分野別セミナー

Isogenies are morphism between elliptic curves. Their computation is of importance in cryptography for at least two reasons. Firstly, they can be used to count the number of points on elliptic curves, which is used to discard some of them from cryptographic applications. Secondly, some propositions of quantum-resistant cryptosystems, following De Feo, Jao, and Plut's 2011 article, rely directly on isogenies.


To tackle this problem, computations over the p-adic numbers can be a decisive tool. The field of p-adic numbers is a field extension of the rational numbers analogous to the real numbers, but much more suited for arithmetic applications. As for the real numbers, computation over the p-adic numbers has to be done at finite precision.


With X. Caruso and D. Roe, we have provided a new method to handle precision over p-adics that relies on differentials and first-order approximation. In a joint work with P. Lairez, we apply this method to tackle the problem of the computation of isogenies between elliptic curves.