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, although the lack of exercises makes this book more suitable for researchers.
Algebraic curves are the main ingredient in elliptic curve cryptography (ECC), which is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Elliptic curves are also used in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic curve factorization.
Most of the literature involving both algebraic curves and cryptography is focused on ECC , but algebraic curves goes far beyond elliptic curve or public key cryptography. The authors of this book make an exhaustive review on some other topics where algebraic curves, mainly in higher genus, are important as well. After three introductory chapters, they address the problems in secret sharing schemes, authentication codes, frameproof codes, key distribution schemes, broadcast encryption, and sequences. It is worthwhile to mention that algebraic curves come up essentially (but not only) as an application of algebraic geometric codes to every topic.
Table of contents:
1) Introduction to Algebraic Curves; 1.1) Plane Curves. 1.2) Algebraic Curves and Their Function Fields. 1.3) Smooth Curves. 1.4) Riemann-Roch Theorem. 1.5) Rational Points and Zeta Functions.
2) Introduction to Error-Correcting Codes: 2.1) Introduction. 2.2) Linear Codes. 2.3) Bounds. 2.4) Algebraic Geometry Codes. 2.5) Asymptotic Behavior of Codes.
3) Elliptic Curves and Their Applications to Cryptography: 3.1) Basic Introduction. 3.2) Maps between Elliptic Curves. 3.3) The Group \(E(\mathbb F_q)\) and Its Torsion Subgroups. 3.4) Computational Considerations on Elliptic Curves. 3.5) Pairings on an Elliptic Curve. 3.6) Elliptic Curve Cryptography
4) Secret Sharing Schemes: 4.1) The Shamir Threshold Scheme. 4.2) Other Threshold Schemes. 4.3) General Secret Sharing Schemes. 4.4) Information Rate. 4.5) Quasi-Perfect Secret Sharing Schemes. 4.6) Linear Secret Sharing Schemes. 4.7) Multiplicative Linear Secret Sharing Schemes. 4.8) Secret Sharing from Error-Correcting Codes. 4.9) Secret Sharing from Algebraic Geometry Codes
5) Authentication Codes: 5.1) Authentication Codes. 5.2) Bounds of A-Codes. 5.3) A-Codes and Error-Correcting Codes. 5.4) Universal Hash Families and A-Codes. 5.5) A-Codes from Algebraic Curves. 5.6) Linear Authentication Codes
6) Frameproof Codes: 6.1) Introduction. 6.2) Constructions of Frameproof Codes without Algebraic Geometry. 6.3) Asymptotic Bounds and Constructions from Algebraic Geometry. 6.4) Improvements to the Asymptotic Bound
7) Key Distribution Schemes: 7.1) Key Predistribution. 7.2) Key Predistribution Schemes with Optimal Information Rates. 7.3) Linear Key Predistribution Schemes. 7.4) Key Predistribution Schemes from Algebraic Geometry. 7.5) Key Predistribution Schemes from Cover-Free Families. 7.6) Perfect Hash Families and Algebraic Geometry
8) Broadcast Encryption and Multicast Security: 8.1) One-Time Broadcast Encryption. 8.2) Multicast Re-Keying Schemes. 8.3) Re-Keying Schemes with Dynamic Group Controllers
8.4) Some Applications from Algebraic Geometry
9) Sequences: 9.1) Introduction. 9.2) Linear Feedback Shift Register Sequences. 9.3) Constructions of Almost Perfect Sequences. 9.4) Constructions of Multisequences. 9.5) Sequences with Low Correlation and Large Linear Complexity.