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2018 | Book

Design of Digital Chaotic Systems Updated by Random Iterations

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About this book

This brief studies the general problem of constructing digital chaotic systems in devices with finite precision from low-dimensional to high-dimensional settings, and establishes a general framework for composing them. The contributors demonstrate that the associated state networks of digital chaotic systems are strongly connected. They then further prove that digital chaotic systems satisfy Devaney’s definition of chaos on the domain of finite precision. The book presents Lyapunov exponents, as well as implementations to show the potential application of digital chaotic systems in the real world; the authors also discuss the basic advantages and practical benefits of this approach.

The authors explore the solutions to dynamic degradation (including short cycle length, decayed distribution and low linear complexity) by proposing novel modelling methods and hardware designs for two different one-dimensional chaotic systems, which satisfy Devaney’s definition of chaos. They then extend it to a higher-dimensional digital-domain chaotic system, which has been used in image-encryption technology. This ensures readers do not encounter large differences between actual and theoretical chaotic orbits through small errors.

Digital Chaotic Systems serves as an up-to-date reference on an important research topic for researchers and students in control science and engineering, computing, mathematics and other related fields of study.

Table of Contents

Frontmatter
Chapter 1. An Introduction to Digital Chaotic Systems Updated by Random Iterations
Abstract
The objective of this first chapter is to introduce the so-called digital chaotic systems updated by random iterations and to present the latest developments in this field of research. Basic notations and terminologies are also provided for the sake of completeness.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 2. Integer Domain Chaotic Systems (IDCS)
Abstract
The approach followed to build integer domain chaotic systems (IDCS) is presented in this chapter, and its mathematical chaos behavior demonstrated using the Devaney’s definition. Then, an analog-digital hybrid circuit is designed, to realize in practice a first basic IDCS. In this circuit design, the strategy generation is achieved thanks to a sample-hold circuit and a decoder circuit, so as to convert the uniform noise signal into a random sequence, which plays a key role in the circuit implementation. The proposed systematic methodology is further validated by experimental observations.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 3. Chaotic Bitwise Dynamical Systems (CBDS)
Abstract
In this chapter, we first recall the basic concept of real domain chaotic systems (RDCS) and integer domain chaotic systems (IDCS). Let \(N \in \{1,2,\ldots \}\) be a positive integer, \(\mathbb {B} = \{0,1\}\) denote the set of Boolean numbers with its usual algebraic structure, and \(\mathbb {B}^N\) the set of binary vectors of size N.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 4. One-Dimensional Digital Chaotic Systems (ODDCS)
Abstract
In this chapter, the structure of one-dimensional digital chaotic systems (ODDCS) in digital devices with finite precision is summarized, and the general framework of composing ODDCS is established.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 5. Higher-Dimensional Digital Chaotic Systems (HDDCS)
Abstract
Traditionally, chaotic systems are built on the domain of infinite precision in mathematics. However, quantization is inevitable for digital devices, which causes dynamical degradation. To cope with this problem, many methods were proposed, such as perturbing chaotic states and cascading multiple chaotic systems. This chapter aims at developing a novel methodology to design higher-dimensional digital chaotic systems (HDDCS) on the domain of finite precision. The proposed system is based on the chaos generation strategy controlled by random sequences. It is proven to satisfy Devaney’s definition of chaos. Also, we calculate the Lyapunov exponents for HDDCS. The application of HDDCS in image encryption is demonstrated via the field programmable gate array (FPGA) platform. As each operation of HDDCS is executed in the same fixed precision, no quantization loss occurs. Therefore, it provides a perfect solution to the dynamical degradation of digital chaos.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 6. Investigating the Statistical Improvements of Various Chaotic Iterations-Based PRNGs
Abstract
The previous chapters have recalled our various theoretical approaches using chaotic dynamical systems for pseudorandom number generations (PRNGs). They have been theoretically studied and hardware implementations have been designed. We now recall various software algorithms that can be proposed using these chaotic dynamical systems, and we evaluate their statistical behaviors using the three well-known batteries of tests in TestU01 already presented in this book.
Qianxue Wang, Simin Yu, Christophe Guyeux
Chapter 7. Conclusions
Abstract
To solve the degradation of chaotic dynamical properties caused by limitation of finite-precision presentation and quantization, this book developed digital chaotic systems updated by random iterations from low- to high-dimensional settings, utilizing the chaos generation strategy controlled by random sequences. After having recalled the bases of iterative systems, chaotic iterations, and the mathematical theory of chaos, the way to study iterative systems in this mathematical framework has then been explained, and previously obtained chaos conditions have then been listed.
Qianxue Wang, Simin Yu, Christophe Guyeux
Backmatter
Metadata
Title
Design of Digital Chaotic Systems Updated by Random Iterations
Authors
Dr. Qianxue Wang
Prof. Simin Yu
Prof. Christophe Guyeux
Copyright Year
2018
Electronic ISBN
978-3-319-73549-8
Print ISBN
978-3-319-73548-1
DOI
https://doi.org/10.1007/978-3-319-73549-8