Computational and Algorithmic Problems in Finite Fields

by Igor Shparlinski

Overview

'Et moi, ...- si j'avait su comment en revenir. je One service mathematics bas rendemI !be n'y semis point a1J6.' human race. It bas put common sense back JulesVeme where it belongs. on tile topmost sbelf next to tile dusty canister labelled 'discarded nonsense'. The series is divergent; therefore we may be Eric T.BeIl able to do something with il O. Heaviside Mathematics is a tool for thought. A highly necessary tool in a world where both feedback and nonlineari- ties abound. Similarly, all kinds of pans of mathematics serve as tools for other pans and for other sci- ences. Applying a simple rewriting rule to the quote on the right above one finds such statements as: 'One ser- vice topology has rendered mathematical physics ... '; 'One service logic has rendered computer science ... '; 'One service category theory has rendered mathematics .. .'. All arguably true. And all statements obtainable this way fonn pan of the raison d' 8tre of this series.

Product Details

ISBN-13:	9789401047968
Publisher:	Springer Netherlands
Publication date:	10/29/2012
Series:	Mathematics and its Applications , #88
Edition description:	1992
Pages:	240
Product dimensions:	6.30(w) x 9.45(h) x 0.02(d)

Table of Contents

1. Polynomial Factorization.- 1. Univariate factorization.- 2. Multivariate factorization.- 3. Other polynomial decompositions.- 2. Finding irreducible and primitive polynomials.- 1. Construction of irreducible polynomials.- 2. Construction of primitive polynomials.- 3. The distribution of irreducible and primitive polynomials.- 1. Distribution of irreducible and primitive polynomials.- 2. Irreducible and primitive polynomials of a given height and weight.- 3. Sparse polynomials.- 4. Applications to algebraic number fields.- 4. Bases and computation in finite fields.- 1. Construction of some special bases for finite fields.- 2. Discrete logarithm and Zech’s logarithm.- 3. Polynomial multiplication and multiplicative complexity in finite fields.- 4. Other algorithms in finite fields.- 5. Coding theory and algebraic curves.- 1. Codes and points on algebraic curves.- 2. Codes and exponential sums.- 3. Codes and lattice packings and coverings.- 6. Elliptic curves.- 1. Some general properties.- 2. Distribution of primitive points on elliptic curves.- 7. Recurrent sequences in finite fields and leyelic linear codes.- 1. Distribution of values of recurrent sequences.- 2. Applications of recurrent sequences.- 3. Cyclic codes and recurrent sequences.- 8. Finite fields and discrete mathematics.- 1. Cryptography and permutation polynomials.- 2. Graph theory, combinatorics, Boolean functions.- 3. Enumeration problems in finite fields.- 9. Congruences.- 1. Optimal coefficients and pseudo-random numbers.- 2. Residues of exponential functions.- 3. Modular arithmetic.- 4. Other applications.- 10. Some related problems.- 1. Integer factorization, primality testing and the greatest common divisor.- 2. Computational algebraic number theory.- 3. Algebraic complexity theory.- 4.Polynomials with integer coefficients.- Appendix 1.- Appendix 2.- Appendix 3.- Addendum.- References.