Elliptic Curves: Number Theory and Cryptography, Second Edition, 2nd Edition (Hardback) book cover

Elliptic Curves

Number Theory and Cryptography, Second Edition, 2nd Edition

By Lawrence C. Washington

Chapman and Hall/CRC

531 pages | 20 B/W Illus.

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Hardback: 9781420071467
pub: 2008-04-03
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pub: 2008-04-03
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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.


    … 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

    Table of Contents



    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


    Singular Curves

    Elliptic Curves mod n


    Torsion Points

    Division Polynomials

    The Weil Pairing

    The Tate–Lichtenbaum Pairing

    Elliptic Curves over Finite Fields


    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


    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


    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


    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


    The Complex Theory

    The Algebraic Theory

    Vélu’s Formulas

    Point Counting


    Hyperelliptic Curves

    Basic Definitions


    Cantor’s Algorithm

    The Discrete Logarithm Problem

    Zeta Functions

    Elliptic Curves over Finite Fields

    Elliptic Curves over Q

    Fermat’s Last Theorem


    Galois Representations

    Sketch of Ribet’s Proof

    Sketch of Wiles’s Proof







    Exercises appear at the end of each chapter.

    About the Series

    Discrete Mathematics and Its Applications

    Learn more…

    Subject Categories

    BISAC Subject Codes/Headings:
    COMPUTERS / Security / General
    MATHEMATICS / Number Theory
    MATHEMATICS / Combinatorics