Reducing lattice bases by means of approximations

Abstract

Let L be a k-dimensional lattice in ℝm with basis B = (b1,…,bk). Let A = (a1,…,ak) 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 = (t1,…,tk) ∈ GL(k, ℤ) such that Ati ≤ siλi(L(A)) where L(A) is the lattice generated by the columns of A, λi(L(A)) is the i-th successive minimum of that lattice and si ≥ 1, 1 ≤ i ≤ k. For c > 0 we determine which precision of A is necessary to guarantee that Bti ≤ (1+c)siλi(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.