Secure Communications based on Chaotic Systems by Mohamed Haroun B.Sc., Alexandria ...

122 Pages·2015·5.77 MB·English

Secure Communications based on Chaotic Systems
by
Mohamed Haroun
B.Sc., Alexandria University, 1999
M.Sc., Alexandria University, 2009
A Dissertation Submitted in Partial Fulfillment of the
Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of Electrical and Computer Engineering
(cid:13)c Mohamed Haroun, 2015
University of Victoria
All rights reserved. This dissertation may not be reproduced in whole or in part, by
photocopying or other means, without the permission of the author.
ii
Secure Communications based on Chaotic Systems
by
Mohamed Haroun
B.Sc., Alexandria University, 1999
M.Sc., Alexandria University, 2009
Supervisory Committee
Dr. T. Aaron Gulliver, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Mihai Sima, Departmental Member
(Department of Electrical and Computer Engineering)
Dr. Andrew Rowe, Outside Member
(Department of Mechanical Engineering)
iii
Supervisory Committee
Dr. T. Aaron Gulliver, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Mihai Sima, Departmental Member
(Department of Electrical and Computer Engineering)
Dr. Andrew Rowe, Outside Member
(Department of Mechanical Engineering)
ABSTRACT
This dissertation provides methods to utilize chaos efficiently in secure communi-
cations. Chaos has many desirable characteristics such as ergodicity and sensitivity
to initial conditions, and is considered an ideal candidate for use in cryptography
and secure communications. On the other hand, it suffers from sensitivity to noise
and fading if it is used for physical layer transmission, and errors due to the finite
precision of the numerical algorithms in digital systems. This limits the use of chaos
in cryptographic applications. Accordingly, this dissertation proposes new algorithms
to enhance the security of modern communication systems using chaos. The focus
is on developing chaotic cryptosystems for wireless systems that are reliable, secure,
and have good performance.
iv
Contents
Supervisory Committee ii
Abstract iii
Table of Contents iv
List of Tables vii
List of Figures viii
List of Abbreviations xiv
Acknowledgements xv
1 Introduction 1
1.1 Chaos Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Chaos in Secure Communications . . . . . . . . . . . . . . . . . . . . 3
1.3 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Low Complexity Discrete Multi-dimensional Chaotic Generators 9
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Difference Equations of the New Class of Discrete Chaotic Generators 10
2.3 Performance and Synchronization . . . . . . . . . . . . . . . . . . . . 13
2.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3 A New 3D Chaotic Cipher for Encrypting Two Data Streams Si-
multaneously 23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
v
3.3 The New Cipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Cipher Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.5 Security and Performance Analysis . . . . . . . . . . . . . . . . . . . 30
3.5.1 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . 31
3.5.2 Differential analysis . . . . . . . . . . . . . . . . . . . . . . . . 32
3.5.3 Attacks on the system . . . . . . . . . . . . . . . . . . . . . . 32
3.5.4 Key space analysis and brute-force attack . . . . . . . . . . . 34
3.5.5 Key sensitivity . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.5.6 Lorenz attacks . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.6 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.7 Effect of Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4 Real-time Image Encryption using a Three-Dimensional Discrete
Dual Chaotic Cipher 41
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2 The Proposed Lorenz Dual Chaotic Cipher . . . . . . . . . . . . . . . 42
4.3 Cipher Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.4 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
4.4.1 Key space analysis and brute-force attack . . . . . . . . . . . 46
4.4.2 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . 46
4.4.3 Differential analysis . . . . . . . . . . . . . . . . . . . . . . . . 47
4.4.4 Lorenz attacks . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.4.5 Execution time . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.5 FPGA Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5 Secret Key Generation using Chaotic Signals over Frequency Se-
lective Fading Channels 54
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
5.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
5.3 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.1 Probing the channel . . . . . . . . . . . . . . . . . . . . . . . 58
5.3.2 Threshold selection . . . . . . . . . . . . . . . . . . . . . . . . 59
5.3.3 Information reconciliation . . . . . . . . . . . . . . . . . . . . 60
vi
