Weak Random Sources Research Articles

Amplifying the Randomness of Weak Sources Correlated With Devices

The problem of device-independent randomness amplification against no-signaling adversaries has so far been studied under the assumption that the weak source of randomness is uncorrelated with the (quantum) devices used in the amplification procedure. In this work, we relax this assumption, and reconsider the original protocol of Colbeck and Renner using a Santha-Vazirani (SV) source. To do so, we introduce an SV-like condition for devices, namely that any string of SV source bits remains weakly random conditioned upon any other bit string from the same SV source and the outputs obtained when this further string is input into the devices. Assuming this condition, we show that a quantum device using a~singlet state to violate the chained Bell inequalities leads to full randomness in the asymptotic scenario of a large number of settings, for a restricted set of SV sources (with $0 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.0144$). We also study a device-independent protocol that allows for correlations between the sequence of boxes used in the protocol and the SV source bits used to choose the particular box from whose output the randomness is obtained. Assuming the SV-like condition for devices, we show that the honest parties can achieve amplification of the weak source, for the parameter range $0 \leq \varepsilon<0.0132$, against a class of attacks given as a mixture of product box sequences, made of extremal no-signaling boxes, with additional symmetry conditions. Composable security proof against this class of attacks is provided.

Device-independent randomness amplification with a single device

Expansion and amplification of weak randomness with untrusted quantum devices has recently become a very fruitful topic of research. Here we contribute with a procedure for amplifying a single weak random source using tri-partite GHZ-type entangled states. If the quality of the source reaches a fixed threshold $R=\frac{1}{4}\log_{2}(10)$, perfect random bits can be produced. This technique can be used to extract randomness from sources that can't be extracted neither classically, nor by existing procedures developed for Santha-Vazirani sources. Our protocol works with a single fault-free device decomposable into three non-communicating parts, that is repeatedly reused throughout the amplification process.

Review of Deterministic Extraction from weak random sources by Ariel Gabizon

Review of Deterministic Extraction from weak random sources by Ariel Gabizon

Quantum to Classical Randomness Extractors

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 a natural question 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. As the first application, 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. As the second application, we resolve the central open question in the noisy-storage model [Wehner et al., PRL 100, 220502 (2008)] by linking security to the quantum capacity of the adversary's storage device.

Encryption with weakly random keys using quantum ciphertext

The lack of perfect randomness can cause significant problems in securing communication between two parties. McInnes and Pinkas \cite{McInnesPinkas-ImpossibilityofPrivate-1991} proved that unconditionally secure encryption is impossible when the key is sampled from a weak random source. The adversary can always gain some information about the plaintext, regardless of the cryptosystem design. Most notably, the adversary can obtain full information about the plaintext if he has access to just two bits of information about the source (irrespective on length of the key). In this paper we show that for every weak random source there is a cryptosystem with a classical plaintext, a classical key, and a quantum ciphertext that bounds the adversary's probability $p$ to guess correctly the plaintext strictly under the McInnes-Pinkas bound, except for a single case, where it coincides with the bound. In addition, regardless of the source of randomness, the adversary's probability $p$ is strictly smaller than $1$ as long as there is some uncertainty in the key (Shannon/min-entropy is non-zero). These results are another demonstration that quantum information processing can solve cryptographic tasks with strictly higher security than classical information processing.

Two-Source Extractors Secure Against Quantum Adversaries

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 (1988) two-source inner product extractor and its multi-bit variant by Dodis et al. (RANDOM'04). 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.

On The Randomness Complexity of Property Testing

We initiate a general study of the randomness complexity of property testing, aimed at reducing the randomness complexity of testers without (significantly) increasing their query complexity. One concrete motivation for this study is provided by the observation that the product of the randomness and query complexity of a tester determine the actual query complexity of implementing a version of this tester that utilizes a weak source of randomness (through a randomness-extractor). We present rather generic upper and lower bounds on the randomness complexity of property testing and study in depth the special case of testing bipartiteness in two standard property testing models.

Competitive algorithms and lower bounds for online randomized call control in cellular networks

AbstractWe address an important communication issue arising in wireless cellular networks that utilize frequency division multiplexing (FDM) technology. In such networks, many users within the same geographical region (cell) can communicate simultaneously with other users of the network using distinct frequencies. The spectrum of the available frequencies is limited; thus, efficient solutions to the call control problem are essential. The objective of the call control problem is, given a spectrum of available frequencies and users that wish to communicate, to maximize the benefit, i.e., the number of users that communicate without signal interference. We consider cellular networks of reuse distance k ≥ 2 and we study the online version of the problem using competitive analysis. In cellular networks of reuse distance 2, the previously best known algorithm that beats the lower bound of 3 on the competitiveness of deterministic algorithms, works on networks with one frequency, achieves a competitive ratio against oblivious adversaries, which is between 2.469 and 2.651, and uses a number of random bits at least proportional to the size of the network. We significantly improve this result by presenting a series of simple randomized algorithms that have competitive ratios significantly smaller than 3, work on networks with arbitrarily many frequencies, and use only a constant number of random bits or a comparable weak random source. The best competitiveness upper bound we obtain is 16/7 using only four random bits. In cellular networks of reuse distance k&gt; 2, we present simple randomized online call control algorithms with competitive ratios, which significantly beat the lower bounds on the competitiveness of deterministic ones and use only O(log k) random bits. Also, we show new lower bounds on the competitiveness of online call control algorithms in cellular networks of any reuse distance. In particular, we show that no online algorithm can achieve competitive ratio better than 2, 25/12, and 2.5, in cellular networks with reuse distance k ε {2, 3, 4}, k = 5, and k ≥ 6, respectively. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

