Katz, J., Sahai, A., & Waters, B. (2008, April). Predicate encryption supporting disjunctions, polynomial equations, and inner products. In annual international conference on the theory and applications of cryptographic techniques (pp. 146-162). Springer, Berlin, Heidelberg.
Abstract
Predicate encryption is a new paradigm generalizing, among other things, identity-based encryption. In a predicate encryption scheme, secret keys correspond to predicates and ciphertexts are associated with attributes; the secret key SK f corresponding to a predicate f can be used to decrypt a ciphertext associated with attribute I if and only if f(I) = 1. Constructions of such schemes are currently known for relatively few classes of predicates.
We construct such a scheme for predicates corresponding to the evaluation of inner products over 
(for some large integer N). This, in turn, enables constructions in which predicates correspond to the evaluation of disjunctions, polynomials, CNF/DNF formulae, or threshold predicates (among others). Besides serving as a significant step forward in the theory of predicate encryption, our results lead to a number of applications that are interesting in their own right.