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arXiv:2106.02766 · IEEE Transactions on Information Theory

Quantum Measurement Adversary

Divesh Aggarwal, Naresh Goud Boddu, Rahul Jain, Maciej Obremski

Published Sep 8, 2023• 5 citations
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Abstract

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 ($\mathsf {qma}$), which generates side information using entanglement and on post-measurement; and 2) the quantum communication adversary ($\mathsf {qca}$), which generates side information using entanglement and communication between multiple sources. We show that: 1)$\mathsf {qma}$is the strongest adversary among all the known adversaries, in the sense that the side information of all other adversaries can be generated by$\mathsf {qma}$; 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).

DOI
10.1109/tit.2023.3313149

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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.
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.
In privacy amplification, two mutually trusted parties aim to amplify the secrecy of an initial shared secret XXX in order to establish a shared private key KKK 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 1/21/21/2. 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.
Dodis and Wichs [DW09] introduced the notion of a non-malleable extractor to study the problem of privacy amplification with an active adversary. A non-malleable extractor is a much stronger version of a strong extractor. Given a weakly-random string x and a uniformly random seed y as the inputs, the non-malleable extractor nmExt has the property that nmExt(x, y) appears uniform even given y as well as nmExt(x,A(y)), for an arbitrary function A with A(y) 6= y. Dodis and Wichs showed that such an object can be used to give optimal privacy amplification protocols with an active adversary. Previously, there are only two known explicit constructions of non-malleable extractors [DLWZ11, CRS11]. Both constructions only work for (n, k)-sources with k > n/2 and thus they also only achieve optimal privacy amplification protocols for k > n/2. In this paper, we give the first construction of non-malleable extractors for k 0, and a conditional construction that can potentially achieve k = αn for any constant α > 0. Using our non-malleable extractor, we also obtain the first optimal privacy amplification protocol for min-entropy k = (1/2 − δ)n, with an active adversary. Our constructions mainly use the bilinear property of the inner product function, and involve appropriate encodings of the seed y as well as ideas from additive combinatorics. ∗Partially supported by NSF Grants CCF-0634811, CCF-0916160, THECB ARP Grant 003658-0113-2007, and a Simons postdoctoral fellowship. ISSN 1433-8092 Electronic Colloquium on Computational Complexity, Report No. 166 (2011)
We construct several explicit quantum secure non-malleable-extractors. All the quantum secure non-malleable-extractors we construct are based on the constructions by Chattopadhyay, Goyal and Li [2015] and Cohen [2015]. 1) We construct the first explicit quantum secure non-malleable-extractor for (source) min-entropy k≥poly(log⁡(nε))k \geq \textsf{poly}\left(\log \left( \frac{n}ε \right)\right)k≥poly(log(εn​)) (nnn is the length of the source and εεε is the error parameter). Previously Aggarwal, Chung, Lin, and Vidick [2019] have shown that the inner-product based non-malleable-extractor proposed by Li [2012] is quantum secure, however it required linear (in nnn) min-entropy and seed length. Using the connection between non-malleable-extractors and privacy amplification (established first in the quantum setting by Cohen and Vidick [2017]), we get a 222-round privacy amplification protocol that is secure against active quantum adversaries with communication poly(log⁡(nε))\textsf{poly}\left(\log \left( \frac{n}ε \right)\right)poly(log(εn​)), exponentially improving upon the linear communication required by the protocol due to [2019]. 2) We construct an explicit quantum secure 222-source non-malleable-extractor for min-entropy k≥n−nΩ(1)k \geq n- n^{Ω(1)}k≥n−nΩ(1), with an output of size nΩ(1)n^{Ω(1)}nΩ(1) and error 2−nΩ(1)2^{- n^{Ω(1)}}2−nΩ(1). 3) We also study their natural extensions when the tampering of the inputs is performed ttt-times. We construct explicit quantum secure ttt-non-malleable-extractors for both seeded (t=dΩ(1)t=d^{Ω(1)}t=dΩ(1)) as well as 222-source case (t=nΩ(1)t=n^{Ω(1)}t=nΩ(1)).
Privacy amplification is the task by which two cooperating parties transform a shared weak secret, about which an eavesdropper may have side information, into a uniformly random string uncorrelated from the eavesdropper. Privacy amplification against passive adversaries, where it is assumed that the communication is over a public but authenticated channel, can be achieved in the presence of classical as well as quantum side information by a single-message protocol based on strong extractors. In 2009 Dodis and Wichs devised a two-message protocol to achieve privacy amplification against active adversaries, where the public communication channel is no longer assumed to be authenticated, through the use of a strengthening of strong extractors called non-malleable extractors which they introduced. Dodis and Wichs only analyzed the case of classical side information. We consider the task of privacy amplification against active adversaries with quantum side information. Our main result is showing that the Dodis-Wichs protocol remains secure in this scenario provided its main building block, the non-malleable extractor, satisfies a notion of quantum-proof non-malleability which we introduce. We show that an adaptation of a recent construction of non-malleable extractors due to Chattopadhyay et al. is quantum proof, thereby providing the first protocol for privacy amplification that is secure against active quantum adversaries. Our protocol is quantitatively comparable to the near-optimal protocols known in the classical setting.