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Randomness is a vital resource for modern day information processing, especially for cryptography. A wide range of applications critically rely on abundant, high quality random numbers generated securely. Here we show how to expand a random seed at an exponential rate without trusting the underlying quantum devices. Our approach is secure against the most general adversaries, and has the following new features: tolerating a constant level of implementation imprecision, requiring only a unit size quantum memory per device component for the honest implementation, and allowing a large natural class of constructions. In conjunct with a recent work by Chung, Shi and Wu (QIP 2014), it leads to robust unbounded expansion using just 2 multi-part devices. It can also be adapted for distributing cryptographic keys securely. The proof begins with a known protocol and proceeds by showing that the Renyi divergence of the outputs of the protocol (for a specific bounding operator) decreases linearly as the protocol iterates. At the heart of the proof are a new uncertainty principle on quantum measurements, and a method for simulating trusted measurements with untrusted devices. A full version of this paper containing additional results developed after the conference submission is available as arXiv:1402.0489.

Robust protocols for securely expanding randomness and distributing keys using untrusted quantum devices