wireless ad hoc networking

144 Wireless Ad Hoc Networking
(KDC). Every sensor verifies the authenticity of the list using the KDC’s public
key. It then removes keys in the revocation list from its key ring. The
revocation does not take extra storage. If the number of revoked sensors is
big, the connectivity of the network may degrade significantly. In this case,
the central controller can initiate rekeying and restore the connectivity of
the network. Since sensors being compromised is a rare event, it can be
expected that only after a long time, a network needs rekeying.
Another type of key predistribution is the key space approach. Instead of
relying on independent key sets, a key space approach uses some mathematical
space definition, which we will discuss later. Two types of key
spaces have been proposed. The first type is based on a matrix. The second
type is based on a bivariate polynomial. Both key space schemes exhibit
a threshold property. There exists a positive constant λ. When there are
less than λ compromised sensors, the rest of the network is still perfectly
secure. But once λ sensors are compromised, all the links in the network
are no longer secure.
The storage and computation overhead of key space approaches are
proportional to λ. In a large sensor network, using the key space scheme is
not scalable due to a possible larger number of sensors being compromised.
To make the key space approach scalable and still preserve the threshold
property, both schemes propose to use multiple key spaces. The multiple
key spaces approach is the combination of the key space approach and random
key predistribution. In the multiple key spaces approach, each sensor
is randomly assigned a set of key spaces. So each sensor can establish a
secure connection with any other sensor sharing the same key space.
Since any two sensors within the same key space can establish a secure
connection, the connectivity of the multiple key spaces approach is on the
same level with that of random key predistribution. However, because of
the threshold property, compromising a few sensors no longer affects any
other part of the network.
The matrix-based key space is based on the following idea. First, a primitive
element s from a finite field GF (q) is chosen, where q is the smallest
prime larger than the key size. The following matrix G of size (λ + 1) × N
is generated:
⎡
⎤
1 1 1 ... 1
s s 2 s 3 ... s N
G =
s 2 (
s
2 ) 2 (
s
3 ) 2 (
... s
N ) 2
⎢
⎥
⎣
.
s λ (
s
2 ) λ (
s
3 ) λ (
... s
N ) ⎦
λ
The jth column of G, which is denoted by G(j), is distributed to
sensor j. The key distribution center also generates ω random symmetric
matrices D 1 , D 2 , ..., D ω of size (λ + 1) × (λ + 1). Then each tuple