Ebook: Finite Fields: Theory and Computation: The Meeting Point of Number Theory, Computer Science, Coding Theory and Cryptography

This book is mainly devoted to some computational and algorithmic problems in finite fields such as, for example, polynomial factorization, finding irreducible and primitive polynomials, the distribution of these primitive polynomials and of primitive points on elliptic curves, constructing bases of various types and new applications of finite fields to other areas of mathematics. For completeness we in­ clude two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number gener­ ators, modular arithmetic, etc.) and computational number theory (primality testing, factoring integers, computation in algebraic number theory, etc.). The problems considered here have many applications in Computer Science, Cod­ ing Theory, Cryptography, Numerical Methods, and so on. There are a few books devoted to more general questions, but the results contained in this book have not till now been collected under one cover. In the present work the author has attempted to point out new links among different areas of the theory of finite fields. It contains many very important results which previously could be found only in widely scattered and hardly available conference proceedings and journals. In particular, we extensively review results which originally appeared only in Russian, and are not well known to mathematicians outside the former USSR.

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This book provides an exhaustive survey of the most recent achievements in the theory and applications of finite fields and in many related areas such as algebraic number theory, theoretical computer science, coding theory and cryptography. Topics treated include polynomial factorization over finite fields, the finding and distribution of irreducible primitive and other special polynomials, constructing special bases of extensions of finite fields, curves and exponential sums, and linear recurrent sequences. Besides a general overview of the area, its results and methods, it suggests a number of interesting research problems of various levels of difficulty. The volume concludes with an impressive bibliographical section containing more than 2300 references.
Audience: This work will be of interest to graduate students and researchers in field theory and polynomials, number theory, symbolic computation, symbolic/algebraic manipulation, and coding theory.

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This book provides an exhaustive survey of the most recent achievements in the theory and applications of finite fields and in many related areas such as algebraic number theory, theoretical computer science, coding theory and cryptography. Topics treated include polynomial factorization over finite fields, the finding and distribution of irreducible primitive and other special polynomials, constructing special bases of extensions of finite fields, curves and exponential sums, and linear recurrent sequences. Besides a general overview of the area, its results and methods, it suggests a number of interesting research problems of various levels of difficulty. The volume concludes with an impressive bibliographical section containing more than 2300 references.
Audience: This work will be of interest to graduate students and researchers in field theory and polynomials, number theory, symbolic computation, symbolic/algebraic manipulation, and coding theory.
Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-11
Links Flowchart....Pages 13-15
Polynomial Factorization....Pages 17-44
Finding Irreducible and Primitive Polynomials....Pages 45-64
The Distribution of Irreducible, Primitive and Other Special Polynomials and Matrices....Pages 65-98
Bases and Computation in Finite Fields....Pages 99-148
Coding Theory and Algebraic Curves....Pages 149-213
Elliptic Curves....Pages 215-237
Recurrence Sequences in Finite Fields and Cyclic Linear Codes....Pages 239-264
Finite Fields and Discrete Mathematics....Pages 265-324
Congruences....Pages 325-359
Some Related Problems....Pages 361-401
Back Matter....Pages 403-528

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This book provides an exhaustive survey of the most recent achievements in the theory and applications of finite fields and in many related areas such as algebraic number theory, theoretical computer science, coding theory and cryptography. Topics treated include polynomial factorization over finite fields, the finding and distribution of irreducible primitive and other special polynomials, constructing special bases of extensions of finite fields, curves and exponential sums, and linear recurrent sequences. Besides a general overview of the area, its results and methods, it suggests a number of interesting research problems of various levels of difficulty. The volume concludes with an impressive bibliographical section containing more than 2300 references.
Audience: This work will be of interest to graduate students and researchers in field theory and polynomials, number theory, symbolic computation, symbolic/algebraic manipulation, and coding theory.
Content:
Front Matter....Pages i-xiv
Introduction....Pages 1-11
Links Flowchart....Pages 13-15
Polynomial Factorization....Pages 17-44
Finding Irreducible and Primitive Polynomials....Pages 45-64
The Distribution of Irreducible, Primitive and Other Special Polynomials and Matrices....Pages 65-98
Bases and Computation in Finite Fields....Pages 99-148
Coding Theory and Algebraic Curves....Pages 149-213
Elliptic Curves....Pages 215-237
Recurrence Sequences in Finite Fields and Cyclic Linear Codes....Pages 239-264
Finite Fields and Discrete Mathematics....Pages 265-324
Congruences....Pages 325-359
Some Related Problems....Pages 361-401
Back Matter....Pages 403-528
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