Pseudorandom number generation using chaotic true orbits of the Bernoulli map.

Area of Science:

• Number Theory
• Chaos Theory
• Cryptography

Background:

• Pseudorandom number generators (PRNGs) are crucial for simulations and cryptography.
• Existing PRNGs often face challenges with true randomness and sequence duplication.
• The Bernoulli map offers a theoretical basis for chaotic dynamics.

Purpose of the Study:

• To create a PRNG that precisely simulates chaotic orbits.
• To develop a seed selection strategy for generating distinct pseudorandom binary sequences.
• To validate the randomness properties of the generated sequences.

Main Methods:

• Utilizing the Bernoulli map on quadratic algebraic integers for orbit computation.
• Implementing an equidistant seed selection method within the unit interval.
• Performing rigorous statistical testing on generated sequences.

Main Results:

• The PRNG accurately computes chaotic true orbits.
• The seed selection method ensures uniform distribution and non-coinciding sequence tails.
• Statistical tests confirm good randomness properties of the generated sequences.

Conclusions:

• The proposed PRNG offers a robust method for generating high-quality pseudorandom binary sequences.
• This approach enhances the reliability of random number generation in computational applications.
• The mathematical foundation in chaotic dynamics provides a strong basis for randomness.