Reducing lattice bases by means of approximations

Let L be a k-dimensional lattice in IR m with basis B = (b 1 ; : : : ; b k). 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 = (t 1 ; : : : ; t k) 2 GL(k; Z Z) such that At i sii(L(A)) where L(A) is the lattice generated by the columns of A, i(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 Bt i (1+c)sii(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 eeected almost as fast as such reductions of an integer lattice basis.