Isogenies of Elliptic Curves and their Application to Cryptography

Isogenies of elliptic curves, or more generally of abelian varieties, are surjective homomorphism having finite kernel and they play an important role in the study of arithmetic and geometric properties of elliptic curves. Moreover in recent years there has been an increasing interest in isogeny of elliptic curves from cryptographers. The main reason lies in quantum computer as Luca de Feo eloquently puts it: ''The main reason for this is the sudden realization by the cryptographic community of the very possibly near arrival of a general purpose quantum computer. While the capabilities of such futuristic machine would render all of Elliptic Curves Cryptography and Pairing Based Cryptography suddenly worthless, Isogeny Based Cryptography seems to resist much better to the cryptanalytic powers of the quantum computer'' (Mathematics of Isogeny Based Cryptography, arXiv:1711.04062).

The first week of the school will be devoted to lay down the basis for the more advanced courses of the second week: Algebraic Number Theory, Finite Fields, Elliptic Curves, Graph theory (split between the 2 weeks). All this courses are introductory and there will have exercise/training session. In the second week we have one introductory course Elliptic curves over Finite fields and their Endomorphisms Rings, and two advanced courses Isogenies of Elliptic curves and Isogeny based cryptography. For the courses of the second week we plan to have hands on training sessions (i.e. on computers) as well as exercises sessions.

Official language of the school: English