Lattice-based FHE as secure as PKE

Abstract

We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of O(n[superscript 1.5+ε])-approximation for lattice problems (such as GapSVP) under quantum reductions for any ε 〉 0 (or O(n[superscript 2+ε])-approximation under classical reductions). This matches the best known hardness for "regular" (non-homomorphic) lattice based public-key encryption up to the ε factor. A number of previous methods had hit a roadblock at quasipolynomial approximation. (As usual, a circular security assumption can be used to achieve a non-leveled FHE scheme.)