15), 2645–2652 (2008).
[CrossRef]

2007 (3)

T. Xiang, K. W. Wong, and X. F. Liao, “Selective image encryption using a spatiotemporal chaotic system,” Chaos 17(2), 023115 (2007).
[CrossRef] [PubMed]

H. S. Kwok and W. K. S. Tang, “A fast image encryption system based on chaotic maps with finite precision representation,” Chaos Solitons Fractals 32(4), 1518–1529 (2007).
[CrossRef]

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

2006 (2)

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[CrossRef]

G. Alvarez and S. Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcat. Chaos 16(8), 2129–2151 (2006).
[CrossRef]

2005 (2)

S. G. Lian, J. S. Sun, and Z. Q. Wang, “A block cipher based on a suitable use of the chaotic standard map,” Chaos Solitons Fractals 26(1), 117–129 (2005).
[CrossRef]

F. Belkhouche, I. Gokcen, and U. Qidwai, “Chaotic gray-level image transformation,” J. Electron. Imaging 14(4), 043001 (2005).
[CrossRef]

2004 (2)

G. R. Chen, Y. B. Mao, and C. K. Chui, “A symmetric image encryption scheme based on 3D chaotic cat maps,” Chaos Solitons Fractals 21(3), 749–761 (2004).
[CrossRef]

Y. B. Mao, G. R. Chen, and S. G. Lian, “A novel fast image encryption scheme based on 3D chaotic baker maps,” Int. J. Bifurcat. Chaos 14(10), 3613–3624 (2004).
[CrossRef]

1998 (1)

J. Fridrich, “Symmetric ciphers based on two-dimensional chaotic maps,” Int. J. Bifurcat. Chaos 8(6), 1259–1284 (1998).
[CrossRef]

1974 (1)

F. Rannou, “Numerical study of discrete plane area-preserving map,” Astron. Astrophys. 31, 289–301 (1974).

Abbas, A. M.

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

Ahadpour, S.

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

Akhavan, A.

S. Behnia, A. Akhshani, H. Mahmodi, and A. Akhavan, “A novel algorithm for image encryption based on mixture of chaotic maps,” Chaos Solitons Fractals 35(2), 408–419 (2008).
[CrossRef]

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

Akhshani, A.

S. Behnia, A. Akhshani, H. Mahmodi, and A. Akhavan, “A novel algorithm for image encryption based on mixture of chaotic maps,” Chaos Solitons Fractals 35(2), 408–419 (2008).
[CrossRef]

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

Allah, O. S. F.

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

Alvarez, G.

G. Alvarez and S. Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcat. Chaos 16(8), 2129–2151 (2006).
[CrossRef]

Behnia, S.

S. Behnia, A. Akhshani, H. Mahmodi, and A. Akhavan, “A novel algorithm for image encryption based on mixture of chaotic maps,” Chaos Solitons Fractals 35(2), 408–419 (2008).
[CrossRef]

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

Belghith, S.

R. Rhouma, S. Meherzi, and S. Belghith, “OCML-based colour image encryption,” Chaos Solitons Fractals 40(1), 309–318 (2009).
[CrossRef]

Belkhouche, F.

F. Belkhouche, I. Gokcen, and U. Qidwai, “Chaotic gray-level image transformation,” J. Electron. Imaging 14(4), 043001 (2005).
[CrossRef]

Borujeni, S. E.

S. E. Borujeni and M. Eshghi, “Chaotic image encryption design using Tompkins-Paige algorithm,” Math. Probl. Eng. 2009, 762652 (2009).

Chen, G. R.

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
[CrossRef]

Y. B. Mao, G. R. Chen, and S. G. Lian, “A novel fast image encryption scheme based on 3D chaotic baker maps,” Int. J. Bifurcat. Chaos 14(10), 3613–3624 (2004).
[CrossRef]

G. R. Chen, Y. B. Mao, and C. K. Chui, “A symmetric image encryption scheme based on 3D chaotic cat maps,” Chaos Solitons Fractals 21(3), 749–761 (2004).
[CrossRef]

Chen, Z. Q.

T. G. Gao and Z. Q. Chen, “A new image encryption algorithm based on hyper-chaos,” Phys. Lett. A 372(4), 394–400 (2008).
[CrossRef]

Chui, C. K.

