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Lattices defined as modules over algebraic rings or orders have garneredinterest recently, particularly in the fields of cryptography and codingtheory. Whilst there exist many attempts to generalise the conditions for LLLreduction to such lattices, there do not seem to be any attempts so far togeneralise stronger notions of reduction such as Minkowski, HKZ and BKZreduction. Moreover, most lattice reduction methods for modules over algebraicrings involve applying traditional techniques to the embedding of the moduleinto real space, which distorts the structure of the algebra. In this paper, wegeneralise some classical notions of reduction theory to that of free modulesdefined over an order. Moreover, we extend the definitions of Minkowski, HKZand BKZ reduction to that of such modules and show that bases reduced in thismanner have vector lengths that can be bounded above by the successive minimaof the lattice multiplied by a constant that depends on the algebra and thedimension of the module. In particular, we show that HKZ reduced bases arepolynomially close to the successive minima of the lattice in terms of themodule dimension. None of our definitions require the module to be embedded andthus preserve the structure of the module.

Reduction Theory of Algebraic Modules and their Successive Minima