  In this work, we develop upper bounds for key rates for device-independent
quantum key distribution (DI-QKD) protocols and devices. We study the reduced
cc-squashed entanglement and show that it is a convex functional. As a result,
we show that the convex hull of the currently known bounds is a tighter upper
bound on the device-independent key rates of standard CHSH-based protocol. We
further provide tighter bounds for DI-QKD key rates achievable by any protocol
applied to the CHSH-based device. This bound is based on reduced relative
entropy of entanglement optimized over decompositions into local and non-local
parts. In the dynamical scenario of quantum channels, we obtain upper bounds
for device-independent private capacity for the CHSH based protocols. We show
that the device-independent private capacity for the CHSH based protocols on
depolarizing and erasure channels is limited by the secret key capacity of
dephasing channels.
