Problems and new solutions in the boolean domain

482 Pages·2016·3.579 MB·English

by  Steinbach, Bernd

Problems and New
Solutions in the
Boolean Domain

Problems and New
Solutions in the
Boolean Domain
Edited by
Bernd Steinbach
Problems and New Solutions in the Boolean Domain

Edited by Bernd Steinbach

This book first published 2016

Cambridge Scholars Publishing

Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK

British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library

Copyright © 2016 by Bernd Steinbach and contributors

All rights for this book reserved. No part of this book may be reproduced,
stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without
the prior permission of the copyright owner.

ISBN (10): 1-4438-8947-4
ISBN (13): 978-1-4438-8947-6
Contents
List of Figures ............................................ xi
List of Tables .............................................. xv
Foreword ................................................... xvii
Preface ..................................................... xix
Acknowledgments .........................................xxvii
List of Abbreviations ..................................... xxxi
I Methods, Algorithms, and Programs 1
1 General Methods ...................................... 3
1.1 A Vector Space Method for Boolean Networks . . . . . 3
1.1.1 A Vector Space Method for Boolean Networks 3
1.1.2 History of Switching Theory. . . . . . . . . . . 7
1.1.3 Useful Topics in Linear Algebra . . . . . . . . . 9
1.1.4 Vector Space Information Models . . . . . . . . 13
1.1.5 Switching Network Transfer Matrices. . . . . . 16
1.1.6 Transfer Matrices of Switching Circuits . . . . 19
1.1.7 Switching Network Simulation . . . . . . . . . 26
1.1.8 Switching Network Justification . . . . . . . . . 33
1.1.9 Algorithms and Implementation . . . . . . . . 38
1.2 Solving Combinatorial Problems . . . . . . . . . . . . 51
1.2.1 Boolean Equations . . . . . . . . . . . . . . . . 51
1.2.2 Solution With Regard to Variables . . . . . . . 57
1.2.3 Set-Related Problems . . . . . . . . . . . . . . 59
vi Contents
1.2.4 Graph-Related Problems. . . . . . . . . . . . . 64
1.2.5 Rule-Based Problems . . . . . . . . . . . . . . 65
1.2.6 Combinatorial Design . . . . . . . . . . . . . . 68
1.2.7 Extremely Complex Boolean Problems . . . . . 70
1.2.8 Summary and Comments . . . . . . . . . . . . 75
1.3 Simplification of Extremely Large Expressions . . . . . 76
1.3.1 An Application in High Energy Physics . . . . 76
1.3.2 The History of Simplification . . . . . . . . . . 78
1.3.3 Methods of Simplification . . . . . . . . . . . . 79
1.3.4 Monte Carlo Tree Search . . . . . . . . . . . . 81
1.3.5 Nested Monte Carlo Search . . . . . . . . . . . 88
1.3.6 Simulated-Annealing-UCT. . . . . . . . . . . . 90
1.3.7 Consequences for High Energy Physics . . . . . 94
1.3.8 An Outlook on Expression Simplification . . . 94
1.4 Novel Polynomial Expansions of Symmetric Functions 96
1.4.1 Preliminaries and Background . . . . . . . . . 96
1.4.2 Main Definitions . . . . . . . . . . . . . . . . . 98
1.4.3 Generation of the Carrier Vector . . . . . . . . 100
1.4.4 Generation of the Reduced Spectrum. . . . . . 106
1.4.5 The Complexity of the Combinatorial Method 110
1.4.6 Discussion of the Reached Improvements. . . . 111
2 Efficient Calculations ................................ 117
2.1 XBOOLE-CUDA - Fast Calculations on the GPU. . . 117
2.1.1 Challenges for Boolean Calculations . . . . . . 117
2.1.2 The Concepts Realized in XBOOLE . . . . . . 118
2.1.3 Parallel and Serial Architectures . . . . . . . . 126
2.1.4 Efficient GPU Programming. . . . . . . . . . . 131
2.1.5 Implementation XBOOLE-CUDA . . . . . . . 133
2.1.6 Evaluation of XBOOLE-CUDA . . . . . . . . . 137
2.1.7 Recommendations and Future Work . . . . . . 146
2.2 Efficient Computing of the Gibbs Dyadic Derivatives . 150
2.2.1 Walsh and Dyadic Analysis . . . . . . . . . . . 150
2.2.2 Gibbs Dyadic Derivatives . . . . . . . . . . . . 151
2.2.3 Computing the Gibbs Dyadic Derivatives . . . 154
2.2.4 Comparison of Methods and Algorithms . . . . 165
2.3 Understanding Randomized Algorithms Performance . 167
2.3.1 Observations from Experiments . . . . . . . . . 167
2.3.2 The Role of Interpretations . . . . . . . . . . . 170
2.3.3 Stochastic Models . . . . . . . . . . . . . . . . 172
Contents vii
2.3.4 The EM Algorithm . . . . . . . . . . . . . . . . 172
2.3.5 The Performance of the ABC Tool . . . . . . . 177
2.3.6 Randomly Valued MAX-3SAT Formulas . . . . 180
2.3.7 Simulated Annealing . . . . . . . . . . . . . . . 183
2.3.8 Summary of Interpretations . . . . . . . . . . . 184
II Applications 187
3 Several Aspects of Security ......................... 189
3.1 Fast Network Intrusion Detection Systems . . . . . . . 189
3.1.1 Background on Intrusion Detection Systems . . 189
