On sublattice determinants in reduced bases

TL;DR

This paper establishes inequalities on sublattice determinants in LLL-reduced bases, demonstrating that LLL reduction produces sublattices with small determinants and providing new bounds on vector norms.

Contribution

It generalizes existing inequalities on shortest vectors to sublattices and introduces new upper bounds on the product of the first few vectors' norms.

Findings

Inequalities on determinants of sublattices in LLL-reduced bases

LLL-reduction finds sublattices with small determinants

New upper bounds on the product of the first few vectors' norms

Abstract

We prove several inequalities on the determinants of sublattices in LLL-reduced bases. They generalize the inequalities on the length of the shortest vector proven by Lenstra, Lenstra, and Lovasz, and show that LLL-reduction finds not only a short vector, but more generally, sublattices with small determinants. We also prove new upper bounds on the product of the norms of the first few vectors.