Role of Symmetry and Geometry in a chaotic Pseudo-Random Bit Generator

In this work, Pseudo-Random Bit Generation (PRBG) based on 2D chaotic mappings of logistic type is considered. The sequences generated with two Pseudorandom Bit Generators (PRBGs) of this type are statistically tested and the computational effectiveness of the generators is estimated. The role played by the symmetry and the geometrical properties of the underlying chaotic attractors is also explored. Considering these PRBGs valid for cryptography, the size of the available key spaces are calculated. Additionally, a novel mechanism called ‘symmetry-swap’ is introduced in order to enhance the PRBG algorithm. It is shown that it can increase the degrees of freedom of the key space, while maintaining the speed and performance in the PRBG.

💡 Research Summary

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The paper investigates pseudo‑random bit generation (PRBG) using two‑dimensional chaotic maps of logistic type, focusing on how the symmetry and geometric properties of the underlying attractors can be exploited to improve randomness, security, and efficiency. The authors first review the state of the art in chaotic PRBGs, noting that most existing schemes rely on one‑dimensional maps or on arbitrary two‑dimensional maps without a systematic analysis of their structural features. Building on Madhekar‑Suneel’s 2006 work that employed the Hénon map, the authors propose a generalized framework that can be applied to any 2‑D chaotic system possessing a symmetric coupling.

Two specific maps are examined: (1) the canonical Hénon map (xₖ₊₁ = a·xₖ² + yₖ + 1, yₖ₊₁ = b·xₖ) with parameters a = 1.4, b = 0.3, and (2) a pair of symmetrically coupled logistic maps introduced in a previous study. For each map the phase space is divided into four sub‑spaces (quadrants) by thresholds τₓ and τᵧ, which are set to the medians of the first 10 000 iterates of x and y respectively. Points falling into a sub‑space are encoded as a two‑bit symbol