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arXiv:2304.09643

Finite Device-Independent Extraction of a Block Min-Entropy Source against Quantum Adversaries

Ravishankar Ramanathan

Published Apr 19, 2023• 0 citations
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Abstract

The extraction of randomness from weakly random seeds is a problem of central importance with multiple applications. In the device-independent setting, this problem of quantum randomness amplification has been mainly restricted to specific weak sources of Santha-Vazirani type, while extraction from the general min-entropy sources has required a large number of separated devices which is impractical. In this paper, we present a device-independent protocol for amplification of a single min-entropy source (consisting of two blocks of sufficiently high min-entropy) using a device consisting of two spatially separated components and show a proof of its security against general quantum adversaries.

DOI
10.48550/arxiv.2304.09643

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The extraction of randomness from weakly random seeds is a problem of central importance with multiple applications. In the device-independent setting, this problem of quantum randomness amplification has been mainly restricted to specific weak sources of Santha-Vazirani type, while extraction from the general min-entropy sources has required a large number of separated devices which is impractical. In this paper, we present a device-independent protocol for amplification of a single min-entropy source (consisting of two blocks of sufficiently high min-entropy) using a device consisting of two spatially separated components and show a proof of its security against general quantum adversaries.
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.
Expansion and amplification of weak randomness play a crucial role in many security protocols. Using quantum devices, such procedure is possible even without trusting the devices used, by utilizing correlations between outcomes of parts of the devices. We show here how to extract random bits with an arbitrarily low bias from single arbitrarily weak min-entropy block source in a device independent setting. To do this we use Mermin devices that exhibit superclassical correlations. The number of devices used scales polynomially in the length of the random sequence nnn. Our protocol is robust; it can tolerate devices that malfunction with probability decreasing polynomially in nnn at the cost of minor increase in the number of devices used.
Secret and perfect randomness is an essential resource in cryptography. Yet, it is not even clear that such exists. It is well known that the tools of classical computer science do not allow us to create secret and perfect randomness from a single weak public source. Quantum physics, on the other hand, allows for such a process, even in the most paranoid cryptographic sense termed “device-independent quantum cryptography”. We propose and prove the security of a new device-independent protocol that takes any single public Santha-Vazirani source as input and creates a secret close to uniform string in the presence of a quantum adversary. Our work is the first to achieve randomness amplification with all the following properties: (1) amplification and “privatization” of a public Santha-Vazirani source with arbitrary bias (2) the use of a device with only two components (3) non-vanishing extraction rate and (4) maximal noise tolerance. In particular, this implies that our protocol is the first protocol that can possibly be implemented with reachable parameters. We achieve these by combining three new tools: a particular family of Bell inequalities, a proof technique to lower bound entropy in the device-independent setting, and a framework for quantum-proof multi-source extractors.
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.
In this paper we design a protocol to extract random bits with an arbitrarily low bias from a single arbitrarily weak min-entropy block source in a device independent setting. The protocol employs Mermin devices that exhibit super-classical correlations. Number of devices used scales polynomially in the length of the block n, containing entropy of at least two bits. Our protocol is robust, it can tolerate devices that malfunction with a probability dropping polynomially in n at the cost of constant increase of the number of devices used.
In this paper we design a protocol to extract random bits with an arbitrarily low bias from a single arbitrarily weak min-entropy block source in a device independent setting. The protocol employs Mermin devices that exhibit super-classical correlations. Number of devices used scales polynomially in the length of the block n, containing entropy of at least two bits. Our protocol is robust, it can tolerate devices that malfunction with a probability dropping polynomially in n at the cost of constant increase of the number of devices used.