Optimizing Chaotic Behavior: Systematic Shifting and Operations for Robust 1-D Chaotic Maps

In this work, we present a systematic framework to optimize robust 1-D chaotic maps, focusing on expanding the uninterrupted chaotic region and enhancing chaotic properties throughout the entire parameter space. To achieve these objectives, we propose three distinct techniques, each involving systematic manipulations of chaotic seed maps. These manipulations include shifts and operations such as multiplication and division, which result in significant improvements in their chaotic characteristics. The effectiveness of the proposed methods is demonstrated through a comprehensive analysis using bifurcation plots, the maximum Lyapunov exponent, the correlation coefficient, Shannon entropy, the average Lyapunov exponent, and the chaotic ratio. The results illustrate the attainment of an extensive and uninterrupted chaotic range, alongside enhanced chaotic behavior due to the application of shifted maps. Additionally, in this work we also investigate the impact of combining general shifted maps with halfway-shifted maps, showing that their product leads to further improvements in chaotic properties and their division widens the chaotic ratio. In the last proposed method, the combined product division (CPD) maps achieve the highest overall performance, attaining a maximum Lyapunov exponent of 1.3567 and an average Lyapunov exponent of 1.3498, while maintaining a perfect chaotic ratio (CR = 1) across the entire parameter space. To demonstrate hardware feasibility, some of the proposed maps were also implemented on an FPGA, and the results were compared with MATLAB R2022b simulations. The close match between the two validates the practicality of implementing these chaotic systems in hardware. The proposed techniques have potential applications in areas such as random number generation, chaos-based cryptography, and secure communication, among others. To prove this, the optimized maps are leveraged to design a chaos-based pseudo-random number generator (PRNG) that passes statistical tests including NIST SP 800-22 (all 15 sub-tests passed) and TestU01 (38/38 Rabbit, 17/17 Alphabit, 102/102 BlockAlphabit), validating cryptographic-grade randomness.