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Discrete Dynamical Systems and Chaotic Machines: Theory and ApplicationsSuurem pilt
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For computer scientists, especially those in the security field, the use of chaos has been limited to the computation of a small collection of famous but unsuitable maps that offer no explanation of why chaos is relevant in the considered contexts. Discrete Dynamical Systems and Chaotic Machines: Theory and Applications shows how to make finite machines, such as computers, neural networks, and wireless sensor networks, work chaotically as defined in a rigorous mathematical framework. Taking into account that these machines must interact in the real world, the authors share their research results on the behaviors of discrete dynamical systems and their use in computer science.





Covering both theoretical and practical aspects, the book presents:



















Key mathematical and physical ideas in chaos theory





Computer science fundamentals, clearly establishing that chaos properties can be satisfied by finite state machines





Concrete applications of chaotic machines in computer security, including pseudorandom number generators, hash functions, digital watermarking, and steganography





Concrete applications of chaotic machines in wireless sensor networks, including secure data aggregation and video surveillance











Until the authors recent research, the practical implementation of the mathematical theory of chaos on finite machines raised several issues. This self-contained book illustrates how chaos theory enables the study of computer security problems, such as steganalysis, that otherwise could not be tackled. It also explains how the theory reinforces existing cryptographically secure tools and schemes.
List of Figures
xi
List of Tables
xiii
Preface xv
Symbol Description xvii
I An introduction to chaos
1(12)
1 Classical Examples by Way of Introduction
3(10)
1.1 Historical Context
3(1)
1.2 Feigenbaum's Bifurcation
4(4)
1.2.1 An iterative process
4(2)
1.2.2 Feigenbaum's bifurcation
6(1)
1.2.2.1 Presentation
6(1)
1.2.2.2 Parameter between 0 and 0.75
6(1)
1.2.2.3 Parameter between 0.75 and 1.25
6(1)
1.2.2.4 Parameter between 1.25 and 1.368
6(1)
1.2.2.5 Summary
7(1)
1.2.2.6 Parameter between 1.368 and 1.401
7(1)
1.2.2.7 Parameter between 1.401 and 2
7(1)
1.3 The Logistic Map
8(1)
1.3.1 Definition of the logistic map
8(1)
1.3.2 Behavior of the sequence
8(1)
1.3.3 Sensitivity to the initial conditions
8(1)
1.4 The Lorenz System
9(4)
II The mathematical theory of chaos
13(32)
2 Definitions and Notations
15(6)
2.1 Topological and Metrical Spaces
15(2)
2.1.1 Topological spaces, open sets, and neighborhoods
15(1)
2.1.1.1 First definitions
15(1)
2.1.1.2 Examples of topologies
16(1)
2.1.2 Distances, metrical spaces
16(1)
2.2 Compactness and Completeness
17(1)
2.2.1 Compactness
17(1)
2.2.1.1 Preliminaries
17(1)
2.2.1.2 Compactness of topological and metric spaces
17(1)
2.2.2 Completeness
18(1)
2.3 Continuity
18(1)
2.4 Discrete Dynamical Systems
19(2)
3 Devaney's Formulation of Chaos
21(16)
3.1 Periodicity, Stability, and Regularity
22(1)
3.1.1 Periodic and stable points
22(1)
3.1.2 Regular systems
22(1)
3.2 Simplification of Discrete Dynamical Systems
23(4)
3.2.1 Invariant subsystems
23(1)
3.2.2 (Un)decomposability and transitivity
