Lattices have received considerable attention as a potential
source of computational hardness to be used in cryptography,
after a breakthrough result of [Ajtai, STOC 1996]
connecting the average-case and worst-case complexity of various
lattice problems. The purpose of this paper is twofold. On the
expository side, we present a rigorous self contained proof of
results along the lines of Ajtai's seminal work. At the same
time, we explore to what extent Ajtai's original results can be
quantitatively improved. As a by-product, we define a random
class of lattices such that computing short nonzero vectors in
the class with non-negligible probability is at least as hard as
approximating the length of the shortest nonzero vector in
*any* n-dimensional lattice within worst-case
approximation factors *gamma(n)=n ^{3} omega(sqrt{log
n log log n})*. This improves previously known best
connection factor