The Average Update Energy Cost

In this section, we examine the dependency of the average update energy cost when the key assignment structure is a tree of degree α with N leaves (a multicast group size of N members) on α, N , and derive an upper bound for E_Ave.

2.5.1 Dependency of EAve on the group size N , the tree degree α, and the network topology

Let ˜EMi denote the energy expenditure for updating the compromised keys after the deletion

of the ith _{member. Also, let p(M}

i) denote the probability for member Mi to leave the

multicast group. We define the average key update energy, EAve, required for key update

after a member deletion as:

EAve = N

X

i=1

GC r₂ r₄ r₃ r₁ r_|TR| 1 2 3 4 5 6 7 9 8 N N-1 Sub-group SG i Multicast Group MG (a) (b) (c)

Figure 2.4: (a) Theorem 1: When a message is sent to all members of M G, all relay nodes T R = {r1, r2, . . . , r|T R|} have to transmit. When a message is sent to any sub-group SGi ⊆ M G, a subset T Ri ⊆ T R of relay nodes need to transmit. (b) Theorem 2: A single

transmission of power (dGC,Mf)

γmax _{reaches all members of M G with one hop, resulting in}

routing tree R, (c) The optimal multicast routing tree R∗ _{always requires less power than} R, by definition.

EAve depends on the energy ˜EMi required to deliver key updates if each member Mi were

to be deleted from the multicast group M G. Regardless of which member is deleted, the

GC needs to transmit (α log_αN − 1) key update messages [123]. These messages are routed

to different sub-groups SG_i ⊆ M G, where i = 1 . . . (α log_αN − 1) . Letting ER

X denote the

energy expenditure for sending an identical message to every member of the sub-group

X ⊆ M G via the routing tree R, the energy expenditure ˜E_Mi for updating keys after the deletion of member Mi∈ M G is:

˜

EMi =

α log_XαN −1 i=1

E_SGR _i. (2.6)

The ˜EMi depends on the routing tree R and cannot be analytically expressed unless a

specific realization of R is provided. Hence, we derive an upper bound on ˜E_Mi, making use of a sequence of properties of the WBA. To do so, we first prove the following theorem: Theorem 2.5. The energy required to deliver a message to a sub-group of members SGi ⊆ M G via a routing tree R, cannot be greater than the energy required to deliver a message

to all members of M G, via the same routing tree R.

E_SGR _i ≤ E_{M G}R , ∀ SGi⊆ M G. (2.7)

Proof. Let T R = {r1, r2, . . . , r|T R|} denote the set of all relay nodes in the multicast routing

tree R utilized by the multicast group M G. In order to deliver a single message to every member of the multicast group M G, every node in T R has to transmit. Hence,

E_{M G}R = X

ri∈T R

Eri, (2.8)

where Eri denotes the energy required for transmission of one message by relay node ri.

In order to deliver a message to a sub-group SGi ⊆ M G, a subset T Ri ⊆ T R of the relay

nodes has to transmit (no more than the total number of relay nodes can transmit). Hence,

|T Ri| ≤ |T R|, ∀SGi⊆ M G. The energy to deliver a message to a sub-group SGi is:

E_SGR _i =X

T Ri

E_ri ≤X T R

E_ri = E_{M G}R , ∀ SG_i ⊆ M G (2.9)

In figure 2.4(a) we illustrate Theorem 1. To deliver a message to all members of M G, all relay nodes need to transmit. However, in order to deliver a message to a sub-group SGi

no more than the whole set T R of relay nodes may transmit. Hence, the energy required to deliver a message to any sub-group of M G, is no greater than the energy required to deliver a message to all members of M G.

