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.