Certification of the QR factor R, and of lattice basis reducedness

TL;DR

This paper introduces a rigorous algorithm for certifying whether a lattice basis is LLL-reduced by verifying the QR factor R, providing reliable certificates with minimal additional computational cost, and demonstrating its effectiveness through experiments.

Contribution

The paper presents a novel, efficient method for certifying lattice basis reduction and QR factorization accuracy using floating point error bounds, enhancing reliability of lattice reduction algorithms.

Findings

The certification algorithm requires only six times the cost of QR factorization.

It successfully certifies LLL-reduced bases in experiments.

The method reliably detects non-reduced bases or insufficient precision.

Abstract

Given a lattice basis of n vectors in Z^n, we propose an algorithm using 12n^3+O(n^2) floating point operations for checking whether the basis is LLL-reduced. If the basis is reduced then the algorithm will hopefully answer ''yes''. If the basis is not reduced, or if the precision used is not sufficient with respect to n, and to the numerical properties of the basis, the algorithm will answer ''failed''. Hence a positive answer is a rigorous certificate. For implementing the certificate itself, we propose a floating point algorithm for computing (certified) error bounds for the entries of the R factor of the QR matrix factorization. This algorithm takes into account all possible approximation and rounding errors. The cost 12n^3+O(n^2) of the certificate is only six times more than the cost of numerical algorithms for computing the QR factorization itself, and the certificate may be implemented using matrix library routines only. We report experiments that show that for a reduced basis of adequate dimension and quality the certificate succeeds, and establish the effectiveness of the certificate. This effectiveness is applied for certifying the output of fastest existing floating point heuristics of LLL reduction, without slowing down the whole process.