Cryptography An Introduction

Learning about cryptography requires examining fundamental issues about information security. Questions abound, ranging from “From whom are we protecting ourselves?” and “How can we measure levels of security?” to “What are our opponent's capabilities?” and “What are their goals?” Answering these questions requires an understanding of basic cryptography. This book, written by R...

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Learning about cryptography requires examining fundamental issues about information security. Questions abound, ranging from “From whom are we protecting ourselves?” and “How can we measure levels of security?” to “What are our opponent's capabilities?” and “What are their goals?” Answering these questions requires an understanding of basic cryptography. This book, written by Russian cryptographers, explains those basics.

Chapters are independent and can be read in any order. The introduction gives a general description of all the main notions of modern cryptography: a cipher, a key, security, an electronic digital signature, a cryptographic protocol, etc. Other chapters delve more deeply into this material. The final chapter presents problems and selected solutions from “Cryptography Olympiads for (Russian) High School Students”.

This is an English translation of a Russian textbook. It is suitable for advanced high school students and undergraduates studying information security. It is also appropriate for a general mathematical audience interested in cryptography.

Cover 1
Title 2
Copyright 3
Contents 4
Preface 8
Chapter 1. Main Notions 12
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Cover 1
Title 2
Copyright 3
Contents 4
Preface 8
Chapter 1. Main Notions 12
§1. Introduction 12
§2. The subject of cryptography 14
§3. Mathematical basis 21
§4. New directions 24
§5. Conclusion 30
Chapter 2. Cryptography and Complexity Theory 32
§1. Introduction 32
§2. Cryptography and the P ≠ NP conjecture 35
§3. One-way functions 37
§4. Pseudorandom generators 40
§5. Zero-knowledge proofs 43
Chapter 3. Cryptographic Protocols 50
§1. Introduction 50
§2. Integrity. Authentication and electronic signature protocols 53
§3. Untraceability. Electronic money 71
§4. Coin flipping by telephone protocols 79
§5. More about secret sharing 85
§6. Playing building blocks, or Election protocols 88
§7. Beyond standard assumptions. Confidential message transmission 94
§8. In place of a conclusion 97
Chapter 4. Algorithmic Problems of Number Theory 98
§1. Introduction 98
§2. The RSA cryptosystem 100
§3. Complexity of number-theoretic algorithms 104
§4. How to distinguish between a composite and a prime number 110
§5. How to construct large prime numbers 113
§6. How to test primality of a large number 116
§7. How to factorize a composite number 121
§8. Discrete logarithms 125
§9. Conclusion 131
Chapter 5. Mathematics of Secret Sharing 132
§1. Introduction 132
§2. Secret sharing for arbitrary access structures 134
§3. Linear secret sharing 138
§4. Ideal secret sharing and matroids 140
Chapter 6. Cryptography Olympiads for High School Students 146
§1. Introduction 146
§2. Substitution ciphers 150
§3. Transposition ciphers 163
§4. Periodic polyalphabetic substitution ciphers 170
§5. Problems 176
§6. Answers, hints, solutions 195
Bibliography 236
Back Cover 241
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