Dense and smooth lattices in any genus

The Lattice Isomorphism Problem (LIP) was recently introduced as a new hardness assumption for post-quantum cryptography. The strongest known efficiently computable invariant for LIP is the genus of a lattice. To instantiate LIP-based schemes one often requires the existence of a lattice that (1) lies in some fixed genus, and (2) has some good geometric properties such as a high packing density or a small smoothness parameter.

In this talk I will show that such lattices exist. In particular, building upon classical results by Siegel (1935), we will see that essentially any genus contains a lattice with a close to optimal packing density, smoothing parameter and covering radius.