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GROUP THEORETIC CRYPTOGRAPHY C8369_FM.indd 1 3/2/15 11:30 AM CHAPMAN & HALL/CRC CRYPTOGRAPHY AND NETWORK SECURITY Series Editors Douglas R. Stinson and Jonathan Katz Published Titles Lidong Chen and Guang Gong, Communication System Security Shiu-Kai Chin and Susan Older, Access Control, Security, and Trust: A Logical Approach M. Jason Hinek, Cryptanalysis of RSA and Its Variants Antoine Joux, Algorithmic Cryptanalysis Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptography, Second Edition Sankar K. Pal, Alfredo Petrosino, and Lucia Maddalena, Handbook on Soft Computing for Video Surveillance Burton Rosenberg, Handbook of Financial Cryptography and Security María Isabel González Vasco and Rainer Steinwandt, Group Theoretic Cryptography C8369_FM.indd 2 3/2/15 11:30 AM Chapman & Hall/CRC CRYPTOGRAPHY AND NETWORK SECURITY GROUP THEORETIC CRYPTOGRAPHY María Isabel González Vasco Universidad Rey Juan Carlos Madrid, Spain Rainer Steinwandt Florida Atlantic University Boca Raton, FL C8369_FM.indd 3 3/2/15 11:30 AM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150312 International Standard Book Number-13: 978-1-58488-837-6 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume 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 holders 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 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 photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To our families TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Contents List of Figures xi Symbols xiii Preface xv I Preliminaries 1 1 Mathematical background 3 1.1 Algebraic structures in a nutshell . . . . . . . . . . . . . . . 3 1.2 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Summary and further reading . . . . . . . . . . . . . . . . . 13 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Basics on complexity 17 2.1 Complexity classes . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Asymptotic notation and examples . . . . . . . . . . . . . . 20 2.3 Summary and further reading . . . . . . . . . . . . . . . . . 22 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Cryptology: An introduction 25 3.1 A short historical overview . . . . . . . . . . . . . . . . . . . 25 3.1.1 Historical encryption schemes . . . . . . . . . . . . . . 25 3.1.2 Public-key cryptography . . . . . . . . . . . . . . . . . 28 3.2 Modern cryptology . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Summary and further reading . . . . . . . . . . . . . . . . . 34 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 II Public-Key Encryption 37 4 Provable security guarantees 39 4.1 Public-key encryption revisited . . . . . . . . . . . . . . . . . 39 4.2 Characterizing secure public-key encryption . . . . . . . . . . 42 4.3 One-way functions and random oracles . . . . . . . . . . . . 53 4.4 The general Bellare–Rogaway construction . . . . . . . . . . 57 4.5 IND-CCA security with an Abelian group: RSA-OAEP . . . 62 4.6 One-way functions from non-Abelian groups? . . . . . . . . . 64 4.7 Summary and further reading . . . . . . . . . . . . . . . . . 68 vii viii Contents 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Public-key encryption in the standard model 73 5.1 The Cramer–Shoup encryption scheme from 1998 . . . . . . 73 5.2 Going beyond: Tools . . . . . . . . . . . . . . . . . . . . . . 80 5.2.1 Projective hash families . . . . . . . . . . . . . . . . . 80 5.2.2 Subset membership problems . . . . . . . . . . . . . . 83 5.2.3 Hash proof systems. . . . . . . . . . . . . . . . . . . . 84 5.3 General Cramer–Shoup encryption scheme . . . . . . . . . . 85 5.4 A concrete instantiation . . . . . . . . . . . . . . . . . . . . . 87 5.5 Projective hash families from (non-Abelian) groups . . . . . 88 5.5.1 Group action systems . . . . . . . . . . . . . . . . . . 88 5.5.2 Group action projective hash families . . . . . . . . . 92 5.6 Summary and further reading . . . . . . . . . . . . . . . . . 94 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Public-key encryption using infinite groups 97 6.1 The word problem in finitely presented groups . . . . . . . . 