  The nonlocal behavior of quantum mechanics can be used to generate guaranteed
fresh randomness from an untrusted device that consists of two nonsignalling
components; since the generation process requires some initial fresh randomness
to act as a catalyst, one also speaks of randomness expansion. Colbeck and Kent
proposed the first method for generating randomness from untrusted devices,
however, without providing a rigorous analysis. This was addressed subsequently
by Pironio et al. [Nature 464 (2010)], who aimed at deriving a lower bound on
the min-entropy of the data extracted from an untrusted device, based only on
the observed non-local behavior of the device. Although that article succeeded
in developing important tools towards the acquired goal, it failed in putting
the tools together in a rigorous and correct way, and the given formal claim on
the guaranteed amount of min-entropy needs to be revisited. In this paper we
show how to combine the tools provided by Pironio et al., as to obtain a
meaningful and correct lower bound on the min-entropy of the data produced by
an untrusted device, based on the observed non-local behavior of the device.
Our main result confirms the essence of the improperly formulated claims of
Pironio et al., and puts them on solid ground. We also address the question of
composability and show that different untrusted devices can be composed in an
alternating manner under the assumption that they are not entangled. This
enables for superpolynomial randomness expansion based on two untrusted yet
unentangled devices.
