  The Eschenauer--Gligor (EG) random key predistribution scheme has been widely
recognized as a typical approach to secure communications in wireless sensor
networks (WSNs). However, there is a lack of precise probability analysis on
the reliable connectivity of WSNs under the EG scheme. To address this, we
rigorously derive the asymptotically exact probability of $k$-connectivity in
WSNs employing the EG scheme with unreliable links represented by independent
on/off channels, where $k$-connectivity ensures that the network remains
connected despite the failure of any $(k-1)$ sensors or links. Our analytical
results are confirmed via numerical experiments, and they provide precise
guidelines for the design of secure WSNs that exhibit a desired level of
reliability against node and link failures.
