the full version of the extended abstract that appears in the proceedings of CT-RSA 2016. A constrained pseudorandom function F : K×X → Y for a family T ⊆ 2X of subsets of X is a function where for any key k ∈ K and set S ∈ T one can efficiently compute a constrained key kS which allows to evaluate F(k, ·) on all inputs x ∈ S, while even given this key, the outputs on all inputs x /∈ S look random. At Asiacrypt’13 Boneh and Waters gave a construction which supports the most general set family so far. Its keys kC are defined for sets decided by