Description
Book SynopsisOffering graduate students with the necessary theoretical tools for applying algebraic geometry to information theory, this title covers primary applications in coding theory and cryptography. It includes a discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields.
Trade Review"Coding theory has a rapidly growing corpus of knowledge, and now appears explicitly in several classifications in the MSC. [This] book ... is certainly an important addition to the literature in this area and a serious candidate for becoming one of the standard textbooks in related courses."--Cicero Carvalho, Mathematical Reviews
Table of ContentsPreface 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
