Texts in Theoretical Computer Science An EATCS Series

Transcription

1 Texts in Theoretical Computer Science An EATCS Series Editors: W. Brauer G. Rozenberg A. Salomaa On behalf of the European Association for Theoretical Computer Science (EATCS) Advisory Board: G. Ausiello M. Broy C.S. Calude A. Condon D. Harel J. Hartmanis T. Henzinger J. Hromkovic N. Jones T. Leighton M. Nivat C. Papadimitriou D. Scott

2 Jörg Rothe Complexity Theory and Cryptology An Introduction to Cryptocomplexity With 63 Figures and 56 Tables 123

3 Author Prof. Dr. Jörg Rothe Universität Düsseldorf Institut für Informatik Universitätsstr Düsseldorf, Germany rothe@cs.uni-duesseldorf.de Series Editors Prof. Dr. Wilfried Brauer Institut für Informatik der TUM Boltzmannstrasse Garching, Germany Brauer@informatik.tu-muenchen.de Prof. Dr. Grzegorz Rozenberg Leiden Institute of Advanced Computer Science University of Leiden Niels Bohrweg CA Leiden, The Netherlands rozenber@liacs.nl Prof. Dr. Arto Salomaa Turku Centre for Computer Science Lemminkäisenkatu 14 A Turku, Finland asalomaa@utu.fi Library of Congress Control Number: ACM Computing Classification (1998): E.3, F. 1.3, F.2.2 ISBN Springer Berlin Heidelberg New York ISBN Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springeronline.com Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: KünkelLopka, Heidelberg Typesetting: Camera ready by author Production: LE-TeX Jelonek, Schmidt & Vöckler GbR, Leipzig Printed on acid-free paper 45/3142/YL

4 To Irene, Paula, and Ella

5 Preface This book is an accessible introduction to complexity theory and cryptology, two closely related areas in theoretical computer science. Based on courses taught at Heinrich-Heine-Universität Düsseldorf and Friedrich-Schiller-Universität Jena since 1996, this textbook is written mainly for undergraduate and graduate students in computer science, mathematics, and engineering. Researchers, teachers, and practitioners working in these fields will also find this book a comprehensive, up-to-date, research-focused guide to central topics in cryptocomplexity. Chapter 1 provides more details about this book, including suggestions for how to use it and a brief outline of its chapters. Acknowledgments I am deeply indebted to Gerd Wechsung who encouraged me to write this book. I am also much indebted to Tobias Riege, Holger Spakowski, and Gerd Wechsung for carefully proofreading parts of the book and for their invaluable help, corrections, and suggestions. The remaining errors are my responsibility because I couldn t persuade any of the proofreaders to take responsibility for them. For their generous advice, help, and support over the past decade and for hosting numerous wonderful, inspiring, and prolific research visits at the University of Rochester and the Rochester Institute of Technology, I am deeply indebted to Lane A. Hemaspaandra and Edith Hemaspaandra. I also thank all my fellow researchers, coauthors, and colleagues: Lane and Edith Hemaspaandra, Alina Beygelzimer, Bernd Borchert, Judy Goldsmith, André Große, Christopher M. Homan, Zhigen Jiang, Mitsunori Ogihara, Kari Pasanen, Rajesh P. N. Rao, Tobias Riege, Amitabh Saxena, Holger Spakowski, Jörg Vogel, Osamu Watanabe, and Gerd Wechsung. Some of the research described in this book has been done jointly with them. I very much appreciate the help and advice of my friends in Rochester, NY, and Atlanta, GA, who had to read some of the stories in this book. I am grateful to Kathleen and Charles Landers-Appell, Mette Stromnes and Dave Lutz, Narin Hassan and Mark Leibert, and, last but not least, Jodi Beckwith and Stefan Cohen.

