Modified Logistic Maps for Cryptographic Application
ABSTRACT
In this paper, definition and properties of logistic map along with orbit and bifurcation diagrams, Lyapunov exponent, and its histogram are considered. In order to expand chaotic region of Logistic map and make it suitable for cryptography, two modified versions of Logistic map are proposed. In the First Modification of Logistic map (FML), vertical symmetry and transformation to the right are used. In the Second Modification of Logistic (SML) map, vertical and horizontal symmetry and transformation to the right are used. Sensitivity of FML to initial condition is less and sensitivity of SML map to initial condition is more than the others. The total chaotic range of SML is more than others. Histograms of Logistic map and SML map are identical. Chaotic range of SML map is fivefold of chaotic range of Logistic map. This property gave more key space for cryptographic purposes.
In this paper, definition and properties of logistic map along with orbit and bifurcation diagrams, Lyapunov exponent, and its histogram are considered. In order to expand chaotic region of Logistic map and make it suitable for cryptography, two modified versions of Logistic map are proposed. In the First Modification of Logistic map (FML), vertical symmetry and transformation to the right are used. In the Second Modification of Logistic (SML) map, vertical and horizontal symmetry and transformation to the right are used. Sensitivity of FML to initial condition is less and sensitivity of SML map to initial condition is more than the others. The total chaotic range of SML is more than others. Histograms of Logistic map and SML map are identical. Chaotic range of SML map is fivefold of chaotic range of Logistic map. This property gave more key space for cryptographic purposes.
Cite this paper
Borujeni, S. and Ehsani, M. (2015) Modified Logistic Maps for Cryptographic Application. Applied Mathematics, 6, 773-782. doi: 10.4236/am.2015.65073.
Borujeni, S. and Ehsani, M. (2015) Modified Logistic Maps for Cryptographic Application. Applied Mathematics, 6, 773-782. doi: 10.4236/am.2015.65073.
References
[1] Alligood, K., Sauer, T. and Yorke, J. (1996) Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York. [2] Strogatz, S. (1994) Nonlinear Dynamics and Chaos. Perseus Books, Cambridge. [3] Schuster, H.G. and Just, W. (2005) Deterministic Chaos: An Introduction. 4th Edition, WILEY-VCH Verlag GmbH, Weinheim.
http://dx.doi.org/10.1002/3527604804 [4] Li, S., Li, Q., Li, W., Mou, X. and Cai, Y. (2001) Statistical Properties of Digital Piecewise Linear Chaotic Maps and Their Roles in Cryptography and Pseudo-Random Coding. Cryptography and Coding, 2260, 205-221.
http://dx.doi.org/10.1007/3-540-45325-3_19 [5] Addabbo, T., Alioto, M., Bernardi, S., Fort, A., Rocchi, S. and Vignoli, V. (2004) The Digital Tent Map: Performance Analysis and Optimized Design as a Source of Pseudo-Random Bits. Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference, IMTC 04, 2, 1301-1304. [6] Addabbo, T., Alioto, M., Bernardi, S., Fort, A., Rocchi, S. and Vignoli, V. (2004) Hardware-Efficient PRBGs Based on 1-D Piecewise Linear Chaotic Maps. Proceedings of the 11th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2004, 13-15 December 2004, 242-245. [7] Pareek, N., Patidar, V. and Sud, K. (2010) A Random Bit Generator Using Chaotic Maps. International Journal of Network Security, 10, 32-38. [8] Shastry, M., Nagaraj, N. and Vaidya, P. (2006) The B-Exponential Map: A Generalization of the Logistic Map, and Its Applications in Generating Pseudo-Random Numbers. eprint arXiv.org:cs/0607069. [9] Basios, V., Forti, G.L. and Gilbert, T. (2009) Statistical Properties of Time-Reversible Triangular Maps of the Square. Journal of Physics A: Mathematical and Theoretical, 42, 1-13.
