Lattice Problems, Gauge Functions and Parameterized Algorithms

TL;DR

This paper introduces new parameterized algorithms for the subspace avoiding problem in lattices, utilizing AKS sieving, with improved approximation and exact solutions based on subspace dimension and gauge functions.

Contribution

It presents novel parameterized algorithms for SAP using AKS sieving, including approximation and exact algorithms that depend on subspace dimension and gauge functions.

Findings

A randomized (1+ε)-approximation algorithm with runtime 2^{O(n)}.(1/ε)^k.

A 2^{O(n+k log k)} exact algorithm for SAP in any ℓ_p norm.

Lower bound of Ω(2^n) on query complexity for AKS sieving-based SVP algorithms.

Abstract

Given a k-dimensional subspace M\subseteq \R^n and a full rank integer lattice L\subseteq \R^n, the \emph{subspace avoiding problem} SAP is to find a shortest vector in L\setminus M. Treating k as a parameter, we obtain new parameterized approximation and exact algorithms for SAP based on the AKS sieving technique. More precisely, we give a randomized $(1+\epsilon)$-approximation algorithm for parameterized SAP that runs in time 2^{O(n)}.(1/\epsilon)^k, where the parameter k is the dimension of the subspace M. Thus, we obtain a 2^{O(n)} time algorithm for \epsilon=2^{-O(n/k)}. We also give a 2^{O(n+k\log k)} exact algorithm for the parameterized SAP for any \ell_p norm. Several of our algorithms work for all gauge functions as metric with some natural restrictions, in particular for all \ell_p norms. We also prove an \Omega(2^n) lower bound on the query complexity of AKS sieving based exact algorithms for SVP that accesses the gauge function as oracle.