Functional Encryption for Quadratic Functions, and Applications to Predicate Encryption

We present a functional encryption scheme based on standard assumptions where ciphertexts are associated with a tuple of values (x1, . . . , xn) ∈ Zp , secret keys are associated with a degree-two polynomial, and the decryption of a ciphertext ct(x1,...,xn)∈Znp with a secret key skP∈Zp[X1,...,Xn],deg(P )≤2 recovers P (x1, . . . , xn), where the ciphertext contains only O(n) group elements. Our scheme, which achieves selective security based on pairings, also yields a new predicate encryption scheme that supports degree-two polynomial evaluation, generalizing both [26] and [10].