The Full-Domain Hash (FDH) signature scheme  forms one the most basic usages of random oracles. It works with a family F of trapdoor permutations (TDP), where the signature of m is computed as f -1(h(m)) (here f ∈ R F and h is modelled as a random oracle). It is known to be existentially unforgeable for any TDP family F , although a much tighter security reduction is known for a restrictive class of TDP's [10,14] - namely, those induced by a family of claw-free permutations (CFP) pairs. The latter result was shown  to match the best possible "black-box" security reduction in the random oracle model, irrespective of the TDP family F (e.g., RSA) one might use. In this work we investigate the question if it is possible to instantiate the random oracle h with a "real" family of hash functions H such that the corresponding schemes can be proven secure in the standard model, under some natural assumption on the family T. Our main result rules out the existence of such instantiations for any assumption on T which (1) is satisfied by a family of random permutations; and (2) does not allow the attacker to invert f ∈ R F on an a-priori unbounded number of points. Moreover, this holds even if the choice of H can arbitrarily depend on f. As an immediate corollary, we rule out instantiating FDH based on general claw-free permutations, which shows that in order to prove the security of FDH in the standard model one must utilize significantly more structure on F than what is sufficient for the best proof of security in the random oracle model.