Faster Lattice Enumeration

TL;DR

This paper introduces obtuse bases, which can be computed efficiently and enable exponential speedups in lattice enumeration algorithms for finding shortest vectors, improving lattice reduction techniques.

Contribution

The paper defines obtuse bases and shows they can be computed in polynomial time, leading to faster lattice enumeration algorithms for shortest vector problems.

Findings

Obtuse bases can be computed in O(n^4) time.

Using obtuse bases makes lattice enumeration exponentially faster.

Extreme pruning algorithm can be applied on obtuse 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. 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.