Doubly spatial encryption from DBDH

Functional encryption is an emerging paradigm for public-key encryption which enables fine-grained control of access to encrypted data. Doubly-spatial encryption (DSE) captures all functionalities that we know how to realize via pairings-based assumptions, including (H)IBE, IPE, NIPE, CP-ABE and KP-ABE. In this paper, we propose a construction of DSE from the decisional bilinear Diffie-Hellman (DBDH) assumption. This also yields the first non-zero inner product encryption (NIPE) scheme based on DBDH. Quite surprisingly, we know how to realize NIPE and DSE from stronger assumptions in bilinear groups but not from the basic DBDH assumption. Along the way, we present a novel algebraic characterization of no instances for the DSE functionality, which we use crucially in the proof of security.