Cellular Automata Pseudorandom Sequence Generation by Smarak Acharya BE, Visvesvaraya ...

60 Pages·2017·0.5 MB·English

Cellular Automata Pseudorandom Sequence Generation
by
Smarak Acharya
BE, Visvesvaraya Technological University, 2011
A Thesis Submitted in Partial Fulfillment of the
Requirements for the Degree of
MASTER OF APPLIED SCIENCE
in the Department of Electrical and Computer Engineering
(cid:13)c Smarak Acharya, 2017
University of Victoria
All rights reserved. This thesis may not be reproduced in whole or in part, by
photocopying or other means, without the permission of the author.
ii
Cellular Automata Pseudorandom Sequence Generation
by
Smarak Acharya
BE, Visvesvaraya Technological University, 2011
Supervisory Committee
Dr. T. Aaron Gulliver, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Daler Rakhmatov, Departmental Member
(Department of Electrical and Computer Engineering)
iii
Supervisory Committee
Dr. T. Aaron Gulliver, Supervisor
(Department of Electrical and Computer Engineering)
Dr. Daler Rakhmatov, Departmental Member
(Department of Electrical and Computer Engineering)
ABSTRACT
Pseudorandom sequences have many applications in fields such as wireless commu-
nication, cryptography and built-in self test of integrated circuits. Maximal length
sequences (m-sequences) are commonly employed pseudorandom sequences because
they have ideal randomness properties like balance, run and autocorrelation. How-
ever, the linear complexity of m-sequences is poor. This thesis considers the use of
one-dimensional Cellular Automata (CA) to generate pseudorandom sequences that
have high linear complexity and good randomness. The properties of these sequences
are compared with those of the corresponding m-sequences to determine their suit-
ability.
iv
Contents
Supervisory Committee ii
Abstract iii
Table of Contents iv
List of Tables vi
List of Figures viii
Acknowledgements ix
Dedication x
1 Introduction 1
1.1 LFSRs and m-Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2 Cellular Automata System for Pseudorandom Sequence Generation 13
2.1 The 1D CA Evaluation System . . . . . . . . . . . . . . . . . . . . . 15
2.2 Filtering Criteria . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
3 Results and Analysis 20
3.1 Initial Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2 Filter Results for n = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Filter Results for n = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Filter Results for n = 5 and n = 6 . . . . . . . . . . . . . . . . . . . . 24
3.5 Observations for n up to 6 and Results for n = 7 and 8 . . . . . . . . 25
3.6 Comparison with m-sequences . . . . . . . . . . . . . . . . . . . . . . 28
v
3.7 Results for n > 8 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.8 Execution Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4 Conclusions 46
4.1 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
Bibliography 49
vi
List of Tables
Table 1.1 LFSR State Transitions . . . . . . . . . . . . . . . . . . . . . . . 6
Table 1.2 Rule 30 State Table . . . . . . . . . . . . . . . . . . . . . . . . . 9
Table 1.3 Rule 90 State Table . . . . . . . . . . . . . . . . . . . . . . . . . 9
Table 3.1 All-zero Sequences Produced by Even Rules for SV = 0 and
n = 3,4,5,6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Table 3.2 Linear Complexity versus Observed Cell for n = 3, RR = 15 and
RC = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Table 3.3 Linear Complexity versus Observed Cell for n = 4, RR = 169
and RC = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Table 3.4 Linear Complexity versus Observed Cell for n = 5, RR = 146
and RC = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Table 3.5 Linear Complexity versus Observed Cell for n = 6, RR = 89 and
RC = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Table 3.6 Maximum Complexity versus Size for SV = 3 and RC = 2 . . . 24
Table 3.7 Sequences produced by Complementary Rules for n = 3,4 . . . . 25
Table 3.8 Sequences produced by Duplicate Linear Rules for n = 4 . . . . 26
Table 3.9 Initial Filtered Parameters LR, RC and RR for n = 3 . . . . . . 27
Table 3.10Results for LR = 3, RC = 2 and RR = 99 for n = 3 . . . . . . . 27
Table 3.11Results for LR = 6, RC = 2 and RR = 18 for n = 3 . . . . . . . 28
