Elliptic Curves
Number Theory and Cryptography
1st Edition
Published on 28. May 2003
Book
Hardback
440 pages
978-1-58488-365-4 (ISBN)
Description
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.
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.
More details
Series
Language
English
Place of publication
United Kingdom
Publishing group
Taylor & Francis Ltd
Target group
College/higher education
Senior-level undergraduate and graduate students in mathematics and computer science
Illustrations
21 s/w Abbildungen
1000 equations; 21 Illustrations, black and white
Dimensions
Height: 235 mm
Width: 156 mm
Weight
748 gr
ISBN-13
978-1-58488-365-4 (9781584883654)
Copyright in bibliographic data and cover images is held by Nielsen Book Services Limited or by the publishers or by their respective licensors: all rights reserved.
Schweitzer Classification
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Chapman & Hall/CRC
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Content
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' s Proof
APPENDICES
Number Theory
Groups
Fields
REFERENCES
INDEX
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' s Proof
APPENDICES
Number Theory
Groups
Fields
REFERENCES
INDEX