Being a permutation is also orthogonal to one-wayness in quantum world: Impossibilities of quantum one-way permutations from one-wayness primitives

Abstract In this paper, three impossible results for one-way permutations in the quantum world are obtained. The first one is the impossibility of fully black-box reduction from one-way permutations to one-way functions in a quantum setting. The two-oracle method is the main technique adopted in our proof that was originally proposed by Hsiao and Reyzin, and extended later to a quantum setting by Hosoyamada and Yamakawa. Informally this technique involves a pair of oracles, and in our proof, one oracle is chosen as a random function, and the other one is devised for breaking the security of quantum one-way permutation. We show, according to the first oracle, that there is an oracle-aided circuit that is a quantum-secure one-way function. However, we also show, according to the second oracle, that there is no one-way permutation relative to those oracles. It hence claims the impossibility in the quantum setting to construct one-way permutations from one-way functions (even if both input and output are classical). The rest two impossibilities are essential enhancements of the first impossibility: We successfully obtain that it is impossible, in the quantum world, to obtain one-way permutations in a black-box manner from injective adaptive one-way functions, and (respectively) even from injective adaptive trapdoor functions. In order to reach the latter two results, we manage to get over the obstacle of finding the inverse of random injective functions even given the partial invert oracle in the quantum setting. As a result, we get a quantum lower bound of inverting random injective functions in random case. That should be of independent interest in this paper, which might, to authors' knowledge, be the first quantum lower bound of inverting random injective functions given the partial invert oracle.