Automorphic Forms and Cryptographic Lattice Problems
Vortrag von Samed Düzlü
Sprecher eingeladen von: Prof. Dr. Joachim Rosenthal
Datum: 09.04.25 Zeit: 15.15 - 16.15 Raum: Y27H28
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.