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The Full-Domain Hash (FDH) signature scheme forms [3] one the most basic usages of random oracles. It works with a family $\mathcal{F}$ of trapdoor permutations (TDP), where the signature of m is computed as f−−1(h(m)) (here ${f} \in_{\mathcal{R}} \mathcal{F}$ and h is modelled as a random oracle). It is known to be existentially unforgeable for any TDP family $\mathcal{F}$ [3], 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 [11] to match the best possible “black-box” security reduction in the random oracle model, irrespective of the TDP family $\mathcal{F}$ (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 $\mathcal{H}$ such that the corresponding schemes can be proven secure in the standard model, under some natural assumption on the family $\mathcal{F}$. Our main result rules out the existence of such instantiations for any assumption on $\mathcal{F}$ which (1) is satisfied by a family of random permutations; and (2) does not allow the attacker to invert ${f} \in_{\mathcal{R}} \mathcal{F}$ on an a-priori unbounded number of points. Moreover, this holds even if the choice of $\mathcal{H}$ 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 $\mathcal{F}$ than what is sufficient for the best proof of security in the random oracle model.

On the Generic Insecurity of the Full Domain Hash