Protocols for Secure Communication in Wireless Sensor Networks

4.2. Random Key Pre-Distribution 105
The link keys define the authentication graph that exists “on top” of the
communication graph, in which the edges are determined by radio connectivity.
(The vertices of both graphs correspond to the sensor nodes.) If we allow
link keys only between nodes that are spatially close, i.e. between which exists
a radio link, the authentication graph is a subgraph of the communication graph.
If we also allow link keys between remote nodes, the edgeds in the authentication
graph become arbitrarily distributed, being constrained only by the ability
of pairs of nodes to establish link keys. In the former case, the authentication
graph is geometric, while in the latter case, it is closer to a random graph.
Nodes with few neighbours may not be able to connect to any neighbour at
all. A small number of nodes being disconnected from the main component of
the network should not pose a problem in practice. As a matter of fact, it can
be expected that some nodes will fail to function during the course of normal
operation, and the network will have to deal with such failures anyway. Some
application scenarios may depend on a network that is densely connected, with
each node having many neighbours to talk to. For other scenarios, a lightly
connected network may suffice. It is part of deployment planning to determine
the necessary and economically feasible density of the network.
A random predistribution scheme yields a certain probability pc with which
two nodes are able to establish a link key. This probability depends both on
the key pool size and the key ring size. While the key ring size is usually
constrained by the available memory in the nodes, the size of the key pool can
be varied arbitrarily, since it exists in full only in the key distribution center.
As defined in section 4.2.1, let S be the key pool size and m the number
of root keys in a node’s key ring. In order to determine the probability pc,
we first determine the probability with which two nodes share exactly i keys
(0 ≤ i ≤ m), which is
Pr[i shared] =
S S−i
i
2(m−i) 2(m−i) m−i
S 2
m
(4.1)
Chan et al. [39] have given a derivation of this expression. Here, we give a
slightly modified explanation of its terms. The numerator designates the number
of possibilities to select two key rings which have exactly i common ele-
ments:
• We assume i common keys. There are S i ways to select i items from the
key pool.
• There are m − i distinct keys left to be drawn for each node from the key
pool, so 2(m − i) keys in total. The i keys that already have been deter-