G. R. Chen, Y. B. Mao, and C. K. Chui, “A symmetric image encryption scheme based on 3D chaotic cat maps,” Chaos Solitons Fractals 21(3), 749–761 (2004).
[CrossRef]

Cui, M. G.

X. J. Tong and M. G. Cui, “Image encryption scheme based on 3D baker with dynamical compound chaotic sequence cipher generator,” Signal Process. 89(4), 480–491 (2009).
[CrossRef]

Eftekhari-Moghadam, A. M.

S. Mazloom and A. M. Eftekhari-Moghadam, “Color image encryption based on coupled nonlinear chaotic map,” Chaos Solitons Fractals 42(3), 1745–1754 (2009).
[CrossRef]

Elashry, I. F.

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

El-Rabaie, S.

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

El-Samie, F. E. A.

I. F. Elashry, O. S. F. Allah, A. M. Abbas, S. El-Rabaie, and F. E. A. El-Samie, “Homomorphic image encryption,” J. Electron. Imaging 18(3), 033002 (2009).
[CrossRef]

Eshghi, M.

S. E. Borujeni and M. Eshghi, “Chaotic image encryption design using Tompkins-Paige algorithm,” Math. Probl. Eng. 2009, 762652 (2009).

Fridrich, J.

J. Fridrich, “Symmetric ciphers based on two-dimensional chaotic maps,” Int. J. Bifurcat. Chaos 8(6), 1259–1284 (1998).
[CrossRef]

Fu, C.

X. Ma, C. Fu, W. M. Lei, and S. Li, “A novel chaos-based image encryption scheme with an improved permutation process,” Int. J. Adv. Comput. Technol. 3(5), 223–233 (2011).
[CrossRef]

Gao, T. G.

T. G. Gao and Z. Q. Chen, “A new image encryption algorithm based on hyper-chaos,” Phys. Lett. A 372(4), 394–400 (2008).
[CrossRef]

Gokcen, I.

F. Belkhouche, I. Gokcen, and U. Qidwai, “Chaotic gray-level image transformation,” J. Electron. Imaging 14(4), 043001 (2005).
[CrossRef]

Huang, C. K.

C. K. Huang and H. H. Nien, “Multi chaotic systems based pixel shuffle for image encryption,” Opt. Commun. 282(11), 2123–2127 (2009).
[CrossRef]

Kwok, B. S. H.

K. W. Wong, B. S. H. Kwok, and C. H. Yuen, “An efficient diffusion approach for chaos-based image encryption,” Chaos Solitons Fractals 41(5), 2652–2663 (2009).
[CrossRef]

K. W. Wong, B. S. H. Kwok, and W. S. Law, “A fast image encryption scheme based on chaotic standard map,” Phys. Lett. A 372(15), 2645–2652 (2008).
[CrossRef]

Kwok, H. S.

H. S. Kwok and W. K. S. Tang, “A fast image encryption system based on chaotic maps with finite precision representation,” Chaos Solitons Fractals 32(4), 1518–1529 (2007).
[CrossRef]

Law, W. S.

K. W. Wong, B. S. H. Kwok, and W. S. Law, “A fast image encryption scheme based on chaotic standard map,” Phys. Lett. A 372(15), 2645–2652 (2008).
[CrossRef]

Lei, W. M.

X. Ma, C. Fu, W. M. Lei, and S. Li, “A novel chaos-based image encryption scheme with an improved permutation process,” Int. J. Adv. Comput. Technol. 3(5), 223–233 (2011).
[CrossRef]

Li, S.

X. Ma, C. Fu, W. M. Lei, and S. Li, “A novel chaos-based image encryption scheme with an improved permutation process,” Int. J. Adv. Comput. Technol. 3(5), 223–233 (2011).
[CrossRef]

G. Alvarez and S. Li, “Some basic cryptographic requirements for chaos-based cryptosystems,” Int. J. Bifurcat. Chaos 16(8), 2129–2151 (2006).
[CrossRef]

Li, Z. Q.

F. Y. Sun, S. T. Liu, Z. Q. Li, and Z. W. Lü, “A novel image encryption scheme based on spatial chaos map,” Chaos Solitons Fractals 38(3), 631–640 (2008).
[CrossRef]

Lian, S. G.

