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. Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields Includes applications to coding theory and cryptography Covers the latest advances in algebraic-geometry codes Features applications to cryptography not treated in other books Cover......Page 1 Title......Page 4 Copyright......Page 5 Contents......Page 8 Preface......Page 10 1.1 Structure of Finite Fields......Page 14 1.2 Algebraic Closure of Finite Fields......Page 17 1.3 Irreducible Polynomials......Page 20 1.4 Trace and Norm......Page 22 1.5 Function Fields of One Variable......Page 25 1.6 Extensions of Valuations......Page 38 1.7 Constant Field Extensions......Page 40 2.1 Affine and Projective Spaces......Page 43 2.2 Algebraic Sets......Page 50 2.3 Varieties......Page 57 2.4 Function Fields of Varieties......Page 63 2.5 Morphisms and Rational Maps......Page 69 3.1 Nonsingular Curves......Page 81 3.2 Maps Between Curves......Page 89 3.3 Divisors......Page 93 3.4 Riemann-Roch Spaces......Page 97 3.5 Riemann's Theorem and Genus......Page 100 3.6 The Riemann-Roch Theorem......Page 102 3.7 Elliptic Curves......Page 108 3.8 Summary: Curves and Function Fields......Page 117 4.1 Zeta Functions......Page 118 4.2 The Hasse-Weil Theorem......Page 128 4.3 Further Bounds and Asymptotic Results......Page 135 4.4 Character Sums......Page 140 5.1 Background on Codes......Page 160 5.2 Algebraic-Geometry Codes......Page 164 5.3 Asymptotic Results......Page 168 5.4 NXL and XNL Codes......Page 187 5.5 Function-Field Codes......Page 194 5.6 Applications of Character Sums......Page 200 5.7 Digital Nets......Page 205 6.1 Background on Cryptography......Page 219 6.2 Elliptic-Curve Cryptosystems......Page 223 6.3 Hyperelliptic-Curve Cryptography......Page 227 6.4 Code-Based Public-Key Cryptosystems......Page 231 6.5 Frameproof Codes......Page 236 6.6 Fast Arithmetic in Finite Fields......Page 246 A.1 Topological Spaces......Page 254 A.2 Krull Dimension......Page 257 A.3 Discrete Valuation Rings......Page 258 Bibliography......Page 262 D......Page 270 L......Page 271 S......Page 272 Z......Page 273

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.

• Introduces graduate students and advanced undergraduates to the foundations of algebraic geometry for applications to information theory
• Provides the first detailed discussion of the interplay between projective curves and algebraic function fields over finite fields
• Includes applications to coding theory and cryptography
• Covers the latest advances in algebraic-geometry codes
• Features applications to cryptography not treated in other books