Topics in geometry, coding theory and cryptography

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.

A. Garcia and H. StichtenothH. Niederreiter and H. Wang and C. XingC. Guneri and F. OzbudakA. Topuzoglu and A. WinterhofR. Murty and I. Shparlinski

Foreword	p. vii
1 Explicit Towers of Function Fields over Finite Fields	p. 1
1 Introduction	p. 1
2 Towers and Codes	p. 5
3 Genus and Splitting Rate of a Tower	p. 16
4 Explicit Tame Towers	p. 24
5 Explicit Wild Towers	p. 31
6 Miscellaneous Results	p. 47
References	p. 55
2 Function Fields over Finite Fields and Their Applications to Cryptography	p. 59
1 Introduction	p. 59
2 Applications to Combinatorial Cryptography	p. 60
3 Applications to Stream Ciphers and Linear Complexity	p. 89
References	p. 99
3 Artin-Schreier Extensions and Their Applications	p. 105
1 Introduction	p. 105
2 Artin-Schreier Extensions	p. 107
3 Cyclic Codes and Their Weights	p. 111
4 Trace Codes	p. 120
5 Maximal Function Fields	p. 126
References	p. 130
4 Pseudorandom Sequences	p. 135
1 Introduction	p. 135
2 Linear Complexity and Linear Complexity Profile	p. 137
3 Autocorrelation and Related Distribution Measures for Binary Sequences	p. 154
4 Discrepancy and Uniform Distribution	p. 157
References	p. 162
5 Group Structure of Elliptic Curves over Finite Fields and Applications	p. 167
1 Introduction	p. 167
2 Group Structure	p. 171
3 Applications to Cryptography	p. 180
References	p. 187
Appendix Algebraic Function Fields	p. 195
About the Authors	p. 199

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