Module lattices and their shortest vectors

Abstract

We study the shortest vector lengths in module lattices over arbitrary number fields, with an emphasis on cyclotomic fields. In particular, we sharpen the techniques of arXiv:2308.15275v2 to establish improved results for the variance of the number of lattice vectors of bounded Euclidean norm in a random module lattice. We then derive tight probabilistic bounds for the shortest vector lengths for several notions of random module lattice.

20 pages. This article is an improved and augmented version of some of the results in arXiv:2402.10305