Robustification of a One-Dimensional Generic Sigmoidal Chaotic Map with Application of True Random Bit Generation

Abstract

The search for generation approaches to robust chaos has received considerable attention due to potential applications in cryptography or secure communications. This paper is of interest regarding a 1-D sigmoidal chaotic map, which has never been distinctly investigated. This paper introduces a generic form of the sigmoidal chaotic map with three terms, i.e., x n+1 = ∓Af NL (Bx n) ± Cx n ± D, where A, B, C, and D are real constants. The unification of modified sigmoid and hyperbolic tangent (tanh) functions reveals the existence of a "unified sigmoidal chaotic map" generically fulfilling the three terms, with robust chaos partially appearing in some parameter ranges. A simplified generic form, i.e., x n+1 = ∓f NL (Bx n) ± Cx n , through various S-shaped functions, has recently led to the possibility of linearization using (i) hardtanh and (ii) signum functions. This study finds a linearized sigmoidal chaotic map that potentially offers robust chaos over an entire range of parameters. Chaos dynamics are described in terms of chaotic waveforms, histogram, cobweb plots, fixed point, Jacobian, and a bifurcation structure diagram based on Lyapunov exponents. As a practical example, a true random bit generator using the linearized sigmoidal chaotic map is demonstrated. The resulting output is evaluated using the NIST SP800-22 test suite and TestU01.

A family of robust chaotic S-unimodal maps based on the Gaussian function

“…Similarly, Patra and Banerjee [14], in 2018, investigated robust chaos in a 3D piecewise linear map and derived sufficiency conditions for homoclinic and heteroclinic intersections. Jiteurtragool [15], in 2018, presented simplified forms of generic sigmoidal chaotic maps and a linearized sigmoidal chaotic map exhibiting chaotic behaviour over a wide parameter range. Hua et al [16], in 2017, devised an effective approach for creating n-dimensional hyperchaotic Cat maps, tailored to desired levels of complexity.…”