TABLE OF CONTENTS: Preface ix
Chapter 1: Finite Fields and Function Fields 1
1.1 Structure of Finite Fields 1
1.2 Algebraic Closure of Finite Fields 4
1.3 Irreducible Polynomials 7
1.4 Trace and Norm 9
1.5 Function Fields of One Variable 12
1.6 Extensions of Valuations 25
1.7 Constant Field Extensions 27
Chapter 2: Algebraic Varieties 30
2.1 Affine and Projective Spaces 30
2.2 Algebraic Sets 37
2.3 Varieties 44
2.4 Function Fields of Varieties 50
2.5 Morphisms and Rational Maps 56
Chapter 3: Algebraic Curves 68
3.1 Nonsingular Curves 68
3.2 Maps Between Curves 76
3.3 Divisors 80
3.4 Riemann-Roch Spaces 84
3.5 Riemann's Theorem and Genus 87
3.6 The Riemann-Roch Theorem 89
3.7 Elliptic Curves 95
3.8 Summary: Curves and Function Fields 104
Chapter 4: Rational Places 105
4.1 Zeta Functions 105
4.2 The Hasse-Weil Theorem 115
4.3 Further Bounds and Asymptotic Results 122
4.4 Character Sums 127
Chapter 5: Applications to Coding Theory 147
5.1 Background on Codes 147
5.2 Algebraic-Geometry Codes 151
5.3 Asymptotic Results 155
5.4 NXL and XNL Codes 174
5.5 Function-Field Codes 181
5.6 Applications of Character Sums 187
5.7 Digital Nets 192
Chapter 6: Applications to Cryptography 206
6.1 Background on Cryptography 206
6.2 Elliptic-Curve Cryptosystems 210
6.3 Hyperelliptic-Curve Cryptography 214
6.4 Code-Based Public-Key Cryptosystems 218
6.5 Frameproof Codes 223
6.6 Fast Arithmetic in Finite Fields 233
A Appendix 241
A.1 Topological Spaces 241
A.2 Krull Dimension 244
A.3 Discrete Valuation Rings 245
Bibliography 249
Index 257
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