Special Issue on Randomness and Complexity

Special Issue on Randomness and Complexity

Constructing Locally Computable Extractors and Cryptosystems in the Bounded-Storage Model

We consider the problem of constructing randomness extractors that are locally computable; that is, read only a small number of bits from their input. As recently shown by Lu locally computable extractors directly yield secure private-key cryptosystems in Maurer’s bounded-storage model. We suggest a general “sample-then-extract” approach to constructing locally computable extractors: use essentially any randomness-efficient sampler to select bits from the input and then apply any extractor to the selected bits. Plugging in known sampler and extractor constructions, we obtain locally computable extractors, and hence cryptosystems in the bounded-storage model, whose parameters improve upon previous constructions. We also provide lower bounds showing that the parameters we achieve are nearly optimal. The correctness of the sample-then-extract approach follows from a fundamental lemma of Nisan and Zuckerman, which states that sampling bits from a weak random source roughly preserves the min-entropy rate. We also present a refinement of this lemma, showing that the min-entropy rate is preserved up to an arbitrarily small additive loss, whereas the original lemma loses a logarithmic factor.

Computing with Very Weak Random Sources

We give an efficient algorithm to extract randomness from a very weak random source using a small additional number t of truly random bits. Our work extends that of Nisan and Zuckerman [ J. Comput. System Sci., 52 (1996), pp. 43--52] in that t remains small even if the entropy rate is well below constant. A key application of this is in running randomized algorithms using such a very weak source of randomness. For any fixed $\gamma > 0$, we show how to simulate RP algorithms in time $n^{O(\log n)}$ using the output of a \ds with min-entropy $R^\gamma$. Such a weak random source is asked once for $R$ bits; it outputs an $R$-bit string according to any probability distribution that places probability at most $2^{-R^\gamma}$ on each string. If $\gamma > 1/2$, our simulation also works for BPP; for $\gamma > 1-1/(k+1)$, our simulation takes time $n^{O(\logk n)}$ (log(k) is the logarithm iterated k times). We also give a polynomial-time BPP simulation using Chor--Goldreich sources of min-entropy $R^{\Omega(1)}$, which is optimal. We present applications to time-space tradeoffs, expander constructions, and to the hardness of approximation. Of independent interest is our randomness-efficient Leftover Hash Lemma, a key tool for extracting randomness from weak random sources.

Weak Random Sources, Hitting Sets, and BPP Simulations

We show how to simulate any BPP algorithm in polynomial time by using a weak random source of r bits and min-entropy $r^{\gamma}$ for any $\gamma >0$. This follows from a more general result about sampling with weak random sources. Our result matches an information-theoretic lower bound and solves a question that has been open for some years. The previous best results were a polynomial time simulation of RP [M. Saks, A. Srinivasan, and S. Zhou, Proc. 27th ACM Symp. on Theory of Computing, 1995, pp. 479--488] and a quasi-polynomial time simulation of BPP [A. Ta-Shma, Proc. 28th ACM Symp. on Theory of Computing, 1996, pp. 276--285]. Departing significantly from previous related works, we do not use extractors; instead, we use the OR-disperser of Saks, Srinivasan, and Zhou in combination with a tricky use of hitting sets borrowed from [Andreev, Clementi, and Rolim, J. ACM, 45 (1998), pp. 179--213].

Simulating BPP using a general weak random source

We show how to simulate BPP and approximation algorithms in polynomial time using the output from a δ-source. A δ-source is a weak random source that is asked only once forR bits, and must output anR-bit string according to some distribution that places probability no more than 2−δR on any particular string. We also give an application to the unapproximability of MAX CLIQUE.

Unbiased Bits from Sources of Weak Randomness and Probabilistic Communication Complexity

A new model for weak random physical sources is presented. The new model strictly generalizes previous models (e.g., the Santha and Vazirani model [27]). The sources considered output strings according to probability distributions in which no single string is too probable.The new model provides a fruitful viewpoint on problems studied previously such as: • Extracting almost-perfect bits from sources of weak randomness. The question of possibility as well as the question of efficiency of such extraction schemes are addressed. • Probabilistic communication complexity. It is shown that most functions have linear communication complexity in a very strong probabilistic sense. • Robustness of BPP with respect to sources of weak randomness (generalizing a result of Vazirani and Vazirani [32], [33]).

Inexpensive Weak Random Light Source for Timing Photomultiplier Channels

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