A Novel Hierarchical Identity-Based Fully Homomorphic Encryption Scheme from Lattices.

Hierarchical identity-based fully homomorphic encryption (HIBFHE) scheme is a powerful scheme, as it aggregates the advantages of both fully homomorphic encryption and hierarchical identity-based encryption systems. In recent years, the construction of HIBFHE schemes were mainly based on lattices due to their conjectured resistance against quantum cryptanalysis, however, which makes these cryptosystems further unpractical. The first hierarchical identity-based fully homomorphic encryption scheme was presented by Gentry, Sahai and Waters (CRYPTO 2013). Their scheme however works with a not well performed trapdoor and delegation algorithm; that is, the trapdoor is conceptually and algorithmically complex, and the delegation algorithm’s performance is sensitive with the lattice dimension. In this work, we substantially improve their work by using a novel trapdoor function and its relevant algorithms. Specifically, we first use that construct an efficient algorithm for sampling-invertible matrix, based on this we construct a novel delegation algorithm which can keep the lattice dimension unchanged upon delegation. Building on this result, we first construct a more efficient hierarchical identity-based encryption scheme, and then transform it to HIBFHE scheme by using eigenvector method. Under the hardness of Learning with Errors problem, the resulting scheme can be proven secure in the standard model. To the best of our knowledge, this is the first HIBFHE scheme in fixed dimension.