Random Oracle Reducibility

We discuss a reduction notion relating the random oracles in two cryptographic schemes A and B. Basically, the random oracle of scheme B reduces to the one of scheme A if any hash function instantiation of the random oracle (possibly still oracle based) which makes A secure also makes B secure. In a sense, instantiating the random oracle in scheme B is thus not more demanding than the one for scheme A. If, in addition, the standard cryptographic assumptions for scheme B are implied by the ones for scheme A, we can conclude that scheme B actually relies on weaker assumptions. Technically, such a conclusion cannot be made given only individual proofs in the random oracle model for each scheme. The notion of random oracle reducibility immediately allows to transfer an uninstantiability result from an uninstantiable scheme B to a scheme A to which the random oracle reduces. We are nonetheless mainly interested in the other direction as a mean to establish hierarchically ordered random-oracle based schemes in terms of security assumptions. As a positive example, we consider the twin Diffie-Hellman (DH) encryption scheme of Cash et al. (Journal of Cryptology, 2009), which has been shown to be secure under the DH assumption in the random oracle scheme. It thus appears to improve over the related hashed ElGamal encryption scheme which relies on the random oracle model and the strong DH assumption where the adversary also gets access to a decisional DH oracle. As explained above, we complement this belief by showing that the random oracle in the twin DH scheme actually reduces to the one of the hashed ElGamal encryption scheme. We finally discuss further random oracle reductions between common signature schemes like GQ, PSS, and FDH.