Rational points on curves over finite fields : theory and applications / Harald Niederreiter, Chaoping Xing.

Format:

Book

Author/Creator:

Niederreiter, Harald, 1944-

Contributor:

Xing, Chaoping, 1963-

Series:

London Mathematical Society lecture note series ; 285.

London Mathematical Society lecture note series ; 285

Language:

English

Subjects (All):

Curves, Algebraic.

Finite fields (Algebra).

Coding theory.

Physical Description:

x, 245 pages : illustrations ; 23 cm.

Place of Publication:

Cambridge ; New York : Cambridge University Press, 2001.

Summary:

Discussion of theory and applications of algebraic curves over finite fields with many rational points.

Contents:

1 Background on Function Fields 1

1.1 Riemann-Roch Theorem 1

1.2 Divisor Class Groups and Ideal Class Groups 6

1.3 Algebraic Extensions and the Hurwitz Formula 10

1.4 Ramification Theory of Galois Extensions 14

1.5 Constant Field Extensions 20

1.6 Zeta Functions and Rational Places 26

2 Class Field Theory 36

2.1 Local Fields 36

2.2 Newton Polygons 38

2.3 Ramification Groups and Conductors 39

2.4 Global Fields 44

2.5 Ray Class Fields and Hilbert Class Fields 47

2.6 Narrow Ray Class Fields 50

2.7 Class Field Towers 55

3 Explicit Function Fields 62

3.1 Kummer and Artin-Schreier Extensions 62

3.2 Cyclotomic Function Fields 65

3.3 Drinfeld Modules of Rank 1 72

4 Function Fields with Many Rational Places 76

4.1 Function Fields from Hilbert Class Fields 76

4.2 Function Fields from Narrow Ray Class Fields 82

4.2.1 The First Construction 82

4.2.2 The Second Construction 92

4.2.3 The Third Construction 94

4.3 Function Fields from Cyclotomic Fields 108

4.4 Explicit Function Fields 113

5 Asymptotic Results 122

5.1 Asymptotic Behavior of Towers 122

5.2 The Lower Bound of Serre 126

5.3 Further Lower Bounds for A(q[superscript m]) 133

5.4 Explicit Towers 136

5.5 Lower Bounds on A(2), A(3), and A(5) 138

6 Applications to Algebraic Coding Theory 141

6.1 Goppa's Algebraic-Geometry Codes 141

6.2 Beating the Asymptotic Gilbert-Varshamov Bound 150

6.3 NXL Codes 156

6.4 XNL Codes 160

6.5 A Propagation Rule for Linear Codes 164

7 Applications to Cryptography 170

7.1 Background on Stream Ciphers and Linear Complexity 170

7.2 Constructions of Almost Perfect Sequences 177

7.3 A Construction of Perfect Hash Families 184

7.4 Hash Families and Authentication Schemes 186

8 Applications to Low-Discrepancy Sequences 191

8.1 Background on (t, m, s)-Nets and (t, s)-Sequences 191

8.2 The Digital Method 197

8.3 A Construction Using Rational Places 203

8.4 A Construction Using Arbitrary Places 212

A Curves and Their Function Fields 219

A.1 Transcendence Degree 219

A.2 Affine Spaces 219

A.3 Projective Spaces 220

A.4 Affine Varieties 222

A.5 Projective Varieties 224

A.6 Projective Curves 225.

Notes:

Includes bibliographical references (pages 227-239) and index.

ISBN:

0521665434

OCLC:

46402143