Ling / Wang / Xing Algebraic Curves in Cryptography

1. Auflage 2013
ISBN: 978-1-4200-7947-0
Verlag: Taylor & Francis
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 340 Seiten

Reihe: Discrete Mathematics and Its Applications

The reach of algebraic curves in cryptography goes far beyond elliptic curve or public key cryptography yet these other application areas have not been systematically covered in the literature. Addressing this gap, Algebraic Curves in Cryptography explores the rich uses of algebraic curves in a range of cryptographic applications, such as secret sharing, frameproof codes, and broadcast encryption.

Suitable for researchers and graduate students in mathematics and computer science, this self-contained book is one of the first to focus on many topics in cryptography involving algebraic curves. After supplying the necessary background on algebraic curves, the authors discuss error-correcting codes, including algebraic geometry codes, and provide an introduction to elliptic curves. Each chapter in the remainder of the book deals with a selected topic in cryptography (other than elliptic curve cryptography). The topics covered include secret sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. Chapters begin with introductory material before featuring the application of algebraic curves.

Autoren/Hrsg.

Ling, San

Wang, Huaxiong

Xing, Chaoping

Fachgebiete

• Mathematik | Informatik Mathematik Mathematik Allgemein Diskrete Mathematik, Kombinatorik
• Interdisziplinäres Wissenschaften Wissenschaften: Forschung und Information Informationstheorie, Kodierungstheorie
• Mathematik | Informatik Mathematik Algebra
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Kryptologie, Informationssicherheit
• Mathematik | Informatik Mathematik Geometrie Algebraische Geometrie
• Mathematik | Informatik EDV | Informatik Daten / Datenbanken Informationstheorie, Kodierungstheorie

Inhaltsverzeichnis

Introduction to Algebraic Curves

Plane Curves

Algebraic Curves and Their Function Fields

Smooth Curves

Riemann-Roch Theorem

Rational Points and Zeta Functions

Introduction to Error-Correcting Codes

Introduction

Linear Codes

Bounds

Algebraic Geometry Codes

Asymptotic Behavior of Codes

Elliptic Curves and Their Applications to Cryptography

Basic Introduction

Maps between Elliptic Curves

The Group E(Fq) and Its Torsion Subgroups

Computational Considerations on Elliptic Curves

Pairings on an Elliptic Curve
Elliptic Curve Cryptography

Secret Sharing Schemes

The Shamir Threshold Scheme

Other Threshold Schemes
General Secret Sharing Schemes
Information Rate
Quasi-Perfect Secret Sharing Schemes

Linear Secret Sharing Schemes

Multiplicative Linear Secret Sharing Schemes

Secret Sharing from Error-Correcting Codes

Secret Sharing from Algebraic Geometry Codes

Authentication Codes

Authentication Codes

Bounds of A-Codes
A-Codes and Error-Correcting Codes

Universal Hash Families and A-Codes

A-Codes from Algebraic Curves

Linear Authentication Codes

Frameproof Codes

Introduction

Constructions of Frameproof Codes without Algebraic Geometry
Asymptotic Bounds and Constructions from Algebraic Geometry
Improvements to the Asymptotic Bound

Key Distribution Schemes

Key Predistribution

Key Predistribution Schemes with Optimal Information Rates
Linear Key Predistribution Schemes

Key Predistribution Schemes from Algebraic Geometry

Key Predistribution Schemes from Cover-Free Families
Perfect Hash Families and Algebraic Geometry

Broadcast Encryption and Multicast Security

One-Time Broadcast Encryption
Multicast Re-Keying Schemes
Re-Keying Schemes with Dynamic Group Controllers

Some Applications from Algebraic Geometry

Sequences

Introduction

Linear Feedback Shift Register Sequences

Constructions of Almost Perfect Sequences
Constructions of Multisequences

Sequences with Low Correlation and Large Linear Complexity

Bibliography

Index

Über Autor(innen)

San Ling is a professor in the Division of Mathematical Sciences, School of Physical and Mathematical Sciences at Nanyang Technological University. He received a PhD in mathematics from the University of California, Berkeley. His research interests include the arithmetic of modular curves and application of number theory to combinatorial designs, coding theory, cryptography, and sequences.

Huaxiong Wang is an associate professor in the Division of Mathematical Sciences at Nanyang Technological University. He is also an honorary fellow at Macquarie University. He received a PhD in mathematics from the University of Haifa and a PhD in computer science from the University of Wollongong, Australia. His research interests include cryptography, information security, coding theory, combinatorics, and theoretical computer science.

Chaoping Xing is a professor at Nanyang Technological University. He received a PhD from the University of Science and Technology of China. His research focuses on the areas of algebraic curves over finite fields, coding theory, cryptography, and quasi-Monte Carlo methods.