3 research outputs found
Pseudorandom sequence generation using binary cellular automata
Tezin basılısı İstanbul Şehir Üniversitesi Kütüphanesi'ndedir.Random numbers are an integral part of many applications from computer simulations,
gaming, security protocols to the practices of applied mathematics and physics. As
randomness plays more critical roles, cheap and fast generation methods are becoming a
point of interest for both scientific and technological use.
Cellular Automata (CA) is a class of functions which attracts attention mostly due to the
potential it holds in modeling complex phenomena in nature along with its discreteness
and simplicity. Several studies are available in the literature expressing its potentiality
for generating randomness and presenting its advantages over commonly used random
number generators.
Most of the researches in the CA field focus on one-dimensional 3-input CA rules. In
this study, we perform an exhaustive search over the set of 5-input CA to find out the
rules with high randomness quality. As the measure of quality, the outcomes of NIST
Statistical Test Suite are used.
Since the set of 5-input CA rules is very large (including more than 4.2 billions of rules),
they are eliminated by discarding poor-quality rules before testing.
In the literature, generally entropy is used as the elimination criterion, but we preferred
mutual information. The main motive behind that choice is to find out a metric for
elimination which is directly computed on the truth table of the CA rule instead of the
generated sequence. As the test results collected on 3- and 4-input CA indicate, all rules
with very good statistical performance have zero mutual information. By exploiting this
observation, we limit the set to be tested to the rules with zero mutual information. The
reasons and consequences of this choice are discussed.
In total, more than 248 millions of rules are tested. Among them, 120 rules show out-
standing performance with all attempted neighborhood schemes. Along with these tests,
one of them is subjected to a more detailed testing and test results are included.
Keywords: Cellular Automata, Pseudorandom Number Generators, Randomness TestsContents
Declaration of Authorship ii
Abstract iii
Öz iv
Acknowledgments v
List of Figures ix
List of Tables x
1 Introduction 1
2 Random Number Sequences 4
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Theoretical Approaches to Randomness . . . . . . . . . . . . . . . . . . . 5
2.2.1 Information Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2.2 Complexity Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2.3 Computability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3 Random Number Generator Classification . . . . . . . . . . . . . . . . . . 7
2.3.1 Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.3.2 Non-Physical TRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.3 Pseudorandom Number Generators . . . . . . . . . . . . . . . . . . 10
2.3.3.1 Generic Design of Pseudorandom Number Generators . . 10
2.3.3.2 Cryptographically Secure Pseudorandom Number Gener- ators . . . . . . . . . . . . . .11
2.3.4 Hybrid Random Number Generators . . . . . . . . . . . . . . . . . 13
2.4 A Comparison between True and Pseudo RNGs . . . . . . . . . . . . . . . 14
2.5 General Requirements on Random Number Sequences . . . . . . . . . . . 14
2.6 Evaluation Criteria of PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.7 Statistical Test Suites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.8 NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.1 Hypothetical Testing . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.8.2 Tests in NIST Test Suite . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.1 Frequency Test . . . . . . . . . . . . . . . . . . . . . . . . 20
2.8.2.2 Block Frequency Test . . . . . . . . . . . . . . . . . . . . 20
2.8.2.3 Runs Test . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.8.2.4 Longest Run of Ones in a Block . . . . . . . . . . . . . . 21
2.8.2.5 Binary Matrix Rank Test . . . . . . . . . . . . . . . . . . 21
2.8.2.6 Spectral Test . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.8.2.7 Non-overlapping Template Matching Test . . . . . . . . . 22
2.8.2.8 Overlapping Template Matching Test . . . . . . . . . . . 22
2.8.2.9 Universal Statistical Test . . . . . . . . . . . . . . . . . . 23
2.8.2.10 Linear Complexity Test . . . . . . . . . . . . . . . . . . . 23
