Elliptic Curves: Number Theory and Cryptography / Edition 2

Hardcover (Print)
Used and New from Other Sellers
Used and New from Other Sellers
from $92.35
Usually ships in 1-2 business days
(Save 16%)
Other sellers (Hardcover)
  • All (8) from $92.35   
  • New (3) from $99.28   
  • Used (5) from $92.35   


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.

Read More Show Less

Editorial Reviews

From the Publisher
… 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

Read More Show Less

Product Details

  • ISBN-13: 9781420071467
  • Publisher: Taylor & Francis
  • Publication date: 4/2/2008
  • Series: Discrete Mathematics and Its Applications Series , #43
  • Edition description: REV
  • Edition number: 2
  • Pages: 536
  • Sales rank: 1,134,849
  • Product dimensions: 6.10 (w) x 9.30 (h) x 1.30 (d)

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.

Read More Show Less

Customer Reviews

Be the first to write a review
( 0 )
Rating Distribution

5 Star


4 Star


3 Star


2 Star


1 Star


Your Rating:

Your Name: Create a Pen Name or

Barnes & Noble.com Review Rules

Our reader reviews allow you to share your comments on titles you liked, or didn't, with others. By submitting an online review, you are representing to Barnes & Noble.com that all information contained in your review is original and accurate in all respects, and that the submission of such content by you and the posting of such content by Barnes & Noble.com does not and will not violate the rights of any third party. Please follow the rules below to help ensure that your review can be posted.

Reviews by Our Customers Under the Age of 13

We highly value and respect everyone's opinion concerning the titles we offer. However, we cannot allow persons under the age of 13 to have accounts at BN.com or to post customer reviews. Please see our Terms of Use for more details.

What to exclude from your review:

Please do not write about reviews, commentary, or information posted on the product page. If you see any errors in the information on the product page, please send us an email.

Reviews should not contain any of the following:

  • - HTML tags, profanity, obscenities, vulgarities, or comments that defame anyone
  • - Time-sensitive information such as tour dates, signings, lectures, etc.
  • - Single-word reviews. Other people will read your review to discover why you liked or didn't like the title. Be descriptive.
  • - Comments focusing on the author or that may ruin the ending for others
  • - Phone numbers, addresses, URLs
  • - Pricing and availability information or alternative ordering information
  • - Advertisements or commercial solicitation


  • - By submitting a review, you grant to Barnes & Noble.com and its sublicensees the royalty-free, perpetual, irrevocable right and license to use the review in accordance with the Barnes & Noble.com Terms of Use.
  • - Barnes & Noble.com reserves the right not to post any review -- particularly those that do not follow the terms and conditions of these Rules. Barnes & Noble.com also reserves the right to remove any review at any time without notice.
  • - See Terms of Use for other conditions and disclaimers.
Search for Products You'd Like to Recommend

Recommend other products that relate to your review. Just search for them below and share!

Create a Pen Name

Your Pen Name is your unique identity on BN.com. It will appear on the reviews you write and other website activities. Your Pen Name cannot be edited, changed or deleted once submitted.

Your Pen Name can be any combination of alphanumeric characters (plus - and _), and must be at least two characters long.

Continue Anonymously

    If you find inappropriate content, please report it to Barnes & Noble
    Why is this product inappropriate?
    Comments (optional)