SoK: Learning with Errors, Circular Security, and Fully Homomorphic Encryption

All known constructions of fully homomorphic encryption (FHE) schemes from the learning with errors (LWE) assumption require the encryption schemes to be circular secure. A long-standing open problem in the study of FHE schemes is to demonstrate evidence for their circular security. In this work, we systematize the flavors of circular security required for a number of FHE constructions, formulate circular security conjectures, show search-to-decision reductions for them, and pose several open problems.