Lyapunov Exponent Enhancement in Chaotic Maps with Uniform Distribution Modulo One Transformation

Abstract

Most of the chaotic maps are not suitable for chaos-based cryptosystems due to their narrow chaotic parameter range and lacking of strong unpredictability. This work presents a nonlinear transformation approach for Lyapunov exponent enhancement and robust chaotification in discrete-time chaotic systems for generating highly independent and uniformly distributed random chaotic sequences. The outcome of the new chaotic systems can directly be used in random number and random bit generators without any post-processing algorithms for various information technology applications. The proposed Lyapunov exponent enhancement based chaotic maps are analyzed with Lyapunov exponents, bifurcation diagrams, entropy, correlation and some other statistical tests. The results show that excellent random features can be accomplished even with one-dimensional chaotic maps with the proposed approach.

Exploiting Dynamic Vector-Level Operations and a 2D-Enhanced Logistic Modular Map for Efficient Chaotic Image Encryption

“…Given the limitations of classical chaotic systems, there are also many researchers dedicated to creating novel chaotic systems that can better fulfill the requirements of image encryption [ 16 , 21 , 23 , 24 , 25 , 26 , 27 , 28 , 29 ]. In [ 23 ], Hua et al suggested a two-dimensional (2D) modular chaotification system (2D-MCS) to enhance the chaotic performance of existing maps.…”

“…In [ 23 ], Hua et al suggested a two-dimensional (2D) modular chaotification system (2D-MCS) to enhance the chaotic performance of existing maps. By introducing two coupling parameters and the modulo one transformation, Ablay [ 24 ] proposed a novel LE-enhanced chaotification model. This model can convert any two one-dimensional (1D) chaotic maps into 2D chaotic maps with uniform trajectory distributions and better chaotic performance.…”

A Family of 1D Chaotic Maps without Equilibria

“…However, the newly created systems are often computationally complicated (e.g., they require the use of appropriate numerical methods to generate solutions), or it is difficult to precisely determine for which parameters chaos occurs. Furthermore, in the literature, works on the so-called chaotification can be found, that is, methods of constructing new systems or improving the properties of already existing chaotic systems [16][17][18].…”

A Chaotification Model Based on Modulo Operator and Secant Functions for Enhancing Chaos

“…To do so, the goal is to prove that a chaotification model can achieve higher Lyapunov exponent values than the existing chaotic maps, and verify that with numerical experiments. Such examples are to combine any map with a cosine function (Natiq et al 2019), a sine function (Hua et al 2018), a sine and cosecant functions , a cascade sine operation (Wu 2021), an internal perturbation model (Dong et al 2021), a remainder operation addition (Moysis et al 2022b), or the modulo operator, which has been shown to be effective in improving chaotic behavior (Ablay 2022;Zhang et al 2022).…”