Algebraic geometry has found fascinating applications to coding theory and cryptography in the last few decades. This book aims to provide the necessary theoretical background for reading the contemporary literature on these applications. An aspect that we emphasize, as it is very useful for the applications, is the interplay between nonsingular projective curves over finite fields and global function fields. This correspondence is well known and frequently employed by researchers, but nevertheless it is difficult to find detailed proofs of the basic facts about this correspondence in the expository literature. One contribution of our book is to fill this gap by giving complete proofs of these results.

We also want to offer the reader a taste of the applications of algebraic geometry, and in particular of algebraic curves over finite fields, to coding theory and cryptography. Several books, among them our earlier book Rational Points on Curves over Finite Fields: Theory and Applications, have already treated such applications. Accordingly, besides presenting standard topics such as classical algebraic-geometry codes, we have also selected material that cannot be found in other books, partly because it is of recent origin.

As a reflection of the above aims, the book splits into two parts. The first part, consisting of Chapters 1 to 4, develops the theory of algebraic varieties, of algebraic curves, and of their function fields, with the emphasis gradually shifting to global function fields. The second part consists of Chapters 5 and 6 and describes applications to coding theory and cryptography, respectively. The book is written at the level of advanced undergraduates and first-year graduate students with a good background in algebra.

We are grateful to our former Ph.D. students David Mayor and Ayineedi Venkateswarlu for their help with typesetting and proofreading.We also thank Princeton University Press for the invitation to write this book.