Finite fields: structure and arithmetics.

*(English)*Zbl 0779.11058
Mannheim: B. I.-Wissenschaftsverlag. 344 p. (1993).

The theory of finite fields has seen spectacular progress in the last ten years, and this is also reflected by the number of textbooks and monographs that have been devoted to this subject recently. The present book concentrates on advanced topics and is not meant as an introduction. In the choice of the material there is some intersection with the book of A. Menezes et al. [Applications of finite fields (Kluwer, Boston, 1993; reviewed below)], especially as far as the treatment of bases of finite fields is concerned, but there is also plenty of material here that cannot be found in other books.

After a brief tour d’horizon of the basics of finite fields in Chapter 1, the realm of the constructive theory of finite fields is entered right away in Chapter 2. Various constructions of irreducible, selfreciprocal irreducible, and primitive polynomials are presented and discrete and Zech logarithms are discussed. In this connection it should be noted that the term “Zech logarithm” is historically inaccurate and should be replaced by “Jacobi logarithm”: the work of Zech dates from 1849, but this logarithm was already introduced and tabulated by Jacobi in 1837 [see C. G. J. Jacobi, Monatsber. Königl. Akad. Wiss. Berlin 1837, 127-136].

Chapters 3, 4, and 5 are devoted to special bases of finite fields, such as normal bases, optimal normal bases, and self-dual bases. This is currently a hot topic, and the reader is brought all the way to the frontier of research. The principal motivation in the selection of special bases is the speed-up of finite field arithmetic, and the author does an excellent job explaining the advantages of various bases. Linear feedback shift-register sequences are treated in Chapter 6. In addition to the classical theory, interesting connections of these sequences with cyclic difference sets are discussed. There is also a very attractive section on an approach to the linear complexity from the point of view of the discrete Fourier transform. Characters of finite fields, Gaussian sums over finite fields, and their applications are the topics of the last chapter. A bibliography of 15 pages, a list of symbols, and a subject index conclude the book.

The material is well structured and presented in a clear and efficient manner. Occasional examples illustrate the concepts and results very nicely. The author’s background as a combinatorialist shines through in the selection of the applied material and in some side remarks. The book can be highly recommended for researchers and graduate students in pure and applied algebra and number theory. Although it contains no exercises, it is well suited for self-study because of its careful expository style.

After a brief tour d’horizon of the basics of finite fields in Chapter 1, the realm of the constructive theory of finite fields is entered right away in Chapter 2. Various constructions of irreducible, selfreciprocal irreducible, and primitive polynomials are presented and discrete and Zech logarithms are discussed. In this connection it should be noted that the term “Zech logarithm” is historically inaccurate and should be replaced by “Jacobi logarithm”: the work of Zech dates from 1849, but this logarithm was already introduced and tabulated by Jacobi in 1837 [see C. G. J. Jacobi, Monatsber. Königl. Akad. Wiss. Berlin 1837, 127-136].

Chapters 3, 4, and 5 are devoted to special bases of finite fields, such as normal bases, optimal normal bases, and self-dual bases. This is currently a hot topic, and the reader is brought all the way to the frontier of research. The principal motivation in the selection of special bases is the speed-up of finite field arithmetic, and the author does an excellent job explaining the advantages of various bases. Linear feedback shift-register sequences are treated in Chapter 6. In addition to the classical theory, interesting connections of these sequences with cyclic difference sets are discussed. There is also a very attractive section on an approach to the linear complexity from the point of view of the discrete Fourier transform. Characters of finite fields, Gaussian sums over finite fields, and their applications are the topics of the last chapter. A bibliography of 15 pages, a list of symbols, and a subject index conclude the book.

The material is well structured and presented in a clear and efficient manner. Occasional examples illustrate the concepts and results very nicely. The author’s background as a combinatorialist shines through in the selection of the applied material and in some side remarks. The book can be highly recommended for researchers and graduate students in pure and applied algebra and number theory. Although it contains no exercises, it is well suited for self-study because of its careful expository style.

Reviewer: H.Niederreiter (Wien)

##### 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) |

11T06 | Polynomials over finite fields |

94A55 | Shift register sequences and sequences over finite alphabets in information and communication theory |

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

11T24 | Other character sums and Gauss sums |