Finite Fields and Coding Theory

► Daniel Panario - Carleton University, Canada

This course is an introduction to finite fields, emphasizing their structure and applications to coding theory. Topics include linear codes, cyclic codes, and Reed–Solomon codes, illustrating how the structure of finite fields enables efficient encoding and decoding techniques.

Function Fields and Algebraic Curves over Finite Fields

► Luciane Quoos - Univeridade Federal do Rio de Janeiro, Brazil 

► Daniele Bartoli - Univeristà degli Studi di Perugia, Italy

This course will explore the deep connections between function fields and algebraic curves over finite fields. Topics include foundations of the theory of algebraic function fields, such as places, divisors, the Riemann–Roch theorem, extensions of algebraic function fields, as well as preliminaries of algebraic curves over finite fields, the Hasse–Weil bound, maximal curves, automorphism groups and Weierstrass points. 

The course also includes hands-on experience with MAGMA to explore computational aspects of function fields and algebraic curves.

Elliptic Curves and applications to Cryptography

► Maria Corte-Real Santos - ENS de Lyon, France

► Travis Morrison - Virginia Tech, United States

This course will first introduce the fundamental mathematics of elliptic curves. It will then discuss cryptosystems based on their arithmetic properties, starting from the first ones proposed by Koblitz and Miller in 1985, up to the most recent constructions based on isogenies in the context of post-quantum cryptography.

The course also includes practical sessions using SageMath to implement cryptographic protocols based on elliptic curves.

Algebraic Geometric Codes

► Beth Malmskog - Colorado College, United States 

► Alain Couvreur - INRIA Saclay, France

This course will focus on the construction and properties of algebraic geometric codes, covering codes defined over curves and possibly higher dimensional varieties. Topics will include evaluation codes, their parameters, decoding, and local properties of AG codes.

Code-based Cryptography
► Alain Couvreur - INRIA Saclay, France

This course will cover the key concepts of code-based encryption and digital signatures, focusing on well-known systems like the McEliece cryptosystem. Students will learn about the security assumptions underlying these systems, such as the hardness of decoding random linear codes, and will explore both the advantages and challenges of using code-based cryptography compared to other post-quantum approaches.