On Sublattice Determinants in Reduced Bases

Lenstra, Lenstra, and Lov´asz 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 m be an LLL-reduced basis of the lattice L, and d1, . . . , dk arbitrary linearly independent vectors in L. Then kb1 k ≤ 2 (n−k)/2+(k−1)/4 (detL(d1, . . . , dk))1/k , (1) detL(b1, . . . , bk) ≤ 2 k(n−k)/2 detL(d1, . . . , dk), (2) detL(b1, . . . , bk) ≤ 2 k(n−k)/4 (detL) k/n , (3) kb1 k · · · kbk k ≤ 2 k(n−k)/2+k(k−1)/4 detL(d1, . . . , dk), (4) kb1 k · · · kbk k ≤ 2 k(n−1)/4 (det L) k/n . (5) In the most general setting, we prove: Theorem 2. Let b1, . . . , bn ∈ R m 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 k(n−j)/2+k(j−k)/4 (detL(d1, . . . , dj ))k/j , (6) kb1 k · · · kbk k ≤ 2 k(n−j)/2+k(j−1)/4 (