Increasing Average Period Lengths by Switching of Robust Chaos Maps in Finite Precision

TL;DR

This paper explores how switching between robust chaos maps can increase average period lengths in finite precision systems, enhancing chaotic cryptography, and introduces a new generalized Logistic map with robust chaos for improved pseudo-random number generation.

Contribution

The paper introduces a novel generalized Logistic map exhibiting robust chaos and demonstrates that switching between such maps increases average period lengths in finite precision, benefiting cryptography.

Findings

Switching between chaotic maps yields longer periodic orbits.

The proposed pseudo-random generator passes statistical randomness tests.

Robust chaos maps prevent the existence of attracting periodic orbits.

Abstract

Grebogi, Ott and Yorke (Phys. Rev. A 38(7), 1988) have investigated the effect of finite precision on average period length of chaotic maps. They showed that the average length of periodic orbits ($T$) of a dynamical system scales as a function of computer precision ($\epsilon$) and the correlation dimension ($d$) of the chaotic attractor: $T \sim \epsilon^{-d/2}$. In this work, we are concerned with increasing the average period length which is desirable for chaotic cryptography applications. Our experiments reveal that random and chaotic switching of deterministic chaotic dynamical systems yield higher average length of periodic orbits as compared to simple sequential switching or absence of switching. To illustrate the application of switching, a novel generalization of the Logistic map that exhibits Robust Chaos (absence of attracting periodic orbits) is first introduced. We then propose a pseudo-random number generator based on chaotic switching between Robust Chaos maps which is found to successfully pass stringent statistical tests of randomness.