Improving the pseudo-randomness properties of chaotic maps using deep-zoom

Area of Science:

• Chaos Theory
• Dynamical Systems
• Computational Mathematics

Background:

• Discrete-time chaotic maps are fundamental in understanding complex systems.
• Existing methods for analyzing chaotic maps have limitations in exploring their intricate dynamics.
• The potential of chaotic maps for applications like pseudo-random number generation is an active research area.

Purpose of the Study:

• To propose a generalized method for composing new orbits from a given chaotic map.
• To investigate the phenomenon of rapid randomization in these newly composed orbits.
• To develop and validate a Pseudo-Random Number Generator (PRNG) based on this method.

Main Methods:

• A 'deep-zoom' approach using k-digits to the right of the decimal separator of chaotic map points.
• Composition of new orbits from underlying chaotic maps.
• Rigorous statistical testing of the proposed PRNG using DIEHARD and NIST test suites.
• Graphical analyses including time-evolution, bifurcation diagrams, Lyapunov exponents, Poincaré diagrams, and frequency distributions.

Main Results:

• A generalized method for generating new chaotic orbits was successfully developed.
• Rapid randomization was observed, making new orbits statistically indistinguishable from original ones.
• The proposed k-logistic map-based PRNG demonstrated high-quality pseudo-randomness, passing rigorous statistical tests.
• Performance of the new PRNG was found to be comparable to the Mersenne Twister.

Conclusions:

• The proposed method offers a novel way to explore and enhance chaotic dynamics.
• Simple chaotic maps, like the logistic map, can be effectively utilized for high-performance PRNGs.
• This research opens avenues for developing new cryptographic tools and improving random number generation techniques.