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.
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.
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 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. 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.
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.
How to generate provably true randomness with minimal assumptions? This question is important not only for the efficiency and the security of information processing, but also for understanding how extremely unpredictable events are possible in Nature. All current solutions require special structures in the initial source of randomness, or a certain independence relation among two or more sources. Both types of assumptions are impossible to test and difficult to guarantee in practice. Here we show how this fundamental limit can be circumvented by extractors that base security on the validity of physical laws and extract randomness from untrusted quantum devices. In conjunction with the recent work of Miller and Shi (arXiv:1402:0489), our physical randomness extractor uses just a single and general weak source, produces an arbitrarily long and near-uniform output, with a close-to-optimal error, secure against all-powerful quantum adversaries, and tolerating a constant level of implementation imprecision. The source necessarily needs to be unpredictable to the devices, but otherwise can even be known to the adversary. Our central technical contribution, the Equivalence Lemma, provides a general principle for proving composition security of untrusted-device protocols. It implies that unbounded randomness expansion can be achieved simply by cross-feeding any two expansion protocols. In particular, such an unbounded expansion can be made robust, which is known for the first time. Another significant implication is, it enables the secure randomness generation and key distribution using public randomness, such as that broadcast by NIST's Randomness Beacon. Our protocol also provides a method for refuting local hidden variable theories under a weak assumption on the available randomness for choosing the measurement settings.
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)-close to uniformly distributed and private, where ε is the robustness parameter of the self-test and m 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.
We present the first complete implementation of a randomness and privacy amplification protocol based on Bell tests. This allows the building of device-independent random number generators which output provably unbiased and private numbers, even if using an uncharacterised quantum device potentially built by an adversary. Our generation rates are linear in the runtime of the quantum device and the classical randomness post-processing has quasi-linear complexity -- making it efficient on a standard personal laptop. The statistical analysis is tailored for real-world quantum devices, making it usable as a quantum technology today.
We then showcase our protocol on the quantum computers from the IBM-Q experience. Although not purposely built for the task, we show that quantum computer can run faithful Bell tests by adding minimal assumptions. At a high level, these amount to trusting that the quantum device was not purposely built to trick the user, but otherwise remains mostly uncharacterised. In this semi-device-independent manner, our protocol generates provably private and unbiased random numbers on today's quantum computers.
In privacy amplification, two mutually trusted parties aim to amplify the secrecy of an initial shared secret X in order to establish a shared private key K 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 1/2. 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.