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

Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRBS). 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 ap...

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Vydáno v:IEEE transactions on computers Ročník 71; číslo 2; s. 364 - 377
Hlavní autoři: Li, Chengqing, Tan, Kai, Feng, Bingbing, Lu, Jinhu
Médium: Journal Article
Jazyk:angličtina
Vydáno: New York IEEE 01.02.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Témata:
Arithmetic
Cats
chaotic cryptography
cycle structure
Domains
Dynamic structural analysis
Extraterrestrial measurements
Fixed point arithmetic
Fixed points (mathematics)
generalized Cat map
Iterative methods
Logistics
period distribution
Periodic structures
PRBS
PRNS
Pseudorandom binary sequences
pseudorandom number sequence
Space vehicles
Transforms
Transient analysis
ISSN:0018-9340, 1557-9956
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Shrnutí:Chaotic dynamics is an important source for generating pseudorandom binary sequences (PRBS). 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 article 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 <inline-formula><tex-math notation="LaTeX">e</tex-math> <mml:math><mml:mi>e</mml:mi></mml:math><inline-graphic xlink:href="li-ieq1-3051387.gif"/> </inline-formula> 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 can serve as a benchmark for studying the dynamics of the variants of the Cat map in any domain. In addition, the used methodology can be used to evaluate randomness of PRBS generated by iterating any other maps.
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ISSN:0018-9340
1557-9956
DOI:10.1109/TC.2021.3051387