Lattices with Symmetry

TL;DR

This paper presents a deterministic polynomial-time algorithm for deciding if a symmetric lattice has an orthonormal basis, leveraging advanced algebraic and number theory techniques, especially effective for lattices with high symmetry.

Contribution

It introduces a new algorithm that efficiently determines orthonormal bases in symmetric lattices, improving over the lack of algorithms for large-rank lattices without symmetry.

Findings

Algorithm runs in polynomial time for symmetric lattices

Uses algebraic number theory and lattice reduction techniques

Effective for high-symmetry, large-rank lattices

Abstract

For large ranks, there is no good algorithm that decides whether a given lattice has an orthonormal basis. But when the lattice is given with enough symmetry, we can construct a provably deterministic polynomial-time algorithm to accomplish this, based on the work of Gentry and Szydlo. The techniques involve algorithmic algebraic number theory, analytic number theory, commutative algebra, and lattice basis reduction.