Date of Original Version
Abstract or Description
In wireless sensor networks (WSNs), the Eschenauer–Gligor (EG) key pre-distribution scheme is a widely recognized way to secure communications. Although the connectivity properties of secure WSNs with the EG scheme have been extensively investigated, few results address physical transmission constraints. These constraints reflect real–world implementations of WSNs in which two sensors have to be within a certain distance from each other to communicate. In this paper, we present zero–one laws for connectivity in WSNs employing the EG scheme under transmission constraints. These laws improve recent results ,  significantly and help specify the critical transmission ranges for connectivity. Our analytical findings, which are also confirmed via numerical experiments, provide precise guidelines for the design of secure WSNs in practice. In addition to secure WSNs, our theoretical results are also applied to frequency hopping of wireless networks, as discussed in detail..
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