  We initiate a systematic study to provide upper bounds on device-independent
key, secure against a non-signaling adversary (NSDI). We employ the idea of
"squashing" on the secrecy monotones and show that squashed secrecy monotones
are the upper bounds on the NSDI key. Our technique for obtaining upper bounds
is based on the non-signaling analog of quantum purification: the complete
extension. As an important instance of an upper bound, we construct a measure
of nonlocality called "squashed nonlocality". Using this bound, we identify
numerically a certain domain of two binary inputs and two binary outputs
non-local devices for which the squashed nonlocality is zero. Therefore one can
not distill secure-key from these non-local devices via a considered (standard)
class of operations. Showing a connection of our approach to [New J. Phys.,
8:126, 2006] we provide, to our knowledge, the tightest known upper bound in
the (3,2,2,2) scenario. Moreover, we formulate a security condition, equivalent
to known ones, for the considered class of protocols. To achieve this, we
introduce a non-signaling norm that constitutes an analogy to the trace norm
used in the security condition of the quantum key distribution.
