Public Key Cryptography : Applications and Attacks

This book covers public-key cryptography, describing in depth all major public-key cryptosystems in current use, including ElGamal, RSA, Elliptic Curve, and digital signature schemes. It explains the underlying mathematics needed to build these schemes, and examines the most common techniques used in attacking them. Illustrated with many examples, the book provides a solid foundation for professionals in government, cloud service providers, and large enterprises using public-key systems to secure their data. It is also useful for those taking the CISSP exam (Certified Information Systems Security Professional).Content: Chapter 1 Introduction (pages 1–11): Chapter 2 Congruence Equations (pages 13–43): Chapter 3 The Elgamal Scheme (pages 45–58): Chapter 4 The RSA Scheme (pages 59–79): Chapter 5 Elliptic Curve Cryptography (pages 81–101): Chapter 6 Digital Signatures (pages 103–131): Chapter 7 Primality Testing (pages 133–150): Chapter 8 Factoring Methods (pages 151–173): Content: Preface xii 0 Introduction 1 0.1 The Meaning of the Word Cryptography 2 0.2 Symmetric Key Cryptography 2 0.3 Public Key (Asymmetric) Cryptography 5 0.4 Key Establishment 8 0.5 Cryptography more than just Hiding Secrets 9 0.6 Standards 10 0.7 Attacks 11 1 Congruence Equations 13 1.1 Congruence Arithmetic 13 1.1.1 Computer Examples 17 1.1.2 Problems 18 1.2 The Euclidean Algorithm Finding Inverses 19 1.2.1 Computer Examples 26 1.2.2 Problems 27 1.3 Discrete Logarithms and Diffie-Hellman Key Exchange 27 1.3.1 Computer Examples 34 1.3.2 Problems 35 1.4 Attacking the Discrete Logarithm 37 1.4.1 Computer Examples 45 1.4.2 Problems 46 2 The ElGamal Scheme 49 2.1 Primitive Roots 49 2.1.1 Computer Examples 55 2.1.2 Problems 55 2.2 The ElGamal Scheme 56 2.2.1 Computer Examples 58 2.2.2 Problems 60 2.3 Security of the ElGamal Scheme 62 2.3.1 Computer Examples 64 2.3.2 Problems 64 3 The RSA Scheme 67 3.1 Euler's Theorem 67 3.1.1 Computer Examples 71 3.1.2 Problems 71 3.2 The RSA Algorithm 71 3.2.1 Computer Examples 76 3.2.2 Problems 78 3.3 RSA Security 79 3.3.1 Computer Examples 85 3.3.2 Problems 87 3.4 Implementing RSA 88 3.4.1 Computer Examples 90 3.4.2 Problems 91 4 Elliptic Curve Cryptography 93 4.1 Elliptic Curves and Elliptic Curve Groups 93 4.1.1 Computer Examples 102 4.1.2 Problems 103 4.2 Elliptic Curve Cryptography 104 4.2.1 Computer Examples 109 4.2.2 Problems 112 4.3 The Elliptic Curve Factoring Scheme 113 4.3.1 Computer Examples 115 4.3.2 Problems 116 5 Digital Signatures 117 5.1 Hash Functions 117 5.1.1 Computer Examples 129 5.1.2 Problems 130 5.2 Digital Signature Schemes 132 5.2.1 Computer Examples 139 5.2.2 Problems 141 5.3 Attacks on Digital Signatures 142 5.3.1 Computer Examples 150 5.3.2 Problems 152 6 Primality Testing 155 6.1 Fermat's Approach and Wilson's Theorem 156 6.1.1 Computer Examples 157 6.1.2 Problems 158 6.2 The Miller-Selfridge-Rabin Primality Test 158 6.2.1 Computer Examples 164 6.2.2 Problems 164 6.3 True Primality Tests 165 6.3.1 Computer Examples 168 6.4 Mersenne Primes and the Lucas-Lehmer Test 169 6.4.1 Computer Examples 171 6.4.2 Problems 171 6.5 Primes is in P 171 6.5.1 Computer Examples 175 6.5.2 Problems 176 7 Factoring Methods 179 7.1 Fermat Again 180 7.1.1 Computer Examples 183 7.1.2 Problems 185 7.2 The Quadratic Sieve 185 7.2.1 Computer Examples 187 7.2.2 Problems 189 7.3 Pollard's p - 1 and rho Methods 189 7.3.1 Computer Examples 193 7.3.2 Problems194 7.4 Continued Fractions and Factoring194 7.4.1 Computer Examples 200 7.4.2 Problems 203 Appendix: Solutions to Problems 207 References 231 Index 236 Notation 239

Complete coverage of the current major public key cryptosystems their underlying mathematics and the most common techniques used in attacking them Public Key Cryptography: Applications and Attacks introduces and explains the fundamentals of public key cryptography and explores its application in all major public key cryptosystems in current use, including ElGamal, RSA, Elliptic Curve, and digital signature schemes. It provides the underlying mathematics needed to build and study these schemes as needed, and examines attacks on said schemes via the mathematical problems on which they are based – such as the discrete logarithm problem and the difficulty of factoring integers. The book contains approximately ten examples with detailed solutions, while each chapter includes forty to fifty problems with full solutions for odd-numbered problems provided in the Appendix. Public Key Cryptography: •Explains fundamentals of public key cryptography •Offers numerous examples and exercises •Provides excellent study tools for those preparing to take the Certified Information Systems Security Professional (CISSP) exam •Provides solutions to the end-of-chapter problems

Public Key Cryptography provides a solid background for anyone who is employed by or seeking employment with a government organization, cloud service provider, or any large enterprise that uses public key systems to secure data.

