To a reader wanting to learn about algebraic geometry, there are many choices of books one can turn to, each of which offers its own spin on both the choice of topics and the presentation. Harald Niederreiter and Chaoping Xing have both written a number of research papers in coding theory, and as such this is the point of view they offer to the reader of their new book Algebraic Geometry in Coding Theory and Cryptography.
The topics covered in the first four chapters are the topics one would expect from most books in algebraic geometry: function fields, varieties, morphisms, curves, divisors, Riemann-Roch spaces, Zeta functions and the Hasse-Weil theorem. Where this book is different from other books is in the final two chapters, which are dedicated to the applications to coding theory and cryptography. The authors start at a basic level with coding theory, defining what a code is and discussing in detail how the curves and divisors and Riemann-Roch spaces of the earlier sections can be used in order to construct codes that have particularly good parameters. In the chapter on cryptography, the authors again start with very basic material on public key cryptosystems and the Diffie-Hellman problem before moving on to discuss cryptosystems based on elliptic and hyperelliptic curves and their divisors. The final sections look at encryption schemes such as the McEliece and Niederreiter (yes, the same Niederreiter) cryptosystems, which use coding theory and algebraic geometric codes in order to encrypt information.
The book lacks the quantity of examples and exercises one might want from a textbook, and there are other books that cover much of the same material (Stichtenoth's Algebraic Function Fields and Codes is a particular favorite of this reviewer, for example). That said, I have found myself reaching for Niederreiter and Xing's book several times in recent weeks, as the exposition in the book is clear and it serves as a nice reference for the material it covers. The bibliography is copious, and the topics covered are well-chosen. This book would make a fine addition to any library or to the shelves of an algebraic geometer wanting to learn some coding theory or vice versa.
Darren Glass is an associate professor of mathematics at Gettysburg College. He can be reached at firstname.lastname@example.org