Arithmetic, Coding and Cryptography

The scientific program will feature a foundational course in algebraic number theory, with a special focus on non-maximal orders that are used in algorithmic applications such as the number fields sieve, and also arise as endomorphism rings of CM elliptic curves. Key topics are the decomposition of primes, Frobenius elements, and the computation of units and class groups. In parallel, the theory of elliptic curves will be examined in depth, covering, among others, their group structure, torsion points, isogenies, endomorphism rings, the Weil pairing, the Mordell-Weil theorem, elliptic curves over finite fields, which are essential concepts for the applications in cryptography.
The course on isogeny-based cryptography will begin by exploring the algorithmic aspects of elliptic curves and isogenies, using these foundations to introduce simple elliptic curve-based cryptographic schemes, such as the Diffie-Hellman key exchange and the Charles-Goren-Lauter hash function. It will then move to more advanced topics, including the Deuring correspondence and the class group action of complex multiplication, and the cryptographic applications that emerge from these ideas. Additionally, coding theory will be approached from an algebraic-geometric perspective, with particular emphasis on the construction of error-correcting linear codes. Students will also explore modern cryptographic protocols, learning how elliptic curves and isogenies play a central role in designing secure communication systems. Hands-on computational sessions using tools like GP Pari or Sage will provide practical experience, enabling students to implement cryptographic algorithms and perform elliptic curve computations.