Algebraic Geometry in Coding Theory and Cryptography

By: Harald Niederreiter, Chaoping Xing

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Hardcover | 11 October 2009

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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

Industry Reviews

"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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You Can Find This Book In

Non-Fiction Reference, Information & Interdisciplinary Subjects Research & Information Information theory Earth Sciences, Geography, Environment, Planning Geography Computing & I.T.Computer Science Mathematics Geometry Algebraic Geometry Number Theory

Combinatorics & Graph Theory Coding Theory & Cryptology Computer Security Data Encryption

Preface	p. ix
Finite Fields and Function Fields	p. 1
Structure of Finite Fields	p. 1
Algebraic Closure of Finite Fields	p. 4
Irreducible Polynomials	p. 7
Trace and Norm	p. 9
Function Fields of One Variable	p. 12
Extensions of Valuations	p. 25
Constant Field Extensions	p. 27
Algebraic Varieties	p. 30
Affine and Projective Spaces	p. 30
Algebraic Sets	p. 37
Varieties	p. 44
Function Fields of Varieties	p. 50
Morphisms and Rational Maps	p. 56
Algebraic Curves	p. 68
Nonsingular Curves	p. 68
Maps Between Curves	p. 76
Divisors	p. 80
Riemann-Roch Spaces	p. 84
Riemann's Theorem and Genus	p. 87
The Riemann-Roch Theorem	p. 89
Elliptic Curves	p. 95
Summary: Curves and Function Fields	p. 104
Rational Places	p. 105
Zeta Functions	p. 105
The Hasse-Weil Theorem	p. 115
Further Bounds and Asymptotic Results	p. 122
Character Sums	p. 127
Applications to Coding Theory	p. 147
Background on Codes	p. 147
Algebraic-Geometry Codes	p. 151
Asymptotic Results	p. 155
NXL and XNL Codes	p. 174
Function-Field Codes	p. 181
Applications of Character Sums	p. 187
Digital Nets	p. 192
Applications to Cryptography	p. 206
Background on Cryptography	p. 206
Elliptic-Curve Cryptosystems	p. 210
Hyperelliptic-Curve Cryptography	p. 214
Code-Based Public-Key Cryptosystems	p. 218
Frameproof Codes	p. 223
Fast Arithmetic in Finite Fields	p. 233
Appendix	p. 241
Topological Spaces	p. 241
Krull Dimension	p. 244
Discrete Valuation Rings	p. 245
Bibliography	p. 249
Index	p. 257
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Algebraic Geometry in Coding Theory and Cryptography

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