Basis Reduction over Algebraic Lattices

Lattices, in the geometric sense, are an infinite set of discrete points in space with a certain structure. Not only are these objects useful mathematically, but they are also of great importance for coding and cryptographic purposes. When used for the purpose of cryptography, breaking the cryptographic scheme is as hard as solving the closest vector problem (CVP), which goes as follows: "Given a lattice and a vector, which lattice point is closest to the vector?" The CVP has been well studied for geometric lattices, however research remains fruitful for generalisations of geometric lattices, such as "ideal lattices" and "module lattices" which have more of an algebraic flavour than geometric lattices. As these structures seem to be the prime candidates for practical lattice cryptosystems, it makes sense to investigate the CVP (or for ideal lattices, the shortest-generator problem) for ideal and module lattices. Certain questions will be addressed in order to further research into this field, such as recovering close integers over so-called "norm-Euclidean domains", which could be approached by analysing the problem in a geometric setting using hyperbolae at lattice points.

This research aligns with EPSRC research areas of Number theory and Theoretical Computer Science.