http://dx.doi.org/10.1088/1751-8113/42/3/035102 [10] Huang, W. (2005) Characterizing Chaotic Processes That Generate Uniform Invariant Density. Chaos, Solitons & Fractals, 25, 449-460.
http://dx.doi.org/10.1016/j.chaos.2004.11.016 [11] Anikin, V., Arkadaksky, S., Kuptsov, S., Remizov, A. and Vasilenko, L. (2008) Lyapunov Exponent for Chaotic 1D Maps with Uniform Invariant Distribution. Bulletin of the Russian Academy of Sciences: Physics, 72, 1684-1688.
http://dx.doi.org/10.3103/S106287380812023X [12] Huang, W. (2005) Constructing an Opposite Map to a Specified Chaotic Map. Nonlinearity, 18, 1375-1391.
http://dx.doi.org/10.1088/0951-7715/18/3/022
[1] Alligood, K., Sauer, T. and Yorke, J. (1996) Chaos: An Introduction to Dynamical Systems. Springer-Verlag, New York. [2] Strogatz, S. (1994) Nonlinear Dynamics and Chaos. Perseus Books, Cambridge. [3] Schuster, H.G. and Just, W. (2005) Deterministic Chaos: An Introduction. 4th Edition, WILEY-VCH Verlag GmbH, Weinheim.
http://dx.doi.org/10.1002/3527604804 [4] Li, S., Li, Q., Li, W., Mou, X. and Cai, Y. (2001) Statistical Properties of Digital Piecewise Linear Chaotic Maps and Their Roles in Cryptography and Pseudo-Random Coding. Cryptography and Coding, 2260, 205-221.
http://dx.doi.org/10.1007/3-540-45325-3_19 [5] Addabbo, T., Alioto, M., Bernardi, S., Fort, A., Rocchi, S. and Vignoli, V. (2004) The Digital Tent Map: Performance Analysis and Optimized Design as a Source of Pseudo-Random Bits. Proceedings of the 21st IEEE Instrumentation and Measurement Technology Conference, IMTC 04, 2, 1301-1304. [6] Addabbo, T., Alioto, M., Bernardi, S., Fort, A., Rocchi, S. and Vignoli, V. (2004) Hardware-Efficient PRBGs Based on 1-D Piecewise Linear Chaotic Maps. Proceedings of the 11th IEEE International Conference on Electronics, Circuits and Systems, ICECS 2004, 13-15 December 2004, 242-245. [7] Pareek, N., Patidar, V. and Sud, K. (2010) A Random Bit Generator Using Chaotic Maps. International Journal of Network Security, 10, 32-38. [8] Shastry, M., Nagaraj, N. and Vaidya, P. (2006) The B-Exponential Map: A Generalization of the Logistic Map, and Its Applications in Generating Pseudo-Random Numbers. eprint arXiv.org:cs/0607069. [9] Basios, V., Forti, G.L. and Gilbert, T. (2009) Statistical Properties of Time-Reversible Triangular Maps of the Square. Journal of Physics A: Mathematical and Theoretical, 42, 1-13.
http://dx.doi.org/10.1088/1751-8113/42/3/035102 [10] Huang, W. (2005) Characterizing Chaotic Processes That Generate Uniform Invariant Density. Chaos, Solitons & Fractals, 25, 449-460.
http://dx.doi.org/10.1016/j.chaos.2004.11.016 [11] Anikin, V., Arkadaksky, S., Kuptsov, S., Remizov, A. and Vasilenko, L. (2008) Lyapunov Exponent for Chaotic 1D Maps with Uniform Invariant Distribution. Bulletin of the Russian Academy of Sciences: Physics, 72, 1684-1688.
http://dx.doi.org/10.3103/S106287380812023X [12] Huang, W. (2005) Constructing an Opposite Map to a Specified Chaotic Map. Nonlinearity, 18, 1375-1391.
http://dx.doi.org/10.1088/0951-7715/18/3/022