On Sublattice Determinants in Reduced Bases

Date

2009-04-04

Author

Patakı, Gabor
Tural, Mustafa Kemal

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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 (d

URI

https://hdl.handle.net/11511/79533
https://arxiv.org/pdf/0804.4014.pdf

Conference Name

AMS 2009 Spring Southeastern Section Meeting, 4 - 05 Nisan 2009

Collections

Department of Industrial Engineering, Conference / Seminar

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Citation Formats

G. Patakı and M. K. Tural, “On Sublattice Determinants in Reduced Bases,” presented at the AMS 2009 Spring Southeastern Section Meeting, 4 - 05 Nisan 2009, Amerika Birleşik Devletleri, 2009, Accessed: 00, 2021. [Online]. Available: https://hdl.handle.net/11511/79533.