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ACM Symposium on Theory of Computing (STOC)

Infinite randomness expansion with a constant number of devices

Matthew Coudron, Henry Yuen

Published May 31, 2014• 44 citations
Publisher

Abstract

We present a device-independent randomness expansion protocol, involving only a constant number of non-signaling quantum devices, that achieves infinite expansion: starting with m bits of uniform private randomness, the protocol can produce an unbounded amount of certified randomness that is exp(--Ω(m1/3))-close to uniform and secure against a quantum adversary. The only parameters which depend on the size of the input are the soundness of the protocol and the security of the output (both are inverse exponential in m). This settles a long-standing open problem in the area of randomness expansion and device-independence.

DOI
10.1145/2591796.2591873

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We present a device-independent randomness expansion protocol, involving only a constant number of non-signaling quantum devices, that achieves infinite expansion: starting with m bits of uniform private randomness, the protocol can produce an unbounded amount of certified randomness that is exp(--Ω(m1/3))-close to uniform and secure against a quantum adversary. The only parameters which depend on the size of the input are the soundness of the protocol and the security of the output (both are inverse exponential in m). This settles a long-standing open problem in the area of randomness expansion and device-independence.
We present a device-independent randomness expansion protocol, involving only a constant number of non-signaling quantum devices, that achieves \emph{infinite expansion}: starting with mmm bits of uniform private randomness, the protocol can produce an unbounded amount of certified randomness that is exp⁡(−Ω(m1/3))\exp(-Ω(m^{1/3}))exp(−Ω(m1/3))-close to uniform and secure against a quantum adversary. The only parameters which depend on the size of the input are the soundness of the protocol and the security of the output (both are inverse exponential in mmm). This settles a long-standing open problem in the area of randomness expansion and device-independence. The analysis of our protocols involves overcoming fundamental challenges in the study of \emph{adaptive} device-independent protocols. Our primary technical contribution is the design and analysis of device-independent protocols which are \emph{Input Secure}; that is, their output is guaranteed to be secure against a quantum eavesdropper, \emph{even if the input randomness was generated by that same eavesdropper}! The notion of Input Security may be of independent interest to other areas such as device-independent quantum key distribution.
We introduce a protocol through which a pair of quantum mechanical devices may be used to generate n bits of true randomness from a seed of O(log n) uniform bits. The bits generated are certifiably random based only on a simple statistical test that can be performed by the user, and on the assumption that the devices obey the no-signaling principle. No other assumptions are placed on the devices' inner workings. A modified protocol uses a seed of O(log^3 n) uniformly random bits to generate nnn bits of true randomness even conditioned on the state of a quantum adversary who may have had prior access to the devices, and may be entangled with them.
Randomness is an important resource for many applications, from gambling to secure communication. However, guaranteeing that the output from a candidate random source could not have been predicted by an outside party is a challenging task, and many supposedly random sources used today provide no such guarantee. Quantum solutions to this problem exist, for example a device which internally sends a photon through a beam-splitter and observes on which side it emerges, but, presently, such solutions require the user to trust the internal workings of the device. Here we seek to go beyond this limitation by asking whether randomness can be generated using untrusted devices---even ones created by an adversarial agent---while providing a guarantee that no outside party (including the agent) can predict it. Since this is easily seen to be impossible unless the user has an initially private random string, the task we investigate here is private randomness expansion. We introduce a protocol for private randomness expansion with untrusted devices which is designed to take as input an initially private random string and produce as output a longer private random string. We point out that private randomness expansion protocols are generally vulnerable to attacks that can render the initial string partially insecure, even though that string is used only inside a secure laboratory; our protocol is designed to remove this previously unconsidered vulnerability by privacy amplification. We also discuss extensions of our protocol designed to generate an arbitrarily long random string from a finite initially private random string. The security of these protocols against the most general attacks is left as an open question.
In quantum cryptography, secure randomness expansion involves using a short private string of random bits to generate a longer one, even in the presence of an adversary who may have access to quantum resources. In this work, we demonstrate that local contextuality-based self-tests are sufficient to construct a randomness expansion protocol that is secure against computationally unbounded quantum adversaries. Our protocol is based on self-testing from non-contextuality inequalities and we prove that our scheme asymptotically produces secure random numbers which are O(mε)\mathcal{O}(m\sqrtε)O(mε​)-close to uniformly distributed and private, where εεε is the robustness parameter of the self-test and mmm is the length of the generated random bit string. Our protocol is semi-device-independent in the sense that it inherits any assumptions necessary for the underlying self-test.
Recent randomness expansion protocols have been proposed which are able to generate an unbounded amount of randomness from a finite amount of truly random initial seed. One such protocol, given by Miller and Shi, uses a pair of non-signaling untrusted quantum mechanical devices. These play XOR games with inputs given by the user in order to generate an output. Here we present an analysis of the required seed size, giving explicit upper bounds for the number of initial random bits needed to jump-start the protocol. The bits output from such a protocol are ε\varepsilonε-close to uniform even against quantum adversaries. Our analysis yields that for a statistical distance of ε=10−1\varepsilon=10^{-1}ε=10−1 and ε=10−6\varepsilon=10^{-6}ε=10−6 from uniformity, the number of required bits is smaller than 225,000 and 715,000, respectively; in general it grows as O(log⁡1ε)O(\log\frac{1}{\varepsilon})O(logε1​).
Measurements on entangled quantum systems necessarily yield outcomes that are intrinsically unpredictable if they violate a Bell inequality. This property can be used to generate certified randomness in a device-independent way, i.e., without making detailed assumptions about the internal working of the quantum devices used to generate the random numbers. Furthermore these numbers are also private; i.e., they appear random not only to the user but also to any adversary that might possess a perfect description of the devices. Since this process requires a small initial random seed to sample the behavior of the quantum devices and to extract uniform randomness from the raw outputs of the devices, one usually speaks of device-independent randomness expansion. The purpose of this paper is twofold. First, we point out that in most real, practical situations, where the concept of device independence is used as a protection against unintentional flaws or failures of the quantum apparatuses, it is sufficient to show that the generated string is random with respect to an adversary that holds only classical side information; i.e., proving randomness against quantum side information is not necessary. Furthermore, the initial random seed does not need to be private with respect to the adversary, provided that it is generated in a way that is independent from the measured systems. The devices, however, will generate cryptographically secure randomness that cannot be predicted by the adversary, and thus one can, given access to free public randomness, talk about private randomness generation. The theoretical tools to quantify the generated randomness according to these criteria were already introduced in S. Pironio et al. [Nature (London) 464, 1021 (2010)], but the final results were improperly formulated. The second aim of this paper is to correct this inaccurate formulation and therefore lay out a precise theoretical framework for practical device-independent randomness generation.