On sublattice determinants in reduced bases

Lenstra, Lenstra, and Lovász in [7] proved several inequalities showing that the vectors in an LLL-reduced basis are short, and near orthogonal. Here we present generalizations, from which with k = 1, and k = n we can recover their inequalities: Theorem 1. Let b1, . . . , bn ∈ R be an LLL-reduced basis of the lattice L, and d1, . . . , dk arbitrary linearly independent vectors in L. Then ‖b1 ‖ ≤ 2(detL(d1, . . . , dk)), (1) detL(b1, . . . , bk) ≤ 2 detL(d1, . . . , dk), (2) detL(b1, . . . , bk) ≤ 2(detL), (3) ‖b1 ‖ · · · ‖bk ‖ ≤ 2 detL(d1, . . . , dk), (4) ‖b1 ‖ · · · ‖bk ‖ ≤ 2(detL). (5) In the most general setting, we prove: Theorem 2. Let b1, . . . , bn ∈ R be an LLL-reduced basis of the lattice L, 1 ≤ k ≤ j ≤ n, and d1, . . . , dj arbitrary linearly independent vectors in L. Then detL(b1, . . . , bk) ≤ 2(detL(d1, . . . , dj)) , (6) ‖b1‖ · · · ‖bk ‖ ≤ 2(detL(d1, . . . , dj)) . (7) Mathematics subject classification codes: 11H06, 52C07 ∗Department of Statistics and Operations Research, UNC Chapel Hill, [email protected], [email protected]