The Shortest Vector in a Lattice is Hard to Approximate to within Some Constant

Abstract

We show that approximating the shortest vector problem (in any $\ell_p$ norm) to within any constant factor less than $\sqrt[p]2$ is hard for NP under reverse unfaithful random reductions with inverse polynomial error probability. In particular, approximating the shortest vector problem is not in RP (random polynomial time), unless NP equals RP. We also prove a proper NP-hardness result (i.e., hardness under deterministic many-one reductions) under a reasonable number theoretic conjecture on the distribution of square-free smooth numbers. As part of our proof, we give an alternative construction of Ajtai's constructive variant of Sauer's lemma that greatly simplifies Ajtai's original proof.