Reduction Theory of Algebraic Modules and their Successive Minima

TL;DR

This paper extends classical lattice reduction notions like Minkowski, HKZ, and BKZ to algebraic modules, providing bounds on basis vector lengths relative to successive minima without embedding the modules into real space.

Contribution

It generalizes reduction theories to modules over algebraic rings, preserving module structure and establishing bounds on reduced bases relative to successive minima.

Findings

HKZ reduced bases are polynomially close to successive minima

New definitions of reduction do not require module embedding

Bounds depend on algebra and module dimension

Abstract

Lattices defined as modules over algebraic rings or orders have garnered interest recently, particularly in the fields of cryptography and coding theory. Whilst there exist many attempts to generalise the conditions for LLL reduction to such lattices, there do not seem to be any attempts so far to generalise stronger notions of reduction such as Minkowski, HKZ and BKZ reduction. Moreover, most lattice reduction methods for modules over algebraic rings involve applying traditional techniques to the embedding of the module into real space, which distorts the structure of the algebra. In this paper, we generalise some classical notions of reduction theory to that of free modules defined over an order. Moreover, we extend the definitions of Minkowski, HKZ and BKZ reduction to that of such modules and show that bases reduced in this manner have vector lengths that can be bounded above by the successive minima of the lattice multiplied by a constant that depends on the algebra and the dimension of the module. In particular, we show that HKZ reduced bases are polynomially close to the successive minima of the lattice in terms of the module dimension. None of our definitions require the module to be embedded and thus preserve the structure of the module.