Instantiating the Random Oracle Using a Random Real

In modern cryptography, the random oracle model is widely used as an imaginary framework in which the security of a cryptographic scheme is discussed. Since the random oracle is an imaginary object, even if the security of a cryptographic scheme is proved in the random oracle model, the random oracle has to be instantiated using a concrete cryptographic hash function such as the SHA hash functions if we want to use the scheme in the real world. However, it is not clear how much the instantiation can maintain the security originally proved in the random oracle model, nor is it clear whether the random oracle can be instantiated somehow while keeping the original security. In the present paper we investigate this problem, and consider the instantiation of the random oracle by a random real. Here, a random real is an individual infinite binary string which is classified as “random”, and not a random variable. It plays a central role in the field of algorithmic randomness. Algorithmic randomness enables us to classify an individual infinite binary string into random or not. We show that the security proved in the random oracle model is firmly maintained after instantiating it by a random real. The results of this paper are based only on the definition of the security of a cryptographic scheme, and do not depend on specific schemes.