Toward Basing Fully Homomorphic Encryption on Worst-Case Hardness

Abstract

AbstractGentry proposed a fully homomorphic public key encryption scheme that uses ideal lattices. He based the security of his scheme on the hardness of two problems: an average-case decision problem over ideal lattices, and the sparse (or “low-weight”) subset sum problem (SSSP).We provide a key generation algorithm for Gentry’s scheme that generates ideal lattices according to a “nice” average-case distribution. Then, we prove a worst-case / average-case connection that bases Gentry’s scheme (in part) on the quantum hardness of the shortest independent vector problem (SIVP) over ideal lattices in the worst-case. (We cannot remove the need to assume that the SSSP is hard.) Our worst-case / average-case connection is the first where the average-case lattice is an ideal lattice, which seems to be necessary to support the security of Gentry’s scheme.KeywordsPrime IdealFull VersionIdeal LatticeHomomorphic EncryptionShort VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.