Section: New Results

Elliptic curve and Abelian varieties cryptology

Participant : Damien Robert.

In [22], E. Milio and D. Robert describe an algorithm to evaluate in quasi-linear time Hilbert modular functions in dimension 2, and also how to recover in time quasi-linear the period matrix from the value of the function. They apply this theory to the modular functions $j(\tau /\beta )$ and $\theta (\tau /\beta )$ where $\beta$ is a totally real positive number of the quadratic real field corresponding to the Hilbert surface to construct modular polynomials parametrizing cyclic isogenies between principally polarised abelian varieties. This extends the construction of classical modular polynomials but allow to have much smaller polynomials, which allow to compute them up to norm $\ell =91$ rather than $\ell =7$ in dimension 2 for classical polynomials.

In [20], Dudeanu, Alina and Jetchev, Dimitar and Robert, Damien and Vuille, Marius describe an algorithm to compute cyclic isogenies from their kernels. This extends the work of [10] from isogenies with maximal isotropic kernels for the Weil pairing to cyclic isogenies, using real multiplication. Such isogenies are indispensable to fully explore the isogeny graph and will be able to speed up a lot of algorithms that needs isogenous curves, like the CRT method for class polynomials.