We give a random class of lattices in Zn whose elements can be generated together with a short vector in them so that, if there is a probabilistic polynomial time algorithm which finds a short vector in a random lattice with a probability of at least ~ then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in ev-e~g lattice in Zn with a probability exponentially close to one. [...] Key Method (2) Find the shortest nonzero vector in an n-dimensional lattice…Expand Abstract