Computational and Combinatorial Group Theory and Cryptography

Transcription

1 582 Computational and Combinatorial Group Theory and Cryptography AMS Special Sessions: Computational Algebra, Groups, and Applications April 30 May 1, 2011 University of Nevada, Las Vegas, NV Mathematical Aspects of Cryptography and Cyber Security September 10 11, 2011 Cornell University, Ithaca, NY Benjamin Fine Delaram Kahrobaei Gerhard Rosenberger Editors American Mathematical Society

2 Computational and Combinatorial Group Theory and Cryptography AMS Special Sessions: Computational Algebra, Groups, and Applications April 30 May 1, 2011 University of Nevada, Las Vegas, NV Mathematical Aspects of Cryptography and Cyber Security September 10 11, 2011 Cornell University, Ithaca, NY Benjamin Fine Delaram Kahrobaei Gerhard Rosenberger Editors

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4 582 Computational and Combinatorial Group Theory and Cryptography AMS Special Sessions: Computational Algebra, Groups, and Applications April 30 May 1, 2011 University of Nevada, Las Vegas, NV Mathematical Aspects of Cryptography and Cyber Security September 10 11, 2011 Cornell University, Ithaca, NY Benjamin Fine Delaram Kahrobaei Gerhard Rosenberger Editors American Mathematical Society Providence, Rhode Island

5 EDITORIAL COMMITTEE Dennis DeTurck, Managing Editor Michael Loss Kailash Misra Martin J. Strauss 2000 Mathematics Subject Classification. Primary 20-XX, 68-XX. Library of Congress Cataloging-in-Publication Data Computational and combinatorial group theory and cryptography : AMS Special Session on Computational Algebra, Groups, and Applications, April 30 May 1, 2011, University of Nevada, Las Vegas, Nevada : AMS Special Session on Mathematical Aspects of Cryptography and Cyber Security, September 10 11, 2011, Cornell University, Ithaca, New York / Benjamin Fine, Delaram Kahrobaei, Gerhard Rosenberger, editors. p. cm. (Contemporary mathematics ; volume 582) Includes bibliographical references. ISBN (alk. paper) 1. Group theory Congresses. 2. Cryptography Congresses. I. Fine, Benjamin (1948 ), editor of compilation. II. Kahrobaei, Delaram (1975 ), editor of compilation. III. Rosenberger, Gerhard, editor of compilation. IV. AMS Special Session on Computational Algebra, Groups, and Applications (2011 : University of Nevada). V. AMS Special Session on Mathematical Aspects of Cryptography and Cyber Security (2011 : Cornell University) QA174.C dc Copying and reprinting. Material in this book may be reproduced by any means for educational and scientific purposes without fee or permission with the exception of reproduction by services that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given. This consent does not extend to other kinds of copying for general distribution, for advertising or promotional purposes, or for resale. Requests for permission for commercial use of material should be addressed to the Acquisitions Department, American Mathematical Society, 201 Charles Street, Providence, Rhode Island , USA. Requests can also be made by to reprint-permission@ams.org. Excluded from these provisions is material in articles for which the author holds copyright. In such cases, requests for permission to use or reprint should be addressed directly to the author(s). (Copyright ownership is indicated in the notice in the lower right-hand corner of the first page of each article.) c 2012 by the American Mathematical Society. All rights reserved. The American Mathematical Society retains all rights except those granted to the United States Government. Copyright of individual articles may revert to the public domain 28 years after publication. Contact the AMS for copyright status of individual articles. Printed in the United States of America. The paper used in this book is acid-free and falls within the guidelines established to ensure permanence and durability. Visit the AMS home page at

