Abelian varieties and cryptography
GERHARD FREY
Institut für Experimentelle Mathematik,
Universituisburg-Essen, 45326 Essen, Germany.
TONY SHASKA
Department of Mathematics and Statistics,
Oakland University, Rochester, MI 48309, USA.
Abstract. The main purpose of this paper is to give the latest developments in the theory of abelian varieties and their usage in cryptography. In the first part we provide the necessary mathematical background on abelian varieties, their torsion points, Honda-Tate theory, Galois representations, with emphasis on Jacobian varieties and hyperelliptic Jacobians. In the second part we focus on applications of abelian varieties on cryptography and treating separately, elliptic curve cryptography, genus 2 and 3 cryptography, including Diffie-Hellman Key Exchange, index calculus in Picard groups, isogenies of Jacobians via correspondences and applications to discrete logarithms. Several open problems and new directions are suggested.
2010 Mathematics Subject Classification. 14H10,14H45.
Contents
Contents
Preface
Part 1. Abelian varieties
 1. Definitions and basic properties
 2. Endomorphisms and isogenies
 3. Projective Curves and Jacobian Varieties
 4. Applications of the Riemann-Roch Theorem
 5. Modular curves
Part 2. Cryptography
 6. Diffie-Hellman Key Exchange
 7. Index calculus in Picard groups
 8. Isogenies of Jacobians via correspondences and applications to discrete logarithms
 9. Genus 3 curves and cryptography
 10. Genus 2 curves and cryptography
 11. Elliptic curve cryptography
References
Preface
There has been a continued interest on Abelian varieties in mathematics during the last century. Such interest is renewed in the last few years, mostly due to applications of abelian varieties in cryptography. In these notes we give a brief introduction to the mathematical background on abelian varieties and their applications on cryptography with the twofold aim of introducing abelian varieties to the experts in cryptography and introducing methods of cryptography to the mathematicians working in algebraic geometry and related areas.
A word about cryptography
Information security will continue to be one of the greatest challenges of the modern world with implications in technology, politics, economy, and every aspect of everyday life. Developments and drawbacks of the last decade in the area will continue to put emphasis on searching for safer and more efficient crypto-systems. The idea and lure of the quantum computer makes things more exciting, but at the same time frightening.
There are two main methods to achieve secure transmission of information: secret-key cryptography (symmetric-key) and public-key cryptography (asymmetric-key). The main disadvantage of symmetric-key cryptography is that a shared key must be exchanged beforehand in a secure way. In addition, managing keys in a large public network becomes a very complex matter. Public-key cryptography is used as a complement to secret-key cryptography for signatures of authentication or key-exchange. There are two main methods used in public-key cryptography, namely RSA and the discrete logarithm problem (DLP) in cyclic groups of prime order which are embedded on abelian varieties. The last method is usually referred to as curve-based cryptography.
In addition, there is always the concern about the post-quantum world. What will be the crypto-systems which can resist the quantum algorithms? Should we develop such systems now? There is enthusiasm in the last decade that some aspects of curve-based cryptography can be adapted successfully to the post-quantum world. Supersingular Isogeny Diffie-Hellman (SIDH), for example, is based on isogenies of supersingular elliptic curves and is one of the promising schemes for post-quantum cryptography. Isogenies of hyperelliptic Jacobians of dimension 2 or 3 have also been studied extensively in the last decade and a lot of progress has been made. In this paper we give an overview of recent developments in these topics.
Audience
Computer security and cryptography courses for mathematics and computer science majors are being introduced in all major universities. Curve-based cryptography has become a big part of such courses and a popular area even among professional mathematicians who want to get involved in cryptography. The main difficulty that these newcomers is the advanced mathematical background needed to be introduced to curve-based cryptography.
Our target audience is advanced graduate students and researchers from mathematics or computer science departments who work with curve-based cryptography. Many researchers from other areas of mathematics who want to learn about abelian varieties and their use in cryptography will find these notes useful.
For the full paper click here:
2018-1: Curves, Jacobians, and cryptography, G. Frey, T. Shaska, Contemporary Math. 724Â 279--344Â (2019)