Algebraic Geometry in Coding Theory and Cryptography 1st edition by Harald Niederreiter, Chaoping Xing ISBN 0817647457 9780691102887

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ISBN 10: 0817647457
ISBN 13: 9780691102887
Author: Harald Niederreiter, Chaoping Xing

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

Algebraic Geometry in Coding Theory and Cryptography 1st Table of contents:

Preface

• Overview of the Book
• Intended Audience
• Acknowledgments

Chapter 1: Introduction
1.1 Algebraic Geometry and Its Applications
1.2 Overview of Coding Theory
1.3 Cryptography and Its Relationship with Algebraic Geometry
1.4 A Brief History of Algebraic Geometry in Coding Theory

Chapter 2: Basics of Algebraic Geometry
2.1 Varieties, Affine and Projective Geometry
2.2 Field Extensions and Algebraic Curves
2.3 Divisors and Rational Functions
2.4 The Riemann-Roch Theorem
2.5 Singularities and Smooth Varieties

Chapter 3: Algebraic Curves and Codes
3.1 Linear Codes and Their Geometric Interpretation
3.2 Geometric Interpretation of Hamming Codes
3.3 Algebraic Curves and Their Application in Coding Theory
3.4 The Connection Between Curves and Codes
3.5 Gilbert-Varshamov Bound and Geometric Bounds for Codes

Chapter 4: The Theory of Error-Correcting Codes
4.1 Error-Correcting Codes: Basic Concepts
4.2 Linear Codes, Cyclic Codes, and BCH Codes
4.3 Geometric Codes and Their Properties
4.4 The Bounds on Code Size and Error-Correction Capability
4.5 Examples of Algebraic-Geometric Codes

Chapter 5: Algebraic-Geometry Codes
5.1 Definition and Construction of Algebraic Geometry Codes
5.2 The Goppa Codes
5.3 Decoding Algorithms for Algebraic Geometry Codes
5.4 The Performance of Algebraic Geometry Codes
5.5 Applications of Algebraic Geometry Codes

Chapter 6: Cryptographic Systems Based on Algebraic Geometry
6.1 Introduction to Public-Key Cryptosystems
6.2 RSA Cryptosystem and Algebraic Geometry
6.3 Elliptic Curve Cryptography
6.4 The Role of Algebraic Curves in Cryptography
6.5 Applications of Algebraic Geometry in Modern Cryptography

Chapter 7: Algebraic Curves over Finite Fields
7.1 Curves over Finite Fields and Their Properties
7.2 Applications in Cryptography
7.3 Curve Tracing and Classification
7.4 The Role of Finite Fields in Algebraic Geometry

Chapter 8: Goppa Codes and Their Applications in Cryptography
8.1 Goppa’s Construction of Codes
8.2 The Properties of Goppa Codes
8.3 Applications of Goppa Codes in Cryptography
8.4 The Complexity of Goppa-Based Cryptosystems

Chapter 9: The Coding-Theoretic Approach to Cryptography
9.1 Code-Based Cryptography
9.2 Security of Code-Based Cryptosystems
9.3 Attacks on Code-Based Cryptosystems
9.4 The Role of Algebraic Geometry in Enhancing Cryptographic Security

Chapter 10: Recent Developments in Algebraic Geometry Codes and Cryptography
10.1 New Advances in Algebraic Geometry Codes
10.2 Research in Code-Based Cryptography
10.3 New Techniques for Decoding Algebraic Geometry Codes
10.4 Algebraic Geometry and Its Future Directions in Cryptography

Appendices

• A: Basic Concepts in Algebraic Geometry
• B: Mathematical Background and Notation
• C: Further Reading and References

Index

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