Faster Fully Homomorphic Encryption

Abstract

We describe two improvements to Gentry’s fully homomorphic scheme based on ideal lattices and its analysis: we provide a more aggressive analysis of one of the hardness assumptions (the one related to the Sparse Subset Sum Problem) and we introduce a probabilistic decryption algorithm that can be implemented with an algebraic circuit of low multiplicative degree. Combined together, these improvements lead to a faster fully homomorphic scheme, with a Õ(λ 3.5) bit complexity per elementary binary add/mult gate, where λ is the security parameter. These improvements also apply to the fully homomorphic schemes of Smart and Vercauteren [PKC’2010] and van Dijk et al. [Eurocrypt’2010].