Computational and Algorithmic Problems in Finite Fields (Mathematics and its Applications, 88)

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. Read more... Abstract: 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 Front Matter....Pages i-xii Introduction....Pages 1-6 Polynomial Factorization....Pages 7-20 Finding Irreducible and Primitive Polynomials....Pages 21-29 The Distribution of Irreducible and Primitive Polynomials....Pages 30-48 Bases and Computation in Finite Fields....Pages 49-71 Coding Theory and Algebraic Curves....Pages 72-98 Elliptic Curves....Pages 99-108 Recurrent Sequences in Finite Fields and Cyclic Linear Codes....Pages 109-121 Finite Fields and Discrete Mathematics....Pages 122-138 Congruences....Pages 139-152 Some Related Problems....Pages 153-160 Back Matter....Pages 161-240