I have an $n$-dimensional axis-aligned hypercube and a lattice generated by $n$ linearly independent vectors, both in $\mathbb{Z}^n$. My wish is to find all points of the lattice which fall in the interior of the hypercube.
Currently I accomplish this by reducing the lattice using LLL, using the reduced basis to find a small parallelepiped entirely containing the hypercube, then iterating through every lattice point within this parallelepiped, checking if each is contained within the desired region.
Of course, this is a very naive way of doing things, and generates a lot of unnecessary work for even semi-large $n$. The question is then what the most efficient (checking the least number of lattice points) way of finding all points of a (reduced) lattice contained within a hypercube is. If I only want one point does it get easier?