Topics in Geometry, Coding Theory and Cryptography / Edition 1

Topics in Geometry, Coding Theory and Cryptography / Edition 1

by Arnaldo Garcia
     
 

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… See more details below

Overview

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.

Product Details

ISBN-13:
9789048173457
Publisher:
Springer Netherlands
Publication date:
11/29/2010
Series:
Algebra and Applications Series, #6
Edition description:
Softcover reprint of hardcover 1st ed. 2007
Pages:
201
Product dimensions:
6.14(w) x 9.21(h) x 0.45(d)

Table of Contents


Foreword     vii
Explicit Towers of Function Fields over Finite Fields   A. Garcia   H. Stichtenoth     1
Introduction     1
Towers and Codes     5
Genus and Splitting Rate of a Tower     16
Explicit Tame Towers     24
Explicit Wild Towers     31
Miscellaneous Results     47
References     55
Function Fields over Finite Fields and Their Applications to Cryptography   H. Niederreiter   H. Wang   C. Xing     59
Introduction     59
Applications to Combinatorial Cryptography     60
Applications to Stream Ciphers and Linear Complexity     89
References     99
Artin-Schreier Extensions and Their Applications   C. Guneri   F. Ozbudak     105
Introduction     105
Artin-Schreier Extensions     107
Cyclic Codes and Their Weights     111
Trace Codes     120
Maximal Function Fields     126
References     130
Pseudorandom Sequences   A. Topuzoglu   A. Winterhof     135
Introduction     135
Linear Complexity and Linear ComplexityProfile     137
Autocorrelation and Related Distribution Measures for Binary Sequences     154
Discrepancy and Uniform Distribution     157
References     162
Group Structure of Elliptic Curves over Finite Fields and Applications   R. Murty   I. Shparlinski     167
Introduction     167
Group Structure     171
Applications to Cryptography     180
References     187
Algebraic Function Fields     195
About the Authors     199

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