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Note for Cryptography And Network Security - CNS by pentareddy n

  • Cryptography And Network Security - CNS
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Preface Second Edition Lecture notes of a class given during the summer term 2017 at the University of Kaiserslautern. The notes are based on lecture notes by Mohamed Barakat and Timo Hanke [BH12] (see also below). Other good sources and books are, for example, [Buc04, Sch95, MVO96]. Many thanks to Raul Epure for proofreading and suggestions to improve the lecture notes. First Edition These lecture notes are based on the course “Kryptographie” given by Timo Hanke at RWTH Aachen University in the summer semester of 2010. They were amended and extended by several topics, as well as translated into English, by Mohamed Barakat for his course “Cryptography” at TU Kaiserslautern in the winter semester of 2010/11. Besides the literature given in the bibliography section, our sources include lectures notes of courses held by Michael Cuntz, Florian Heß, Gerhard Hiß and Jürgen Müller. We would like to thank them all. Mohamed Barakat would also like to thank the audience of the course for their helpful remarks and questions. Special thanks to Henning Kopp for his numerous improvements suggestions. Also thanks to Jochen Kall who helped locating further errors and typos. Daniel Berger helped me with subtle formatting issues. Many thanks Daniel. i

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Contents Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i i Contents ii 1 Introduction 1 2 Basic Concepts 2.1 Quick & Dirty Introduction to Complexity Theory . . . . . . . . . . . . . . . . . . . . . 2.2 Underlying Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Investigating Security Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 5 7 11 3 Modes of Ciphers 3.1 Block Ciphers . . . . . . . . . . . . . . 3.2 Modes of Block Ciphers . . . . . . . . 3.3 Stream Ciphers . . . . . . . . . . . . . 3.4 A Short Review of Historical Ciphers . . . . 13 13 14 23 25 4 Information Theory 4.1 A Short Introduction to Probability Theory . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Perfect Secrecy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 27 31 35 5 Pseudorandom Sequences 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Linear recurrence equations and pseudorandom bit generators 5.3 Finite fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Statistical tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Cryptographically secure pseudorandom bit generators . . . . . . . . . 47 47 48 53 62 66 6 Modern Symmetric Block Ciphers 6.1 Feistel cipher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Data Encryption Standard (DES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Advanced Encryption Standard (AES) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 70 71 74 7 Candidates of One-Way Functions 7.1 Complexity classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Squaring modulo n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 77 78 . . . . . . . . ii . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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CONTENTS 7.3 7.4 7.5 7.6 7.7 iii Quadratic residues . . . . . . . . . . Square roots . . . . . . . . . . . . . . One-way functions . . . . . . . . . . Trapdoors . . . . . . . . . . . . . . . . The Blum-Goldwasser construction 8 Public Key Cryptosystems 8.1 RSA . . . . . . . . . . . . 8.2 ElGamal . . . . . . . . . . 8.3 The Rabin cryptosystem 8.4 Security models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 81 83 84 85 . . . . 86 86 90 91 93 9 Primality tests 95 9.1 Probabilistic primality tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 9.2 Deterministic primality tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 10 Integer Factorization 10.1 Pollard’s p − 1 method 10.2 Pollard’s ρ method . . 10.3 Fermat’s method . . . 10.4 Dixon’s method . . . . 10.5 The quadratic sieve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 103 104 105 105 107 11 Elliptic curves 11.1 The projective space . . . . . . . . . 11.2 The group structure (E, +) . . . . . 11.3 Elliptic curves over finite fields . . . 11.4 Lenstra’s factorization method . . . 11.5 Elliptic curves cryptography (ECC) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 109 114 120 124 126 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Attacks on the discrete logarithm problem 128 12.1 Specific attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 12.2 General attacks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 13 Digital signatures 13.1 Basic Definitions & Notations . . . . . . . . 13.2 Signatures using OWF with trapdoors . . 13.3 Hash functions . . . . . . . . . . . . . . . . 13.4 Signatures using OWF without trapdoors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 132 133 134 135 A Some analysis 137 A.1 Real functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Bibliography 138

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