The Cycle Structure Of Digital Chaotic Maps

Iterating chaotic map in computer is a typical method to generate pseudorandom number sequences(PRNS).Considering every chaotic state as a node and and the mapping relationship between any pair of nodes as a directed edge,the state-mapping network(SMN)of a chaotic map can be established.Different chaotic maps have a different network structure of the SMN.There exists only cycle in the SMN of some chaotic maps.This means that any state will return to itself by iterating some times.Essentially,iterating a chaotic map is actually a walk in the corresponding SMN from an initial node.By studying the relationship between the structure of SMN and the arithmetic precision,the randomness of the sequences generated by iterating a chaotic map can be effectively revealed.This thesis focuses on the structure of two types of chaotic maps whose SMN owns cycle only.The intrinsic properties are investigated by analyzing the relationship between the distribution of the period(length)of cycles with the arithmetic precision.First,the basic framework and classical methods of studying digital chaotic dynamics with complex networks are reviewed.Then,the properties of the SMN of the Chebyshev polynomials in the finite prime domain are analyzed.How the cycle structure of the SMN changes with increase of fixed-point arithmetic precision is revealed.In addition,the effect of the floating-point operation mode on the SMN is disclosed also.By diagonalizing the transformation matrix of the two-dimensional discrete Cat map,the explicit expression of iterating the map any times is given.Based on this point,its real cycle structure in any binary arithmetic domain is disclosed.The subtle rules on how the cycles(itself and its distribution)change with the arithmetic precision are elaborately investigated and proved.The regular and beautiful patterns of Cat map demonstrated in a computer adopting fixed-point arithmetic are rigorously proved and experimentally verified.The research results of this thesis vividly demonstrate the security defects of pseudo-random sequences generating by iterating a chaotic map.In addition,the used methodology can be used to study the structure and randomness evaluation of PRNS generated by iterating any other maps.