6 VIII Preface I gratefully acknowledge inspiring discussions with Klaus Ambos-Spies, Sigurd Assing, Harald Hempel, Uwe Schöning, Andreas Stelzer, Dietrich Stoyan, Klaus W. Wagner, and Gerd Wechsung. For providing a pleasant work environment, I thank my colleagues at the Computer Science departments in Düsseldorf and Jena: Volker Aurich, Stefan Conrad, Gabor Erdélyi, André Große, Arndt von Haeseler, Harald Hempel, Maren Hinrichs, Dieter Kratsch, Gerhard Lischke, Martin Mauve, Haiko Müller, Tobias Riege, Holger Spakowski, Jörg Vogel, Egon Wanke, and Gerd Wechsung. The support of the technical and secretarial staff in Düsseldorf is also gratefully acknowledged. Thanks to Claudia Forstinger, Claudia Kiometzis, Guido Königstein, Berthold Nöckel, Marga Potthoff, Janus Tomaschewski, and Lutz Voigt. I thank Claude Crépeau for his kind permission to use his original Alice and Bob design in Figure 1.1. For generous advice and professional support, I am grateful to the Springer series editors and staff, in particular to Wilfried Brauer, Grzegorz Rozenberg, Arto Salomaa, Alfred Hofmann, Ingeborg Mayer, and Ronan Nugent, and to Julia Merz for the cover design. The support of the DFG under grant RO 1202/9-1 and of the DAAD and the NSF under grant NSF-INT /DAAD-315-PPP-gü-ab, which funded my research projects during the planning and writing of this book, is also gratefully acknowledged. Above all, I thank my wife, Irene, and my daughters, Paula and Ella, for their love, advice, encouragement, and support. In particular, I thank Irene for her little dragon, ring, and grail, and I thank Paula and Ella for starring in some of the stories told in this book. Düsseldorf, December 2004 Jörg Rothe

7 Contents Preface...VII 1 Introduction to Cryptocomplexity Foundations of Computer Science and Mathematics Algorithmics Formal Languages and Recursive Function Theory Logic Propositional Logic PredicateLogic Algebra,NumberTheory,andGraphTheory AlgebraandNumberTheory Permutation Groups GraphTheory Probability Theory ExercisesandProblems SummaryandBibliographicRemarks Foundations of Complexity Theory TasksandAimsofComplexityTheory ComplexityMeasuresandClasses Speed-Up,Compression,andHierarchyTheorems Between Logarithmic and Polynomial Space Reducibilities and Completeness Many-One Reducibilities, Hardness, and Completeness NL-Completeness NP-Completeness InsideNP PversusNPandtheGraphIsomorphismProblem The Berman Hartmanis Isomorphism Conjecture and One-WayFunctions

8 X Contents 3.7 ExercisesandProblems SummaryandBibliographicRemarks Foundations of Cryptology TasksandAimsofCryptology SomeClassicalCryptosystemsandTheirCryptanalysis Substitution and Permutation Ciphers AffineLinearBlockCiphers BlockandStreamCiphers PerfectSecrecy Shannon s Theorem and Vernam s One-Time Pad EntropyandKeyEquivocation ExercisesandProblems SummaryandBibliographicRemarks Hierarchies Based on NP BooleanHierarchyoverNP Polynomial Hierarchy Parallel Access to NP ABriefDigressiontoSocialChoiceTheory Determining Young Winners Is Complete for Parallel Access to NP QueryHierarchiesoverNP The Boolean Hierarchy Collapsing the Polynomial Hierarchy AlternatingTuringMachines TheLowandtheHighHierarchywithinNP ExercisesandProblems SummaryandBibliographicRemarks Randomized Algorithms and Complexity Classes The Satisfiability Problem of Propositional Logic DeterministicTimeComplexity Probabilistic Time Complexity Probabilistic Polynomial-Time Classes PP, RP, and ZPP: Monte Carlo and Las Vegas Algorithms BPP: Bounded-Error Probabilistic Polynomial Time QuantifiersandArthur-MerlinGames QuantifiersandBPP Arthur-MerlinHierarchy CountingClasses GraphIsomorphismandLowness GraphIsomorphismIsintheLowHierarchy GraphIsomorphismIsinSPP ExercisesandProblems SummaryandBibliographicRemarks

9 Contents XI 7 RSA Cryptosystem, Primality, and Factoring RSA RSAPublic-KeyCryptosystem RSADigitalSignatureScheme Primality Tests FermatTest Miller Rabin Test Solovay StrassenTest Primality Is in P Factoring TrialDivision Pollard salgorithm QuadraticSieve Other Factoring Methods Security of RSA: Possible Attacks and Countermeasures ExercisesandProblems SummaryandBibliographicRemarks Other Public-Key Cryptosystems and Protocols Diffie HellmanandtheDiscreteLogarithmProblem DiffieandHellman ssecret-keyagreementprotocol DiscreteLogarithmandtheDiffie HellmanProblem ElGamal sprotocols ElGamal spublic-keycryptosystem ElGamal sdigitalsignaturescheme SecurityofElGamal sprotocols Rabin spublic-keycryptosystem Rabin scryptosystem SecurityofRabin ssystem Arthur-MerlinGamesandZero-Knowledge MerkleandHellman spublic-keycryptosystem Rabi,Rivest,andSherman sprotocols ExercisesandProblems SummaryandBibliographicRemarks List of Figures List of Tables References Index...444