KSII Transactions on Internet and Information Systems (TIIS)

Korean Society for Internet Information (한국인터넷정보학회)
권호 표지
DOI QR코드

DOI QR Code

An Image Encryption Scheme Based on Concatenated Torus Automorphisms

  • Mao, Qian (Department of Optical-Electrical and Computer Engineering, University of Shanghai for Science and Technology) ;
  • Chang, Chin-Chen (Department of Computer Science and Information Engineering, Asia University) ;
  • Wu, Hsiao-Ling (Department of Information Engineering and Computer Science, Feng Chia University)
  • Received : 2013.01.08
  • Accepted : 2013.05.18
  • Published : 2013.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

A novel, chaotic map that is based on concatenated torus automorphisms is proposed in this paper. As we know, cat map, which is based on torus automorphism, is highly chaotic and is often used to encrypt information. But cat map is periodic, which decreases the security of the cryptosystem. In this paper, we propose a novel chaotic map that concatenates several torus automorphisms. The concatenated mechanism provides stronger chaos and larger key space for the cryptosystem. It is proven that the period of the concatenated torus automorphisms is the total sum of each one's period. By this means, the period of the novel automorphism is increased extremely. Based on the novel, concatenated torus automorphisms, two application schemes in image encryption are proposed, i.e., 2D and 3D concatenated chaotic maps. In these schemes, both the scrambling matrices and the iteration numbers act as secret keys. Security analysis shows that the proposed, concatenated, chaotic maps have strong chaos and they are very sensitive to the secret keys. By means of concatenating several torus automorphisms, the key space of the proposed cryptosystem can be expanded to $2^{135}$. The diffusion function in the proposed scheme changes the gray values of the transferred pixels, which makes the periodicity of the concatenated torus automorphisms disappeared. Therefore, the proposed cryptosystem has high security and they can resist the brute-force attacks and the differential attacks efficiently. The diffusing speed of the proposed scheme is higher, and the computational complexity is lower, compared with the existing methods.

