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**Finite fields: theory and computation. The meeting point of number theory, computer science, coding theory and cryptography.**
*(English)*
Zbl 0967.11052

Mathematics and its Applications (Dordrecht). 477. Dordrecht: Kluwer Academic Publishers. xiv, 528 p. (1999).

The theory of finite fields has become increasingly important in the last twenty years. On the one hand there are classical algebraic and number theoretic problems related to finite fields and on the other hand finite fields have many modern applications in computer science, coding theory and cryptography. This excellent book surveys the most recent achievements in the theory and applications of finite fields and is not meant as an introduction. (For an introduction see the masterpiece of R. Lidl and H. Niederreiter [Finite fields, Encyclopedia of Mathematics and Its Applications. 20. Cambridge: Cambridge Univ. Press (1996; Zbl 0866.11069)].) The book is evidently an extension of the author’s book “Computational and algorithmic problems in finite fields” [Mathematics and Its Applications. Soviet Series. 88. Dordrecht: Kluwer Academic Publishers (1992; Zbl 0780.11064)]. The following table of contents can only give a glance of the importance of the book.

1. Polynomial factorization

2. Finding irreducible and primitive polynomials

3. The distribution of irreducible, primitive and other special polynomials and matrices

4. Bases and computations in finite fields

5. Coding theory and algebraic curves

6. Elliptic curves

7. Recurrence sequences in finite fields and cyclic linear codes

8. Finite fields and discrete mathematics

9. Congruences

10. Some related problems (primality testing, integer factorization, lattice basis reduction, algorithmic algebraic number theory, integer polynomials, algebraic complexity theory).

The book suggests numerous open problems and concludes with more than 3000 references. Consequently, it is essential for each researcher in finite field theory and related areas.

1. Polynomial factorization

2. Finding irreducible and primitive polynomials

3. The distribution of irreducible, primitive and other special polynomials and matrices

4. Bases and computations in finite fields

5. Coding theory and algebraic curves

6. Elliptic curves

7. Recurrence sequences in finite fields and cyclic linear codes

8. Finite fields and discrete mathematics

9. Congruences

10. Some related problems (primality testing, integer factorization, lattice basis reduction, algorithmic algebraic number theory, integer polynomials, algebraic complexity theory).

The book suggests numerous open problems and concludes with more than 3000 references. Consequently, it is essential for each researcher in finite field theory and related areas.

Reviewer: Arne Winterhof (Wien)

### MSC:

11Txx | Finite fields and commutative rings (number-theoretic aspects) |

11-02 | Research exposition (monographs, survey articles) pertaining to number theory |

11Y16 | Number-theoretic algorithms; complexity |

11T71 | Algebraic coding theory; cryptography (number-theoretic aspects) |

94A60 | Cryptography |

94B27 | Geometric methods (including applications of algebraic geometry) applied to coding theory |

11T06 | Polynomials over finite fields |

11T23 | Exponential sums |

11T55 | Arithmetic theory of polynomial rings over finite fields |

65C10 | Random number generation in numerical analysis |

68Q25 | Analysis of algorithms and problem complexity |

68W30 | Symbolic computation and algebraic computation |

11Y40 | Algebraic number theory computations |

14G05 | Rational points |

11G20 | Curves over finite and local fields |

11Y05 | Factorization |

11Y11 | Primality |