A Course in Cryptography

343 Pages·2019·2.607 MB·English

by  Heiko Knospe

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40 The UNDERGRADUATE TEXTS
SERIES
Pure and Applied
Th is book provides a compact course in modern cryptography. Th e mathemat- Sally
ical foundations in algebra, number theory and probability are presented with a
focus on their cryptographic applications. Th e text provides rigorous defi nitions
and follows the provable security approach. Th e most relevant cryptographic
schemes are covered, including block ciphers, stream ciphers, hash functions,
message authentication codes, public-key encryption, key establishment, digital
signatures and elliptic curves. Th e current developments in post-quantum
cryptography are also explored, with separate chapters on quantum computing,
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lattice-based and code-based cryptosystems. A Course in

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Many examples, fi gures and exercises, as well as SageMath (Python) computer
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code, help the reader to understand the concepts and applications of modern
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Cryptography
cryptography. A special focus is on algebraic structures, which are used in many
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cryptographic constructions and also in post-quantum systems. Th e essential s
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mathematics and the modern approach to cryptography and security prepare
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the reader for more advanced studies. n

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Th e text requires only a fi rst-year course in mathematics (calculus and linear
Heiko Knospe
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algebra) and is also accessible to computer scientists and engineers. Th is book is
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suitable as a textbook for undergraduate and graduate courses in cryptography p
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For additional information
and updates on this book, visit
www.ams.org/bookpages/amstext-40
AMSTEXT/40
T h e Th is series was founded by the highly respected
SERIES mathematician and educator, Paul J. Sally, Jr.
Sally
2-color cover: PMS 432 (Gray) and PMS 300 (Blue) Not yet adjusted page # 336 pages • Backspace 1 3/8" • Trim Size: 7" x 10"
A Course in
Cryptography
Heiko Knospe
T h e UNDERGRADUATE TEXTS • 40
SERIES
Pure and Applied
Sally
A Course in
Cryptography
Heiko Knospe
EDITORIAL COMMITTEE
Gerald B. Folland (Chair) Steven J. Miller
Jamie Pommersheim Serge Tabachnikov
2010 Mathematics Subject Classification. Primary 94A60;
Secondary 68P25,81P94, 11T71.
For additional informationand updates on this book, visit
www.ams.org/bookpages/amstext-40
Library of Congress Cataloging-in-Publication Data
Names: Knospe,Heiko,1966–author.
Title: Acourseincryptography/HeikoKnospe.
Description: Providence,RhodeIsland: AmericanMathematicalSociety,[2019]|Series: Pureand
appliedundergraduatetexts;volume40|Includesbibliographicalreferencesandindex.
Identifiers: LCCN2019011732|ISBN9781470450557(alk. paper)
Subjects: LCSH:Cryptography–Textbooks. |Codingtheory–Textbooks. |Ciphers–Textbooks. |
AMS: Information and communication, circuits – Communication, information – Cryptogra-
phy. msc | Computer science – Theory of data – Data encryption. msc | Quantum theory –
