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# Elliptic Curves: Number Theory and Cryptography / Edition 1

Elliptic Curves: Number Theory and Cryptography / Edition 1 available in Hardcover, NOOK Book

**Temporarily Out of Stock Online**

## Overview

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

## Product Details

ISBN-13: | 9781584883654 |
---|---|

Publisher: | Taylor & Francis |

Publication date: | 03/15/2003 |

Series: | Discrete Mathematics and Its Applications Series |

Edition description: | Older Edition |

Pages: | 440 |

Product dimensions: | 6.12(w) x 9.25(h) x 1.10(d) |

## Read an Excerpt

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

## First Chapter

Elliptic curves have played an increasingly important role in number theory and related fields over the last several decades, most notably in areas such as cryptography, factorization, and the proof of Fermat's Last Theorem. However, most books on the subject assume a rather high level of mathematical sophistication, and few are truly accessible to senior undergraduate or beginning graduate students. Assuming only a modest background in elementary number theory, groups, and fields, Elliptic Curves: Number Theory and Cryptography introduces both the cryptographic and number theoretic sides of elliptic curves, interweaving the theory of elliptic curves with their applications. The author introduces elliptic curves over finite fields early in the treatment, leading readers directly to the intriguing cryptographic applications, but the book is structured so that readers can explore the number theoretic aspects independently if desired. By side-stepping algebraic geometry in favor an approach based on basic formulas, this book clearly demonstrates how elliptic curves are used and opens the doors to higher-level studies. Elliptic Curves offers a solid introduction to the mathematics and applications of elliptic curves that well prepares its readers to tackle more advanced problems in cryptography and number theory.

