Topics in Geometry, Coding Theory and Cryptography

This book contains five surveys on different topics related to coding theory. The chapters are largely independent from each other and one can browse freely through them. One should not expect an introductory book on coding theory. The choice of the editors is to present topics that do not appear in other books or survey articles; some acquaintance with the basics of coding theory is thus desirable (for general background on coding theory via function fields, one can refer to [1 ]). However, the book does not require any special knowledge in algebraic geometry beyond divisors and the Riemann-Roch theorem. Each chapter begins with an introduction providing motivation for the material included. A short appendix, collecting the basic definitions on function fields, concludes the book.

A linear code is a subspace of a finite-dimensional vector space over a finite field. Around 1980, V.D.Goppa [2] found how to construct ‘good’ codes using the rational points on algebraic curves. It is therefore interesting to study function fields with certain prescribed properties. Since then, function fields have been used to get better codes than those found using classical approaches [3 ]. This book surveys some of the recent applications of function fields to coding theory, many of which are not to be found in other surveys.

The first chapter deals with towers of function fields (over finite fields), where each intermediate extension is finite and separable and the genus tends to infinity. The goal is to investigate the asymptotic behavior of the number of rational places in such towers, especially in comparison with the growth of the genus. This chapter contains many proofs of results whose interest goes beyond the applications to coding theory. Also, several explicit examples are treated.

The third chapter focuses on extensions of a special type: the authors show how to use the Artin-Schreier extensions (given by solutions of x^p – x = 0) to obtainbounds on the minimum distance of cyclic codes (i.e., codes that are invariant under cyclic permutations of the coordinates).

It is a major problem in cryptography to ensure integrity of sensitive data, in particular to protect copyrighted material. The second chapter of the book explains how to construct authentication codes, frameproof codes, and hash families. The necessary background on generation of random numbers and on the group structure of elliptic curves is provided, respectively, in chapters 4 and 5.

The book is well-written and I think it may be valuable both for researchers and for students with background in algebra (especially finite fields, Galois theory, function fields, divisors). Anyway, this book should be used as a complement, after a course on coding theory. In fact, the main interest of this book is that it focuses on topics not covered in other books on coding theory. One may use [1] and [4 ] as companion books, containing more classical material.

References

[1] H.Stichtenoth, Algebraic function fields and codes, Springer-Verlag, Berlin (1993).

[2] V.D.Goppa, "Codes on algebraic curves," Soviet Math. Dokl., Vol. 24, 170-172 (1981)

[3] M.A. Tsafsman, S.G.Vladut, T. Zink, "Modular curves, Shimura curves and Goppa codes better than the Varshamov-Gilbert bound," Math. Nachr., Vol. 109, 21-28 (1982)

[4] I.F.Blake, G.Seroussi, N.Smart, Elliptic curves in cryptography, London Math. Soc. Lecture Notes Series, Vol. 265, Cambridge University Press (1999)

Fabio Mainardi earned a PhD in Mathematics at the University of Paris 13. His research interests are Iwasawa theory, p-adic L-functions and the arithmetic of automorphic forms. He may be reached at mainardi2002@yahoo.fr.