  Device-independent quantum cryptography allows security even if the devices
used to execute the protocol are untrusted - whether this is due to unknown
imperfections in the implementation, or because the adversary himself
constructed them to subvert the security of the protocol. While
device-independence has seen much attention in the domain of quantum key
distribution, relatively little is known for general protocols. Here we
introduce a new model for device-independence for two-party protocols and
position verification in the noisy-storage model. For the first time, we show
that such protocols are secure in the most general device-independent model in
which the devices may have arbitrary memory, states and measurements. In our
analysis, we make use of a slight modification of a beautiful new tool
developed in [arXiv:1607.01796] called "Entropy Accumulation Theorem". What's
more, the protocols we analyze use only simple preparations and measurements,
and can be realized using any experimental setup able to perform a CHSH Bell
test. Specifically, security can be attained for any violation of the CHSH
inequality, where a higher violation merely leads to a reduction in the amount
of rounds required to execute the protocol.
