Quantum Security of Cryptographic Primitives

We call quantum security the area of IT security dealing with scenarios where one or more parties have access to quantum hardware. This encompasses both the fields of post-quantum cryptography (that is, traditional cryptography engineered to be resistant against quantum adversaries), and quantum cryptography (that is, security protocols designed to be natively run on a quantum infrastructure, such as quantum key distribution). In this work, we propose the first systematic classification of quantum security scenarios, and for each of them we recall the main tools and results, as well as presenting new ones. We achieve this goal by identifying four distinct quantum security classes, or domains, each of them encompassing the security notions and constructions related to a particular scenario. We start with the class QS0, which is `classical cryptography' (meaning that no quantum scenario is considered). Regarding post-quantum cryptography, we introduce the class QS1, where we discuss in detail the problems arising when designing a classical cryptographic object meant to be resistant against adversaries with local quantum computing power, and we provide a classification of the possible quantum security reductions in this scenario when considering provable security. In respect to hybrid classical-quantum models, in the security class QS2 we discuss in detail the possible scenarios where these scenarios arise, and what a correct formalization should be in terms of quantum oracle access. Finally, in the class QS3 we consider all those cryptographic constructions designed to run natively on quantum hardware. We believe that the framework we introduce in this work will be a valuable tool for the scientific community in addressing the challenges arising when formalizing sound constructions and notions of security in the quantum world.