2.8.2.11 Serial Test . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.8.2.12 Approximate Entropy Test . . . . . . . . . . . . . . . . . 24
2.8.2.13 Cumulative Sums Test . . . . . . . . . . . . . . . . . . . . 24
2.8.2.14 Random Excursions Test . . . . . . . . . . . . . . . . . . 24
2.8.2.15 Random Excursions Variant Test . . . . . . . . . . . . . . 25
3 Cellular Automata 26 3.1 History of Cellular Automata . . . . . . . . . . . . . . . . . . . . . . . .26
3.1.1 von Neumann’s Work . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.1.2 Conway’s Life . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.1.3 Wolfram’s Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.2 Cellular Automata and the Definitive Parameters . . . . . . . . . . . . . . 31
3.2.1 Lattice Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
3.2.2 Cell Content . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.3 Guiding Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.2.4 Neighborhood Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.3 A Formal Definition of Cellular Automata . . . . . . . . . . . . . . . . . . 37
3.4 Elementary Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.5 Rule Families . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
3.6 Producing Randomness via Cellular Automata . . . . . . . . . . . . . . . 42
3.6.1 CA-Based PRNGs . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.6.2 Balancedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.3 Mutual Information . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.6.4 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 Test Results 47 4.1 Output of a Statistical Test . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.2 Testing Strategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
4.3 Interpretation of the Test Results . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Rate of success over all trials . . . . . . . . . . . . . . . . . . . . . 49
4.3.2 Distribution of P-values . . . . . . . . . . . . . . . . . . . . . . . . 50
4.4 Testing over a big space of functions . . . . . . . . . . . . . . . . . . . . . 50
4.5 Our Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
4.6 Results and Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.6.1 Change in State Width . . . . . . . . . . . . . . . . . . . . . . . . 53
4.6.2 Change in Neighborhood Scheme . . . . . . . . . . . . . . . . . . . 53
4.6.3 Entropy vs. Statistical Quality . . . . . . . . . . . . . . . . . . . . 58
4.6.4 Mutual Information vs. Statistical Quality . . . . . . . . . . . . . . 60
4.6.5 Entropy vs. Mutual Information . . . . . . . . . . . . . . . . . . . 62
4.6.6 Overall Test Results of 4- and 5-input CA . . . . . . . . . . . . . . 6
4.7 The simplest rule: 1435932310 . . . . . . . . . . . . . . . . . . . . . . . . . 68
5 Conclusion 74
A Test Results for Rule 30 and Rule 45 77
B 120 Rules with their Shortest Boolean Formulae 80
Bibliograph
Esprit '91. Proceedings of the annual Esprit conference. Brussels, 25-29 November 1991. EUR 13853 EN
My Struggles with the Block Universe
This document is the second installment of three in the Cerro Grande Fire
Series. Like its predecessor arXiv:quant-ph/0105039, "Notes on a Paulian Idea,"
it is a collection of letters written to various friends and colleagues, most
of whom regularly circuit this archive. The unifying theme of all the letters
is that each has something to do with the quantum. Particularly, the collection
chronicles the emergence of Quantum Bayesianism as a robust view of quantum
theory, eventually evolving into the still-more-radical "QBism" (with the B
standing for no particular designation anymore), as it took its most
distinctive turn away from various Copenhagen Interpretations. Included are
many anecdotes from the history of quantum information theory: for instance,
the story of the origin of the terms "qubit" and "quantum information" from
their originator's own mouth, a copy of a rejection letter written by E. T.
Jaynes for one of Rolf Landauer's original erasure-cost principle papers, and
much more. Specialized indices are devoted to historical, technical, and
philosophical matters. More roundly, the document is an attempt to provide an
essential ingredient, unavailable anywhere else, for turning QBism into a live
option within the vast spectrum of quantum foundational thought.Comment: CAUTION, do not unthinkingly print from a printer: 2,349 pages, 4
indices, 6 figures, with extensive hyperlinking. Foreword by M. Schlosshauer,
edited by B. C. Stacey. v2: more footnotes, fewer typo