24(1)
3.2.2.1 Decomposability
24(1)
3.2.2.2 Transitivity
24(1)
3.2.2.3 Consequences of the transitivity property
24(1)
3.2.2.4 Transitivity in compact spaces
25(1)
3.2.3 Stronger formulations of transitivity
26(1)
3.2.3.1 Total transitivity
26(1)
3.2.3.2 Strong transitivity
26(1)
3.2.3.3 Topological mixture
26(1)
3.2.4 Perfect discrete dynamical systems
26(1)
3.3 Stability, Sensitivity, and Expansiveness
27(2)
3.3.1 Stability and instability
27(1)
3.3.2 Sensitivity to the initial conditions
28(1)
3.3.3 Expansiveness
28(1)
3.3.3.1 Definition
28(1)
3.3.3.2 Example
29(1)
3.3.3.3 The case of perfect systems
29(1)
3.4 Chaos as Defined by Devaney (1989)
29(1)
3.5 Examples of Chaotic Systems
30(3)
3.5.1 The angle-doubling
30(1)
3.5.2 The tent map
31(1)
3.5.3 Arnold's cat map (1968)
32(1)
3.6 Topological and Metrical Conjugacies
33(4)
3.6.1 The topological semi-conjugacy
33(1)
3.6.1.1 Definition
33(1)
3.6.1.2 Utility of semi-conjugacy
34(1)
3.6.1.3 Example of use
34(1)
3.6.2 Topological conjugacy
35(1)
3.6.2.1 Definitions
35(1)
3.6.2.2 Properties preserved by conjugacy
35(2)
4 Other Formulations of Chaos
37(8)
4.1 The Lyapunov Exponent
37(1)
4.2 Topological and Metrical Entropy
38(7)
4.2.1 Original definition of topological entropy
38(1)
4.2.1.1 Introduction
38(1)
4.2.1.2 Entropy of an open cover
39(1)
4.2.1.3 The topological entropy
39(1)
4.2.1.4 Some examples
40(1)
4.2.2 Definition using separated sets
40(1)
4.2.2.1 Separated points
40(1)
4.2.2.2 Separated sets
41(1)
4.2.2.3 Topological entropy
41(1)
4.2.3 Definition using covering sets
42(1)
4.2.3.1 (n, e)-covers
42(1)
4.2.3.2 Topological entropy
42(1)
4.2.4 Properties of the topological entropy
43(2)
III From theory to practice
45(26)
5 A Fundamental Tool: The Chaotic Iterations
47(20)
5.1 Introducing the Chaotic Iterations
47(1)
5.2 Chaotic Iterations as Devaney's Chaos
48(5)
5.2.1 The new topological space
48(1)
5.2.1.1 Defining the iteration function and the phase space
48(1)
5.2.1.2 Cardinality of X
49(1)
5.2.1.3 A new distance
49(1)
5.2.1.4 Continuity of the iteration function
50(1)
5.2.2 Discrete chaotic iterations as topological chaos
51(1)
5.2.2.1 Regularity
51(1)
5.2.2.2 Transitivity
52(1)
5.2.2.3 Devaney's chaos
53(1)
5.3 Topological Properties of Chaotic Iterations
53(5)
5.3.1 Topological mixing
53(1)
5.3.2 Quantitative measures
54(1)
5.3.2.1 Sensitivity
54(1)
5.3.2.2 Expansiveness
55(1)
5.3.3 Topological Entropy
56(1)
5.3.3.1 Compactness study
56(1)
5.3.3.2 Topological entropy
57(1)
5.4 Characterization
58(2)
5.5 The Lyapunov Exponent
60(7)
5.5.1 The phase space is an interval of the real line
60(1)
5.5.1.1 Toward a topological semiconjugacy
60(2)
5.5.1.2 Chaotic iterations described as a real function
62(1)
5.5.2 Evaluation of the Lyapunov Exponent
63(4)
6 Theoretical Proofs of Chaotic Machines
67(4)
6.1 Chaotic Turing Machines
67(1)
6.2 Practical Issues
68(3)
6.2.1 A program with a chaotic behavior
68(1)
6.2.2 The practical case of finite strategies
69(2)
IV Applications of chaos in the computer science field
71(118)
7 Information Security
73(76)
7.1 Steganography and Digital Watermarking
74(24)
7.1.1 Introduction
74(1)
7.1.2 The proposed approach compared to existing ones
75(1)
7.1.2.1 Related work
75(1)
7.1.2.2 Contributions of the topological approach
76(1)
7.1.3 Chaos for data hiding security
77(1)
7.1.3.1 State-of-the-art
77(1)
7.1.3.2 Topological security
78(1)
