Randomness Amplification under Minimal Fundamental Assumptions on the Devices

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 (${\mathbf{u}}^{*}$, ${\mathbf{x}}^{*}$) for which the conditional probability $P({\mathbf{x}}^{*}|{\mathbf{u}}^{*})$ 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.

Figure 1

Protocol for device-independent randomness amplification from a single device with two no-signaling components.

Figure 2

An illustration of the protocol for randomness amplification using two no-signaling components. The bits from the SV source (black arrows) are used as inputs (${\mathbf{u}}_j^1$, ${\mathbf{u}}_j^2$) for the $j$th run of the two spatially separated devices, with $1\le j\le n$, and the corresponding outputs (${\mathbf{x}}_j^1$, ${\mathbf{u}}_j^2$) are obtained. The inputs and outputs of all the $n$ runs ($u$, $x$) are subjected to two tests: a Bell test for the violation of a specific Bell inequality and a (partial) tomographic test counting a specific number of input-output pairs (${\mathbf{u}}^{*}$, ${\mathbf{x}}^{*}$). If both tests are passed (denoted by ACC), the outputs $x$ (orange arrows) are hashed together with further $n$ bits $t$ from the SV source using an extractor.