Hardness of Approximating the Shortest Vector Problem in Lattices

Let p > 1 be any fixed real. We show that assuming NP RP, there is no polynomial time algorithm that approximates the Shortest Vector Problem (SVP) in p norm within a constant factor. Under the stronger assumption NP RTIME(2poly(log n) ), we show that there is no polynomial-time algorithm with approximation ratio 2(log n)1/2where n is the dimension of the lattice and > 0 is an arbitrarily small constant. We first give a new (randomized) reduction from Closest Vector Problem (CVP) to SVP that achieves some constant factor hardness. The reduction is based on BCH Codes. Its advantage is that the SVP instances produced by the reduction behave well under the augmented tensor product, a new variant of tensor product that we introduce. This enables us to boost the hardness factor to 2(log n)1/2. Categories and Subject Descriptors: F.2 [Theory of Computation]: Analysis of Algorithms and Problem Complexity General Terms: Algorithms, Theory Additional Key Words and Phrases: Approximation...

• Type: Conference
• Year: 2004
• Where: FOCS
• Authors: Subhash Khot
• Keywords: Constant Factor, FOCS 2004, Shortest Vector Problem, Theoretical Computer Science, Vector Problem