Design of digital chaotic systems updated by random iterations

Description

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.;Intro; Preface; References; Acknowledgements; Contents; Abbreviations; 1 An Introduction to Digital Chaotic Systems Updated by Random Iterations; 1.1 General Presentation; 1.2 Mathematical Definitions of Chaos; 1.2.1 Approaches Similar to Devaney; 1.2.2 Li -- Yorke Approach; 1.2.3 Topological Entropy Approach; 1.2.4 Lyapunov Exponent; 1.3 TestU01; 1.4 Plan of This Book; References; 2 Integer Domain Chaotic Systems (IDCS); 2.1 Description of IDCS; 2.1.1 Real Domain Chaotic Systems (RDCS); 2.1.2 IDCS; 2.2 Proof of Chaos for IDCS; 2.2.1 Dense Periodic Points; 2.2.2 Transitive Property. Intro Preface References Acknowledgements Contents Abbreviations 1 An Introduction to Digital Chaotic Systems Updated by Random Iterations 1.1 General Presentation 1.2 Mathematical Definitions of Chaos 1.2.1 Approaches Similar to Devaney 1.2.2 Li -- Yorke Approach 1.2.3 Topological Entropy Approach 1.2.4 Lyapunov Exponent 1.3 TestU01 1.4 Plan of This Book References 2 Integer Domain Chaotic Systems (IDCS) 2.1 Description of IDCS 2.1.1 Real Domain Chaotic Systems (RDCS) 2.1.2 IDCS 2.2 Proof of Chaos for IDCS 2.2.1 Dense Periodic Points 2.2.2 Transitive Property. 2.2.3 Further Investigations of the Chaotic Behavior of IDCS2.2.4 Relationship Between Iterative Input and Output 2.3 Network Analysis of the State Space of IDCS 2.3.1 The Corresponding State Transition Diagram and Its Connectivity Analysis for IDCS with N = 3 2.3.2 The Corresponding State Transition Diagram and Its Connectivity Analysis for IDCS with N = 4 2.4 Circuit Implementation of IDCS References 3 Chaotic Bitwise Dynamical Systems (CBDS) 3.1 Improvements of Chaotic Bitwise Dynamical Systems (CBDS) 3.2 Proof of Chaos for CBDS 3.2.1 Dense Periodic Points. 3.2.2 Transitive Property3.3 Uniformity 3.4 TestU01 Statistical Test Results 3.5 FPGA-Based Realization of CBDS References 4 One-Dimensional Digital Chaotic Systems (ODDCS) 4.1 The Structure of One-Dimensional Digital Chaotic Systems 4.1.1 The Conventional Iterative Update Mechanism 4.1.2 The Iterative Update Mechanism Controlled by Random Sequences 4.2 The Connection Between a Chaotic System and Its Strongly Connected Network 4.2.1 Transitive Property of ODDCS 4.2.2 Dense Periodic Points of ODDCS 4.2.3 Chaotic System and Its Strongly Connected Network. 4.3 Lyapunov Exponents of a Class of ODDCS4.3.1 General Expression of Equivalent Decimal for GF 4.3.2 Mathematical Expression for G(y)y 4.3.3 Estimating the Lyapunov Exponents Reference 5 Higher-Dimensional Digital Chaotic Systems (HDDCS) 5.1 Design of HDDCS 5.1.1 Higher-Dimensional Integer Domain Chaotic Systems (HDDCS) 5.1.2 Description of HDDCS 5.1.3 Comparative Study of RDCS, IDCS, CBDS, and HDDCS 5.1.4 Network Analysis of the State Space of HDDCS 5.2 Chaotic Performance of HDDCS 5.2.1 Dense Periodic Points of HDDCS 5.2.2 Transitive Property of HDDCS. 5.3 Lyapunov Exponents of a Class of HDDCS5.3.1 General Expression of Equivalent Decimal for GF 5.3.2 Mathematical Expression for gi(y1,y2,Â#x83 , ym)yj 5.3.3 Estimating the Lyapunov Exponents 5.4 FPGA-Based Real-Time Application of 3D-DCS 5.4.1 Design of 3D-DCS in FPGA 5.4.2 Design of the FPGA-Based Hardware System for Image Encryption and Decryption 5.4.3 FPGA-Based Implementation Result for Image Encryption and Decryption References 6 Investigating the Statistical Improvements of Various Chaotic Iterations-Based PRNGs 6.1 Various Algorithms for Pseudorandom Number Generation.