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Publisher:

Chapman & Hall/CRC

Publication Date:

2013

Number of Pages:

321

Format:

Hardcover

Series:

Discrete Mathematics and Its Applications

Price:

79.95

ISBN:

9781420079463

Category:

Textbook

[Reviewed by , on ]

Felipe Zaldivar

09/28/2013

The use of algebraic curves over finite fields in coding theory (Goppa codes) and in cryptography (Koblitz and Miller, for the case of elliptic curves) has attracted a lot of attention to a field that was the domain of algebraic geometers and number theorists, for the benefit of the field itself and for the actual or potential applications to everyday tasks. Many textbooks and monographs have been published in this now-crowded field, focusing either on coding theory or cryptography, where in the last case the focus usually is on the highly developed elliptic curve cryptography.

The main goal of the book under review is to call attention and address applications of algebraic curves of higher genus to some important topics in cryptography. The book starts by recalling some basic facts on the geometry and arithmetic of algebraic curves over finite fields. This is followed by a brief discussion of algebraic codes in chapter two, including a few pages on algebraic geometry codes, with references to (two of) the authors previous book on the subject for details. Next, a long chapter three is devoted to elliptic curve cryptography.

After these preliminaries, the remaining chapters treat some selected topics in cryptography such as secret sharing schemes, authentication codes, frameproof codes, key distribution systems, broadcast encryption and sequences. It must be said that the exposition in these chapters is more detailed, covering the algebraic and combinatorial constructions in detail, with the algebraic curves applications coming mainly in the form of examples obtained from algebraic geometry codes, usually in the last few sections of the corresponding chapter, with the exception of the last one. The book is filled with examples to illustrate the various constructions and, assuming a basic knowledge of combinatorics and algebraic geometry it is almost self-contained. However, for its use a textbook the instructor must provide the exercises, since the book comes with no exercises at all.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fz@xanum.uam.mx.

**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*(F_{q}) 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**

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