Obtuse Lattice Bases

TL;DR

This paper introduces obtuse bases in lattice reduction, showing they simplify shortest vector problems and accelerate algorithms, with implementation and testing on small bases demonstrating practical benefits.

Contribution

It defines obtuse bases, proves any lattice basis can be transformed into one, and shows this transformation speeds up shortest vector algorithms.

Findings

Obtuse bases allow all coefficients of shortest vectors to be positive.

Transforming bases to obtuse form exponentially speeds up lattice enumeration.

Implemented algorithm successfully tested on small 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.