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.
Highlights
In order to explain simple chaotic dynamical systems, one-dimensional map is used
Sensitivity of Second Modified Logistic (SML) map to initial condition is observed in the figure
It is appearing that histograms of Logistic map and SML map are identical, while First Modified Logistic (FML) map could not perform acceptable result
Summary
In order to explain simple chaotic dynamical systems, one-dimensional map is used. Tent, Bernoulli and Logistic maps are common examples of them. Logistic map is generally used in most of cryptosystems and pseudo random generators. It is used in chaos-based secure communication system and for generations of binary numbers. Mou, and Cai proposed statistical properties of digital piecewise linear chaotic maps and their roles in cryptography and pseudo-random coding [4]. Nagaraj, and Vaidya proposed a generalization of Logistic map, and its applications in generating pseudo-random numbers [8]. Since chaotic range of logistic map is small, its key space is small.
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