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

Gilles Villard

TL;DR

The paper addresses certifying LLL reducedness of a lattice basis by certifying the $R$ factor of the QR decomposition of an integer basis. It introduces a floating-point algorithm to compute certified componentwise error bounds for $|\widetilde{R}-R|$ in the $R$ factor, using Sun92-type perturbation analysis and a Cholesky interpretation to bound errors under IEEE 754 arithmetic. The reducedness certificate combines fast numerical QR factorization with a certified bound $F$ and a verification of the properness and Lovász conditions, at a total cost of about $12n^3+O(n^2)$ operations, roughly six times the cost of a plain QR factorization. Experiments show the certificate is effective for adequately dimensioned and well-conditioned bases and can certify outputs of fast floating-point LLL heuristics without slowing the overall reduction process, thereby enabling reliable, rigorous reducedness checks in practice.

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.

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

TL;DR

The paper addresses certifying LLL reducedness of a lattice basis by certifying the factor of the QR decomposition of an integer basis. It introduces a floating-point algorithm to compute certified componentwise error bounds for in the factor, using Sun92-type perturbation analysis and a Cholesky interpretation to bound errors under IEEE 754 arithmetic. The reducedness certificate combines fast numerical QR factorization with a certified bound and a verification of the properness and Lovász conditions, at a total cost of about operations, roughly six times the cost of a plain QR factorization. Experiments show the certificate is effective for adequately dimensioned and well-conditioned bases and can certify outputs of fast floating-point LLL heuristics without slowing the overall reduction process, thereby enabling reliable, rigorous reducedness checks in practice.

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.

Paper Structure

This paper contains 17 sections, 4 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Error bounds computation and verification algorithms
  3. Perturbation analyses and bounds for the $QR$ factorization
  4. Strict componentwise bounds for the $R$ factor
  5. $QR$ and Cholesky factorization
  6. The bound on $|\widetilde{R}-R|$
  7. Toward an implementation
  8. Invertibility check of $\widetilde{R}$
  9. Bounding $G$
  10. Bounding the spectral radius of $G$
  11. Bounding $|\widetilde{R}-R|$
  12. Error bounding algorithm for the $QR$ factor $R$
  13. Certified bounds for floating point matrix expressions
  14. Computing an error bound
  15. Computational results
  16. ...and 2 more sections

Key Result

Theorem 4.1

For $B, \widetilde{B} \in {\mathbb R} ^{n \times n}$ symmetric positive definite matrices, let $R$ and $\widetilde{R}$ be the Cholesky factors of $B$ and $\widetilde{B}$. Let $E= \widetilde{B}-B$, and Then if $\rho (G) < 1$ we have

Theorems & Definitions (5)

  • Theorem 4.1
  • Theorem 4.2
  • proof
  • Theorem 6.1
  • Theorem 7.1