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Computational Number Theory and Modern Cryptography

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The only book to provide a unified view of the interplay between computational number theory and cryptography

Computational number theory and modern cryptography are two of the most important and fundamental research fields in information security. In this book, Song Y. Yang combines knowledge of these two critical fields, providing a unified view of the relationships between computational number theory and cryptography. The author takes an innovative approach, presenting mathematical ideas first, thereupon treating cryptography as an immediate application of the mathematical concepts. The book also presents topics from number theory, which are relevant for applications in public-key cryptography, as well as modern topics, such as coding and lattice based cryptography for post-quantum cryptography. The author further covers the current research and applications for common cryptographic algorithms, describing the mathematical problems behind these applications in a manner accessible to computer scientists and engineers.

  • Makes mathematical problems accessible to computer scientists and engineers by showing their immediate application
  • Presents topics from number theory relevant for public-key cryptography applications
  • Covers modern topics such as coding and lattice based cryptography for post-quantum cryptography
  • Starts with the basics, then goes into applications and areas of active research
  • Geared at a global audience; classroom tested in North America, Europe, and Asia
  • Incudes exercises in every chapter
  • Instructor resources available on the book’s Companion Website

Computational Number Theory and Modern Cryptography is ideal for graduate and advanced undergraduate students in computer science, communications engineering, cryptography and mathematics. Computer scientists, practicing cryptographers, and other professionals involved in various security schemes will also find this book to be a helpful reference.

Author(s): Song Y. Yan
Series: Information Security (Wiley)
Edition: 1
Publisher: Wiley
Year: 2013

Language: English
Pages: 433



COMPUTATIONAL NUMBER THEORY AND MODERN CRYPTOGRAPHY



ISBN: 9781118188583



CONTENTS



ABOUT THE AUTHOR



PREFACE



ACKNOWLEDGMENTS



Part I Preliminaries



  1 Introduction

      1.1 What is Number Theory?

      1.2 What is Computation Theory?

      1.3 What is Computational Number Theory?

      1.4 What is Modern Cryptography?

      1.5 Bibliographic Notes and Further Reading

  2 Fundamentals

      2.1 Basic Algebraic Structures

      2.2 Divisibility Theory

      2.3 Arithmetic Functions

      2.4 Congruence Theory

      2.5 Primitive Roots

      2.6 Elliptic Curves

      2.7 Bibliographic Notes and Further Reading



Part II Computational Number Theory



  3 Primality Testing

      3.1 Basic Tests

      3.2 Miller...Rabin Test

      3.3 Elliptic Curve Tests

      3.4 AKS Test

      3.5 Bibliographic Notes and Further Reading

  4 Integer Factorization

      4.1 Basic Concepts

      4.2 Trial Divisions Factoring

      4.3 . and p 1 Methods

      4.4 Elliptic Curve Method

      4.5 Continued Fraction Method

      4.6 Quadratic Sieve

      4.7 Number Field Sieve

      4.8 Bibliographic Notes and Further Reading

  5 Discrete Logarithms

      5.1 Basic Concepts

      5.2 Baby-Step Giant-Step Method

      5.3 Pohlig...Hellman Method

      5.4 Index Calculus

      5.5 Elliptic Curve Discrete Logarithms

      5.6 Bibliographic Notes and Further Reading



Part III Modern Cryptography



  6 Secret-Key Cryptography

      6.1 Cryptography and Cryptanalysis

      6.2 Classic Secret-Key Cryptography

      6.3 Modern Secret-Key Cryptography

      6.4 Bibliographic Notes and Further Reading

  7 Integer Factorization Based Cryptography

      7.1 RSA Cryptography

      7.2 Cryptanalysis of RSA

      7.3 Rabin Cryptography

      7.4 Residuosity Based Cryptography

      7.5 Zero-Knowledge Proof

      7.6 Bibliographic Notes and Further Reading

  8 Discrete Logarithm Based Cryptography

      8.1 Dif"e...Hellman...Merkle Key-Exchange Protocol

      8.2 ElGamal Cryptography

      8.3 Massey...Omura Cryptography

      8.4 DLP-Based Digital Signatures

      8.5 Bibliographic Notes and Further Reading

  9 Elliptic Curve Discrete Logarithm Based Cryptography

      9.1 Basic Ideas

      9.2 Elliptic Curve Dif"e...Hellman...Merkle Key Exchange Scheme

      9.3 Elliptic Curve Massey...Omura Cryptography

      9.4 Elliptic Curve ElGamal Cryptography

      9.5 Elliptic Curve RSA Cryptosystem

      9.6 Menezes...Vanstone Elliptic Curve Cryptography

      9.7 Elliptic Curve DSA

      9.8 Bibliographic Notes and Further Reading



Part IV Quantum Resistant Cryptography



  10 Quantum Computational Number Theory

      10.1 Quantum Algorithms for Order Finding

      10.2 Quantum Algorithms% for Integer% Factorization

      10.3 Quantum Algorithms for Discrete Logarithms

      10.4 Quantum Algorithms for Elliptic Curve Discrete Logarithms

      10.5 Bibliographic Notes and Further Reading

  11 Quantum Resistant Cryptography

      11.1 Coding-Based Cryptography

      11.2 Lattice-Based Cryptography

      11.3 Quantum Cryptography

      11.4 DNA Biological Cryptography

      11.5 Bibliographic Notes and Further Reading



INDEX