Abstract:
We give a random class of n dimensional lattices 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 1/2 then there is also a probabilistic polynomial time algorithm which solves the following three lattice problems in every n dimensional lattice with a probability exponentially close to one. (1) Find the length of a shortest nonzero vector in an n-dimensional lattice, approximately, up to a polynomial factor. (2) Find the shortest nonzero vector in an n-dimensional lattice L where the shortest vector is unique in the sense that any other vector whose length is only a polynomial times longer, is parallel to it. (3) Find a basis in the lattice L whose length is the smallest possible up to a polynomial factor, where the length of a basis is the maximum of the lengths of its elements.

Comment #1 to TR96-007 | 8th February 1996 08:05

Corrected Proof of Lemma 4. Comment on: TR96-007

Abstract:
It has been pointed out by several readers that the proof of Lemma 4 is incorrect. A corrected version of the proof is given here and the sketch of an alternative proof. The lemma states that if there is a linearly independent set with maximal vector length M then there is a basis of length nM.