Finite Fields: Theory and Computation

1. Front Matter

   Pages i-xiv

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2. Introduction

   • Igor E. Shparlinski

   Pages 1-11

3. Links Flowchart

   • Igor E. Shparlinski

   Pages 13-15

4. Polynomial Factorization

   • Igor E. Shparlinski

   Pages 17-44

5. Finding Irreducible and Primitive Polynomials

   • Igor E. Shparlinski

   Pages 45-64

6. The Distribution of Irreducible, Primitive and Other Special Polynomials and Matrices

   • Igor E. Shparlinski

   Pages 65-98

7. Bases and Computation in Finite Fields

   • Igor E. Shparlinski

   Pages 99-148

8. Coding Theory and Algebraic Curves

   • Igor E. Shparlinski

   Pages 149-213

9. Elliptic Curves

   • Igor E. Shparlinski

   Pages 215-237

10. Recurrence Sequences in Finite Fields and Cyclic Linear Codes

    • Igor E. Shparlinski

    Pages 239-264

11. Finite Fields and Discrete Mathematics

    • Igor E. Shparlinski

    Pages 265-324

12. Congruences

    • Igor E. Shparlinski

    Pages 325-359

13. Some Related Problems

    • Igor E. Shparlinski

    Pages 361-401

14. Back Matter

    Pages 403-528

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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.

• Symbol
• algebra
• coding theory
• computer
• computer science
• cryptography
• finite field
• number theory
• theoretical computer science

`...the book can yet be highly recommended as encyclopaedia on these topics, in particular because many Russian papers are incorporated which are hardly accessible.'
Monatschafte für Mathematik, 132:1 (2001)

• School of Mathematics, Physics, Computing and Electronics, Macquarie University, Sydney, Australia

  Igor E. Shparlinski

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