Elliptic Curves

Elliptic Curves: Number Theory and Cryptography, Second Edition

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Features

  • Explores how elliptic curves are used in cryptography and number theory
  • Requires no background in algebraic geometry
  • Discusses both the Weil and Tate–Lichtenbaum pairings
  • Covers elliptic curve cryptosystems, such as pairing-based encryption
  • Introduces three popular packages (Pari, Magma, and Sage) for computing with elliptic curves
  • Summary

    Like its bestselling predecessor, Elliptic Curves: Number Theory and Cryptography, Second Edition develops the theory of elliptic curves to provide a basis for both number theoretic and cryptographic applications. With additional exercises, this edition offers more comprehensive coverage of the fundamental theory, techniques, and applications of elliptic curves.

    New to the Second Edition

  • Chapters on isogenies and hyperelliptic curves
  • A discussion of alternative coordinate systems, such as projective, Jacobian, and Edwards coordinates, along with related computational issues
  • A more complete treatment of the Weil and Tate–Lichtenbaum pairings
  • Doud’s analytic method for computing torsion on elliptic curves over Q
  • An explanation of how to perform calculations with elliptic curves in several popular computer algebra systems
  • Taking a basic approach to elliptic curves, this accessible book prepares readers to tackle more advanced problems in the field. It introduces elliptic curves over finite fields early in the text, before moving on to interesting applications, such as cryptography, factoring, and primality testing. The book also discusses the use of elliptic curves in Fermat’s Last Theorem. Relevant abstract algebra material on group theory and fields can be found in the appendices.

    Table of Contents

    INTRODUCTION
    THE BASIC THEORY
    Weierstrass Equations
    The Group Law
    Projective Space and the Point at Infinity
    Proof of Associativity
    Other Equations for Elliptic Curves
    Other Coordinate Systems
    The j-Invariant
    Elliptic Curves in Characteristic 2
    Endomorphisms
    Singular Curves
    Elliptic Curves mod n
    TORSION POINTS
    Torsion Points
    Division Polynomials
    The Weil Pairing
    The Tate–Lichtenbaum Pairing
    Elliptic Curves over Finite Fields
    Examples
    The Frobenius Endomorphism
    Determining the Group Order
    A Family of Curves
    Schoof’s Algorithm
    Supersingular Curves
    The Discrete Logarithm Problem
    The Index Calculus
    General Attacks on Discrete Logs
    Attacks with Pairings
    Anomalous Curves
    Other Attacks
    Elliptic Curve Cryptography
    The Basic Setup
    Diffie–Hellman Key Exchange
    Massey–Omura Encryption
    ElGamal Public Key Encryption
    ElGamal Digital Signatures
    The Digital Signature Algorithm
    ECIES
    A Public Key Scheme Based on Factoring
    A Cryptosystem Based on the Weil Pairing
    Other Applications
    Factoring Using Elliptic Curves
    Primality Testing
    Elliptic Curves over Q
    The Torsion Subgroup: The Lutz–Nagell Theorem
    Descent and the Weak Mordell–Weil Theorem
    Heights and the Mordell–Weil Theorem
    Examples
    The Height Pairing
    Fermat’s Infinite Descent
    2-Selmer Groups; Shafarevich–Tate Groups
    A Nontrivial Shafarevich–Tate Group
    Galois Cohomology
    Elliptic Curves over C
    Doubly Periodic Functions
    Tori Are Elliptic Curves
    Elliptic Curves over C
    Computing Periods
    Division Polynomials
    The Torsion Subgroup: Doud’s Method
    Complex Multiplication
    Elliptic Curves over C
    Elliptic Curves over Finite Fields
    Integrality of j-Invariants
    Numerical Examples
    Kronecker’s Jugendtraum
    DIVISORS
    Definitions and Examples
    The Weil Pairing
    The Tate–Lichtenbaum Pairing
    Computation of the Pairings
    Genus One Curves and Elliptic Curves
    Equivalence of the Definitions of the Pairings
    Nondegeneracy of the Tate–Lichtenbaum Pairing
    ISOGENIES
    The Complex Theory
    The Algebraic Theory
    Vélu’s Formulas
    Point Counting
    Complements
    Hyperelliptic Curves
    Basic Definitions
    Divisors
    Cantor’s Algorithm
    The Discrete Logarithm Problem
    Zeta Functions
    Elliptic Curves over Finite Fields
    Elliptic Curves over Q
    Fermat’s Last Theorem
    Overview
    Galois Representations
    Sketch of Ribet’s Proof
    Sketch of Wiles’s Proof
    APPENDIX A: NUMBER THEORY
    APPENDIX B: GROUPS
    APPENDIX C: FIELDS
    APPENDIX D: COMPUTER packages
    REFERENCES
    INDEX
    Exercises appear at the end of each chapter.

    Editorial Reviews

    … the book is well structured and does not waste the reader’s time in dividing cryptography from number theory-only information. This enables the reader just to pick the desired information. … a very comprehensive guide on the theory of elliptic curves. … I can recommend this book for both cryptographers and mathematicians doing either their Ph.D. or Master’s … I enjoyed reading and studying this book and will be glad to have it as a future reference.
    —IACR book reviews, April 2010

    Praise for the First Edition
    There are already a number of books about elliptic curves, but this new offering by Washington is definitely among the best of them. It gives a rigorous though relatively elementary development of the theory of elliptic curves, with emphasis on those aspects of the theory most relevant for an understanding of elliptic curve cryptography. … an excellent companion to the books of Silverman and Blake, Seroussi and Smart. It would be a fine asset to any library or collection.
    Mathematical Reviews, Issue 2004e

    Washington … has found just the right level of abstraction for a first book … . Notably, he offers the most lucid and concrete account ever of the perpetually mysterious Shafarevich–Tate group. A pleasure to read! Summing Up: Highly recommended.
    CHOICE, March 2004

    … a nice, relatively complete, elementary account of elliptic curves.
    Bulletin of the AMS

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