Reducing lattice bases by means of approximations
Let L be a k-dimensional lattice in IRm 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) ε GL(k, ℤ) such that Ati ≤ 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 Bt 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.
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