Faster Lattice Enumeration

Abstract

A lattice reduction is an algorithm that transforms the given basis of the lattice to another lattice basis such that problems like finding a shortest vector and closest vector become easier to solve. Some of the famous lattice reduction algorithms are LLL and BKZ reductions. We define a class of bases called \emph{obtuse bases} and show that any lattice basis can be transformed to an obtuse basis in $\mathcal{O}(n^4)$ time. A shortest vector s can be written as $v_1b_1+\cdots+v_nb_n$ where $b_1,\dots,b_n$ are the input basis vectors and $v_1,\dots,v_n$ are integers. When the input basis is obtuse, all these integers can be chosen to be positive for a shortest vector. This property of the obtuse basis makes lattice enumeration algorithm for finding a shortest vector exponentially faster. Moreover, extreme pruning, the current fastest algorithm for lattice enumeration, can be run on an obtuse basis.