Provably Secure Key Establishment Against Quantum Adversaries

At Crypto 2011, some of us had proposed a family of cryptographic protocols for key establishment capable of protecting quantum and classical legitimate parties unconditionally against a quantum eavesdropper in the query complexity model. Unfortunately, our security proofs were unsatisfactory from a cryptographically meaningful perspective because they were sound only in a worst-case scenario. Here, we extend our results and prove that for any ε > 0, there is a classical protocol that allows the legitimate parties to establish a common key after O(N) expected queries to a random oracle, yet any quantum eavesdropper will have a vanishing probability of learning their key after O(N1.5−ε) queries to the same oracle. The vanishing probability applies to a typical run of the protocol. If we allow the legitimate parties to use a quantum computer as well, their advantage over the quantum eavesdropper becomes arbitrarily close to the quadratic advantage that classical legitimate parties enjoyed over classical eavesdroppers in the seminal 1974 work of Ralph Merkle. Along the way, we develop new tools to give lower bounds on the number of quantum queries required to distinguish two probability distributions. This method in itself could have multiple applications in cryptography. We use it here to study average-case quantum query complexity, for which we develop a new composition theorem of independent interest. ∗ A full version of the paper is available at [6], https://arxiv.org/abs/1704.08182. † The work of AB is supported in part by the ERC Advanced Grant MQC. ‡ The work of GB is supported in part by the Canadian Institute for Advanced Research (CIFAR), the Canada Research Chair program, Canada’s Natural Sciences and Engineering Research Council (NSERC) and Québec’s Institut transdisciplinaire d’information quantique. § The work of PH is supported in part by CIFAR and NSERC. ¶ The work of MK is supported in part by EPSRC grant number EP1N003829/1 Verification of Quantum Technology. ‖ The work of SL is supported in part by the European Union Seventh Framework Programme (FP7/20072013) under grant agreement no. 600700 (QALGO) and the French ANR Blanc grant RDAM ANR-12BS02-005. ∗∗The work of LS is supported in part by NSERC discovery grant and discovery accelerator supplements programs. © Aleksandrs Belovs, Gilles Brassard, Peter Høyer, Marc Kaplan, Sophie Laplante, and Louis Salvail; licensed under Creative Commons License CC-BY 12th Conference on the Theory of Quantum Computation, Communication, and Cryptography (TQC 2017). Editor: Mark M. Wilde; Article No. 3; pp. 3:1–3:17 Leibniz International Proceedings in Informatics Schloss Dagstuhl – Leibniz-Zentrum für Informatik, Dagstuhl Publishing, Germany 3:2 Provably Secure Key Establishment Against Quantum Adversarie 1998 ACM Subject Classification A. General Literature