Elliptic curves and their application to cryptograpy

Presentation

Cryptography is a very old disciplin, whose first traces go back to the Antiquity. It is now ubiquitous in the modern world, for the internet, payment orders, email exchanges, credit cards, television, telecommunications... In this lecture, we will focus on the cryptosystems, and their algorithms, based on elliptic curves. These systems are very commonly used, and have overcome in efficiency and security protocols based on classical arithmetic such as RSA. We will also mention problems arising from the imminent arrival of quantum computers (post quantum cryptography). The lecture will be organised in three times.

1. We will develop the elementary theory of elliptic curves on complex numbers, finite fields and mod n. Among the needed technical tool for a proper understanding of the algorithms, we will study the Weil coupling and Hasse theorem. .

2. As application of these abstract notions, we will introduce classical protocoles EDCHE, EDSA, IBE Boneh-Franklin...

3. In TA sessions on computers, we will implement the algorithms, in particular BSGS, MOV, Schoff,... and we will give applications to the factorization of integers.