6 Contents Preface vii Weyl Gröbner Basis Cryptosystems Rashid Ali and Martin Kreuzer 1 A New Look at Finitely Generated Metabelian Groups Gilbert Baumslag, Roman Mikhailov, and Kent E. Orr 21 IA-Automorphisms of Groups with Almost Constant Upper Central Series Marianna Bonanome, Margaret H. Dean, and Marcos Zyman 39 A Proposed Alternative to the Shamir Secret Sharing Scheme Chi Sing Chum, Benjamin Fine, Gerhard Rosenberger, and Xiaowen Zhang 47 Improving Latin Square Based Secret Sharing Schemes Chi Sing Chum and Xiaowen Zhang 51 A Hand-Computation Involving Surface Groups, the Reidemeister-Schreier Rewriting Process and Kurosh Subgroup Theorem Anthony E. Clement 65 Adjunction of Roots in Exponential A-Groups Margaret H. Dean, Stephen Majewicz, and Marcos Zyman 71 Logspace Computations in Coxeter Groups and Graph Groups Volker Diekert, Jonathan Kausch, and Markus Lohrey 77 Collection by Polynomials in Finite p-groups Bettina Eick 95 All Finite Generalized Tetrahedron Groups II Benjamin Fine, Alexander Hulpke, and Gerhard Rosenberger 105 The Classification of One Relator Limit Groups and the Surface Group Conjecture Benjamin Fine and Gerhard Rosenberger 107 Discrimination and Separation in the Metabelian Variety Anthony M. Gaglione, Seymour Lipschutz, and Dennis Spellman 129 v

7 vi CONTENTS A Secret Sharing Scheme Based on Group Presentations and the Word Problem Maggie Habeeb, Delaram Kahrobaei, and Vladimir Shpilrain 143 Authenticated Key Agreement with Key Re-Use in the Short Authenticated Strings Model Stanislaw Jarecki and Nitesh Saxena 151 Publicly Verifiable Secret Sharing Using Non-Abelian Groups Delaram Kahrobaei and Elizabeth Vidaurre 175 A Note on the Hyperbolicity of Strict Pride Groups Matthias Neumann-Brosig 181 An Algorithm to Express Words as a Product of Conjugates of Relators Ellen Ziliak 187

8 Preface This volume consists of contributions by participants and speakers at special sessions at two AMS Conferences. These conferences concerned Computational and Combinatorial Group Theory as well as various aspects of Group-Based Cryptography. One of these sessions was at Cornell University in September of 2011 and the other was the University of Nevada, Las Vegas in May of Over the past twenty years Combinatorial and Infinite Group Theory has been energized by three developments: the emergence of Geometric and Asymptotic Group Theory, the development of Algebraic Geometry over Groups leading to the solution of the Tarski problems, and the development of Group-Based Cryptography. These three areas in turn have had an impact on computational algebra and complexity theory. The papers in this volume, both survey and research, exhibit the tremendous vitality that is at the heart of Group Theory in the beginning of the twenty-first century as well as the diversity of interests in the field. We are grateful to the American Mathematical Society for their help in publication of this volume. In particular, we thank Christine Thivierge for her patience and diligence in assembling this volume. Bejamin Fine Delaram Kahrobaei Gerhard Rosenberger vii

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10 This volume contains the proceedings of the AMS Special Session on Computational Algebra, Groups, and Applications, held April 30 May 1, 2011, at the University of Nevada, Las Vegas, Nevada, and the AMS Special Session on the Mathematical Aspects of Cryptography and Cyber Security, held September 10 11, 2011, at Cornell University, Ithaca, New York. Over the past twenty years combinatorial and infinite group theory has been energized by three developments: the emergence of geometric and asymptotic group theory, the development of algebraic geometry over groups leading to the solution of the Tarski problems, and the development of group-based cryptography. These three areas in turn have had an impact on computational algebra and complexity theory. The papers in this volume, both survey and research, exhibit the tremendous vitality that is at the heart of group theory in the beginning of the twenty-first century as well as the diversity of interests in the field. ISBN CONM/582 CONM 582 Group Theory and Cryptography Fine et al., Editors AMS