Computational and Algorithmic Problems in Finite Fields

By: Igor E. Shparlinski

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256 Pages

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This volume presents an exhaustive treatment of computation and algorithms for finite fields. Topics covered include polynomial factorization, finding irreducible and primitive polynomials, 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 araes of mathematics. For completeness, also included are two special chapters on some recent advances and applications of the theory of congruences (optimal coefficients, congruential pseudo-random number generators, modular arithmetic etc.), and computational number theory (primality testing, factoring integers, computing in algebraic number theory, etc.) The problems considered here have many applications in computer science, coding theory, cryptography, number theory and discrete mathematics. The level of discussion presuppose only a knowledge of the basic facts on finite fields, and the book can be recommended as supplementary graduate text. For researchers and students interested in computational and algorithmic problems in finite fields.

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Non-Fiction Mathematics Number Theory Algebra Mathematical Foundation Set Theory Computing & I.T.Computer Science Databases Data Capture & Analysis

Series Editor's Preface
Preface
Acknowledgements
Notations
Introduction	p. 1
Polynomial Factorization	p. 7
Univariate factorization	p. 7
Multivariate factorization	p. 16
Other polynomial decompositions	p. 20
Finding irreducible and primitive polynomials	p. 21
Construction of irreducible polynomials	p. 21
Construction of primitive polynomials	p. 27
The distribution of irreducible and primitive polynomials	p. 30
Distribution of irreducible and primitive polynomials	p. 30
Irreducible and primitive polynomials of a given height and weight	p. 42
Sparse polynomials	p. 46
Applications to algebraic number fields	p. 47
Bases and computation in finite fields	p. 49
Construction of some special bases for finite fields	p. 49
Discrete logarithm and Zech's logarithm	p. 54
Polynomial multiplication and multiplicative complexity in finite fields	p. 56
Other algorithms in finite fields	p. 64
Coding theory and algebraic curves	p. 72
Codes and points on algebraic curves	p. 72
Codes and exponential sums	p. 86
Codes and lattice packings and coverings	p. 92
Elliptic curves	p. 99
Some general properties	p. 99
Distribution of primitive points on elliptic curves	p. 105
Recurrent sequences in finite fields and cyclic linear codes	p. 109
Distribution of values of recurrent sequences	p. 109
Applications of recurrent sequences	p. 113
Cyclic codes and recurrent sequences	p. 116
Finite fields and discrete mathematics	p. 122
Cryptography and permutation polynomials	p. 122
Graph theory, combinatorics, Boolean functions	p. 129
Enumeration problems in finite fields	p. 136
Congruences	p. 139
Optimal coefficients and pseudo-random numbers	p. 139
Residues of exponential functions	p. 143
Modular arithmetic	p. 148
Other applications	p. 150
Some related problems	p. 153
Integer factorization, primality testing and the greatest common divisor	p. 153
Computational algebraic number theory	p. 155
Algebraic complexity theory	p. 156
Polynomials with integer coefficients	p. 158
Appendix 1	p. 161
Appendix 2	p. 164
Appendix 3	p. 165
Addendum	p. 166
References	p. 191
Index	p. 238
Table of Contents provided by Blackwell. All Rights Reserved.

Computational and Algorithmic Problems in Finite Fields

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Hardcover

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