Algebraic Geometry in Cryptology

Cryptology deals with mathematical techniques for design and analysis of algorithms and protocols for digital security in the presence of malicious adversaries. For example, encryption and digital signatures are used to construct private and authentic communication channels, which are instrumental to secure Internet transactions. 

Some of the most fascinating advances rely on algebraic geometry: Elliptic curves are becoming the new standard for public key primitives and research in elliptic curve cryptography received a recent boost with cryptographic pairings; hyperelliptic curves offer interesting performance for hardware implementations; Goppa codes for curves of genus 0 are used to construct secure code-based cryptosystems; curves over finite fields also find applications in secret sharing schemes and towers of algebraic function fields have an important bearing on strongly multiplicative secret haring schemes with good asymptotic properties. These schemes are highly relevant, as they are the basis for information-theoretically secure multi-party computation.  Depending on time we will also highlight the use of towers of function fields for constructing codes with particularly good error correction properties.

This course provides an introduction to cryptography and to algebraic geometry at a master course level. The main focus is on the diverse applications that algebraic geometry has in cryptology.

2 hours of lecture and 1 hour of exercise.

Basic algebra, number theory, probability theory, discrete
mathematics.

This is mainly aimed at mathematics students (particularly, algebra, and number theory and geometry). Large parts of the course may be well accessible to students of theoretical computer science with a strong (discrete) mathematics background. No prior knowledge of cryptology is required.