In lattice-based cryptography, one of the most popular approaches to
increase efficiency is to attach structures to lattices.
These structures usually come from algebraic number theory:
For instance, taking fractional ideals of number fields become lattices
inside the associated Minkowski space, on which the ring of integers act
by multiplication.
Extending this idea, module lattices are defined by taking submodules
over the ring of integers inside an m-dimensional vector spaces over the
number field.
As the structures involve these algebraic number theoretic notions, it
seems natural to analyze the corresponding lattice problems using number
theoretic techniques.
De Boer et al. (CRYPTO'20) carried out this strategy for ideal lattices:
Fixing a number field, the set of ideal lattices (up to scaling) is a
compact commutative Lie group.
Cryptographic problems on the ideal lattices can be translated to the
study of the space of ideal lattices.
We explain this approach by a slight reformulation in terms of adeles
and ideles.
Then, the natural generalization of the idele class group to GL(m)
corresponds immediately to the class of module lattices of rank m, for
the same fixed number field.
We describe this correspondence and explain the building blocks to
generalize the results of de Boer et al. to module lattices of higher
ranks.
We finish with a partial result and explanation of the obstacles to
conclude an effective solution.