Reducing lattice bases by means of approximations

Abstract

Let L be a k-dimensional lattice in IR^m with basis B = (b ₁,...,b _k). Let A = (a ₁,...,a _k) be a rational approximation to B. Assume that A has rank k and a lattice basis reduction algorithm applied to the columns of A yields a transformation T = (t ₁,...,t _k) ε GL(k, ℤ) such that A t _i ≤ s _i λ _i (L(A)) where L(A) is the lattice generated by the columns of A, λ _i (L(A)) is the i-th successive minimum of that lattice and s _i ≥ 1, 1 ≤ i ≤ k. For c > 0 we determine which precision of A is necessary to guarantee that B t _i ≤ (1+c)s _i λ _i (L), 1 ≤ i ≤ k. As an application it is shown that Korkine-Zolotaref-reduction and LLL-reduction of a non integer lattice basis can be effected almost as fast as such reductions of an integer lattice basis.