S. G. Lian, J. S. Sun, and Z. Q. Wang, “A block cipher based on a suitable use of the chaotic standard map,” Chaos Solitons Fractals 26(1), 117–129 (2005).
[CrossRef]

Y. B. Mao, G. R. Chen, and S. G. Lian, “A novel fast image encryption scheme based on 3D chaotic baker maps,” Int. J. Bifurcat. Chaos 14(10), 3613–3624 (2004).
[CrossRef]

Liao, X. F.

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
[CrossRef]

T. Xiang, K. W. Wong, and X. F. Liao, “Selective image encryption using a spatiotemporal chaotic system,” Chaos 17(2), 023115 (2007).
[CrossRef] [PubMed]

Liu, S. T.

F. Y. Sun, S. T. Liu, Z. Q. Li, and Z. W. Lü, “A novel image encryption scheme based on spatial chaos map,” Chaos Solitons Fractals 38(3), 631–640 (2008).
[CrossRef]

Lü, Z. W.

F. Y. Sun, S. T. Liu, Z. Q. Li, and Z. W. Lü, “A novel image encryption scheme based on spatial chaos map,” Chaos Solitons Fractals 38(3), 631–640 (2008).
[CrossRef]

Ma, X.

X. Ma, C. Fu, W. M. Lei, and S. Li, “A novel chaos-based image encryption scheme with an improved permutation process,” Int. J. Adv. Comput. Technol. 3(5), 223–233 (2011).
[CrossRef]

Mahmodi, H.

S. Behnia, A. Akhshani, H. Mahmodi, and A. Akhavan, “A novel algorithm for image encryption based on mixture of chaotic maps,” Chaos Solitons Fractals 35(2), 408–419 (2008).
[CrossRef]

S. Behnia, A. Akhshani, S. Ahadpour, H. Mahmodi, and A. Akhavan, “A fast chaotic encryption scheme based on piecewise nonlinear chaotic maps,” Phys. Lett. A 366(4-5), 391–396 (2007).
[CrossRef]

Mao, Y. B.

Y. B. Mao, G. R. Chen, and S. G. Lian, “A novel fast image encryption scheme based on 3D chaotic baker maps,” Int. J. Bifurcat. Chaos 14(10), 3613–3624 (2004).
[CrossRef]

G. R. Chen, Y. B. Mao, and C. K. Chui, “A symmetric image encryption scheme based on 3D chaotic cat maps,” Chaos Solitons Fractals 21(3), 749–761 (2004).
[CrossRef]

Mazloom, S.

S. Mazloom and A. M. Eftekhari-Moghadam, “Color image encryption based on coupled nonlinear chaotic map,” Chaos Solitons Fractals 42(3), 1745–1754 (2009).
[CrossRef]

Meherzi, S.

R. Rhouma, S. Meherzi, and S. Belghith, “OCML-based colour image encryption,” Chaos Solitons Fractals 40(1), 309–318 (2009).
[CrossRef]

Nien, H. H.

C. K. Huang and H. H. Nien, “Multi chaotic systems based pixel shuffle for image encryption,” Opt. Commun. 282(11), 2123–2127 (2009).
[CrossRef]

Pareek, N. K.

V. Patidar, N. K. Pareek, and K. K. Sud, “A new substitution-diffusion based image cipher using chaotic standard and logistic maps,” Commun. Nonlinear Sci. Numer. Simul. 14(7), 3056–3075 (2009).
[CrossRef]

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[CrossRef]

Patidar, V.

V. Patidar, N. K. Pareek, and K. K. Sud, “A new substitution-diffusion based image cipher using chaotic standard and logistic maps,” Commun. Nonlinear Sci. Numer. Simul. 14(7), 3056–3075 (2009).
[CrossRef]

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[CrossRef]

Qidwai, U.

F. Belkhouche, I. Gokcen, and U. Qidwai, “Chaotic gray-level image transformation,” J. Electron. Imaging 14(4), 043001 (2005).
[CrossRef]

Rannou, F.

F. Rannou, “Numerical study of discrete plane area-preserving map,” Astron. Astrophys. 31, 289–301 (1974).

Rhouma, R.

