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Algebraic Curves in Cryptography

San Ling, Huaxiong Wang, and Chaoping Xing
Chapman & Hall/CRC
Publication Date: 
Number of Pages: 
Discrete Mathematics and Its Applications
[Reviewed by
Felipe Zaldivar
, on

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

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
Linear Codes
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
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

Linear Feedback Shift Register Sequences
Constructions of Almost Perfect Sequences
Constructions of Multisequences
Sequences with Low Correlation and Large Linear Complexity