Protocols for Secure Communication in Wireless Sensor Networks

118 Chapter 4. Key Establishment
σ 1
15
23
52
σ 2
19
41
46
σ 3
7
34
77
σ 4
31
51 51
σ 5 σ 6 σ 7
28
43
57
Figure 4.5: Attack configurations
After some transformation steps, we get
1
logε 1−
2
( j+2)( j+1)
30
45
63
< t (4.9)
From this equation, we obtain that t grows with approximately the square of
j if a certain resilience has to be maintained. Figure 4.6 illustrates this fact (a
ten-fold increase in the attacker strength roughly requires 100 times the number
of chains to maintain the same resilience).
Example 4. Figure 4.5 shows an example where two nodes establish a key
based on seven hash chains. The white boxes represent the chain values of the
two legitimate nodes. The black boxes represent those of the adversary ( j = 1).
The numbers shown indicate the position indices of the values. The actual
numbers are not important, and it is also irrelevant to which legitimate node
which white box belongs. Only the position of the black box relative to the
lower white box is important. The chain value represented by the lower white
box is the contribution of the hash chain to the shared key. Here, the adversary
is able to construct all but two of these contributions (σ3 and σ7 are secure). It
is sufficient for one contribution to be secure to get a secure link key.
Example 5 (Resilience of multiple hash chains). Let’s assume we expect an
attacker of strength j = 10 and we would like to achieve a resilience of at least
ε = 0.01, i.e. only one out of hundred link keys should be compromised. Using
equation 4.9, we can determine the number of hash chains necessary to achieve
this resilience:
⎡
⎤
t = ⎢
log 0.01
1 −
1
2
(10+2)(10+1)
⎥ = 302
⎥
9
22
49