We initiate the study of multi-source extractors in the quantum world. In this setting, our goal is to extract random bits from two independent weak random sources, on which two quantum adversaries store a bounded amount of information. Our main result is a two-source extractor secure against quantum adversaries, with parameters closely matching the classical case and tight in several instances. Moreover, the extractor is secure even if the adversaries share entanglement. The construction is the Chor-Goldreich [CG88] two-source inner product extractor and its multi-bit variant by Dodis et al. [DEOR04]. Previously, research in this area focused on the construction of seeded extractors secure against quantum adversaries; the multi-source setting poses new challenges, among which is the presence of entanglement that could potentially break the independence of the sources.
We initiate the study of multi-source extractors in the quantum world. In this setting, our goal is to extract random bits from two independent weak random sources, on which two quantum adversaries store a bounded amount of information. Our main result is a two-source extractor secure against quantum adversaries, with parameters closely matching the classical case and tight in several instances. Moreover, the extractor is secure even if the adversaries share entanglement. The construction is the Chor-Goldreich [CG88] two-source inner product extractor and its multi-bit variant by Dodis et al. [DEOR04]. Previously, research in this area focused on the construction of seeded extractors secure against quantum adversaries; the multi-source setting poses new challenges, among which is the presence of entanglement that could potentially break the independence of the sources.
We study the problem of constructing multi-source extractors in the quantum setting, which extract almost uniform random bits against quantum side information collected from several initially independent classical random sources. This is a natural generalization of seeded randomness extraction against quantum side information and classical independent source extraction. 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 is [KK12], where the classical inner-product two-source extractors of [CG88] and [DEOR04] 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 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. We also apply our techniques to two important problems in cryptography and distributed computing --- privacy amplification and network extractor. 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.
Multi-source extractors are functions that extract uniform randomness from multiple (weak) sources of randomness. Quantum multi-source extractors were considered by Kasher and Kempe (2010) (for the quantum independent adversary and the quantum bounded storage adversary), Chung et al. (2014) (for the general entangled adversary) and Arnon-Friedman et al. (2016) (for the quantum Markov adversary). One of the main objectives of this work is to unify all the existing quantum multi-source adversary models. We propose two new models of adversaries: 1) the quantum measurement adversary (qma), which generates side information using entanglement and on post-measurement; and 2) the quantum communication adversary (qca), which generates side information using entanglement and communication between multiple sources. We show that: 1)qmais the strongest adversary among all the known adversaries, in the sense that the side information of all other adversaries can be generated byqma; 2) The (generalized) inner-product function (in fact a general class of two-wise independent functions) continues to work as a good extractor with matching parameters as that of Chor and Goldreich (1985) against classical adversaries; 3) A non-malleable extractor proposed by Li (2012) (against classical adversaries) continues to be secure against quantum side information. This result implies a non-malleable extractor result of Aggarwal et al. (2019) with uniform seed. We strengthen their result via a completely different proof to make the non-malleable extractor of Li secure against quantum side information even when the seed is not uniform; 4) A modification (working with weak local randomness instead of uniform local randomness) of the Dodis and Wichs (2009) protocol for privacy-amplification is secure against active quantum adversaries (those who arbitrarily modify the messages exchanged in the protocol). This strengthens on a recent result due to Aggarwal et al. (2019) which uses uniform local randomness; 5) A tight efficiency lower bound for the (generalized) inner-product function (in fact a general class of two-wise independent functions).
Randomness extractors, widely used in classical and quantum cryptography and other fields of computer science, e.g., derandomization, are functions which generate almost uniform randomness from weak sources of randomness. In the quantum setting one must take into account the quantum side information held by an adversary which might be used to break the security of the extractor. In the case of seeded extractors the presence of quantum side information has been extensively studied. For multi-source extractors one can easily see that high conditional min-entropy is not sufficient to guarantee security against arbitrary side information, even in the classical case. Hence, the interesting question is under which models of (both quantum and classical) side information multi-source extractors remain secure. In this work we suggest a natural model of side information, which we call the Markov model, and prove that any multi-source extractor remains secure in the presence of quantum side information of this type (albeit with weaker parameters). This improves on previous results in which more restricted models were considered or the security of only some types of extractors was shown.
Randomness extractors, widely used in classical and quantum cryptography and other fields of computer science, e.g., derandomization, are functions which generate almost uniform randomness from weak sources of randomness. In the quantum setting one must take into account the quantum side information held by an adversary which might be used to break the security of the extractor. In the case of seeded extractors the presence of quantum side information has been extensively studied. For multi-source extractors one can easily see that high conditional min-entropy is not sufficient to guarantee security against arbitrary side information, even in the classical case. Hence, the interesting question is under which models of (both quantum and classical) side information multi-source extractors remain secure. In this work we suggest a natural model of side information, which we call the Markov model, and prove that any multi-source extractor remains secure in the presence of quantum side information of this type (albeit with weaker parameters). This improves on previous results in which more restricted models were considered and the security of only some types of extractors was shown.
The goal of randomness extraction is to distill (almost) perfect randomness from a weak source of randomness. When the source yields a classical string X, many extractor constructions are known. Yet, when considering a physical randomness source, X is itself ultimately the result of a measurement on an underlying quantum system. When characterizing the power of a source to supply randomness, it is hence natural to ask how much classical randomness we can extract from a quantum system. To tackle this question, we here take on the study of quantum-to-classical randomness extractors (QC-extractors). We provide constructions of QC-extractors based on measurements in a full set of mutually unbiased bases (MUBs), and certain single qubit measurements. The latter are particularly appealing since they are not only easy to implement, but also appear throughout quantum cryptography. We proceed to prove an upper bound on the maximum amount of randomness that we could hope to extract from any quantum state. Some of our QC-extractors almost match this bound. We show two applications of our results. First, we show that any QC-extractor gives rise to entropic uncertainty relations with respect to quantum side information. Such relations were previously only known for two measurements. In particular, we obtain strong relations in terms of the von Neumann (Shannon) entropy as well as the min-entropy for measurements in (almost) unitary two-designs, a full set of MUBs, and single qubit measurements in three MUBs each. Second, we resolve the central open question in the noisy-storage model by linking security to the quantum capacity of the adversary's storage device. More precisely, we show that any two party cryptographic primitives can be implemented securely as long as the adversary's storage device has sufficiently low quantum capacity. Our protocol does not need any quantum storage to implement, and is technologically feasible using present-day technology.
In privacy amplification, two mutually trusted parties aim to amplify the secrecy of an initial shared secret X in order to establish a shared private key K by exchanging messages over an insecure communication channel. If the channel is authenticated the task can be solved in a single round of communication using a strong randomness extractor; choosing a quantum-proof extractor allows one to establish security against quantum adversaries. In the case that the channel is not authenticated, Dodis and Wichs (STOC'09) showed that the problem can be solved in two rounds of communication using a non-malleable extractor, a stronger pseudo-random construction than a strong extractor. We give the first construction of a non-malleable extractor that is secure against quantum adversaries. The extractor is based on a construction by Li (FOCS'12), and is able to extract from source of min-entropy rates larger than 1/2. Combining this construction with a quantum-proof variant of the reduction of Dodis and Wichs, shown by Cohen and Vidick (unpublished), we obtain the first privacy amplification protocol secure against active quantum adversaries.