Lattice-Based Hierarchical Inner Product Encryption

The notion of inner-product encryption (IPE), introduced by Katz, Sahai, and Waters at Eurocrypt 2008, is a generalization of identity-based encryption in which ciphertexts and secret keys are associated to vectors in some finite field. In an IPE scheme, a ciphertext can only be decrypted by a secret key if the vector associated with the latter is orthogonal to that of the ciphertext. In its hierarchical version, first proposed by Okamoto and Takashima (Asiacrypt'09), there exists an additional delegation mechanism which allows users to delegate their decryption capabilities to other users in the system. In this paper, we propose the first construction of a hierarchical inner-product encryption (HIPE) scheme based on lattices assumptions. To achieve this goal, we extend the lattice-based IPE scheme by Agrawal, Freeman, and Vaikuntanathan (Asiacrypt'11) to the hierarchical setting by employing basis delegation technics by Peikert et al. (Eurocrypt' 10) and by Agrawal et al. (Eurocrypt'10). As the underlying IPE scheme, our new scheme is shown to be weak selective secure based on the difficulty of the learning with errors (LWE) problem in the standard model, as long as the total number of levels in the hierarchy is a constant. As an application, we show how our new primitive can be used to build new chosen-ciphertext secure IPE and wildcarded identity-based encryption schemes.