Mathematics in Cyber Research PDF
Preview Mathematics in Cyber Research
Mathematics in Cyber Research Mathematics in Cyber Research Edited by Paul L. Goethals Department of Mathematical Sciences, United States Military Academy, USA Natalie M. Scala College of Business and Economics, Towson University, USA Daniel T. Bennett United States Military Academy, USA First edition published 2022 by CRC Press 6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742 and by CRC Press 4 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN © 2022 Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, LLC Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot as- sume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright hold- ers if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged, please write and let us know so we may rectify it in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including pho- tocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, access www.copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. For works that are not available on CCC, please contact [email protected] Trademark notice: Product or corporate names may be trademarks or registered trademarks and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Goethals, Paul L., editor. | Scala, Natalie M. (Natalie Michele), editor. | Bennett, Daniel T., editor. Title: Mathematics in cyber research / edited by Paul L. Goethals, Department of Mathematical Sciences, United States Military Academy, USA, Natalie M. Scala, College of Business and Economics, Towson University, USA, Daniel T. Bennett, National Renewable Energy Laboratory (NREL), USA. Description: First edition. | London ; Boca Raton : C&H/CRC Press, 2022. | Includes bibliographical references and index. Identifiers: LCCN 2021043509 (print) | LCCN 2021043510 (ebook) | ISBN 9780367374679 (hardback) | ISBN 9781032208046 (paperback) | ISBN 9780429354649 (ebook) Subjects: LCSH: Computer science--Mathematics. | Cyberspace--Mathematics. | Operations research--Mathematics. Classification: LCC QA76.9.M35 M389 2022 (print) | LCC QA76.9.M35 (ebook) | DDC 004.01/51--dc23 LC record available at https://lccn.loc.gov/2021043509 LC ebook record available at https://lccn.loc.gov/2021043510 ISBN: 978-0-367-37467-9 (hbk) ISBN: 978-1-032-20804-6 (pbk) ISBN: 978-0-429-35464-9 (ebk) DOI: 10.1201/9780429354649 Typeset in CMR10 by KnowledgeWorks Global Ltd. Editor’s Note: The views expressed herein do not necessarily represent the views of the U.S. Department of Defense or the U.S. Government. To Dr. B. Rae Cho, Professor Emeritus, and my doctoral advisor at Clemson University. Thank you for initiating and inspiring my curiosity for research. I would not be capable of this achievement without your example, your advice, and your consistent thoughtfulness. —Paul L. Goethals To the Mathematics and Computer Science faculty at John Carroll University, especially Patrick Chen, Barbara D’Ambrosia, Jerry Moreno, Doug Norris, Carl Spitznagel, and David Stenson. Your love of mathematics always inspired me, and your mentorship will never be forgotten. —Natalie M. Scala Contents Preface ix ListofContributors xvii Chapter 1(cid:4) Combinatorics 1 CheyneHomberger Chapter 2(cid:4) Cryptography 53 GretchenL.Matthews,AidanW.Murphy Chapter 3(cid:4) AlgebraicGeometry 97 LubjanaBeshaj Chapter 4(cid:4) Topology 133 SteveHuntsman,JimmyPalladinoandMichaelRobinson Chapter 5(cid:4) DifferentialEquations 171 ParisaFatheddin Chapter 6(cid:4) NetworkScience 207 ElieAlhajjar Chapter 7(cid:4) OperationsResearch 233 PaulL.Goethals,NatalieM.ScalaandNathanielD.Bastian Chapter 8(cid:4) DataAnalysis 267 RaymondR.Hill,DarrylK.Ahner Chapter 9(cid:4) Statistics 303 NitaYodo,MelvinRafi Chapter 10(cid:4) ProbabilityTheory 335 DavidM.Ruth vii viii (cid:4) Contents Chapter 11(cid:4) GameTheory 363 AndrewFielder Chapter 12(cid:4) NumberTheory 393 DaneSkabelund Chapter 13(cid:4) QuantumTheory 421 TravisB.Russell Chapter 14(cid:4) GroupTheory 453 WilliamCocke,Meng-Che‘Turbo’Ho Chapter 15(cid:4) RingTheory 475 Lindsey-KayLauderdale Index 497 Preface INTRODUCTION Since the dawn of the computer, we have observed the continued growth of cy- berspace. The Internet of Things via users, mobile devices, and new technologies, as well as the need for greater interconnectivity among our social, economic, and po- litical systems continue to fuel its expansion. Its importance to business, matters of security, and networking is clear, enabling real-time communication at a global scale. Today’s emphasis on automated systems and robotics, as well as emerging fields such as artificial intelligence, suggest that this space will only continue to increase its role and significance to society in future years. Of critical importance to the international community today are cyber topics such as the security of information and data, the protection of infrastructure, and the defense of networks or automated processes. The growth of cybercrime along with the perceived increase in the threat of an attack or breach of a system has created an environment of general uncertainty and concern. To complicate matters, protecting information is frequently characterized by dichotomies such as security versus privacy or offensive versus defensive action. Moreover, the precise advantages and benefits that cyberspace offers—connectivity, speed of transmission, and open access are in many ways the factors that produce the greatest disadvantages and vulnerabilities in many of our systems and processes. In the last decade, both scholars and practitioners have sought novel ways to ad- dress the problem of cybersecurity. Innovative outcomes have included applications such as blockchain as well as creative methods for cyber forensics, software devel- opment, and intrusion prevention. Accompanying these technological advancements, discussiononcybermattersatnationalandinternationallevelshasfocusedprimarily on the topics of law, policy, and strategy. The objective of these efforts is typically to promote security by establishing agreements among stakeholders on regulatory activities. Varying levels of investment in cyberspace, however, come with varying levels of risk; in some ways, this can translate directly to the degree of emphasis for pushing substantial change. At the very foundation or root of cyberspace systems and processes are tenets and rules governed by principles in mathematics. Topics such as encrypting or de- cryptingfiletransmissions,modelingnetworks,performingdataanalysis,quantifying uncertainty, measuring risk, and weighing decisions or adversarial courses of action represent a very small subset of activities highlighted by mathematics. To facilitate education and a greater awareness of the role of mathematics in cyber systems and processes, a description of research in this area is needed. This ties directly to the ix
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