Adaptively Secure Puncturable Pseudorandom Functions in the Standard Model

We study the adaptive security of constrained PRFs in the standard model. We initiate our exploration with puncturable PRFs. A puncturable PRF family is a special class of constrained PRFs, where the constrained key is associated with an element $$x'$$ in the input domain. The key allows evaluation at all points $$x\ne x'$$. We show how to build puncturable PRFs with adaptive security proofs in the standard model that involve only polynomial loss to the underlying assumptions. Prior work had either super-polynomial loss or applied the random oracle heuristic. Our construction uses indistinguishability obfuscation and DDH-hard algebraic groups of composite order. More generally, one can consider a t-puncturable PRF: PRFs that can be punctured at any set of inputs S, provided the size of S is less than a fixed polynomial. We additionally show how to transform any single puncturable PRF family to a t-puncturable PRF family, using indistinguishability obfuscation.