Analyzing the period distribution of digital chaos with graph theory

Abstract

The related characteristics of chaos are closely related to the basic concepts of confusion and diffusion in cryptography, which makes chaotic systems widely used in chaotic secure communication. However, when a chaotic system is implemented in a hardware device with finite calculation precision, its dynamic characteristics will be degraded to varying degrees due to the different calculation precision of the hardware device. The dynamic degradation of digital chaos will threaten the security of chaotic secure communication. To evaluate the performance of various digital chaotic maps, we proposed a universal period search algorithm (UPSA) that is based on graph theory with the purpose of analyzing the period distribution of digital chaos from a new perspective. The algorithm can calculate the maximal transient length, fixed points and periodic limit cycles (i.e. attractive basins) of digital chaos in finite-precision domains accurately. Furthermore, in order to test the versatility and effectiveness of our proposed algorithm, the periodic distribution and security of several typical digital chaotic maps are analyzed in detail. To a certain degree, UPSA can reduce the difficulty of cryptanalysis of digital chaotic ciphers and understand the network structures of functional graphs of digital chaos.