The continuous hidden subgroup problem and its applications

In this talk, I will introduce a generalization of Shor's factoring algorithm that allows the computation of the S-unit group of a number field in quantum polynomial time algorithm. I will discuss how this in turn allows one to solve other computational number theory problem: computing class groups, finding generators of principal ideals, finding approximate short vectors in ideal lattices, or solving certain norm equations.  Finally, we will discuss the relevance of these methods to the design and the analysis of cryptosystems that resist attacks from quantum computers.