Abstract
A modification of the classic logistic map is proposed, using fuzzy triangular numbers. The resulting map is analysed through its Lyapunov exponent (LE) and bifurcation diagrams. It shows higher complexity compared to the classic logistic map and showcases phenomena, like antimonotonicity and crisis. The map is then applied to the problem of pseudo random bit generation, using a simple rule to generate the bit sequence. The resulting random bit generator (RBG) successfully passes the National Institute of Standards and Technology (NIST) statistical tests, and it is then successfully applied to the problem of image encryption.
Highlights
The field of chaos theory expands to numerous applications related to cryptography, secure communications, engineering, physics, economics, robotics, control, and many more; see, for example, References [1,2,3,4,5] and the references therein
We propose a modified version of the classic logistic map by employing fuzzy triangular numbers to Entropy 2020, 22, 474; doi:10.3390/e22040474
The logistic map was modified through the use of fuzzy triangular numbers, to give a new modified logistic map that exhibits a plethora of chaos related phenomena, for different parameter values
Summary
The field of chaos theory expands to numerous applications related to cryptography, secure communications, engineering, physics, economics, robotics, control, and many more; see, for example, References [1,2,3,4,5] and the references therein. We propose a modified version of the classic logistic map by employing fuzzy triangular numbers to Entropy 2020, 22, 474; doi:10.3390/e22040474 www.mdpi.com/journal/entropy. The idea of passing the values of the logistic map through a fuzzy number is mathematically very simple, yet it leads to a significant improvement of the chaotic behavior of the map, with many chaos related phenomena appearing, like antimonotonicity and crisis. It is important to note that this approach can be applied to any other one-dimensional chaotic system, as well as further modified by considering different types of fuzzy numbers, like trapezoidal, Gaussian, quadratic, exponential, or their combination. The rest of the work is structured as follows: Section 2 presents some preliminaries on fuzzy numbers and the logistic map.
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