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Topics in Geometry, Coding Theory and Cryptography / Edition 1

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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.
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Editorial Reviews

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"This book contains five surveys on different topic related to coding theory. The chapters are largely independent from each other and one can browse freely through them. ... The book is well-written and I think it may be valuable both for researchers and for students with background in algebra. ... In fact, the main interest of this book is that it focuses on topics not covered in other books on coding theory." (Fabio Mainardi, MAA Online, January, 2007)

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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
  • Edition number: 1
  • 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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