FPGA Implementation of PRNGs Based on Chaotic Systems Optimized by DE, GWO, and PSO

The optimization of chaotic systems remains a challenge because the search space of the design parameters can have several orders of magnitude, causing that the corresponding eigenvalues can be very sparse, thus producing unnecessary long simulation times. This imposes the need of estimating the step-size $h$ of the numerical method that discretizes the ordinary differential equations. In this manner, the proposed work shows the optimization of chaotic systems, by applying differential evolution (DE), grey wolf optimization (GWO), and particle swarm optimization (PSO) algorithms. Within the optimization loop, $h$ is estimated taking into account the inverse of the highest eigenvalue, and the total time simulation is estimated by taking the inverse of the lowest eigenvalue. The constraints consider that a chaotic system is simulated only if there exist two complex eigenvalues and if the Fourier transform of the chaotic time series has a spectrum area in a certain threshold. A single-objective function is associated to maximize the Kaplan–Yorke dimension $D_{KY}$, and then PSO, DE, and GWO are executed with the same number of runs, generations, and population individuals. Their performances are compared by Wilcoxon and Levene tests. The best solutions obtained for each optimization algorithm and for each chaotic system are used to implement pseudo-random number generators (PRNG). Finally, the PRNGs that passed NIST and TestU01 tests are implemented into a field-programmable gate array.