  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.
