When proving something to be post-quantum resistant, is it simply that there are no known algorithms to solve a particular assumption, or is there an inherent algebraic structure that provides resistance to a quantum adversary? If the latter, what exactly is this algebraic assumption, and if the former could such an algebraic structure be proven to exist or not?

One such problem I'm familiar with is the hidden subgroup problem https://en.wikipedia.org/wiki/Hidden_subgroup_problem which Shor's algorithm successfully solves. The abelian/non-abelian structure is the main difference, and I'm curious if there's an abstraction that applies to all types of problems.

## Post-Quantum Resistance

*any*secure cryptosystems at all.