## Table of Contents

Exercises

THE BASIC THEORY

Weierstrass Equations

The Group Law

Projective Space and the Point at Infinity

Proof of Associativity

Other Equations for Elliptic Curves

The j-Invariant

Elliptic Curves in Characteristic

Endomorphisms

Singular Curves

Elliptic Curves mod n

Exercises

TORSION POINTS

Torsion Points

Division Polynomials

The Weil Pairing

Exercises

ELLIPTIC CURVES OVER FINITE FIELDS

Examples

The Frobenius Endomorphism

Determining the Group Order

A Family of Curves

Schoof's Algorithm

Supersingular Curves

Exercises

THE DISCRETE LOGARITHM PROBLEM

The Index Calculus

General Attacks on Discrete Logs

The MOV Attack

Anomalous Curves

The Tate-Lichtenbaum Pairing

Other Attacks

Exercises

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

Exercises

OTHER APPLICATIONS

Factoring Using Elliptic Curves

Primality Testing

Exercises

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

Exercises

ELLIPTIC CURVES OVER C

Doubly Periodic Functions

Tori are Elliptic Curves

Elliptic Curves over C

Computing Periods

Division Polynomials

Exercises

COMPLEX MULTIPLICATION

Elliptic Curves over C

Elliptic Curves over Finite Fields

Integrality of j-Invariants

A Numerical Example

Kronecker's Jugendtraum

Exercises

DIVISORS

Definitions and Examples

The Weil Pairing

The Tate-Lichtenbaum Pairing

Computation of the Pairings

Genus One Curves and Elliptic Curves

Exercises

ZETA FUNCTIONS

Elliptic Curves over Finite Fields

Elliptic Curves over Q

Exercises

FERMAT'S LAST THEOREM

Overview

Galois Representations

Sketch of Ribet's Proof

Sketch of Wiles' Proof

APPENDICES

Number Theory

Groups

Fields

REFERENCES

INDEX

## Reading Group Guide

INTRODUCTION

Exercises

THE BASIC THEORY

Weierstrass Equations

The Group Law

Projective Space and the Point at Infinity

Proof of Associativity

Other Equations for Elliptic Curves

The j-Invariant

Elliptic Curves in Characteristic

Endomorphisms

Singular Curves

Elliptic Curves mod n

Exercises

TORSION POINTS

Torsion Points

Division Polynomials

The Weil Pairing

Exercises

ELLIPTIC CURVES OVER FINITE FIELDS

Examples

The Frobenius Endomorphism

Determining the Group Order

A Family of Curves

Schoof's Algorithm

Supersingular Curves

Exercises

THE DISCRETE LOGARITHM PROBLEM

The Index Calculus

General Attacks on Discrete Logs

The MOV Attack

Anomalous Curves

The Tate-Lichtenbaum Pairing

Other Attacks

Exercises

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

Exercises

OTHER APPLICATIONS

Factoring Using Elliptic Curves

Primality Testing

Exercises

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

Exercises

ELLIPTIC CURVES OVER C

Doubly Periodic Functions

Tori are Elliptic Curves

Elliptic Curves over C

Computing Periods

Division Polynomials

Exercises

COMPLEX MULTIPLICATION

Elliptic Curves over C

Elliptic Curves over Finite Fields

Integrality of j-Invariants

A Numerical Example

Kronecker's Jugendtraum

Exercises

DIVISORS

Definitions and Examples

The Weil Pairing

The Tate-Lichtenbaum Pairing

Computation of the Pairings

Genus One Curves and Elliptic Curves

Exercises

ZETA FUNCTIONS

Elliptic Curves over Finite Fields

Elliptic Curves over Q

Exercises

FERMAT'S LAST THEOREM

Overview

Galois Representations

Sketch of Ribet's Proof

Sketch of Wiles' Proof

APPENDICES

Number Theory

Groups

Fields

REFERENCES

INDEX

## Interviews

INTRODUCTION

Exercises

THE BASIC THEORY

Weierstrass Equations

The Group Law

Projective Space and the Point at Infinity

Proof of Associativity

Other Equations for Elliptic Curves

The j-Invariant

Elliptic Curves in Characteristic

Endomorphisms

Singular Curves

Elliptic Curves mod n

Exercises

TORSION POINTS

Torsion Points

Division Polynomials

The Weil Pairing

Exercises

ELLIPTIC CURVES OVER FINITE FIELDS

Examples

The Frobenius Endomorphism

Determining the Group Order

A Family of Curves

Schoof's Algorithm

Supersingular Curves

Exercises

THE DISCRETE LOGARITHM PROBLEM

The Index Calculus

General Attacks on Discrete Logs

The MOV Attack

Anomalous Curves

The Tate-Lichtenbaum Pairing

Other Attacks

Exercises

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

Exercises

OTHER APPLICATIONS

Factoring Using Elliptic Curves

Primality Testing

Exercises

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

Exercises

ELLIPTIC CURVES OVER C

Doubly Periodic Functions

Tori are Elliptic Curves

Elliptic Curves over C

Computing Periods

Division Polynomials

Exercises

COMPLEX MULTIPLICATION

Elliptic Curves over C

Elliptic Curves over Finite Fields

Integrality of j-Invariants

A Numerical Example

Kronecker's Jugendtraum

Exercises

DIVISORS

Definitions and Examples

The Weil Pairing

The Tate-Lichtenbaum Pairing

Computation of the Pairings

Genus One Curves and Elliptic Curves

Exercises

ZETA FUNCTIONS

Elliptic Curves over Finite Fields

Elliptic Curves over Q

Exercises

FERMAT'S LAST THEOREM

Overview

Galois Representations

Sketch of Ribet's Proof

Sketch of Wiles' Proof

APPENDICES

Number Theory

Groups

Fields

REFERENCES

INDEX

## Recipe

Exercises

THE BASIC THEORY

Weierstrass Equations

The Group Law

Projective Space and the Point at Infinity

Proof of Associativity

Other Equations for Elliptic Curves

The j-Invariant

Elliptic Curves in Characteristic

Endomorphisms

Singular Curves

Elliptic Curves mod n

Exercises

TORSION POINTS

Torsion Points

Division Polynomials

The Weil Pairing

Exercises

ELLIPTIC CURVES OVER FINITE FIELDS

Examples

The Frobenius Endomorphism

Determining the Group Order

A Family of Curves

Schoof's Algorithm

Supersingular Curves

Exercises

THE DISCRETE LOGARITHM PROBLEM

The Index Calculus

General Attacks on Discrete Logs

The MOV Attack

Anomalous Curves

The Tate-Lichtenbaum Pairing

Other Attacks

Exercises

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

Exercises

OTHER APPLICATIONS

Factoring Using Elliptic Curves

Primality Testing

Exercises

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

Exercises

ELLIPTIC CURVES OVER C

Doubly Periodic Functions

Tori are Elliptic Curves

Elliptic Curves over C

Computing Periods

Division Polynomials

Exercises

COMPLEX MULTIPLICATION

Elliptic Curves over C

Elliptic Curves over Finite Fields

Integrality of j-Invariants

A Numerical Example

Kronecker's Jugendtraum

Exercises

DIVISORS

Definitions and Examples

The Weil Pairing

The Tate-Lichtenbaum Pairing

Computation of the Pairings

Genus One Curves and Elliptic Curves

Exercises

ZETA FUNCTIONS

Elliptic Curves over Finite Fields

Elliptic Curves over Q

Exercises

FERMAT'S LAST THEOREM

Overview

Galois Representations

Sketch of Ribet's Proof

Sketch of Wiles' Proof

APPENDICES

Number Theory

Groups

Fields

REFERENCES

INDEX