Cryptography and Algorithmic Randomness

TL;DR

This paper uses algorithmic randomness to address the open problem of securely instantiating the random oracle in cryptography, showing that specific computable functions can replace the idealized model without losing security.

Contribution

It demonstrates that for any secure scheme in the random oracle model, a computable function can instantiate the oracle while preserving security, extending to the generic group model.

Findings

Existence of computable functions that instantiate the random oracle securely

Extension of results to the generic group model

Introduction of the concept of effective security

Abstract

The secure instantiation of the random oracle is one of the major open problems in modern cryptography. We investigate this problem using concepts and methods of algorithmic randomness. In modern cryptography, the random oracle model is widely used as an imaginary framework in which the security of a cryptographic scheme is discussed. In the random oracle model, the cryptographic hash function used in a cryptographic scheme is formulated as a random variable uniformly distributed over all possibility of the function, called the random oracle. The main result of this paper is to show that, for any secure signature scheme in the random oracle model, there exists a specific computable function which can instantiate the random oracle while keeping the security originally proved in the random oracle model. In modern cryptography the generic group model is used also for a similar purpose to the random oracle model. We show that the same results hold for the generic group model. In the process of proving the results, we introduce the notion of effective security, demonstrating the importance of this notion in modern cryptography.