Indefiniteness makes lattice reduction easier

TL;DR

This paper explores indefinite lattice reduction, demonstrating that it can produce better reduced forms than previously thought, with approximation quality depending on the lattice's signature rather than its dimension.

Contribution

It shows that indefinite lattice reduction can yield significantly improved representations and that the approximation factor is influenced by the lattice's signature.

Findings

Indefinite lattice reduction can produce better reduced forms.

The approximation factor depends on the lattice's signature.

Previous assumptions about the limitations of indefinite reduction are challenged.

Abstract

Since the invention of the famous LLL algorithm, lattice reduction has been an extremely useful tool in computational number theory. By construction, the LLL algorithm deals with lattices living in a vector space endowed with a positive definite scalar product. However, it seems quite nature to ask about the indefinite case, where the scalar product is replaced by an arbitrary quadratic form, possibily indefinite. This question was considered independently in two lines of work. One by G{\'a}bor Ivanyos and {\'A}gnes Sz{\'a}nt{\'o} and one by Denis Simon. Both lead to an algorithm that generalizes LLL and whose performance is very similar to LLL, i.e. a polynomial-time algorithm that approximates the shortest vector within an approximation factor exponential in the dimension. Denis Simon achieves an approximation factor close to that of LLL under the assumption that no isotropic vectors arise during reduction. G{\'a}bor Ivanyos and {\'A}gnes Sz{\'a}nt{\'o} show that it is possible to avoid isotropic vectors altogether, at the cost of a somewhat worse approximation factor. In this paper, we revisit the reduction of indefinite lattices and conclude that it can lead to much better reduced representations that previously thought. We also conclude that the approximation factor depends on the signature of the indefinite lattice rather than on its dimension.