## Cryptology ePrint Archive: Report 2013/541

Lattice-Based FHE as Secure as PKE

Zvika Brakerski and Vinod Vaikuntanathan

Abstract: We show that (leveled) fully homomorphic encryption (FHE) can be based on the hardness of $\otild(n^{1.5+\epsilon})$-approximation for lattice problems (such as GapSVP) under quantum reductions for any $\epsilon>0$ (or $\otild(n^{2+\epsilon})$-approximation under classical reductions). This matches the best known hardness for regular'' (non-homomorphic) lattice based public-key encryption up to the $\epsilon$ 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.)

Our approach consists of three main ideas: Noise-bounded sequential evaluation of high fan-in operations; Circuit sequentialization using Barrington's Theorem; and finally, successive dimension-modulus reduction.

Category / Keywords: public-key cryptography / public key encryption, fully homomorphic encryption, lattice based cryptography, LWE