A Relation of Primal-Dual Lattices and the Complexity of Shortest Lattice Vector Problem

We give a simpliied proof of a theorem of Lagarias, Lenstra and Schnorr 17] that the problem of approximating the length of the shortest lattice vector within a factor of Cn, for an appropriate constant C, cannot be NP-hard, unless NP = coNP. We also prove that the problem of ndng a n 1=4-unique shortest lattice vector is not NP-hard under polynomial time many-one reductions, unless the polynomial time hierarchy collapses.