Random Oracles in a Quantum World

TL;DR

This paper explores the security of cryptographic schemes in the quantum-accessible random oracle model, highlighting the importance of quantum considerations and proposing conditions under which classical proofs imply quantum security.

Contribution

It introduces the concept of history-free reductions and demonstrates their role in establishing post-quantum security for lattice-based cryptosystems.

Findings

Classical security does not imply quantum security in the random oracle model.

History-free reductions ensure security in the quantum-accessible random oracle model.

Certain lattice-based proposals are proven secure under these new conditions.

Abstract

The interest in post-quantum cryptography - classical systems that remain secure in the presence of a quantum adversary - has generated elegant proposals for new cryptosystems. Some of these systems are set in the random oracle model and are proven secure relative to adversaries that have classical access to the random oracle. We argue that to prove post-quantum security one needs to prove security in the quantum-accessible random oracle model where the adversary can query the random oracle with quantum states. We begin by separating the classical and quantum-accessible random oracle models by presenting a scheme that is secure when the adversary is given classical access to the random oracle, but is insecure when the adversary can make quantum oracle queries. We then set out to develop generic conditions under which a classical random oracle proof implies security in the quantum-accessible random oracle model. We introduce the concept of a history-free reduction which is a category of classical random oracle reductions that basically determine oracle answers independently of the history of previous queries, and we prove that such reductions imply security in the quantum model. We then show that certain post-quantum proposals, including ones based on lattices, can be proven secure using history-free reductions and are therefore post-quantum secure. We conclude with a rich set of open problems in this area.