Protocols for Secure Communication in Wireless Sensor Networks

4.5. Strengthening Random Key Pre-Distribution 121
from the hash chains, are distributed to the nodes. After determining the common
set of hash chains, these are used to agree on a link key between two nodes
as described above. By combining both key agreement schemes in this way, we
retain the advantages of both. This results in an increased resilience compared
to either one of the schemes.
4.5.1 A Combined Approach
We generalize the key selection function F to include the additional step of
transforming an element of the key pool into a hash chain value. First, we fix
two seeds g1 and g2 that define two different contexts for the pseudo random
sequence generator Ψ and the pseudo random number generator Φ. We will
also need to choose the maximum length of key chains, T , and a hash function
h for generating hash chains. Next, we introduce two new functions F1 and F2
that represent the different phases of the new scheme:
• F1(IDu) = Ψ g1(IDu,1,S,m) = 〈v1,...,vm〉
• F2(IDu,s,i) = h ai(s) for Φ g2(IDu,0,T − 1,m) = 〈a1,...,am〉
This leads to our new definition of the key selection function F:
F(IDu) = 〈F2(IDu,K [F1(IDu)[1]],1),...,F2(IDu,K [F1(IDu)[m]],m)〉
Operationally, this means that, for each node u, the KDC first selects m keys
from the key pool uniformly at random, using the (pseudo-) random sequence
generator Ψ g1. On each of these root keys, it then applies the hash function h
repeatedly, where the number of repetitions is determined by Φ g2, to obtain the
final keys that go into the node’s key ring.
The key establishment between two nodes now proceeds in two steps. First,
the common set of indices into K is determined in the same manner as previously
described. Then ϕ = Φ g2 is used to determine the hash chain positions of
the other node and the link key is established as described in section 4.4.
We will now show that although key establishment based on hash chains
alone yields only small resilience, the resilience of a random key predistribution
scheme is significantly improved through this combined approach. We assume
that hash chains are of sufficient length such that equation (4.7) provides a valid
approximation to hash chain key resilience.
4.5.2 Resilience
The strength of the adversay is now not only determined by the number of
distinct root keys he obtains, but also on their hash chain positions. The more