R. Rhouma, S. Meherzi, and S. Belghith, “OCML-based colour image encryption,” Chaos Solitons Fractals 40(1), 309–318 (2009).
[CrossRef]

Sud, K. K.

V. Patidar, N. K. Pareek, and K. K. Sud, “A new substitution-diffusion based image cipher using chaotic standard and logistic maps,” Commun. Nonlinear Sci. Numer. Simul. 14(7), 3056–3075 (2009).
[CrossRef]

N. K. Pareek, V. Patidar, and K. K. Sud, “Image encryption using chaotic logistic map,” Image Vis. Comput. 24(9), 926–934 (2006).
[CrossRef]

Sun, F. Y.

F. Y. Sun, S. T. Liu, Z. Q. Li, and Z. W. Lü, “A novel image encryption scheme based on spatial chaos map,” Chaos Solitons Fractals 38(3), 631–640 (2008).
[CrossRef]

Sun, J. S.

S. G. Lian, J. S. Sun, and Z. Q. Wang, “A block cipher based on a suitable use of the chaotic standard map,” Chaos Solitons Fractals 26(1), 117–129 (2005).
[CrossRef]

Tang, W. K. S.

H. S. Kwok and W. K. S. Tang, “A fast image encryption system based on chaotic maps with finite precision representation,” Chaos Solitons Fractals 32(4), 1518–1529 (2007).
[CrossRef]

Tong, X. J.

X. J. Tong and M. G. Cui, “Image encryption scheme based on 3D baker with dynamical compound chaotic sequence cipher generator,” Signal Process. 89(4), 480–491 (2009).
[CrossRef]

Wang, Y.

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
[CrossRef]

Wang, Z. Q.

S. G. Lian, J. S. Sun, and Z. Q. Wang, “A block cipher based on a suitable use of the chaotic standard map,” Chaos Solitons Fractals 26(1), 117–129 (2005).
[CrossRef]

Wong, K. W.

K. W. Wong, B. S. H. Kwok, and C. H. Yuen, “An efficient diffusion approach for chaos-based image encryption,” Chaos Solitons Fractals 41(5), 2652–2663 (2009).
[CrossRef]

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
[CrossRef]

K. W. Wong, B. S. H. Kwok, and W. S. Law, “A fast image encryption scheme based on chaotic standard map,” Phys. Lett. A 372(15), 2645–2652 (2008).
[CrossRef]

T. Xiang, K. W. Wong, and X. F. Liao, “Selective image encryption using a spatiotemporal chaotic system,” Chaos 17(2), 023115 (2007).
[CrossRef] [PubMed]

Xiang, T.

Y. Wang, K. W. Wong, X. F. Liao, T. Xiang, and G. R. Chen, “A chaos-based image encryption algorithm with variable control parameters,” Chaos Solitons Fractals 41(4), 1773–1783 (2009).
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K. W. Wong, B. S. H. Kwok, and C. H. Yuen, “An efficient diffusion approach for chaos-based image encryption,” Chaos Solitons Fractals 41(5), 2652–2663 (2009).
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Figures (14)

Fig. 1
Fig. 1

Chaotic orbits of Chirikov standard map for various k. (a) k = 0.5. (b) k = 1.0. (c) k=1.5. (d) k=2.0.

Fig. 2
Fig. 2

The application of the Chirikov standard map. (a) The test grayscale image with 256 × 256 size. (b) The test image after applying the Chirikov standard map once. (c) The test image after applying the Chirikov standard map three times. (d) The test image after applying the Chirikov standard map five times.

Fig. 3
Fig. 3

Diffusion process of conventional chaos-based image cipher.

Fig. 4
Fig. 4

Bidirectional diffusion scheme.

Fig. 5
Fig. 5

Chaotic orbit perturbing scheme.

Fig. 6
Fig. 6

NPCR and UACI test. (a) and (b) are two plain images with only one bit difference at the lower right corner. (c) Cipher image of (a). (d) Cipher image of (b). (e) Differential image between (c) and (d).

Fig. 7
Fig. 7

NPCR and UACI performance of the proposed scheme and conventional scheme. (a) NPCR performance. (b) UACI performance.

Fig. 8
Fig. 8

Histograms of plain-image and cipher-image. (a) Plain-image. (b) Histogram of plain-image. (c) Cipher-image. (d) Histogram of cipher-image.