5.4 Chaotic Signals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.4.1 WSS signals over a fading channel . . . . . . . . . . . . . . . . 62
5.4.2 Chaotic signals over a fading channel . . . . . . . . . . . . . . 63
5.5 Key Bit Generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.6 Synchronization between legitimate users . . . . . . . . . . . . . . . . 69
5.7 Performance analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.7.1 Key generation rate . . . . . . . . . . . . . . . . . . . . . . . . 69
5.7.2 Key randomness . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.7.3 Key disagreement probability . . . . . . . . . . . . . . . . . . 72
5.8 KGR comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.9 Security analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
5.10 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6 Secure OFDM with PAPR Reduction using Chaotic Signals 79
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6.2.1 Selected Mapping (SLM) Technique . . . . . . . . . . . . . . . 81
6.3 The Proposed Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.1 Chaotic phase . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.2 Chaotic SLM . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.3 Quantization of the chaotic phase sequences . . . . . . . . . . 85
6.3.4 Data recovery without side-information . . . . . . . . . . . . . 86
6.3.5 Performance Results . . . . . . . . . . . . . . . . . . . . . . . 87
6.4 Security Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Conclusions 94
vii
List of Tables
Table 2.1 The Lyapunov Spectrum of Some Continuous Attractors . . . . 15
Table 2.2 Discrete Rossler Lyapunov Spectrum (averaged over 10 or 50 Runs) 15
Table 2.3 Discrete Lorenz Lyapunov Spectrum (averaged over 10 or 50 Runs) 16
Table 2.4 Largest Lyapunov Exponent for the Discrete Rossler Generator . 18
Table 2.5 Largest Lyapunov Exponent for the Discrete Lorenz Generator . 18
Table 2.6 Conditional Lyapunov Exponents for Different Drive-Response
Subsystems for the Discrete Lorenz and Rossler Systems . . . . 21
Table 3.1 Parameter Sensitivity, Key Length and Keyspace Size . . . . . . 34
Table 3.2 Encryption/Decryption Speeds for Various Ciphers . . . . . . . 39
Table 4.1 The Key Length based on the Master and Permutation Generator
Sensitivities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Table 4.2 Encryption and Decryption Execution Times for Several Ciphers 51
Table 5.1 Six Tap Static Channel Parameters . . . . . . . . . . . . . . . . 66
Table 5.2 106 Key Bits Generated Using the Proposed Algorithm over a
Static Fading Channel . . . . . . . . . . . . . . . . . . . . . . . 67
Table 5.3 106 Key Bits Generated Using the Proposed Algorithm over a
Time-Varying Fading Channel . . . . . . . . . . . . . . . . . . . 68
Table 5.4 Key Generation Rates for Various Algorithms . . . . . . . . . . 75
Table 5.5 Frequency of the Two Mask Values at Alice and Bob Appearing
in the Five Largest Values at Eve for 10,000 Measurements . . . 77
Table 5.6 Correlation of the Keys Generated at Alice/Bob and Eve, and
the Percentage of Mismatched Bits . . . . . . . . . . . . . . . . 77
Table 6.1 The Euclidean Distances for the M = 8 SLM Chaotic Phase
Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Table 6.2 The Euclidean Distances for the M = 8 SLM Sequences at an
Eavesdropper . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
viii
List of Figures
Figure 1.1 The Rossler attractor. . . . . . . . . . . . . . . . . . . . . . . . 2
Figure 2.1 Discrete implementation of (2.3). . . . . . . . . . . . . . . . . . 11
Figure 2.2 Discrete Lorenz parameter g ranges based on the Lyapunov ex-
ponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
(a) Lyapunov exponent versus g for the u variable . . . . . . . . . 16
1
(b) Lyapunov exponent versus g for the v variable . . . . . . . . . 16
2
(c) Lyapunov exponent versus g for the w variable . . . . . . . . . 16
3
Figure 2.3 Discrete Rossler parameter g ranges based on the Lyapunov ex-
ponents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
(a) Lyapunov exponent versus g for the x variable . . . . . . . . . 17
1
(b) Lyapunov exponent versus g for the y variable . . . . . . . . . 17
1
(c) Lyapunov exponent versus g for the z variable . . . . . . . . . 17
1
Figure 2.4 The discrete Lorenz attractor state space vectors. . . . . . . . . 18
(a) (u,v) space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
(b) (v,w) space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Figure 2.5 The discrete Rossler attractor state space vectors. . . . . . . . . 19
(a) (x,y) space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
(b) (y,z) space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Figure 2.6 Autocorrelation of the (a) continuous Lorenz attractor, and (b)
discrete Lorenz attractor. . . . . . . . . . . . . . . . . . . . . . 19
(a) Autocorrelation of the continuous Lorenz attractor output . . . 19