Using Theorem 2.5, we can bound the energy expenditure for updating keys after the deletion of Mi expressed in (2.6) as:

˜

EMi =

α log_XαN −1 i=1

E_SGR _i ≤ E_{M G}R (α log_αN − 1) . (2.10)

total power routing tree R∗_{, and bound the average key update energy E} Ave as: EAve= N X i=1 p(Mi) ˜EMi ≤ N X i=1 p (Mi) ER ∗ M G(α logαN − 1) ≤ E_{M G}R∗ (α log_αN − 1) N X i=1 p (M_i) ≤ E_{M G}R∗ (α log_αN − 1) . (2.11) The bound in (2.11) has two different components. The first component is the minimum energy E_{M G}R∗ , required for sending a message to the whole multicast group M G. While

ER∗

M G depends on the wireless medium characteristics and the network topology, we can

relax the network topology dependency by bounding E_{M G}R∗ using only the wireless medium characteristics and the size of the deployment region.

Let γmax denote the maximum value of the attenuation factor for the heterogeneous

medium where the network is deployed, and Ttrans denote the duration of the transmission

of one message. Let Mf denote the farthest member from the GC, and let dGC,Mf, the

distance between GC and Mf, denote the radius of the deployment region1. Then, the

following theorem holds:

Theorem 2.6. The energy required to deliver a single message to every member of the

multicast group M G via the minimum total power routing tree R∗ _{is no greater than the} energy required to deliver a single message from the GC to Mf via a one hop transmission and assuming an attenuation factor of γ_max between the GC and M_f.

E_{M G}R∗ ≤ E_MR_f =¡dGC,Mf

¢_γmax

Ttrans. (2.12)

Proof. Let γi denote the attenuation factor in the link between the GC and member Mi,

with γi ≤ γmax ∀Mi ∈ M G. The transmission power required for communication via a one

hop link between Mi and GC is, P (dGC,Mi) = (dGC,Mi)γi, Since dGC,Mi ≤ dGC,Mf, γi ≤

1_{The diameter or the size of the deployment region may also be defined as the maximum physical distance}

between any two nodes of the network. However, such a definition leads to a looser upper bound and is not considered.

γmax, ∀Mi ∈ M G, it follows that:

P (dGC,Mi) = (dGC,Mi)γi ≤ (dGC,Mf)γmax, ∀Mi ∈ M G. (2.13)

Hence, by letting the GC transmit with power (dGC,Mf)γmax, we reach every member Mi∈ M G. This is equivalent to constructing a multicast routing tree R where every member is

connected to the GC with one hop. The power required to deliver a message to all members of M G according to R, cannot be less than the minimum total power obtained by R∗_.

Hence, the energy expenditure to deliver a message to all Mi ∈ M G, via R∗ cannot be

greater than the energy expenditure to deliver the same message to all M_i ∈ M G via R,

E_{M G}R∗ ≤ E_{M G}R ≤¡dGC,Mf

¢_γmax

Ttrans. (2.14)

In figure 2.4(b), the GC transmits with power (d_GC,Mf)γmax_{, thus being able to reach}

every member with one hop. In figure 2.4(c) we show a generic optimal multicast routing tree R∗ _{for the same network as in figure 2.4(b). Since R}∗ _{is optimal it holds that E}R∗

M G ≤

¡

dGC,Mf

¢_γmax

Ttrans. The second component of the bound in (2.11), is the number of update

messages sent by the GC for deleting a member from the multicast group. While the number of messages grows logarithmically with the group size N , and N is not a design parameter, we can calculate the tree degree α∗ _{that minimizes the number of update messages.}

d

dα(α logαN − 1) = 0 ⇒ α

∗ _{= e. (2.15)}

The degree of the tree has to be an integer number and hence, the lowest upper bound for

E_Ave is achieved when α = 3. The lowest upper bound for the average key update energy, independent of the network topology and probability distribution of member deletions is:

E_Ave≤ (α∗log_α∗N ) ¡ d_GC,Mf¢γmax_T trans = (3 log3N ) ¡ d_GC,Mf¢γmax_T trans. (2.16)

2.6 Impact of “Power Proximity” on the Energy Efficiency of Key Manage-