Topics in Geometry, Coding Theory and Cryptography

Arnaldo Garcia, Henning Stichtenoth (auth.), Arnaldo Garcia, Henning Stichtenoth (eds.)9781402053337, 1-4020-5333-9

The theory of algebraic function fields over finite fields has its origins in number theory. However, after Goppa`s discovery of algebraic geometry codes around 1980, many applications of function fields were found in different areas of mathematics and information theory, such as coding theory, sphere packings and lattices, sequence design, and cryptography. The use of function fields often led to better results than those of classical approaches.

This book presents survey articles on some of these new developments. Most of the material is directly related to the interaction between function fields and their various applications; in particular the structure and the number of rational places of function fields are of great significance. The topics focus on material which has not yet been presented in other books or survey articles. Wherever applications are pointed out, a special effort has been made to present some background concerning their use.

Table of contents :
EXPLICIT TOWERS OF FUNCTION FIELDS OVER FINITE FIELDS….Pages 1-58
FUNCTION FIELDS OVER FINITE FIELDS AND THEIR APPLICATIONS TO CRYPTOGRAPHY….Pages 59-104
ARTIN-SCHREIER EXTENSIONS AND THEIR APPLICATIONS….Pages 105-133
PSEUDORANDOM SEQUENCES….Pages 135-166
GROUP STRUCTURE OF ELLIPTIC CURVES OVER FINITE FIELDS AND APPLICATIONS….Pages 167-194