3.1.2 Preliminaries . . . . . . . . . . . . . . . . . . . 190
3.1.3 Regular Expression Matching Hardware . . . . 194
3.1.4 Regular Expression Matching Software . . . . . 204
3.1.5 Discussion and Future Prospects . . . . . . . . 218
3.2 Utilization of Boolean Functions in Cryptography . . . 220
3.2.1 Types of Cryptosystems . . . . . . . . . . . . . 220
3.2.2 Cryptographic Properties of Boolean Functions 223
3.2.3 Boolean Functions in Stream Ciphers . . . . . 229
3.2.4 Boolean Functions in Block Ciphers . . . . . . 233
3.2.5 Exploration of the Nonlinearity . . . . . . . . . 237
3.2.6 Further Research Topics . . . . . . . . . . . . . 239
3.3 Minimization of ESOP Forms for Secure Computation 241
3.3.1 Two Party Secure Computation. . . . . . . . . 241
3.3.2 Exor Sum Of Products. . . . . . . . . . . . . . 245
3.3.3 Secure Computation with ESOPs . . . . . . . . 247
3.3.4 Minimization Algorithm . . . . . . . . . . . . . 248
3.3.5 Experimental Results . . . . . . . . . . . . . . 251
3.4 Determination of Almost Optimal Check Bits . . . . . 255
3.4.1 Error Detection Codes . . . . . . . . . . . . . . 255
3.4.2 Error Model. . . . . . . . . . . . . . . . . . . . 256
3.4.3 Determination of Additional Check Bits . . . . 260
3.4.4 Detection of Double-Bit Errors . . . . . . . . . 263
3.4.5 Additional Check Bit for a Byte-Parity Code . 266
4 Exploration of Properties ............................ 269
4.1 Boolean Function Spectra and Circuit Probabilities . . 269
4.1.1 Spectra of Boolean Functions . . . . . . . . . . 269
4.1.2 Boolean Function Output Probability . . . . . 270
viii Contents
4.1.3 Conditional Output Probabilities . . . . . . . . 271
4.1.4 Boolean Difference, Consensus, and Smoothing 276
4.1.5 Switching Function Spectra . . . . . . . . . . . 279
4.1.6 Calculating the Walsh Spectrum . . . . . . . . 279
4.1.7 Calculating the Reed-Muller Spectrum . . . . . 282
4.1.8 Calculating the Haar Spectrum . . . . . . . . . 283
4.1.9 Switching Function Output Probability . . . . 285
4.2 ROBDD-based Computation of Special Sets . . . . . . 287
4.2.1 Hard Problems . . . . . . . . . . . . . . . . . . 287
4.2.2 Fundamentals of ROBDDs . . . . . . . . . . . 288
4.2.3 ROBDDs for Subsets of Powersets . . . . . . . 291
4.2.4 Sets of Minimal and Maximal Sets . . . . . . . 294
4.2.5 Applications Concerning RelView. . . . . . . . 304
4.2.6 Concluding Remarks . . . . . . . . . . . . . . . 308
4.3 Functions with Bent Reed-Muller Spectra . . . . . . . 309
4.3.1 Background . . . . . . . . . . . . . . . . . . . . 309
4.3.2 Formalisms . . . . . . . . . . . . . . . . . . . . 310
4.3.3 Maiorana Class of Bent Reed Muller Spectra . 316
4.3.4 Special Cases when p=3 . . . . . . . . . . . . 319
4.3.5 Properties which Support Future Research . . 324
III Towards Future Technologies 325
5 Reversible Circuits .................................... 327
5.1 A Framework for Reversible Circuit Complexity . . . . 327
5.1.1 Investigations Relating to Reversible Functions 327
5.1.2 Reversible Boolean Functions . . . . . . . . . . 328
5.1.3 Function Classes and Symmetric Groups . . . . 331
5.1.4 Upper Bounds for Single-Target Gate Circuits 333
5.1.5 Upper Bounds for Toffoli Gate Circuits . . . . 334
5.1.6 Lower Bounds for Toffoli Gate Circuits. . . . . 336
5.1.7 Framework for Complexity Analysis . . . . . . 338
5.1.8 Application: Better than Optimal Embedding . 339
5.1.9 Open Problems and Future Work . . . . . . . . 341
5.2 Gate Count Minimal Reversible Circuits . . . . . . . . 342
5.2.1 A Need for New Reversible Benchmarks . . . . 342
5.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . 343
5.2.3 Sequences of Reversible Functions . . . . . . . 347
5.2.4 Minimal Circuits for Selected Functions . . . . 349
Contents ix
6 Quantum Circuits ...................................... 357
6.1 The Synthesis of a Quantum Circuit . . . . . . . . . . 357
6.1.1 Quantum Computation . . . . . . . . . . . . . 357
6.1.2 Building Blocks . . . . . . . . . . . . . . . . . . 358
6.1.3 First Decomposition of a Unitary Matrix . . . 361
6.1.4 Further Decomposition of a Unitary Matrix . . 363
6.1.5 Synthesizing a Fourier Circuit . . . . . . . . . . 366
6.1.6 Synthesizing a ZU Circuit . . . . . . . . . . . . 367
6.1.7 Summary . . . . . . . . . . . . . . . . . . . . . 368
6.2 Universal Two-Qubit Quantum Gates . . . . . . . . . 369
6.2.1 The NCV Gate Library . . . . . . . . . . . . . 369
6.2.2 The Semi-Classical Two-Qubit Gate Library . 373
6.2.3 A Restricted Two-Qubit Gate Library . . . . . 377
6.2.4 Impact on Toffoli Gates . . . . . . . . . . . . . 379
Bibliography ............................................... 383
List of Authors ............................................ 423
Index of Authors .......................................... 431
Index ........................................................ 433