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Topics in geometry, coding theory and cryptography
Algebra, logic and applications ; 6
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Dordrecht : Springer, 2007

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30000010148874 QA268 T66 2007 Open Access Book

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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. This book presents survey articles on some of these new developments. The topics focus on material which has not yet been presented in other books or survey articles.

Table of Contents

A. Garcia and H. StichtenothH. Niederreiter and H. Wang and C. XingC. Guneri and F. OzbudakA. Topuzoglu and A. WinterhofR. Murty and I. Shparlinski
Forewordp. vii
1 Explicit Towers of Function Fields over Finite Fieldsp. 1
1 Introductionp. 1
2 Towers and Codesp. 5
3 Genus and Splitting Rate of a Towerp. 16
4 Explicit Tame Towersp. 24
5 Explicit Wild Towersp. 31
6 Miscellaneous Resultsp. 47
Referencesp. 55
2 Function Fields over Finite Fields and Their Applications to Cryptographyp. 59
1 Introductionp. 59
2 Applications to Combinatorial Cryptographyp. 60
3 Applications to Stream Ciphers and Linear Complexityp. 89
Referencesp. 99
3 Artin-Schreier Extensions and Their Applicationsp. 105
1 Introductionp. 105
2 Artin-Schreier Extensionsp. 107
3 Cyclic Codes and Their Weightsp. 111
4 Trace Codesp. 120
5 Maximal Function Fieldsp. 126
Referencesp. 130
4 Pseudorandom Sequencesp. 135
1 Introductionp. 135
2 Linear Complexity and Linear Complexity Profilep. 137
3 Autocorrelation and Related Distribution Measures for Binary Sequencesp. 154
4 Discrepancy and Uniform Distributionp. 157
Referencesp. 162
5 Group Structure of Elliptic Curves over Finite Fields and Applicationsp. 167
1 Introductionp. 167
2 Group Structurep. 171
3 Applications to Cryptographyp. 180
Referencesp. 187
Appendix Algebraic Function Fieldsp. 195
About the Authorsp. 199