Hardness of Lattice Problems in `p Norm

We show that for any integer p ≥ 46, the Shortest Vector Problem in `p norm is hard to approximate within factor (4/3)1−45.7/p. We also show a hardness factor of (3/2)1−111.7/p for p ≥ 112. As p grows, these factors approach 4/3 and 3/2 respectively. Both results hold under the assumption NP 6⊆ ZPP. We give a very simple reduction from known hardness results for Hypergraph Independent Set Problem. For certain range of values of p, our results improve upon the hardness factors of 2− δ and p1−δ due to Micciancio [22] and Khot [14] respectively. We hope that our line of work would eventually give a simple proof of hardness of SVP in `2 norm. In the spirit of proving hardness results for lattice problems in `p norms, we also show that an important variant of SVP, the Unique Shortest Vector Problem, is NP-Hard (under randomized reductions) for p ≥ 1. This generalizes a result of Kumar and Sivakumar [16], who showed this for p = 2.