The Graph Structure of the Generalized Discrete Arnold's Cat Map

TL;DR

This paper explicitly formulates the cycle structure of the generalized discrete Arnold's Cat map in binary domains, revealing patterns and behaviors crucial for cryptography and pseudorandom sequence generation.

Contribution

It provides an explicit iteration formula and analyzes the cycle structure and pattern changes of the generalized Cat map across different arithmetic precisions.

Findings

Explicit formulation of the map's iteration in binary domains

Disclosed cycle structures and their dependence on arithmetic precision

Validated patterns through rigorous proofs and experiments

Abstract

Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRNS). Much efforts have been devoted to obtaining period distribution of the generalized discrete Arnold's Cat map in various domains using all kinds of theoretical methods, including Hensel's lifting approach. Diagonalizing the transform matrix of the map, this paper gives the explicit formulation of any iteration of the generalized Cat map. Then, its real graph (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 $e$ are elaborately investigated and proved. The regular and beautiful patterns of Cat map demonstrated in a computer adopting fixed-point arithmetics are rigorously proved and experimentally verified. The results will facilitate research on dynamics of variants of the Cap map in any domain and its effective application in cryptography. In addition, the used methodology can be used to evaluate randomness of PRNS generated by iterating any other maps.