Finding Collisions in a Quantum World: Quantum Black-Box Separation of Collision-Resistance and One-Wayness

Since the celebrated work of Impagliazzo and Rudich (STOC 1989), a number of black-box impossibility results have been established. However, these works only ruled out classical black-box reductions among cryptographic primitives. Therefore, it may be possible to overcome these impossibility results by using quantum reductions. To exclude such a possibility, we have to extend these impossibility results to the quantum setting. In this paper, we study black-box impossibility in the quantum setting. We first formalize a quantum counterpart of fully black-box reduction following the formalization by Reingold, Trevisan and Vadhan (TCC 2004). Then we prove that there is no quantum fully black-box reduction from collision-resistant hash functions to one-way permutations (or even trapdoor permutations). We take both of classical and quantum implementations of primitives into account. This is an extension to the quantum setting of the work of Simon (Eurocrypt 1998) who showed a similar result in the classical setting.