This work contains two major parts: comprehensively studying the security notions of cryptographic hash functions against quantum attacks and the relationships between them; and revisiting whether Merkle-Damgard and related iterated hash constructions preserve the security properties of the compression function in the quantum setting. Specifically, we adapt the seven notions in Rogaway and Shrimpton (FSE'04) to the quantum setting and prove that the seemingly stronger attack model where an adversary accesses a challenger in quantum superposition does not make a difference. We confirm the implications and separations between the seven properties in the quantum setting, and in addition we construct explicit examples separating an inherently quantum notion called collapsing from several proposed properties. Finally, we pin down the properties that are preserved under several iterated hash schemes. In particular, we prove that the ROX construction in Andreeva et al. (Asiacrypt'07) preserves the seven properties in the quantum random oracle model.

Accepted to PQCrypto 2019