Analysing the dynamics of digital chaotic maps via a new period search algorithm

Abstract

The dynamics of chaotic maps are severely inhibited by the limited precision of the computational device that is used to implement them, and thus, their applications in cryptography and secure communications are seriously limited. To evaluate the degree of degradation of digital chaotic maps, we designed a fast period search algorithm (FPSA) that is based on a tree structure for analysing the periodicities of digital chaotic maps from a new perspective. FPSA can calculate the maximal transient length, fixed points and periodic limit cycles of digital chaotic maps in finite-precision domains quickly and accurately. Furthermore, based on this algorithm, the security of 1-D logistic maps and 2-D Arnold maps in engineering applications is analysed to demonstrate the versatility and effectiveness of our proposed algorithm. This algorithm plays an active role in analysing the structures of the functional graphs of digital chaotic maps in digital computers.