Keywords

References

  1. G. Jakimoski and L. Kocarev, "Chaos and cryptography: block encryption ciphers based on chaotic maps," IEEE Transactions on Circuits and Systems I: Fundamental Theory and Applications, vol. 48, no. 2, pp. 163-169, 2001. https://doi.org/10.1109/81.904880
  2. C. E. Shannon, "Communication theory of secrecy systems," Bell System Technical Journal (Bell System Tech.), vol. 28, no. 4, pp. 656-715, 1949. https://doi.org/10.1002/j.1538-7305.1949.tb00928.x
  3. B. Jovic and C. P. Unsworth, "Fast synchronisation of chaotic maps for secure chaotic communications," Electronics Letters, vol. 46, no. 1, pp. 49-50, 2010. https://doi.org/10.1049/el.2010.1532
  4. K. W. Wong, Q. Z. Lin and J. Y. Chen, "Simultaneous arithmetic coding and encryption using chaotic maps," IEEE Transactions on Circuits and Systems II: Express Briefs, vol. 57, no. 2, pp. 146-150, 2010. https://doi.org/10.1109/TCSII.2010.2040315
  5. G. A. Sathishkumar, S. Ramachandran and K. B. Bagan, "Image encryption using random pixel permutation by chaotic mapping," in Proc. of IEEE Symposium on Computers & Informatics (ISCI), pp. 247-251, 2012.
  6. J. Fridrich, "Symmetric ciphers based on two-dimensional chaotic maps," International Journal of Bifurcation and Chaos, vol. 8, no 6, pp. 1259-1284, 1998. https://doi.org/10.1142/S021812749800098X
  7. Z. L. Zhu, W. Zhang, K. W. Wong and H. Yu, "A chaos-based symmetric image encryption scheme using a bit-level permutation," Information Science, vol. 181, no. 6, pp. 1171-1186, 2011. https://doi.org/10.1016/j.ins.2010.11.009
  8. R. S. Ye and H. Q. Huang, "A novel image shuffling and watermarking scheme based on standard map," in Proc. of International Conference on Information Engineering and Computer Science (ICIECS), pp. 1-4, 2009.
  9. V. Patidar, G. Purohit, K. K. Sud and N. K. Pareek, "Image encryption through a novel permutation-substitution scheme based on chaotic standard map," in Proc. of International Workshop on Chaos-Fractals Theories and Applications (IWCFTA), pp. 164-169, 2010.
  10. Y. Sun and G. Y. Wang, "An image encryption scheme based on modified logistic map," in Proc. of Fourth International Workshop on Chaos-Fractals Theories and Applications (IWCFTA), pp. 179-182, 2011.
  11. A. H. Zhu and L. Li, "Improving for chaotic image encryption algorithm based on logistic map," in Proc. of International Conference on Environmental Science and Information Application Technology (ESIAT), vol. 3, pp. 211-214, 2010.
  12. X. Y, Wang, L. Teng and X. Qin, "A novel colour image encryption algorithm based on chaos," Signal Processing, vol. 92, no. 4, pp. 1101-1108, 2012. https://doi.org/10.1016/j.sigpro.2011.10.023
  13. Y. Y. Mao and X. Chen, "An encryption algorithm of chaos based on sine square mapping," in Proc. of Fourth International Symposium on Computational Intelligence and Design (ISCID), pp. 131-134, 2011.
  14. D. E. Goumidi and F. Hachouf, "Modified confusion-diffusion based satellite image cipher using chaotic standard, logistic and sine maps," in Proc. of 2nd European Workshop on Visual Information Processing (EUVIP), pp. 204-209, 2010.
  15. V. I. Arnold and A. Avez, Ergodic Problems of Classical Mechanics, Benjamin, New York, 1968.
  16. P. Akritas, I. E. Antoniou and G. P. Pronko, "On the torus automorphisms: analytic solution, computability and quantization," Chaos, Solitons and Fractals, vol. 12, no. 14, pp. 2805-2814, 2001. https://doi.org/10.1016/S0960-0779(01)00094-7
  17. L. H. Zhu, W. Z. Li, L. J. Liao and H. Li, "A novel algorithm for scrambling digital image based on cat chaotic mapping," in Proc. of International Conference on Intelligent Information Hiding and Multimedia Signal Processing (IIH-MSP), pp. 601-604, 2006.
  18. C. J. Pang, "An image encryption algorithm based on discrete wavelet transform and two dimension cat mapping," in Proc. of International Conference on Networks Security, Wireless Communications and Trusted Computing (NSWCTC), pp. 711-714, 2009.
  19. G. Chen, Y. Mao and C. Chui, "A symmetric image encryption scheme based on 3D chaotic cat maps," Chaos, Solitons, Fractals, vol. 21, no. 3, pp. 749-761, 2004. https://doi.org/10.1016/j.chaos.2003.12.022
  20. C. C. Chang, J. Y. Hsiao and C. L. Chiang, "An image copyright protection scheme based on torus automorphism," in Proc. of the First International Symposium on Cyber Worlds, pp. 217-224, 2002.
  21. F. Chen, K. W. Wong, X. F. Liao and T. Xiang, "Period distribution of generalized discrete arnold cat map for N = pe," IEEE Transactions on Information Theory, vol. 58, no. 1, pp. 445-452, 2012. https://doi.org/10.1109/TIT.2011.2171534
  22. I. Percival and F. Vivaldi, "Arithmetical properties of strongly chaotic motions," Physica D: Nonlinear Phenomena, vol. 25, no. 1-3, pp. 105-130, 1987. https://doi.org/10.1016/0167-2789(87)90096-0

Cited by

  1. An Efficient Chaotic Image Encryption Algorithm Based on Self-adaptive Model and Feedback Mechanism vol.11, pp.3, 2017, https://doi.org/10.3837/tiis.2017.03.029
  2. Non-square colour image scrambling based on two-dimensional Sine-Logistic and Hénon map vol.11, pp.12, 2017, https://doi.org/10.3837/tiis.2017.12.015
  3. Image encryption scheme based on Chaos and DNA plane operations vol.78, pp.18, 2013, https://doi.org/10.1007/s11042-019-07794-9