  The squash operation, or the squashing model, is a useful mathematical tool
for proving the security of quantum key distribution systems using practical
(i.e., non-ideal) detectors. At the present, however, this method can only be
applied to a limited class of detectors, such as the threshold detector of the
Bennett-Brassard 1984 type. In this paper we generalize this method to include
multi-partite measurements, such that it can be applied to a wider class of
detectors. We demonstrate the effectiveness of this generalization by applying
it to the device-independent security proof of the Ekert 1991 protocol, and by
improving the associated key generation rate. For proving this result we use
two physical assumptions, namely, that quantum mechanics is valid, and that
Alice's and Bob's detectors are memoryless.
