Number theory

This activity consists in illustrating various aspects of number theory, including some that can be applied to cryptography. The following themes will be tackled.
- Modular arithmetic: Chinese remainder theorem, quadratic reciprocity.
- Rational quadratic forms: p-adic numbers and local-global principle.
- Analytic methods: zeta function and Dirichlet's theorem.
- Projective cubics: arithmetic properties of elliptic curves.
- Algorithmic number theory: primality tests, factorization.
- Construction of objects relevant to cryptography: elliptic curves and cryptosystems.
The balance between those parts and the details of the program may vary from year to year.

The assessment is based on a written exam, testing the knowledge and understanding of the theoretical notions, the examples and fundamental results, the ability to develop a consistent reasoning, and the mastery of the proof techniques introduced during the lectures.
One or several mini-projects may be proposed during the quadrimester, and contribute to a maximum of 25% of the final grade for the course. This contribution is taken into account only if it is beneficial to the student.