Constructing a 3D Exponential Hyperchaotic Map with Application to PRNG

Abstract

Some weaknesses of 1D chaotic maps, such as lacking of ergodicity, multiple bifurcations, dense periodic windows, and short iteration period, limit their practical applications in cryptography. A higher-dimensional chaotic map with ergodicity can solve these problems. Based on 1D quadratic map, a 3D exponential hyperchaotic map (3D-EHCM) is constructed, and its dynamic behaviors, such as phase diagram, Lyapunov exponent spectrum, Kolmogorov entropy (KE), correlation dimension, approximate entropy and randomness, are analyzed and tested. The results demonstrate that the 3D-EHCM has ergodicity in a larger range of control parameter, and its state points have a longer period. To counteract dynamical degradation and make it suitable for a PRNG, the periodic point detection and random impulsive perturbation are applied to lengthen the aperiodic time sequence, and statistical results demonstrate that a full-period sequence can be obtained.