Algebraic Curves in Cryptography

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.

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 CryptographyBasic Introduction Maps between Elliptic Curves The Group E(Fq) and Its Torsion Subgroups Computational Considerations on Elliptic Curves Pairings on an Elliptic CurveElliptic Curve Cryptography Secret Sharing Schemes The Shamir Threshold Scheme Other Threshold SchemesGeneral Secret Sharing SchemesInformation RateQuasi-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 Boundsof A-CodesA-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 GeometryAsymptotic Bounds and Constructions from Algebraic GeometryImprovements to the Asymptotic Bound Key Distribution Schemes Key Predistribution Key Predistribution Schemes with Optimal Information RatesLinear Key Predistribution Schemes Key Predistribution Schemes from Algebraic Geometry Key Predistribution Schemes from Cover-Free FamiliesPerfect Hash Families and Algebraic Geometry Broadcast Encryption and Multicast Security One-Time Broadcast EncryptionMulticast Re-Keying SchemesRe-Keying Schemes with Dynamic Group Controllers Some Applications from Algebraic Geometry Sequences Introduction Linear Feedback Shift Register Sequences Constructions of Almost Perfect SequencesConstructions of Multisequences Sequences with Low Correlation and Large Linear Complexity Bibliography Index

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.