Practical Functional Encryption for Bilinear Forms

We present a practically efficient functional encryption scheme for the class of functionalities that can be expressed via bilinear forms over the integers. Bilinear forms are a general class of quadratic functions that includes, for instance, multivariate quadratic polynomials. Our realization works over asymmetric bilinear groups and is surprisingly simple, efficient and easy to implement. For instance, in our scheme the public key and each ciphertext consist of 2n+1 and 4n+2 group elements respectively, where n is the dimension of the encrypted vectors, while secret keys are only two group elements. The scheme is proved secure under the standard (adaptive) indistinguishability based security notion of Boneh, Sahai and Waters (TCC 2011). The proof is rather convoluted and relies on the so-called generic bilinear group model. Specifically, our proof comes in two main stages. In a preliminary step, we put forward and prove a new master theorem to argue hardness in the generic bilinear group model of a broad family of interactive decisional problems, which includes the indistinguishability-based security game for our functional encryption scheme. Next, the more technically involved part of the proof consists in showing that our scheme actually fits the requirements of our master theorem.