Fig. 9
Fig. 9

Correlation of horizontal adjacent two pixels. (a) Plain image. (b) Ciphered image.

Fig. 11
Fig. 11

Correlation of diagonal adjacent two pixels. (a) Plain image. (b) Ciphered image.

Fig. 10
Fig. 10

Correlation of vertical adjacent two pixels. (a) Plain image. (b) Ciphered image.

Fig. 12
Fig. 12

Correlation functions of key stream. (a) Autocorrelation function. (b) Cross-correlation function.

Fig. 13
Fig. 13

Key sensitivity test: result 1. (a) Ciphered image using key (K = 512, k = 5.78259581295362, x0 = 0.48729650284971). (b) Ciphered image using key (K = 513, k = 5.78259581295362, x0 = 0.48729650284971). (c) Differential image between (a) and (b). (d) Ciphered image using key (K = 512, k = 5.78259581295363, x0 = 0.48729650284971). (e) Differential image between (a) and (d). (f) Ciphered image using key (K = 512, k = 5.78259581295362, x0 = 0.48729650284972). (g) Differential image between (a) and (f).

Fig. 14
Fig. 14

Key sensitivity test: result 2. (a) Ciphered image using key (K = 768, k = 4.71052756328493, x0 = 0.73195538124604). (b) Deciphered image using key (K = 768, k = 4.71052756328493, x0 = 0.73195538124604). (c) Deciphered image using key (K = 767, k = 4.71052756328493, x0 = 0.73195538124604). (d) Deciphered image using key (K = 768, k = 4.71052756328492, x0 = 0.73195538124604). (e) Deciphered image using key (K = 768, k = 4.71052756328493, x0 = 0.73195538124603).

Tables (3)

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Table 1 Correlation Coefficients of Two Adjacent Pixels in Two Images

Tables Icon

Table 2 Differences between Cipher Images Produced by Slightly Different Keys

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Table 3 Encryption Time and NPCR & UACI Performance of the Proposed and Conventional Schemes

Equations (20)

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{ a i+1 =(a + i b i ) mod 2π, b i+1 =( b i +ksin(a + i b i )) mod 2π,
{ x i+1 =(x + i y i ) mod N, y i+1 =( y i +Ksin 2π x i+1 N ) mod N,
{ x i+1 =(x i y i +Ksin 2π x i N ) mod N, y i+1 =( y i Ksin 2π x i N ) mod N.
c(n)=k(n){[p(n)+k(n)] mod N}c(n1),
x(n+1)= T k ( x n )=cos(k cos 1 x n ), x n [1, 1],
k(n)=mod[floor(((x(n)+1)/2)× 10 14 ),L],
p(n)=[k(n)c(n)c(n1)+Nk(n)] mod N.
NPCR= i=1 W j=1 H D(i,j) W×H ×100%,
D(i,j)={ 0 if P 1 (i,j)= P 2 (i,j), 1 if P 1 (i,j) P 2 (i,j).
NPC R expected =( 1 1 2 log 2 L )×100%,
UACI= 1 W×H [ i=1 W j=1 H | P 1 (i,j) P 2 (i,j) | L1 ]×100%.
UAC I expected = 1 L 2 ( i=1 L1 i(i+1) L1 )×100%.
I diff =| P 1 (i,j) P 2 (i,j) |.
H(K, x 0 ,k)1.156× 10 50 = 2 167 ,
r xy = 1 N i=1 N ( x i x ¯ )( y i y ¯ ) ( 1 N i=1 N ( x i x ¯ ) 2 )( 1 N i=1 N ( y i y ¯ ) 2 ) ,
x ¯ = 1 N i=1 N x i ,
y ¯ = 1 N i=1 N y i ,
H(S)= i=0 2 N 1 P( s i ) log 2 P( s i ) ,
autocorr(k)= i=0 N1 ( x i x ¯ )( x i+k x ¯ ) i=0 N1 ( x i x ¯ ) 2 ,
crosscorr(k)= i=0 N1 ( x i x ¯ )( y ik y ¯ ) i=0 N1 ( x i x ¯ ) 2 i=0 N1 ( y i y ¯ ) 2 ,

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