Jiteurtragool Nattagit, Masayoshi Tachibana, San-Um Wimol
School of Systems Engineering, Electronic and Photonic Engineering, Kochi University of Technology, Tosayamada, Kami City, Kochi 782-8502, Japan.
Center of Excellence in Intelligent Systems Integration, Faculty of Engineering, Thai-Nichi Institute of Technology (TNI), 1771/1, Pattanakarn Rd, Suan Luang, Bangkok 10250, Thailand.
Entropy (Basel). 2018 Feb 20;20(2):136. doi: 10.3390/e20020136.
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) ± x ± , where , , , and 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) ± x, 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.
由于在密码学或安全通信中的潜在应用，对生成鲁棒混沌的方法的研究受到了广泛关注。本文关注的是一种一维S型混沌映射，此前从未有过明确的研究。本文介绍了一种具有三项的S型混沌映射的通用形式，即 = ∓(x) ± x ± ，其中 、 、 、 为实常数。修正Sigmoid函数和双曲正切（tanh）函数的统一揭示了一种“统一S型混沌映射”的存在，该映射通常满足这三项，并且在某些参数范围内部分出现鲁棒混沌。最近，通过各种S型函数得到的简化通用形式，即 = ∓(x) ± x，导致了使用（i）hardtanh函数和（ii）符号函数进行线性化的可能性。本研究发现了一种线性化的S型混沌映射，它有可能在整个参数范围内提供鲁棒混沌。混沌动力学通过混沌波形、直方图、蛛网图、不动点、雅可比矩阵以及基于李雅普诺夫指数的分岔结构图来描述。作为一个实际例子，展示了一种使用线性化S型混沌映射的真随机比特发生器。使用NIST SP800 - 22测试套件和TestU01对生成的输出进行评估。
