Obtuse Lattice Bases

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. We define a class of bases called obtuse bases and show that any lattice basis can be transformed to an obtuse basis. A shortest vector $\mathbf{s}$ can be written as $\mathbf{s}=v_1\mathbf{b}_1+\dots+v_n\mathbf{b}_n$ where $\mathbf{b}_1,\dots,\mathbf{b}_n$ 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 obtuse bases makes the lattice enumeration algorithm for finding a shortest vector exponentially faster. We have implemented the algorithm for making bases obtuse, and tested it some small bases.