97 6.1.1 The encryption scheme of Wagner and Magyarik . . . 98 6.1.2 Polly Cracker . . . . . . . . . . . . . . . . . . . . . . . 102 6.1.3 A successor of the Wagner–Magyarik scheme . . . . . 103 6.2 Using a group that is not finitely presentable? . . . . . . . . 105 6.3 Braid groups in cryptography . . . . . . . . . . . . . . . . . . 108 6.3.1 Basics on braid groups . . . . . . . . . . . . . . . . . . 108 6.3.2 Some computational problems in the braid group B . 111 n 6.4 Summary and further reading . . . . . . . . . . . . . . . . . 117 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 III Secret-Key Encryption 121 7 Block ciphers 123 7.1 Advanced Encryption Standard . . . . . . . . . . . . . . . . 124 7.1.1 Specifying the round function . . . . . . . . . . . . . . 124 7.1.2 Key schedule . . . . . . . . . . . . . . . . . . . . . . . 126 7.1.3 Encryption and decryption with AES . . . . . . . . . 126 7.2 Data Encryption Standard . . . . . . . . . . . . . . . . . . . 129 7.2.1 General structure of DES: A Feistel cipher. . . . . . . 129 7.2.2 Round function of DES . . . . . . . . . . . . . . . . . 132 7.2.3 Key schedule . . . . . . . . . . . . . . . . . . . . . . . 134 7.3 Permutation Group Mappings . . . . . . . . . . . . . . . . . 135 7.4 Modes of operation . . . . . . . . . . . . . . . . . . . . . . . 137 7.4.1 Electronic codebook (ECB) mode. . . . . . . . . . . . 137 7.4.2 Cipher block chaining (CBC) mode. . . . . . . . . . . 138 7.4.3 Cipher feedback (CFB) mode . . . . . . . . . . . . . . 138 7.4.4 Output feedback (OFB) mode . . . . . . . . . . . . . 139 Contents ix 7.4.5 Counter (CTR) mode . . . . . . . . . . . . . . . . . . 140 7.5 Summary and further reading . . . . . . . . . . . . . . . . . 141 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8 Cryptographic hash functions and message authentication codes 145 8.1 Cryptographic hash functions . . . . . . . . . . . . . . . . . . 145 8.2 Deriving a hash function from a block cipher . . . . . . . . . 147 8.3 Cayley hash functions . . . . . . . . . . . . . . . . . . . . . . 149 8.4 Message authentication codes . . . . . . . . . . . . . . . . . . 151 8.4.1 Keyed-Hash Message Authentication Code. . . . . . . 152 8.4.2 Cipher-based Message Authentication Code . . . . . . 153 8.5 Summary and further reading . . . . . . . . . . . . . . . . . 154 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 IV Other Cryptographic Constructions 157 9 Key establishment protocols 159 9.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.1.1 Provable security for key exchange protocols . . . . . 162 9.1.2 A secure construction . . . . . . . . . . . . . . . . . . 166 9.2 Anshel–Anshel–Goldfeld key exchange . . . . . . . . . . . . . 170 9.3 Braid-based key exchange . . . . . . . . . . . . . . . . . . . . 174 9.4 Constructions over matrix groups . . . . . . . . . . . . . . . 176 9.5 Summary and further reading . . . . . . . . . . . . . . . . . 178 9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 10 Signature and identification schemes 181 10.1 Definitions and terminology . . . . . . . . . . . . . . . . . . . 181 10.2 RSA signatures: FDH and PSS . . . . . . . . . . . . . . . . . 184 10.3 Identification schemes . . . . . . . . . . . . . . . . . . . . . . 188 10.4 Summary and further reading . . . . . . . . . . . . . . . . . 191 10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 V Appendix 195 A Solutions to selected exercises 197 A.1 Solutions to selected exercises of Part I . . . . . . . . . . . . 197 A.2 Solutions to selected exercises of Part II . . . . . . . . . . . . 199 A.3 Solutions to selected exercises of Part III . . . . . . . . . . . 202 A.4 Solutions to selected exercises of Part IV . . . . . . . . . . . 203 References 207 Index 225

