  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: cryptographic level of security, tolerating
a constant level of imprecision in the devices, requiring only a unit size
quantum memory per device component for the honest implementation, and allowing
a large natural class of constructions for the protocol. In conjunct with a
recent work by Chung, Shi and Wu, it also leads to robust unbounded expansion
using just 2 multi-part devices. When adapted for distributing cryptographic
keys, our method achieves, for the first time, exponential expansion combined
with cryptographic security and noise tolerance. The proof 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.
