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SIAM Journal on Computing

Deterministic Randomness Extraction from Generalized and Distributed Santha--Vazirani Sources

Salman Beigi, Omid Etesami, Amin Gohari

Published Jan 1, 2017• 8 citations
Publisher

Abstract

A Santha--Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to $s$ “nondegenerate” dice that are $c$-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if $s<c$. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.

DOI
10.1137/15m1027206

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A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs $(a_1, b_1), (a_2, b_2), \ldots$ distributed between two parties, where the first party receives $a_i$'s and the second one receives $b_i$'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.

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Cryptographic protocols are commonly designed and their security proven under the assumption that the protocol parties have access to perfect (uniform) randomness. Physical randomness sources deployed in practical implementations of these protocols often fall short in meeting this assumption, but instead provide only a steady stream of bits with certain high entropy. Trying to ground cryptographic protocols on such imperfect, weaker sources of randomness has thus far mostly given rise to a multitude of impossibility results, including the impossibility to construct provably secure encryption, commitments, secret sharing, and zero-knowledge proofs based solely on a weak source. More generally, indistinguishability-based properties break down for such weak sources. In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirani (SV) sources. The quantification relies on a novel relaxation of indistinguishability by a quantitative parameter. We call the resulting notion dierential indistinguishability in order to reflect its structural similarity to dierential privacy. More concretely, we prove that indistinguishability with uniform randomness implies dierential indistinguishability with weak randomness. We show that if the amount of weak randomness is limited (e.g., by using it only to seed a PRG), all cryptographic primitives and protocols still achieve dierential indistinguishability.

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A Santha--Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for nonbinary sequences. We show that unlike the binary setup of Santha and Vazirani, deterministic randomness extraction in the generalized case is sometimes possible. In particular, if the adversary has access to sss “nondegenerate” dice that are ccc-sided and can choose one die to throw based on the previous realizations of the dice, then deterministic randomness extraction is possible if s<cs<cs<c. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions complement each other in the nondegenerate cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs (a1,b1),(a2,b2),…(a_1, b_1), (a_2, b_2), \ldots(a1​,b1​),(a2​,b2​),… distributed between two parties, where the first party receives aia_iai​'s and the second one receives bib_ibi​'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (nondistributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from independently and identically distributed sources to adversarial sources.
A Santha-Vazirani (SV) source is a sequence of random bits where the conditional distribution of each bit, given the previous bits, can be partially controlled by an adversary. Santha and Vazirani show that deterministic randomness extraction from these sources is impossible. In this paper, we study the generalization of SV sources for non-binary sequences. We show that unlike the binary case, deterministic randomness extraction in the generalized case is sometimes possible. We present a necessary condition and a sufficient condition for the possibility of deterministic randomness extraction. These two conditions coincide in "non-degenerate" cases. Next, we turn to a distributed setting. In this setting the SV source consists of a random sequence of pairs (a1,b1),(a2,b2),…(a_1, b_1), (a_2, b_2), \ldots(a1​,b1​),(a2​,b2​),… distributed between two parties, where the first party receives aia_iai​'s and the second one receives bib_ibi​'s. The goal of the two parties is to extract common randomness without communication. Using the notion of maximal correlation, we prove a necessary condition and a sufficient condition for the possibility of common randomness extraction from these sources. Based on these two conditions, the problem of common randomness extraction essentially reduces to the problem of randomness extraction from (non-distributed) SV sources. This result generalizes results of Gács and Körner, and Witsenhausen about common randomness extraction from i.i.d. sources to adversarial sources.
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, Nature Physics 8, 450-454 (2012), on randomness amplification 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 ≤ ε <(2^((1/12))−1)/2(2^((1/12))+1) ≈ 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 against this attack for the parameter range 0 ≤ ε < 0.0132. We leave the case of a yet more general attack on the amplification protocol as an interesting open problem.
The extraction of randomness from weakly random seeds is a topic of central importance in cryptography. Weak sources of randomness can be considered to be either private or public, where public sources such as the NIST randomness beacon broadcast the random bits once they are generated. The problem of device-independent randomness extraction from weak public sources against no-signalling adversaries has remained open. In this paper, we show protocols for device-independent and one-sided device-independent amplification of randomness from weak public Santha Vazirani (SV) sources that use a finite number of devices and are secure against no-signaling adversaries. Specifically, under the assumption that the device behavior is as prescribed by quantum mechanics the protocols allow for amplification of public εεε-SV sources for arbitrary initial ε∈[0,0.5)ε\in [0,0.5)ε∈[0,0.5). On the other hand, when only the assumption of no-signaling between the components of the device is made, the protocols allow for amplification of a limited set of weak public SV sources.
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 paper, 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 (with0≤ε<(2(1/12)−1)/(2(2(1/12)+1))≈0.01440 \leq \varepsilon < (2^{(1/12)} - 1)/(2(2^{(1/12)} + 1)) \approx 0.01440≤ε<(2(1/12)−1)/(2(2(1/12)+1))≈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 range0≤ε<0.01320 \leq \varepsilon <0.01320≤ε<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.
Recently, the physically realistic protocol amplifying the randomness of Santha-Vazirani sources producing cryptographically secure random bits was proposed; however, for reasons of practical relevance, the crucial question remained open regarding whether this can be accomplished under the minimal conditions necessary for the task. Namely, is it possible to achieve randomness amplification using only two no-signaling components and in a situation where the violation of a Bell inequality only guarantees that some outcomes of the device for specific inputs exhibit randomness? Here, we solve this question and present a device-independent protocol for randomness amplification of Santha-Vazirani sources using a device consisting of two nonsignaling components. We show that the protocol can amplify any such source that is not fully deterministic into a fully random source while tolerating a constant noise rate and prove the composable security of the protocol against general no-signaling adversaries. Our main innovation is the proof that even the partial randomness certified by the two-party Bell test [a single input-output pair (u^{*}, x^{*}) for which the conditional probability P(x^{*}|u^{*}) is bounded away from 1 for all no-signaling strategies that optimally violate the Bell inequality] can be used for amplification. We introduce the methodology of a partial tomographic procedure on the empirical statistics obtained in the Bell test that ensures that the outputs constitute a linear min-entropy source of randomness. As a technical novelty that may be of independent interest, we prove that the Santha-Vazirani source satisfies an exponential concentration property given by a recently discovered generalized Chernoff bound.
Cryptographic protocols are commonly designed and their security proven under the assumption that the protocol parties have access to perfect (uniform) randomness. Physical randomness sources deployed in practical implementations of these protocols often fall short in meeting this assumption, but instead provide only a steady stream of bits with certain high entropy. Trying to ground cryptographic protocols on such imperfect, weaker sources of randomness has thus far mostly given rise to a multitude of impossibility results, including the impossibility to construct provably secure encryption, commitments, secret sharing, and zero-knowledge proofs based solely on a weak source. More generally, indistinguishability-based properties break down for such weak sources. In this paper, we show that the loss of security induced by using a weak source can be meaningfully quantified if the source is bounded, e.g., for the well-studied Santha-Vazirani (SV) sources. The quantification relies on a novel relaxation of indistinguishability by a quantitative parameter. We call the resulting notion dierential indistinguishability in order to reflect its structural similarity to dierential privacy. More concretely, we prove that indistinguishability with uniform randomness implies dierential indistinguishability with weak randomness. We show that if the amount of weak randomness is limited (e.g., by using it only to seed a PRG), all cryptographic primitives and protocols still achieve dierential indistinguishability.