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.
Table of Contents
Introduction to Algebraic Curves
Algebraic Curves and Their Function Fields
Rational Points and Zeta Functions
Introduction to Error-Correcting Codes
Algebraic Geometry Codes
Asymptotic Behavior of Codes
Elliptic Curves and Their Applications to Cryptography
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
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
Bounds of A-Codes
A-Codes and Error-Correcting Codes
Universal Hash Families and A-Codes
A-Codes from Algebraic Curves
Linear Authentication 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 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
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.
"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