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Algebraic Geometry in Coding Theory and Cryptography

This textbook equips graduate students and advanced undergraduates with the necessary theoretical tools for applying algebraic geometry to information theory, and it covers primary applications in coding theory and cryptography. Harald Niederreiter and Chaoping Xing provide the first detailed discussion of the interplay between nonsingular projective curves and algebraic function fields over finite fields. This interplay is fundamental to research in the field today, yet until now no other textbook has featured complete proofs of it. Niederreiter and Xing cover classical applications like algebraic-geometry codes and elliptic-curve cryptosystems as well as material not treated by other books, including function-field codes, digital nets, code-based public-key cryptosystems, and frameproof codes. Combining a systematic development of theory with a broad selection of real-world applications, this is the most comprehensive yet accessible introduction to the field available. It introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory. It provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields. It includes applications to coding theory and cryptography. It covers the latest advances in algebraic-geometry codes. It features applications to cryptography not treated in other books.
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This is a beautifully written volume that gives the necessary background to read the research literature on coding and cryptography based on concepts from curves in algebraic geometries. Both of the authors are outstanding researchers, well known for the clarity and depth of their contributions. This work is a valuable and welcome addition to the literature on coding and cryptography. -- Ian F. Blake, University of British Columbia

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

About the Author

Harald Niederreiter is professor of mathematics and computer science at the National University of Singapore. Chaoping Xing is professor of mathematics at the Nanyang Technological University in Singapore. They are the authors of "Rational Points on Curves over Finite Fields: Theory and Applications".


"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

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