Is there a scheme using these principles that isn't known to be insecure? Or one that has a proof of security by reduction to a known (or supposed) hard problem?

A brief look online makes it sound like all provided schemes have extremely adhoc arguments of security or are known to be broken.

Unfortunately, a reasonable source I found is paywalled: https://link.springer.com/chapter/10.1007/978-981-16-6332-1_...