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Applications of Finite Fields : Recent Economic Thought - A. J. Menezes

Applications of Finite Fields

Recent Economic Thought

Hardcover

Published: 30th November 1992
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The theory of finite fields, whose origins can be traced back to the works of Gauss and Galois, has played a part in various branches of mathematics, in recent years there has been a resurgence of interest in finite fields, and this is partly due to important applications in coding theory and cryptography. Applications of Finite Fields introduces some of these recent developments.
This book focuses attention on some specific recent developments in the theory and applications of finite fields. While the topics selected are treated in some depth, Applications of Finite Fields does not attempt to be encyclopedic. Among the topics studied are different methods of representing the elements of a finite field (including normal bases and optimal normal bases), algorithms for factoring polynomials over finite fields, methods for constructing irreducible polynomials, the discrete logarithm problem and its implications to cryptography, the use of elliptic curves in constructing public key cryptosystems, and the uses of algebraic geometry in constructing good error-correcting codes.
This book is developed from a seminar held at the University of Waterloo. The purpose of the seminar was to bridge the knowledge of the participants whose expertise and interests ranged from the purely theoretical to the applied. As a result, this book will be of interest to a wide range of students, researchers and practitioners in the disciplines of computer science, engineering and mathematics.
Applications of Finite Fields is an excellent reference and may be used as a text for a course on the subject.

Preface
Acknowledgements
Introduction to Finite Fields and Basesp. 1
Bases
The Enumeration of Bases
Applications
Factoring Polynomials over Finite Fieldsp. 17
A Few Basics
Root Finding
Factoring
Factoring Multivariate Polynomials
Construction of Irreducible Polynomialsp. 39
Specific Irreducible Polynomials
Irreducibility of Compositions of Polynomials
Recursive Constructions
Composed Product of Irreducible Polynomials
A General Approach
Normal Basesp. 69
Some Properties of Normal bases
Distribution of Normal Elements
Characterization of N-Polynomials
Construction of Normal Bases
Optimal Normal Basesp. 93
Constructions
Determination of all Optimal Normal Bases
An Open Problem
The Discrete Logarithm Problemp. 115
Applications
The Discrete Logarithm Problem: General Remarks
Square Root Methods
The Pohlig-Hellman Method
The Index Calculus Method
Best Algorithms
Computational Results
Discrete Logarithms and Factoring
Elliptic Curves over Finite Fieldsp. 139
Group Law
The Discriminant and j-Invariant
Curves over K, char(K) [actual symbol not reproducible] 2, 3
Curves over K, char(K) = 2
Group Structure
Supersingular Curves
Elliptic Curve Cryptosystemsp. 151
Singular Elliptic Curves
The Elliptic Curve Logarithm Problem
Implementation
Introduction to Algebraic Geometryp. 173
Affine Varieties
Plane Curves
Projective Varieties
Projective Plane Curves
Dimension of X
Divisors on X
Differentials on X
Algebraic Curves over a Finite Field
Codes From Algebraic Geometryp. 191
Hermitian Codes
Codes From Elliptic Curves
Codes From Elliptic Curves over F[subscript 2m]
Decoding Algebraic Geometric Codes
Appendix - Other Applicationsp. 211
Indexp. 215
Table of Contents provided by Blackwell. All Rights Reserved.

ISBN: 9780792392828
ISBN-10: 0792392825
Series: Recent Economic Thought
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 218
Published: 30th November 1992
Publisher: SPRINGER VERLAG GMBH
Country of Publication: NL
Dimensions (cm): 24.23 x 16.46  x 1.73
Weight (kg): 0.47