The use of algebraic curves over finite fields in coding theory (Goppa codes) and in cryptography (Koblitz and Miller, for the case of elliptic curves) has attracted a lot of attention to a field that was the domain of algebraic geometers and number theorists, for the benefit of the field itself and for the actual or potential applications to everyday tasks. Many textbooks and monographs have been published in this now-crowded field, focusing either on coding theory or cryptography, where in the last case the focus usually is on the highly developed elliptic curve cryptography.
The main goal of the book under review is to call attention and address applications of algebraic curves of higher genus to some important topics in cryptography. The book starts by recalling some basic facts on the geometry and arithmetic of algebraic curves over finite fields. This is followed by a brief discussion of algebraic codes in chapter two, including a few pages on algebraic geometry codes, with references to (two of) the authors previous book on the subject for details. Next, a long chapter three is devoted to elliptic curve cryptography.
After these preliminaries, the remaining chapters treat some selected topics in cryptography such as secret sharing schemes, authentication codes, frameproof codes, key distribution systems, broadcast encryption and sequences. It must be said that the exposition in these chapters is more detailed, covering the algebraic and combinatorial constructions in detail, with the algebraic curves applications coming mainly in the form of examples obtained from algebraic geometry codes, usually in the last few sections of the corresponding chapter, with the exception of the last one. The book is filled with examples to illustrate the various constructions and, assuming a basic knowledge of combinatorics and algebraic geometry it is almost self-contained. However, for its use a textbook the instructor must provide the exercises, since the book comes with no exercises at all.
Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is firstname.lastname@example.org.