Key-Homomorphic Constrained Pseudorandom Functions

Abstract

A pseudorandom function (PRF) is a keyed function \(F : {\mathcal K}\times{\mathcal X}\rightarrow{\mathcal Y}\) where, for a random key \(k\in{\mathcal K}\), the function F(k,·) is indistinguishable from a uniformly random function, given black-box access. A key-homomorphic PRF has the additional feature that for any keys k,k′ and any input x, we have F(k + k′, x) = F(k,x) ⊕ F(k′,x) for some group operations + , ⊕ on \(\mathcal{K}\) and \(\mathcal{Y}\), respectively. A constrained PRF for a family of sets \({\mathcal S} \subseteq \mathcal{P}({\mathcal X})\) has the property that, given any key k and set \(S \in \mathcal{S}\), one can efficiently compute a “constrained” key k S that enables evaluation of F(k,x) on all inputs x ∈ S, while the values F(k,x) for x ∉ S remain pseudorandom even given k S .