Fine-grained hardness of lattice problems

Abstract

A lattice is a discrete set of points that follow a repeating arrangement of points and more formally defined as a discrete additive subgroup (like Zn) of a vector space (usually Rn). The most important computational problems on lattices are the Shortest vector problem (SVP), which is finding the shortest non-zero vector in a lattice and the Closest vector problem (CVP), which is finding the closest lattice vector to a given target vector; both of which are notoriously hard in practice. Even theoretically, NP-hardness is achieved at approximation ratio O(1) for approx-SVP (under random reductions) and O(√n) for approx-CVP. One approach to further explore the complexity of lattice algorithms beyond the coarse distinction of P and NP-hard and make stronger claims about the running time, is to use a collection of different fine-grained reductions (based on some reasonable conjecture). There have been many works based on the analysis of the fine-grained complexity of lattice problems, with varying assumptions such as ETH, SETH, Gap-SETH, etc. One notoriously hard problem, Label Cover and the implications of its hardness on lattice algorithms has not been studied in detail yet. In this thesis, we investigate and observe fine-grained reductions from approx-Label Cover to approx-CVP and other similar lattice problems.