Table 3.12Final Filtered Parameters LR, RC and RR for n = 3 . . . . . . 28
Table 3.13Results for LR = 1, RC = 2 and RR = 122 for n = 3 . . . . . . 29
Table 3.14Results for LR = 4, RC = 2 and RR = 183 for n = 3 . . . . . . 29
Table 3.15Final Filtered Parameters LR, RC and RR for n = 4 . . . . . . 30
Table 3.16Results for LR = 8, RC = 3 and RR = 86 for n = 4 . . . . . . . 30
Table 3.17Results for LR = 9, RC = 3 and RR = 178 for n = 4 . . . . . . 31
Table 3.18Final Filtered Parameters LR, RC and RR for n = 5 . . . . . . 31
Table 3.19Final Filtered Parameters LR, RC and RR for n = 6 . . . . . . 32
vii
Table 3.20Results for LR = 3 RC = 4 and RR = 91 for n = 5 . . . . . . . 34
Table 3.21Results for LR = 46, RC = 2 and RR = 86 for n = 6 (Part 1) . 35
Table 3.22Results for LR = 46, RC = 2 and RR = 86 for n = 6 (Part 2) . 36
Table 3.23Filtered Parameters, Associated Primitive Polynomials and Du-
plicate LRs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
Table 3.24Filtered Parameters LR, RC and RR for n = 7 . . . . . . . . . 37
Table 3.25Filtered Parameters LR, RC and RR for n = 8 . . . . . . . . . 37
Table 3.26Results for LR = 43, RC = 6 and RR = 18 for n = 7 . . . . . . 38
Table 3.27Results for LR = 93, RC = 4 and RR = 225 for n = 8 . . . . . . 39
Table 3.28Comparison of an m-sequence with the CA for n = 3, LR = 6,
SV = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.29Comparison of an m-sequence with the CA for n = 3, LR = 6,
SV = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.30Comparison of an m-sequence with the CA for n = 4, LR = 10,
SV = 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.31Comparison of an m-sequence with the CA for n = 5, LR = 7,
SV = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.32Comparison of an m-sequence with the CA for n = 5, LR = 7,
SV = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.33Comparison of an m-sequence with the CA for n = 6, LR = 45,
SV = 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.34Comparison of an m-sequence with the CA for n = 7, LR = 43,
SV = 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.35Comparison of an m-sequence with the CA for n = 8, LR = 93,
SV = 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
Table 3.36Results for LR = 305, RC = 8 and RR = 163 for n = 9 . . . . . 41
Table 3.37Results for LR = 1008, RC = 4 and RR = 154 for n = 10 . . . . 42
Table 3.38Results for LR = 706, RC = 10 and RR = 86 for n = 11 . . . . 43
Table 3.39Results for LR = 634, RC = 3 and RR = 99 for n = 12 . . . . . 44
Table 3.40Execution Times for n = 3,4,5 and 6 . . . . . . . . . . . . . . . 45
Table 3.41Execution Times for n = 7 and 8 . . . . . . . . . . . . . . . . . 45
viii
List of Figures
Figure 1.1 An example of a binary linear feedback shift register (LFSR). . 3
Figure 1.2 An example of a Galois LFSR. . . . . . . . . . . . . . . . . . . 3
Figure 1.3 Generic n-bit LFSR. . . . . . . . . . . . . . . . . . . . . . . . . 4
Figure 1.4 An example of a 4-bit maximum length LFSR. . . . . . . . . . 6
Figure 1.5 An example of a 1D cellular automaton. . . . . . . . . . . . . . 8
Figure 1.6 An example of a 4-bit 1D CA that produce an m-sequence. . . 11
Figure 2.1 Anexampleoftheconfiguredrulesinthe1DCAevaluationsystem. 15
Figure 2.2 Modules of the 1D CA evaluation system. . . . . . . . . . . . . 17
Figure 2.3 Sequence generated by a CA of size n = 4. . . . . . . . . . . . . 18
Figure 3.1 Maximum linear complexity using linear rules based on primitive
polynomials versus size. . . . . . . . . . . . . . . . . . . . . . . 25
Figure 3.2 An example of complementary rules and a reversed CA. . . . . 26
ix
ACKNOWLEDGEMENTS
MysinceregratitudetomysupervisorDr. T.AaronGulliverforprovidinghisvaluable
guidanceandencouragementduringmystudiesasanMASCstudentattheUniversity
of Victoria. His expertise and knowledge were essential for sucessfully completing my
thesis. His patience and flexibility enabled me to work on my studies and thesis in
a productive and independent manner. I would also like to thank all my friends and
fellow students for their unfailing support professionally and personally throughout
my studies.
x
DEDICATION
I dedicate this thesis to my parents, who have been my pillars of support through
thick and thin in my life.