Complete coverage of the current major public key cryptosystems their underlying mathematics and the most common techniques used in attacking them Public Key Cryptography: Applications and Attacks introduces and explains the fundamentals of public key cryptography and explores its application in all major public key cryptosystems in current use, including ElGamal, RSA, Elliptic Curve, and digital signature schemes. It provides the underlying mathematics needed to build and study these schemes as needed, and examines attacks on said schemes via the mathematical problems on which they are based – such as the discrete logarithm problem and the difficulty of factoring integers. The book contains approximately ten examples with detailed solutions, while each chapter includes forty to fifty problems with full solutions for odd-numbered problems provided in the Appendix. Public Key Cryptography: • Explains fundamentals of public key cryptography • Offers numerous examples and exercises • Provides excellent study tools for those preparing to take the Certified Information Systems Security Professional (CISSP) exam • Provides solutions to the end-of-chapter problems Public Key Cryptography provides a solid background for anyone who is employed by or seeking employment with a government organization, cloud service provider, or any large enterprise that uses public key systems to secure data. In the past few years elliptic curve cryptography has moved from a fringe activity to a major challenger to the dominant RSA/DSA systems. Elliptic curves offer major advances on older systems such as increased speed, less memory and smaller key sizes. As digital signatures become more and more important in the commercial world the use of elliptic curve-based signatures will become all pervasive. This book summarizes knowledge built up within Hewlett-Packard over a number of years, and explains the mathematics behind practical implementations of elliptic curve systems. Due to the advanced nature of the mathematics there is a high barrier to entry for individuals and companies to this technology. Hence this book will be invaluable not only to mathematicians wanting to see how pure mathematics can be applied but also to engineers and computer scientists wishing (or needing) to actually implement such systems. Elliptic Curve Cryptography Is A Major Challenger To The Dominant Rsa/dsa Systems, With Increased Speed, Less Memory Required And Smaller Key Sizes. This Book Summarises Knowledge Built Up Within Hewlett Packard Over A Number Of Years. Ch. I. Introduction -- Ch. Ii. Finite Field Arithmetic -- Ch. Iii. Arithmetic On An Elliptic Curve -- Ch. Iv. Efficient Implementation Of Elliptic Curves -- Ch. V. The Elliptic Curve Discrete Logarithm Problem -- Ch. Vi. Determining The Group Order -- Ch. Vii. Schoof's Algorithm And Extensions -- Ch. Viii. Generating Curves Using Complex Multiplication -- Ch. Ix. Other Applications Of Elliptic Curves -- Ch. X. Hyperelliptic Cryptosystems -- App. A. Curve Examples. I.f. Blake, G. Seroussi, And N.p. Smart. Includes Bibliographical References And Index. This text covers public-key cryptography, describing in depth all major public-key cryptosystems in current use, including ElGamal, RSA, Elliptic Curve, and digital signature schemes. It explains the underlying mathematics needed to build these schemes, and examines the most common techniques used in attacking them Abstract: This book covers public-key cryptography, describing in depth all major public-key cryptosystems in current use, including ElGamal, RSA, Elliptic Curve, and digital signature schemes. It explains the