(b) Autocorrelation of the discrete Lorenz attractor output . . . . . 19
Figure 2.7 Autocorrelation for different values of g. . . . . . . . . . . . . . 20
(a) g = 0.024 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
(b) g = 0.01 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
(c) g = 0.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
ix
Figure 2.8 Synchronization with different initial conditions at the transmit-
ter and receiver using the (a) Rossler discrete attractor state y
as the drive signal, and (b) Lorenz discrete attractor using state
v as the drive signal. . . . . . . . . . . . . . . . . . . . . . . . . 21
(a) Rossler synchronization . . . . . . . . . . . . . . . . . . . . . . 21
(b) Lorenz synchronization . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 3.1 (a) and (b) The original text file and image, (c) the transmitted
signal, (d) the autocorrelation of the transmitted signal, and (e)
and (f) the recovered text file and image. . . . . . . . . . . . . 30
(a) Original text file . . . . . . . . . . . . . . . . . . . . . . . . . . 30
(b) Original bird image . . . . . . . . . . . . . . . . . . . . . . . . . 30
(c) Transmitted signal . . . . . . . . . . . . . . . . . . . . . . . . . 30
(d) Autocorrelation of the transmitted signal . . . . . . . . . . . . . 30
(e) Recovered text file . . . . . . . . . . . . . . . . . . . . . . . . . 30
(f) Recovered bird image . . . . . . . . . . . . . . . . . . . . . . . . 30
Figure 3.2 (a) The original image, (b) the transmitted signal, (c) the auto-
correlation of the transmitted signal, and (d) the recovered image. 31
(a) Original image . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
(b) Transmitted signal . . . . . . . . . . . . . . . . . . . . . . . . . 31
(c) Autocorrelation of the transmitted signal . . . . . . . . . . . . . 31
(d) Recovered image . . . . . . . . . . . . . . . . . . . . . . . . . . 31
Figure 3.3 (a) The cross-correlation of the encrypted signals generated from
two text files with a small difference between them, (b) the au-
tocorrelation of the difference between these encrypted signals,
(c) the cross-correlation of the encrypted signals generated from
two image files with a small difference between them, and (d) the
autocorrelation of the difference between the encrypted signals. 33
(a) Cross-correlation between two text files . . . . . . . . . . . . . . 33
(b) Autocorrelation of the difference . . . . . . . . . . . . . . . . . 33
(c) Cross-correlation between two images . . . . . . . . . . . . . . . 33
(d) Autocorrelation of the difference . . . . . . . . . . . . . . . . . 33
x
Figure 3.4 Decrypted text files using the chaotic cryptosystem with the
parameters changed slightly to (a) U = 0.100001, (b) V =
0 0
0.000001, (c) W = 0.000001, (d) A = 10.001, (e) B = 28.001,
0
(f) C = 2.6677, (g) g = 0.010001, (h) g = 0.010001, and (i)
1 2
g = 0.010001. . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3
(a) U = 0.100001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
0
(b) V = 0.000001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
0
(c) W = 0.000001 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
0
(d) A = 10.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
(e) B = 28.001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
(f) C = 2.6677 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
(g) g = 0.010001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
1
(h) g = 0.010001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2
(i) g = 0.010001 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3
Figure 3.5 (a) and (b) The return maps, (c) the power spectrum, and (d)
the bird image. . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
(a) V return map . . . . . . . . . . . . . . . . . . . . . . . . . . 37
max
(b) V return map . . . . . . . . . . . . . . . . . . . . . . . . . . 37
min
(c) Power spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
(d) Bird image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Figure 4.1 An example of image encryption: (a) the original image, (b) the
encrypted signal, and (c) the recovered image. . . . . . . . . . . 44
(a) Original image . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
(b) Encrypted signal . . . . . . . . . . . . . . . . . . . . . . . . . . 44
(c) Recovered image . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Figure 4.2 The effect of the image (a) on the time-domain chaotic signal,
and (b) the autocorrelation of the chaotic signal. . . . . . . . . 45
(a) Chaotic signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
(b) Autocorrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
Figure 4.3 The autocorrelation of the signal in Fig. 4.1-b. . . . . . . . . . . 47
Figure 4.4 The cross-correlation of the encrypted signals with and without
an injected image. . . . . . . . . . . . . . . . . . . . . . . . . . 47