Reveal the correlation between randomness and Lyapunov exponent of n-dimensional non-degenerate hyper chaotic map

Abstract

In this paper, we investigate the correlation between the Lyapunov exponent (LE) and the randomness in chaotic maps. The recently constructed chaotic map has certain limitations when used in cryptography. These limitations include the lack of ergodicity in phase space, weak positive LE, and poor randomness. To overcome these limitations and reveal the correlation between LE and randomness, we first constructed a generic n-dimensional non-degenerate hyper chaotic map (nD-NDCM) model with adjustable LE through the eigenvalues of the Jacobian matrix. Furthermore, we instantiated 2D-NDCM and 3D-NDCM based on this model, and then we used the bifurcation diagrams, phase diagrams, and LE to analyze their dynamic performance. Additionally, we used NIST SP800-22 and TestU01 to analyze the iterative sequence and determine the corresponding LE interval with good randomness. Utilizing the established correlation, we can directly and feasibly construct any multi-dimensional chaotic map with good randomness.