  Security is an important issue in wireless sensor networks (WSNs), which are
often deployed in hostile environments. The q-composite key predistribution
scheme has been recognized as a suitable approach to secure WSNs. Although the
q-composite scheme has received much attention in the literature, there is
still a lack of rigorous analysis for secure WSNs operating under the
q-composite scheme in consideration of the unreliability of links. One main
difficulty lies in analyzing the network topology whose links are not
independent. Wireless links can be unreliable in practice due to the presence
of physical barriers between sensors or because of harsh environmental
conditions severely impairing communications. In this paper, we resolve the
difficult challenge and investigate k-connectivity in secure WSNs operating
under the q-composite scheme with unreliable communication links modeled as
independent on/off channels, where k-connectivity ensures connectivity despite
the failure of any (k - 1) sensors or links, and connectivity means that any
two sensors can find a path in between for secure communication. Specifically,
we derive the asymptotically exact probability and a zero-one law for
k-connectivity. We further use the theoretical results to provide design
guidelines for secure WSNs. Experimental results also confirm the validity of
our analytical findings.
