Multi-Source Randomness Extractors Against Quantum Side Information, and their Applications

We study the problem of constructing multi-source extractors in the quantum setting, which extract almost uniform random bits against an adversary who collects quantum side information from several initially independent classical random sources. This is a natural generalization of the two much studied problems of seeded randomness extraction against quantum side information, and classical independent source extractors. With new challenges such as potential entanglement in the side information, it is not a prior clear under what conditions do quantum multi-source extractors exist; the only previous work in this setting is [19], where the classical inner-product two-source extractors of [7] and [10] are shown to be quantum secure in the restricted Independent Adversary (IA) Model and entangled Bounded Storage (BS) Model. In this paper we propose a new model called General Entangled (GE) Adversary Model, which allows arbitrary entanglement in the side information and subsumes both the IA model and the BS model. We proceed to show how to construct GE-secure quantum multi-source extractors. To that end, we propose another model called One-sided Adversary (OA) Model, which is weaker than all the above models. Somewhat surprisingly, we establish an equivalence between strong OA-security and strong GE-security. As a result, all classical multi-source extractors can either directly work, or be modified to work in the GE model at the cost of one extra random source. Thus, our constructions essentially match the best known constructions of classical multi-source extractors. This answers several open questions in [19, 8]. We also apply our techniques to two important problems in cryptography and distributed computing — privacy amplification and network extractor. Both problems deal with converting secret weak random sources into secret uniform random bits in a communicating environment, with the presence of a passive adversary who has unlimited computational power and can see every message transmitted. We show that as long as the sources have certain amounts of conditional min-entropy in our GE model (even with entangled quantum side information), we can design very efficient privacy amplification protocols and network extractors.