7.1.3.3 Unpredictability and classes of attacks
79(1)
7.1.4 Topological security of spread-spectrum data hiding schemes
80(1)
7.1.4.1 A first proof of topological security
80(4)
7.1.4.2 Qualitative and quantitative evaluation of spread-spectrum
84(4)
7.1.4.3 Consequences
88(1)
7.1.5 A class of expansive data hiding schemes based on chaotic iterations
89(1)
7.1.5.1 A topologically secure data hiding algorithm
89(3)
7.1.5.2 Illustrative examples
92(4)
7.1.5.3 Properties of the chaotic machine
96(1)
7.1.6 Discussion
97(1)
7.2 Pseudorandom Number Generators
98(44)
7.2.1 Introduction
98(2)
7.2.2 Secured generators
100(1)
7.2.3 Pseudorandom generators based on chaotic iterations (CI PRNG)
100(1)
7.2.3.1 Some well-known pseudorandom generators
101(1)
7.2.3.2 The "Old CI" generator: algorithms and examples
102(7)
7.2.3.3 The "New CI" algorithm
109(6)
7.2.4 Experimental study of the randomness of the proposed generators
115(1)
7.2.4.1 Some well-known statistical batteries of tests for PRNGs
115(7)
7.2.4.2 Test results for some well-known PRNGs
122(4)
7.2.4.3 Test results and comparative analysis for the proposed Old CI
126(3)
7.2.4.4 Test results and comparative analysis for the New CI PRNG
129(4)
7.2.4.5 Further investigations of two chaotic iteration schemes based on the XORshift generator
133(9)
7.3 Hash Functions
142(7)
7.3.1 Introduction
142(1)
7.3.2 A chaotic hash function
142(1)
7.3.2.1 How to obtain E
142(2)
7.3.2.2 How to choose S
144(1)
7.3.2.3 How to construct the digest
144(1)
7.3.3 Application example
145(1)
7.3.4 A chaotic neural network as hash function
146(3)
8 Wireless Sensor Networks
149(40)
8.1 Video Surveillance
150(14)
8.1.1 Introduction
150(1)
8.1.2 Related works
151(2)
8.1.3 Smart threats
153(1)
8.1.3.1 Presentation
153(1)
8.1.3.2 Classification of malicious attacks
154(1)
8.1.3.3 Security levels in CKA
155(1)
8.1.4 Chaos-based scheduling
155(1)
8.1.4.1 Network capabilities
155(1)
8.1.4.2 Deploying the network
156(1)
8.1.4.3 Initialization of the WVSN
156(1)
8.1.4.4 Surveillance
156(1)
8.1.5 Theoretical study
157(1)
8.1.5.1 Scheduling as chaotic iterations
157(1)
8.1.5.2 Complexity
158(1)
8.1.5.3 Coverage
158(1)
8.1.5.4 Security study
158(2)
8.1.6 Simulation results
160(4)
8.1.7 Conclusion and perspectives of the chaos-based surveillance
164(1)
8.2 Secure Aggregation
164(25)
8.2.1 Presentation of the problem
164(2)
8.2.2 Security in sensor networks
166(1)
8.2.2.1 Data confidentiality
166(2)
8.2.2.2 Node authentication
168(1)
8.2.3 Tree-based data aggregation
169(1)
8.2.4 Sensor data encryption using fully homomorphic cryp-tosystem
170(1)
8.2.4.1 Operations over elliptic curves
170(2)
8.2.4.2 Public/private keys generation with ECC
172(1)
8.2.4.3 Encryption and decryption
172(1)
8.2.4.4 Homomorphic properties
173(1)
8.2.4.5 Encryption for sensor networks
174(2)
8.2.4.6 Practical issues
176(1)
8.2.4.7 Security study
177(1)
8.2.4.8 Experimental results
178(2)
8.2.5 Authentication over homomorphic sensor networks
180(1)
8.2.5.1 Information hiding-based authentication
181(3)
8.2.5.2 Security study of Zhang et al. authentication scheme
184(1)
8.2.5.3 Authentication based on chaotic iterations
185(3)
8.2.6 Discussion
188(1)
V Conclusion
189(8)
9 Conclusion
191(6)
9.1 Synthesis
191(3)
9.2 Perspectives
194(3)
Bibliography 197(14)
Index 211
Bahi, Jacques; Guyeux, Christophe
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