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Randomness is an important resource for many applications, from gambling tosecure communication. However, guaranteeing that the output from a candidaterandom source could not have been predicted by an outside party is achallenging task, and many supposedly random sources used today provide no suchguarantee. Quantum solutions to this problem exist, for example a device whichinternally sends a photon through a beam-splitter and observes on which side itemerges, but, presently, such solutions require the user to trust the internalworkings of the device. Here we seek to go beyond this limitation by askingwhether randomness can be generated using untrusted devices---even ones createdby an adversarial agent---while providing a guarantee that no outside party(including the agent) can predict it. Since this is easily seen to beimpossible unless the user has an initially private random string, the task weinvestigate here is private randomness expansion.  We introduce a protocol for private randomness expansion with untrusteddevices which is designed to take as input an initially private random stringand produce as output a longer private random string. We point out that privaterandomness expansion protocols are generally vulnerable to attacks that canrender the initial string partially insecure, even though that string is usedonly inside a secure laboratory; our protocol is designed to remove thispreviously unconsidered vulnerability by privacy amplification. We also discussextensions of our protocol designed to generate an arbitrarily long randomstring from a finite initially private random string. The security of theseprotocols against the most general attacks is left as an open question.

Private Randomness Expansion With Untrusted Devices