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GROUP THEORETIC CRYPTOGRAPHY C8369_FM.indd 1 3/2/15 11:30 AM CHAPMAN & HALL/CRC CRYPTOGRAPHY AND NETWORK SECURITY Series Editors Douglas R. Stinson and Jonathan Katz Published Titles Lidong Chen and Guang Gong, Communication System Security Shiu-Kai Chin and Susan Older, Access Control, Security, and Trust: A Logical Approach M. Jason Hinek, Cryptanalysis of RSA and Its Variants Antoine Joux, Algorithmic Cryptanalysis Jonathan Katz and Yehuda Lindell, Introduction to Modern Cryptography, Second Edition Sankar K. Pal, Alfredo Petrosino, and Lucia Maddalena, Handbook on Soft Computing for Video Surveillance Burton Rosenberg, Handbook of Financial Cryptography and Security María Isabel González Vasco and Rainer Steinwandt, Group Theoretic Cryptography C8369_FM.indd 2 3/2/15 11:30 AM Chapman & Hall/CRC CRYPTOGRAPHY AND NETWORK SECURITY GROUP THEORETIC CRYPTOGRAPHY María Isabel González Vasco Universidad Rey Juan Carlos Madrid, Spain Rainer Steinwandt Florida Atlantic University Boca Raton, FL C8369_FM.indd 3 3/2/15 11:30 AM CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2015 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Version Date: 20150312 International Standard Book Number-13: 978-1-58488-837-6 (eBook - PDF) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume 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 holders 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 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 photocopying, microfilming, and recording, or in any information stor- age or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copy- right.com (http://www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that pro- vides licenses and registration for a variety of users. For organizations that have been granted a photo- copy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com To our families TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk Contents List of Figures xi Symbols xiii Preface xv I Preliminaries 1 1 Mathematical background 3 1.1 Algebraic structures in a nutshell . . . . . . . . . . . . . . . 3 1.2 Finite groups . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Summary and further reading . . . . . . . . . . . . . . . . . 13 1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2 Basics on complexity 17 2.1 Complexity classes . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Asymptotic notation and examples . . . . . . . . . . . . . . 20 2.3 Summary and further reading . . . . . . . . . . . . . . . . . 22 2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3 Cryptology: An introduction 25 3.1 A short historical overview . . . . . . . . . . . . . . . . . . . 25 3.1.1 Historical encryption schemes . . . . . . . . . . . . . . 25 3.1.2 Public-key cryptography . . . . . . . . . . . . . . . . . 28 3.2 Modern cryptology . . . . . . . . . . . . . . . . . . . . . . . 31 3.3 Summary and further reading . . . . . . . . . . . . . . . . . 34 3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 II Public-Key Encryption 37 4 Provable security guarantees 39 4.1 Public-key encryption revisited . . . . . . . . . . . . . . . . . 39 4.2 Characterizing secure public-key encryption . . . . . . . . . . 42 4.3 One-way functions and random oracles . . . . . . . . . . . . 53 4.4 The general Bellare–Rogaway construction . . . . . . . . . . 57 4.5 IND-CCA security with an Abelian group: RSA-OAEP . . . 62 4.6 One-way functions from non-Abelian groups? . . . . . . . . . 64 4.7 Summary and further reading . . . . . . . . . . . . . . . . . 68 vii viii Contents 4.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 5 Public-key encryption in the standard model 73 5.1 The Cramer–Shoup encryption scheme from 1998 . . . . . . 73 5.2 Going beyond: Tools . . . . . . . . . . . . . . . . . . . . . . 80 5.2.1 Projective hash families . . . . . . . . . . . . . . . . . 80 5.2.2 Subset membership problems . . . . . . . . . . . . . . 83 5.2.3 Hash proof systems. . . . . . . . . . . . . . . . . . . . 84 5.3 General Cramer–Shoup encryption scheme . . . . . . . . . . 85 5.4 A concrete instantiation . . . . . . . . . . . . . . . . . . . . . 87 5.5 Projective hash families from (non-Abelian) groups . . . . . 88 5.5.1 Group action systems . . . . . . . . . . . . . . . . . . 88 5.5.2 Group action projective hash families . . . . . . . . . 