Finding a Very Short Lattice Vector in the Extended Search Space

Abstract

The problem of finding a lattice vector approximating a shortest nonzero lattice vector (approximate SVP) is a serious problem that concerns lattices. Finding a lattice vector of the secret key of some lattice-based cryptosys- tems is equivalent to solving some hard approximate SVP. We call such vectors very short vectors (VSVs). Lattice basis reduction is the main tool for finding VSVs. However, the main lattice basis reduction algorithms cannot find VSVs in lattices in dimensions ∼200 or above. Exhaustive search can be considered to be a key technique toward eliminating the limitations with current lattice basis reduction algorithms. However, known methods of carrying out exhaustive searches can only work in relatively low-dimensional lattices. We defined the extended search space (ESS) and experimentally confirmed that exhaustive searches in ESS make it possible to find VSVs in lattices in dimensions ∼200 or above with the parameters computed from known VSVs. This paper presents an extension of our earlier work. We demonstrate the practical effectiveness of our technique by presenting a method of choosing the parameters without known VSVs. We also demonstrate the effectiveness of distributed searches.