A Novel Fibonacci-Sequence-Based Chaotification Model for Enhancing Chaos in 1-D Maps

Chaotic maps have found applications in a wide range of fields. However, many existing one-dimensional (1-D) chaotic maps suffer from limitations. Their smaller chaotic regions restrict their effectiveness in areas like cryptography and nonlinear dynamics modeling, where randomness and security are crucial. This article proposes a novel approach that utilizes a Fibonacci sequence-based 1-D chaotic map model to address these limitations and achieve a more comprehensive chaotic range. The study compares the proposed model against six commonly used 1-D maps: 1) Logistic; 2) Sine; 3) Chebyshev; 4) Quadratic; 5) Simple Quadratic; and 6) Singer. We evaluate the enhanced chaotic maps using various chaos dynamical tests, including bifurcation diagrams, Lyapunov exponents, 2-D and 3-D phase plots, the 0-1 test, cobweb plots, and approximate and sample entropies. The results confirm that all FCM maps consistently exhibit chaotic behavior across the entire control parameter space, with higher positive Lyapunov exponents, and larger approximate and sample entropy values. The 0-1 test yields an indicator value near the ideal value of 1, with linear M-t plots and p-q plots exhibiting chaotic Brownian motion. The cobweb plot for all maps displayed intricate, dense trajectories, while the 2-D and 3-D plots displayed points filling the entire phase space. Additionally, the enhanced FCM map-based pseudorandom bit generator (PRBG) has been designed and assessed for operational efficiency, speed, and security. The findings validate that the PRBG achieves high resource efficiency, and successfully passes all 15 NIST tests for statistical randomness. The comprehensive evaluation results demonstrate the efficacy of the proposed maps and PRBG, making them a promising and preferable choice for various practical applications.