Non-zero Inner Product Encryption with Short Ciphertexts and Private Keys

We describe two constructions of non-zero inner product encryption (NIPE) systems in the public index setting, both having ciphertexts and secret keys of constant size. Both schemes are obtained by tweaking the Boneh-Gentry-Waters broadcast encryption system (Crypto 2005) and are proved selectively secure under previously considered assumptions in groups with a bilinear map. Our first realization builds on prime-order bilinear groups and is proved secure under the Decisional Bilinear Diffie-Hellman Exponent assumption, which is parameterized by the length n of vectors over which the inner product is defined. By moving to composite order bilinear groups, we are able to obtain security under static subgroup decision assumptions following the Deja Q framework of Chase and Meiklejohn (Eurocrypt 2014) and its extension by Wee (TCC 2016). Our schemes are the first NIPE systems to achieve such parameters, even in the selective security setting. Moreover, they are the first proposals to feature optimally short private keys, which only consist of one group element. Our prime-order-group realization is also the first one with a deterministic key generation mechanism.