92 5.6 Summary and further reading . . . . . . . . . . . . . . . . . 94 5.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 6 Public-key encryption using infinite groups 97 6.1 The word problem in finitely presented groups . . . . . . . . 97 6.1.1 The encryption scheme of Wagner and Magyarik . . . 98 6.1.2 Polly Cracker . . . . . . . . . . . . . . . . . . . . . . . 102 6.1.3 A successor of the Wagner–Magyarik scheme . . . . . 103 6.2 Using a group that is not finitely presentable? . . . . . . . . 105 6.3 Braid groups in cryptography . . . . . . . . . . . . . . . . . . 108 6.3.1 Basics on braid groups . . . . . . . . . . . . . . . . . . 108 6.3.2 Some computational problems in the braid group B . 111 n 6.4 Summary and further reading . . . . . . . . . . . . . . . . . 117 6.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 III Secret-Key Encryption 121 7 Block ciphers 123 7.1 Advanced Encryption Standard . . . . . . . . . . . . . . . . 124 7.1.1 Specifying the round function . . . . . . . . . . . . . . 124 7.1.2 Key schedule . . . . . . . . . . . . . . . . . . . . . . . 126 7.1.3 Encryption and decryption with AES . . . . . . . . . 126 7.2 Data Encryption Standard . . . . . . . . . . . . . . . . . . . 129 7.2.1 General structure of DES: A Feistel cipher. . . . . . . 129 7.2.2 Round function of DES . . . . . . . . . . . . . . . . . 132 7.2.3 Key schedule . . . . . . . . . . . . . . . . . . . . . . . 134 7.3 Permutation Group Mappings . . . . . . . . . . . . . . . . . 135 7.4 Modes of operation . . . . . . . . . . . . . . . . . . . . . . . 137 7.4.1 Electronic codebook (ECB) mode. . . . . . . . . . . . 137 7.4.2 Cipher block chaining (CBC) mode. . . . . . . . . . . 138 7.4.3 Cipher feedback (CFB) mode . . . . . . . . . . . . . . 138 7.4.4 Output feedback (OFB) mode . . . . . . . . . . . . . 139 Contents ix 7.4.5 Counter (CTR) mode . . . . . . . . . . . . . . . . . . 140 7.5 Summary and further reading . . . . . . . . . . . . . . . . . 141 7.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 8 Cryptographic hash functions and message authentication codes 145 8.1 Cryptographic hash functions . . . . . . . . . . . . . . . . . . 145 8.2 Deriving a hash function from a block cipher . . . . . . . . . 147 8.3 Cayley hash functions . . . . . . . . . . . . . . . . . . . . . . 149 8.4 Message authentication codes . . . . . . . . . . . . . . . . . . 151 8.4.1 Keyed-Hash Message Authentication Code. . . . . . . 152 8.4.2 Cipher-based Message Authentication Code . . . . . . 153 8.5 Summary and further reading . . . . . . . . . . . . . . . . . 154 8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 IV Other Cryptographic Constructions 157 9 Key establishment protocols 159 9.1 Setting the stage . . . . . . . . . . . . . . . . . . . . . . . . . 159 9.1.1 Provable security for key exchange protocols . . . . . 162 9.1.2 A secure construction . . . . . . . . . . . . . . . . . . 166 9.2 Anshel–Anshel–Goldfeld key exchange . . . . . . . . . . . . . 170 9.3 Braid-based key exchange . . . . . . . . . . . . . . . . . . . . 174 9.4 Constructions over matrix groups . . . . . . . . . . . . . . . 176 9.5 Summary and further reading . . . . . . . . . . . . . . . . . 178 9.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 10 Signature and identification schemes 181 10.1 Definitions and terminology . . . . . . . . . . . . . . . . . . . 181 10.2 RSA signatures: FDH and PSS . . . . . . . . . . . . . . . . . 184 10.3 Identification schemes . . . . . . . . . . . . . . . . . . . . . . 188 10.4 Summary and further reading . . . . . . . . . . . . . . . . . 191 10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 V Appendix 195 A Solutions to selected exercises 197 A.1 Solutions to selected exercises of Part I . . . . . . . . . . . . 197 A.2 Solutions to selected exercises of Part II . . . . . . . . . . . . 199 A.3 Solutions to selected exercises of Part III . . . . . . . . . . . 202 A.4 Solutions to selected exercises of Part IV . . . . . . . . . . . 203 References 207 Index 225

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