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Lemma 1. Assuming that two neighboring ordinary nodes U and V have only a common auxiliary node, if U shares exactly i root keys with V, the common derived keys shared between them must be no more than.

Hence, we have several different strategies to obtain a certain number of common auxiliary nodes according to the application.

In addition, it also shows an interesting phenomenon, that is, increasing the number of common auxiliary nodes does not result in a remarkable decrease of the number of preloaded root keys when the auxiliary node preloads enough number of initial keys.

Note that the hop count is a designed system parameter which determines the number of common auxiliary nodes for two neighboring ordinary nodes.

We will discuss how the hop count along with network degree and radio range affects the number of common auxiliary nodes in Section 5. When an ordinary node wants to establish pairwise key with its neighboring ordinary node, will send all of its root key identifiers and its auxiliary node identifiers to.

For example, when an auxiliary node preloads 3000 initial keys and the number of common auxiliary nodes increases from 2 to 3, the number of root keys required in ordinary node decreases by only 3.29% (from 152 to 147).

Hence there are ways to let every common auxiliary node provide shared initial keys.

If so, they cannot directly establish pairwise key with help of their common auxiliary nodes.

Ordinary node must authenticate each initial key sent from the common auxiliary nodes.

Similarly, increasing the transmission of auxiliary nodes can also effectively increase the number of common auxiliary nodes.

The adversary may impersonate the common auxiliary nodes or use the compromised auxiliary nodes to fabricate initial keys.