Axiomatics,foundations,philosophy–Quantumcryptography. msc|Numbertheory–Finite
fields and commutative rings (number-theoretic aspects) – Algebraic coding theory; cryptog-
raphy. msc
Classification: LCCQA268.K58272019|DDC005.8/24–dc23
LCrecordavailableathttps://lccn.loc.gov/2019011732
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Contents
Preface xiii
GettingStartedwithSageMath 1
0.1. Installation 1
0.2. SageMathCommandLine 2
0.3. BrowserNotebooks 2
0.4. ComputationswithSageMath 3
Chapter1. Fundamentals 7
1.1. Sets,RelationsandFunctions 7
1.2. Combinatorics 14
1.3. ComputationalComplexity 16
1.4. DiscreteProbability 19
1.5. RandomNumbers 23
1.6. Summary 27
Exercises 28
Chapter2. EncryptionSchemesandDefinitionsofSecurity 31
2.1. EncryptionSchemes 32
2.2. PerfectSecrecy 35
2.3. ComputationalSecurity 36
2.4. IndistinguishableEncryptions 37
2.5. EavesdroppingAttacks 39
vii
viii Contents
2.6. ChosenPlaintextAttacks 41
2.7. ChosenCiphertextAttacks 43
2.8. PseudorandomGenerators 45
2.9. PseudorandomFunctions 48
2.10. BlockCiphersandOperationModes 52
2.11. Summary 58
Exercises 58
Chapter3. ElementaryNumberTheory 61
3.1. Integers 61
3.2. Congruences 65
3.3. ModularExponentiation 67
3.4. Summary 69
Exercises 69
Chapter4. AlgebraicStructures 73
4.1. Groups 73
4.2. RingsandFields 81
4.3. FiniteFields 82
4.4. LinearandAffineMaps 92
4.5. Summary 97
Exercises 97
Chapter5. BlockCiphers 101
5.1. ConstructionsofBlockCiphers 101
5.2. AdvancedEncryptionStandard 104
5.3. Summary 111
Exercises 111
Chapter6. StreamCiphers 115
6.1. DefinitionofStreamCiphers 115
6.2. LinearFeedbackShiftRegisters 119
6.3. RC4 128
6.4. Salsa20andChaCha20 130
6.5. Summary 135
Exercises 135
Contents ix
Chapter7. HashFunctions 137
7.1. DefinitionsandSecurityRequirements 137
7.2. ApplicationsofHashFunctions 139
7.3. Merkle-DamgårdConstruction 140
7.4. SHA-1 142
7.5. SHA-2 145
7.6. SHA-3 146
7.7. Summary 149
Exercises 149
Chapter8. MessageAuthenticationCodes 151
8.1. DefinitionsandSecurityRequirements 151
8.2. CBCMAC 154
8.3. HMAC 156
8.4. AuthenticatedEncryption 157
8.5. Summary 161
Exercises 161
Chapter9. Public-KeyEncryptionandtheRSACryptosystem 163
9.1. Public-KeyCryptosystems 163
9.2. PlainRSA 166
9.3. RSASecurity 168
9.4. GenerationofPrimes 170
9.5. EfficiencyofRSA 173
9.6. PaddedRSA 175
9.7. Factoring 177
9.8. Summary 182
Exercises 182
Chapter10. KeyEstablishment 185
10.1. KeyDistribution 186
10.2. KeyExchangeProtocols 186
10.3. Diffie-HellmanKeyExchange 188
10.4. Diffie-HellmanusingSubgroupsofℤ∗ 190
𝑝
10.5. DiscreteLogarithm 192
10.6. KeyEncapsulation 194
10.7. HybridEncryption 197
x Contents
10.8. Summary 200
Exercises 200
Chapter11. DigitalSignatures 203
11.1. DefinitionsandSecurityRequirements 203
11.2. PlainRSASignature 205
11.3. ProbabilisticSignatureScheme 206
11.4. Summary 210
Exercises 210
Chapter12. EllipticCurveCryptography 213
12.1. WeierstrassEquationsandEllipticCurves 213
12.2. EllipticCurveDiffie-Hellman 222
12.3. EfficiencyandSecurityofEllipticCurveCryptography 223
12.4. EllipticCurveFactoringMethod 224
12.5. Summary 227
Exercises 227
Chapter13. QuantumComputing 229
13.1. QuantumBits 230
13.2. MultipleQubitSystems 234
13.3. QuantumAlgorithms 235
13.4. QuantumFourierTransform 241
13.5. Shor’sFactoringAlgorithm 242
13.6. QuantumKeyDistribution 248
13.7. Summary 251
Exercises 251
Chapter14. Lattice-basedCryptography 253
14.1. Lattices 254
14.2. LatticeAlgorithms 260
14.3. GGHCryptosystem 269
14.4. NTRU 271
14.5. LearningwithErrors 276
14.6. Summary 282
Exercises 282