underlying mathematics needed to build these schemes, and examines the most common techniques used in attacking them. Read more... PUBLIC KEY CRYPTOGRAPHY: Applications and Attacks; Contents; Preface; Acknowledgments; List of Figures; 1 INTRODUCTION; 1.1 The Meaning of the Word Cryptography; 1.2 Symmetric Key Cryptography; 1.2.1 Impact of Technology; 1.2.2 Confusion and Diffusion; 1.2.3 DES and AES; 1.3 Public Key (Asymmetric) Cryptography; 1.3.1 Diffie-Hellman Key Exchange; 1.3.2 RSA; 1.3.3 ElGamal; 1.4 Key Establishment; 1.5 Cryptography-More than just Hiding Secrets; 1.5.1 Digital Signatures; 1.5.2 Authentication; 1.5.3 Nonrepudiation; 1.6 Standards; 1.7 Attacks; 2 CONGRUENCE EQUATIONS; 2.1 Congruence Arithmetic 2.1.1 Computer Examples2.1.2 Problems; 2.2 The Euclidean Algorithm-Finding Inverses; 2.2.1 Computer Examples; 2.2.2 Problems; 2.3 Discrete Logarithms and Diffie-Hellman Key Exchange; 2.3.1 Computer Examples; 2.3.2 Problems; 2.4 Attacking the Discrete Logarithm; 2.4.1 Computer Examples; 2.4.2 Problems; 3 THE ELGAMAL SCHEME; 3.1 Primitive Roots; 3.1.1 Computer Examples; 3.1.2 Problems; 3.2 The ElGamal Scheme; 3.2.1 Computer Examples; 3.2.2 Problems; 3.3 Security of the ElGamal Scheme; 3.3.1 Computer Examples; 3.3.2 Problems; 4 THE RSA SCHEME; 4.1 Euler's Theorem; 4.1.1 Computer Examples 6.2.7 Problems6.3 Attacks on Digital Signatures; 6.3.1 The Birthday Attack; 6.3.2 Intruder-in-the-Middle Attack; 6.3.3 Intruder-in-the-Middle and Diffie-Hellman; 6.3.4 An Attack on RSA Signatures: What if Eve Uses Alice's Signature?; 6.3.5 Protecting ElGamal with Signatures; 6.3.6 Oscar Tries to Trick Bob; 6.3.7 Oscar Captures a Message with a Signed Version; 6.3.8 Alice Reuses a Random Number; 6.3.9 Computer Examples; 6.3.10 Problems; 7 PRIMALITY TESTING; 7.1 Fermat's Approach and Wilson's Theorem; 7.1.1 Computer Examples; 7.1.2 Problems; 7.2 The Miller-Selfridge-Rabin Primality Test 5.2.2 An Elliptic Curve ElGamal System5.2.3 Elliptic Curve Key Exchange (ECKE); 5.2.4 Computer Examples; 5.2.5 Problems; 5.3 The Elliptic Curve Factoring Scheme; 5.3.1 Computer Examples; 5.3.2 Problems; 6 DIGITAL SIGNATURES; 6.1 Hash Functions; 6.1.1 SHA-1; 6.1.2 MD5; 6.1.3 The Search for New Hash Functions; 6.1.4 Computer Examples; 6.1.5 Problems; 6.2 Digital Signature Schemes; 6.2.1 Direct Digital Signature Schemes; 6.2.2 Using RSA for Signing; 6.2.3 Using ElGamal for Signing; 6.2.4 Arbitrated Digital Signature Schemes; 6.2.5 The Digital Signature Standard; 6.2.6 Computer Examples In the past few years elliptic curve cryptography has moved from a fringe activity to a major system in the commercial world. This timely work summarizes knowledge gathered at Hewlett-Packard over a number of years and explains the mathematics behind practical implementations of elliptic curve systems. Since the mathematics is advanced, a high barrier to entry exists for individuals and companies new to this technology. Hence, this book will be invaluable not only to mathematicians but also to engineers and computer scientists who want to actually implement such systems. 4.1.2 Problems4.2 The RSA Algorithm; 4.2.1 The RSA Parameters; 4.2.2 Computer Examples; 4.2.3 Problems; 4.3 RSA Security; 4.3.1 Computer Examples; 4.3.2 Problems; 4.4 Implementing RSA; 4.4.1 Computer Examples; 4.4.2 Problems; 5 ELLIPTIC CURVE CRYPTOGRAPHY; 5.1 Elliptic Curves and Elliptic Curve Groups; 5.1.1 What is an Elliptic Curve?; 5.1.2 Operations on Points; 5.1.3 Adding a Point to Its Negative; 5.1.4 Adding a Point to Itself; 5.1.5 Computer Examples; 5.1.6 Problems; 5.2 Elliptic Curve Cryptography; 5.2.1 Elliptic Curve Groups and the Discrete Logarithm Problem