Applications of finite fields.

*(English)*Zbl 0779.11059
The Kluwer International Series in Engineering and Computer Science. 199. Boston: Kluwer Academic Publishers. xi, 218 p. (1993).

Because of applications in so many diverse areas, finite fields have become increasingly important algebraic structures. In addition to many important theoretical problems related to finite fields, they form the fundamental underlying structure in algebraic coding theory for the error-free transmission of information. In cryptology for the secure transmission of information, many cryptographic systems rely on the difficulty of solving various finite field problems. In addition, finite fields play very important roles in combinatorial design theory, pseudorandom number generation, and last but certainly not least, there are many important computational and algorithmic problems related to finite fields which have very important practical applications. Some of these algorithmic problems involve the construction of various kinds of bases for ease of computations involving finite field arithmetic, the factoring of polynomials, and the discrete logarithm problem.

This excellent book grew out of a 10-week seminar on the applications of finite fields held at the University of Waterloo. Each chapter is reasonably self-contained and in addition, the references pertaining to a given chapter are listed at the end of that chapter. Some exercises and research problems are also included with each chapter. The book is divided into the following 10 chapters and an appendix.

1. Introduction to finite fields and bases; 2. Factoring polynomials over finite fields; 3. Construction of irreducible polynomials; 4. Normal bases; 5. Optimal normal bases; 6. The discrete logarithm problem; 7. Elliptic curves over finite fields; 8. Elliptic curve cryptosystems; 9. Introduction to algebraic geometry; 10. Codes from algebraic geometry; Appendix – other applications.

As indicated above this is an excellent book which focuses on recent developments in the theory and applications of finite fields. One of the outstanding points in my view is that the book provides an excellent discussion of a given topic, proving many of the results, but on the other hand not proving every detail so as to bog down the reader with too many details. One can quickly obtain a very recent survey of the topic as well as refer to the references for further details and proofs of additional results. This makes the book an outstanding reference for researchers as well as a fine choice for a text in a graduate level course. The book will be useful to students and professors from mathematics, computer science, and engineering.

This book is very well written, a pleasure to read, and most importantly, it discusses the fascinating subject of various theoretical and applied aspects of finite fields. I strongly recommend the book.

This excellent book grew out of a 10-week seminar on the applications of finite fields held at the University of Waterloo. Each chapter is reasonably self-contained and in addition, the references pertaining to a given chapter are listed at the end of that chapter. Some exercises and research problems are also included with each chapter. The book is divided into the following 10 chapters and an appendix.

1. Introduction to finite fields and bases; 2. Factoring polynomials over finite fields; 3. Construction of irreducible polynomials; 4. Normal bases; 5. Optimal normal bases; 6. The discrete logarithm problem; 7. Elliptic curves over finite fields; 8. Elliptic curve cryptosystems; 9. Introduction to algebraic geometry; 10. Codes from algebraic geometry; Appendix – other applications.

As indicated above this is an excellent book which focuses on recent developments in the theory and applications of finite fields. One of the outstanding points in my view is that the book provides an excellent discussion of a given topic, proving many of the results, but on the other hand not proving every detail so as to bog down the reader with too many details. One can quickly obtain a very recent survey of the topic as well as refer to the references for further details and proofs of additional results. This makes the book an outstanding reference for researchers as well as a fine choice for a text in a graduate level course. The book will be useful to students and professors from mathematics, computer science, and engineering.

This book is very well written, a pleasure to read, and most importantly, it discusses the fascinating subject of various theoretical and applied aspects of finite fields. I strongly recommend the book.

Reviewer: G.L.Mullen (University Park)

##### MSC:

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

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

11T30 | Structure theory for finite fields and commutative rings (number-theoretic aspects) |

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

94A60 | Cryptography |

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

14H52 | Elliptic curves |

11-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to number theory |

11T06 | Polynomials over finite fields |

11G20 | Curves over finite and local fields |