  The imperfection of measurements in real-world scenarios can compromise the
performance of device-independent quantum key distribution (DIQKD) protocols.
In this paper, we focus on a specific DIQKD protocol that relies on the
violation of the Svetlichny's inequality (SI), considering an eavesdropper
utilizing the convex combination attack. Our analysis covers both the
three-party DIQKD case and the general $n$-party scenario. Our main result is
the relationship between the measurement accuracy and the extractable
secret-key rate in all multi-party scenarios. The result demonstrates that as
measurement accuracy improves, the extractable secret-key rate approaches $1$,
reaching its maximum value when the measurement accuracy is perfect. We reveal
that achieving positive extractable secret-key rates requires a threshold
measurement accuracy that is consistently higher than the critical measurement
accuracy necessary to violate the SI. We depict these thresholds for $n$-party
scenarios ranging from $n=3$ to $n=10$, demonstrating that as the number of
parties ($n$) increases, both thresholds exhibit a rapid and monotonic
convergence towards unity. Furthermore, we consider a scenario involving a
non-maximally entangled state with imperfect measurements, where the emission
of the initial GHZ state undergoes noise during transmission, resulting in a
Werner state. The study further quantifies and demonstrates the relationship
between the extractable secret-key rate, the visibility of the Werner state,
and the measurement accuracy, specifically emphasizing the three-party
scenario. This study aims to illuminate the influence of imperfect measurement
accuracy on the security and performance of multi-party DIQKD protocols. The
results emphasize the importance of high measurement accuracy in achieving
positive secret-key rates and maintaining the violation of the SI.
