STUDYING THE PROPERTIES OF PIXELS PERMUTATIONS BASED ON DISCRETIZED STANDARD MAP
In this article, we described specifics of pixels permutations based on the discretized, two-dimensional Chirikov standard map. Some properties of the discretized Chirikov map can be used by an attacker to recover the original images that are studied. For images with dimensions N ´ N the vulnerability of permutations allows for brute force attacks, and shown is the ability of an intruder to restore the original image without setting the value of keys permutations. Presented is also, successful cryptographic attack on the encrypted image through permutation of pixels. It is found that for images with dimension N ´ N the maximum number of combinations is equal to NN-1. A modified Chirikov map was proposed with improved permutation properties, due to the use of two nonlinearities, that increase the keys space to N2!.
discretized standard map; permutation of pixels; key space; precision of computing
Alvarez, G., Li, S. J.: Some Basic Cryptographic Requirements for Chaos-Based Cryptosystems. Inter. Journal of Bif. and Chaos 16(8)/2006, 2129–2151. DOI: https://doi.org/10.1142/S0218127406015970
Argyris A., Syvridis D., Larger L., Annovazzi-Lodi V., Colet P., Fischer I., García-Ojalvo J., Mirasso C.R., Pesquera L., Shore K.A.: Chaos-based communications at high bit rates using commercial fibre-optic links. Nature 438(7066)/2005, 343–346. DOI: https://doi.org/10.1038/nature04275
Arroyo D., Alvarez G., Fernandez V.: A basic framework for the cryptanalysis of digital chaos-based cryptography. Proc. of the 6th International Multi-Conference on Systems, Signals and Devices, Djerba 2009, 58–63. DOI: https://doi.org/10.1109/SSD.2009.4956652
Chirikov B. V.: Research concerning the theory of nonlinear resonance and stochasticity Preprint 267, Institute of Nuclear Physics, Novosibirsk, 1969, (Engl. Trans., CERN Trans. 1971, 71–40).
Fridrich J.: Symmetric Ciphers Based on Two-Dimensional Chaotic Maps. Inter. Journal of Bif. and Chaos 8(6)/1998, 1259–284. DOI: https://doi.org/10.1142/S021812749800098X
Hussain I., Shah T.: Literature survey on nonlinear components and chaotic nonlinear components of block ciphers. Nonlinear Dynamics 74/2013, 869–904. DOI: https://doi.org/10.1007/s11071-013-1011-8
Jolfaei A., Mirghadri A.: An image encryption approach using chaos and stream cipher. Journal of Theoretical and Applied Information Technology 19(2)/2010, 117–125.
Kocarev L., Lian S. (Eds.): Chaos-Based Cryptography Theory, Algorithms and Applications. Springer-Verlag Berlin Heidelberg, 2011. DOI: https://doi.org/10.1007/978-3-642-20542-2
Lian S. G., Sun J., Wang Z.: A block cipher based on a suitable use of chaotic standard map. Chaos, Solitons and Fractals 26(1)/2005, 117–29. DOI: https://doi.org/10.1016/j.chaos.2004.11.096
Lian S., Sun J., Wang Z.: Security analysis of a chaos-based image encryption algorithm. Phisyca A 351(2)/2005, 645–661. DOI: https://doi.org/10.1016/j.physa.2005.01.001
National Institute of Standards and Technology (May 11, 2010). NIST Digital Library of Mathematical Functions. Section 26.4. Retrieved August 30, 2010.
Solak, E., Cokal, C., Yildiz, O.T., Biyikoglu, T.: Cryptanalysis of fridrich’s chaotic image encryption. Int. J. Bifurcation Chaos 20(5), 1405–1413. DOI: https://doi.org/10.1142/S0218127410026563
von Bremen H. F., Udwadia F. E., Proskurowski W.: An efficient QR based method for the computation of Lyapunov exponents. Physica D 101/1997, 1–16. DOI: https://doi.org/10.1016/S0167-2789(96)00216-3
Warren H. S. .: Hacker’s Delight. Addison-Wesley Professional. 2012.
Yuan G., Yorke J. A.: Collapsing of chaos in one dimensional maps. Physica D: Nonlinear Phenomena 136/2000, 18–30. DOI: https://doi.org/10.1016/S0167-2789(99)00147-5
This work is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.