The solutions developed so far do not take into account the routing paths of the routing tree. In this section we develop a solution that relies on the multicast routing tree R for constructing an energy-efficient key distribution tree T . By accumulating information from the routing tables during the route path establishment, the GC can compute the energy
Ei(R), i = 1..N required to unicast a message to each member of the multicast group. Then,
the GC can characterize a node I as inner compared to an outer node O, if EI(R) ≤ EO(R). As an example in Figure 2.8(a), node six is an outer node compared to node seven, but is an inner node compared to node eight. As the total network transmission power increases, one expects more inner nodes to be covered by transmissions to outer nodes.
Assume that node I is an inner node compared to node O, i.e. EI ≤ EO and that by transmitting to O we cover I. The energy expenditure for sending a message to both
I and O is EO if I and O share a common key, and EO+ EI if I and O do not share a common key. Hence, by assigning a common key to I and O we save EI with maximum
savings achieved when EI = EO. Consider for example nodes nine (inner node) and five (outer node) in Figure 2.8(a). By transmitting to node five we cover node nine, due to the
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 ROUTING TREE GC 2 3 4 5 6 7 8 9 3.36 0.98 2.47 28 23.12 9.69 6.41 8.79 M3 M7 M4 M2 M6 M8 M9 M5 K2,3 K1,2 K2,1 K2,2 K0 K1,1 K2,4 K3,1 K3,2 K3,3 K3,4 K3,5 K3,6 K3,7 K3,8 (a) (b) M3 M4 M7 M2 M6 M8 M9 M5 K2,3 K1,2 K2,1 K2,2 K0 K1,1 K2,4 K3,1 K3,2 K3,3 K3,4 K3,5 K3,6 K3,7 K3,8 M3 M5 M4 M8 M2 M9 M7 M6 K2,3 K1,2 K2,1 K2,2 K0 K1,1 K2,4 K3,1 K3,2 K3,3 K3,4 K3,5 K3,6 K3,7 K3,8 (c) (d)
Figure 2.8: (a) The routing paths of a wireless ad hoc network. (b) Key distribution tree built with the Routing-Aware key distribution algorithm. (c) Best possible Key distribution tree. (d) Worst possible key distribution tree
broadcast advantage. Assume that nodes five and nine need to receive a key only common to them and i) they already share a common key, ii) they do not share a common key. In the first case the energy expenditure for sending a key to both five and nine is E{5,9} = 31.45 Energy Units (EU), while in the second case the key has to be unicasted to each node and the required energy is E{6,7}= 58.02 EU.
By assigning common keys to groups of nodes that differ the least in Ei, we save the most energy for sending keys common only to those groups. Consider nodes nine and five in Figure 2.8(a), and assume they already share a common key. We save 26.57 EU for transmitting a key to both of them, which is the highest out of any other possible member pairing. By also assigning a common key to {5, 6, 8, 9} we need only 31.45 EU to update a key to the subgroup, saving 19.46 EU if only pairs {6, 8} and {5, 9} shared a common key and 46.03 EU if there was no key overlap.
If we sort all members according to Ei, i = 1..N in ascending order, we minimize the
energy expenditure difference (Ei+1− Ei) between consecutive members and maximize the
energy savings Ei if transmission to node O covers node I. Therefore, by assigning common
keys to members differing the least in Ei (placing them under the same parent node in
the key distribution tree) we achieve high energy savings. We propose the placement of the multicast members to the leaves of the key distribution tree according to the ascending order of energy expenditure Ei. In figure 2.9 we present our Routing-Aware Key distribution
scheme (RAwKey).
Routing-Aware Key Distribution Scheme (RAwKey)
Step 1: Compute all Ei(R) from the GC to each member of the multicast group.
Step 2: Sort E = {E1, E2, ..., EN} in ascending order.
Step 3: Add members as leaf nodes to the key distribution tree from left to right in the same order as E.
Figure 2.9: The steps of the Routing-Aware Key Distribution scheme (RAwKey).
it an extremely attractive method for key management in secure multicast communications for ad hoc networks.
2.9.1 Application of RAwKey to a sample network
We now illustrate the construct of the key tree for the nine-node network shown in Fig- ure 2.8(a). The GC can communicate with each member of the multicast group by using the routing paths indicated. Sorting the energies for reaching each member of the multi- cast group gives E{M3} < E{M7} < E{M4} < E{M2} < E{M6} < E{M8} < E{M9} < E{M5}. The resulting key distribution tree is shown in Figure 2.8(b). The optimal key distribu- tion tree, obtained by exhaustive searching, is shown in Figure 2.8(c). We can observe that the two trees are almost identical with only members M4 and M7 been interchanged.
The worst possible tree, also obtained through exhaustive search is shown in Figure 2.8(d). The optimal possible tree has EAV EOptim(R, T ) = 62.7 EU, the tree created with RAwKey has
EAV ERAwKey(R, T ) = 63 EU (0.5% worse than the optimal tree) and the worst possible tree has EAV EW orst(R, T ) = 78.3 EU (24.9% worse than the optimal tree).
2.9.2 Complexity of RAwKey
RAwKey requires the computation of the unicast energies to reach every member of the multicast group sorted in ascending order. During the building of the multicast routing tree the GC can acquire the order by which nodes are added to the tree. In the case of SPR the order of adding nodes to the multicast tree is the same as sorting the unicast energies and no further steps are required.
When BIP or MST is used as a routing algorithm, the order by which nodes are added to the multicast tree is not the same as the ascending order of unicast energies. However, the set is almost ordered since nodes requiring less transmit power to be reached are in general added first to the routing tree. Hence, an efficient sorting algorithm for almost sorted data can significantly reduce the sorting time. Bubblesort [7] is known to have very good performance for almost sorted data with O(N ) complexity in the best case (almost sorted sets). The EWMA uses MST as a base algorithm and hence, an almost ordered set
can also be acquired.