An Efficient Image Encryption Algorithm for the Period of Arnold's CAT Map

Deniz Elmacı; Nursin Bas Catak

doi:10.18201/ijisae.2018637935

Authors

Deniz Elmacı Ege University
Nursin Bas Catak Ege University http://orcid.org/0000-0003-1450-5731

DOI:

https://doi.org/10.18201/ijisae.2018637935

Keywords:

Arnold's CAT map, Chaos, Discrete-time dynamical systems, Hyperbolic toral automorphism

Abstract

Arnold's CAT Map (ACM) is a chaotic transformation the 2-dimensional toral automorphism T^2 defined by the mapping /Gamma:T^2 to T^2. There are many applications of ACM in various research areas such as: steganography, encryption of images, texts and watermarks. The transformation of an image is achieved by the randomized order of pixels. After a finite number of repetitions of the transformation, the original image reappears. In this study, encryption of two images is demonstrated together with a proposed algorithm. Moreover, the periodicity of ACM is discussed and an algorithm to change the period of ACM is suggested. The resultant period obtained from the new algorithm is compared with the period obtained from the usual ACM. The results show that the period of the proposed algorithm grows exponentially while the period of ACM has an upper bound.

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References

J. Franks. “Anosov diffeomorphisms" in global analysis, proc. Sympos, Pure math, American Mathematical Society, vol. 14, pp. 61-93, 1968.

Edward R. Scheinerman, “Invitation to Dynamical Systems,” Dept. of Math. Sciences, Johns Hopkins University, USA, 1996.

Timur Karacay, “Determinizm ve Kaos,” 2004.

M.A. Partnof and K. Crum, “Chaos and Arnold's cat map,” 2004.

V.I. Arnold and A. Avez, “Ergodic Problems of Classical Mechanics,” Benjamin, 1968.J. Franks. “Anosov diffeomorphisms" in global analysis, proc. Sympos, Pure math, American Mathematical Society, vol. 14, pp. 61-93, 1968.

J.P. Keating, “Asymptotic properties of the periodic orbits of the cat,” Nonlinearity, vol. 4, pp. 277-307, 1991.

F.J. Dyson and H. Falk. “Period of a discrete cat mapping,” The American Mathematical Monthly, vol. 99, pp. 603-614, 1992.

W. Chen, C. Quan, and C.J. Tay, “Optical color image encryption based on Arnold transform and interference method,” Optics Communications, vol. 282, pp. 3680-3685, 2009.

Z. Liu, L. Xu, T. Liu, H. Chen, P. Li, C. Lin, and S. Liu, “Color image encryption by using Arnold transform and color-blend operation in discrete cosine transform domains,” Optics Communications, vol. 284, pp. 123-128, 2011.

Xiangjun Wu, Haibin Kan, and Jurgen Kurths, “A new colour image encryption scheme based on DNA sequences and multiple improved 1d chaotic maps,” Applied Soft Computing, vol. 37, pp. 24-39, 2015.

A. Soleymani, J.Nordin, and E. Sundararajan, “A chaotic cryptosystem for images based on Henon and Arnold cat map,” Hindawi Scientic World, pp. 1-21, 2004.

Lu Xu, Zhi Li, Jian Li, and Wei Hua, “A novel bit-level image encryption algorithm based on chaotic maps,” Optics and Lasers in Engineering, vol. 78, pp. 17-25, 2016.

Necla Kircali Gursoy and Urfat Nuriyev, “Some Inequalities for Algebra of Fractions and Its Applications,” Advanced Math. Models & Applications, vol. 1 pp. 1-13, 2016.

Ahmed M. Elshamy, Fathi E. Abd El-Samie1, Osama S. Faragallah, Elsayed M. Elshamy, Hala S. El-sayed, S. F. El-zoghdy, Ahmed N. Z. Rashed, Abd El-Naser A. Mohamed, and Ahmad Q. Alhamad, “Optical image cryptosystem using double random phase encoding and Arnold’s Cat map,” Opt Quant Electron, Springer, vol. 48:212, 2016.

Yueping Li, Chunhua Wang, and Hua Chen, “A hyper-chaos-based image encryption algorithm using pixel-level permutation and bit-level permutation,” Optics and Lasers in Engineering, vol. 90, pp. 238-246, 2017.

Chanil Pak, and Lilian Huang, “A new color image encryption using combination of the 1D chaotic map,” Signal Processing, vol 138, pp. 129-137, 2017.

Fredrik Svanstrom, “Properties of a Generalized Arnold's Discrete Cat Map,” M.S. thesis, Dept. Math., Linnaeus University, Sweden, 2014.

F. Chen, K. Wong, X. Liao, and T. Xiang, “Period distribution of generalized discrete Arnold cat map for n=pe.,” Transactions on Information Theory, vol. 58, pp. 445-452, 2012.