School Program
School Activities
The format of a typical day of the summer school will consist of three 60-minute lectures and three 60-minute exercise sessions (total of six hours of academic activities). The lectures will emphasize carefully worked out examples rather than formal proofs. During the exercise sessions, the participants will receive a list of exercises and will work on those exercises in small groups, receiving hints from the instructor if needed. The solutions of the exercises will be discussed at the end of the exercise session. Some of the exercise sessions will focus on computational mathematics; here the participants will be introduced to computational programs such as PARI/GP, SageMath and Magma, and some open ended experimentation will be done using these programs.
In addition to the lectures and exercise sessions, the participants will be divided into groups of 4 or 5 and will work on a project under the guidance of one of the instructors of the school. Special time slots will be allocated for the groups to meet to discuss the projects. Each group will give a 60-minute presentation about their project on the final day of the school.
Description of Courses
Title: Introduction to Cryptography
Instructor: Diana Davidova
Description: This course will introduce symmetric and asymmetric cryptography concepts, their differences, advantages and disadvantages, and give a brief historical overview of cryptography (from the ancient times to WWII and modern cryptosystems). Symmetric cryptography will be discussed in more detail, in particular, block ciphers. The algebraic properties of block ciphers that guarantee their security against different types of attacks will be studied.
Title: Introduction to Elliptic Curves
Instructor: Mihran Papikian
Description: This course will introduce the concept of elliptic curve, the group structure on points of an elliptic curve, isogenies, Galois action on torsion points, elliptic curves over finite fields, Frobenius endomorphism, analytic uniformization over complex numbers, and the Mordell-Weil theorem.
Title: Elliptic Curves and Modular Forms
Instructor: Valentijn Karemaker
Description: Elliptic curves and modular forms are intimately connected, as can be seen for instance in the famous proof of Fermat’s Last Theorem and subsequent research in this direction. The first goal of this course is to give an introduction to the theory of modular forms, starting with modular groups and lattice functions, and only assuming basic complex analysis. The second goal is to indicate some connections between modular forms and elliptic curves, through Hecke operators, L-functions, and modular curves, focussing on explicit examples where these connections become visible.
Title: Elliptic Curves and Cryptography
Instructor: Fabien Pazuki
Description: Elliptic curves over finite fields are key objects in modern cryptography. They are used in two ways: 1) The group structure of their rational points. 2) Isogeny graphs. In both cases, a good understanding of the endomorphisms of elliptic curves is crucial. The goal of the course is to focus on these endomorphisms. One of the main outcomes for the interested student will be to realize that even if the whole cryptographic setting concerns curves over finite fields, one is forced to understand a good share of algebraic number theory and of quaternions algebras.
Title: Computational Aspects of Elliptic Curves
Instructor: Alp Bassa
Description: An overview of various algorithmic aspects of elliptic curves will be given. This will include topics like Lenstra’s elliptic-curve factorization method, pairings on elliptic curves and their applications, computation of isogenies, endomorphisms and torsion, the ECPP algorithm.