Algebraic Curves in Cryptography: 1st Edition (Paperback) book cover

Algebraic Curves in Cryptography

1st Edition

By San Ling, Huaxiong Wang, Chaoping Xing

Chapman and Hall/CRC

340 pages | 15 B/W Illus.

Purchasing Options:$ = USD
New in Paperback: 9781138381414
pub: 2019-06-17
Hardback: 9781420079463
pub: 2013-06-13
eBook (VitalSource) : 9780429187032
pub: 2013-06-13
from $35.50

FREE Standard Shipping!


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.


"This is a self-contained book intended for researchers and graduate students in mathematics and computer science interested in different topics in cryptography involving algebraic curves … The authors of this book make an exhaustive review on some other topics where algebraic curves, mainly in higher genus, are important as well."

Zentralblatt MATH 1282

"The book is written in a user-friendly style, with good coverage of the background, many examples, and a detailed bibliography of over 180 items. It is mainly directed towards graduate students and researchers, but some parts of the book are even accessible for advanced undergraduate students. The book is highly recommended for readers interested in the manifold applications of algebraic curves over finite fields."

—Harald Niederreiter, Mathematical Reviews, March 2014

"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."

—Felipe Zaldivar, MAA Reviews, September 2013

Table of Contents

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



About the Authors

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.

About the Series

Discrete Mathematics and Its Applications

Learn more…

Subject Categories

BISAC Subject Codes/Headings:
COMPUTERS / Security / General
MATHEMATICS / Number Theory
MATHEMATICS / Combinatorics