Protocol Discussion. The common PRF that is employed in Algorithm 1 ensures that the public transmissions are computationally indistin- guishable from random packets, thereby establishing our first key result.
Protocol Discussion. In order to study the dynamics of the proposed extension of our protocol, we simulated a network with 50 clients distributed randomly in a square. The t closest clients to a randomly chosen source define the destination set that establish a given group key. This proximity multicast model is representative of military use cases. | | Figure 7 illustrates how the cumulative number of trans- missions required for master key exchange and group key agreement evolves over time in a 50-client network under the proximity multicast model. The multicast group size is fixed to t = 5 and the number of random keys is set to log2 n/β . Each time a group key is generated, the total number of transmissions required for pairwise and random key agreement is tabulated in addition to those required for group key agreement via Algorithm 1. Initially, there is a sharp increase in the cumulative number of transmissions as pairwise keys are established and random keys propagate via an epidemic model. Over time, the cost of master key exchange is amortized and the slopes of the curves in Figure 7 converge to roughly 2 transmissions per generated group key. Observe in Figure 7 that increasing the infection probability from β = 0.06 to 0.2 decreases the number of transmissions required to establish group keys at steady state but increases the overhead associated with the random key exchange step. Setting β = 0.1 appears to offer a good trade between the steady-state and transient behavior. For comparison, Figure 7 also illustrates the cumulative number of transmissions when the BD protocol is used for group key agreement. Since t = 5, this is simply a line with slope 2t = 10. Observe that after approximately 120 group keys have been generated, the proposed protocol with β = 0.1 becomes more energy-efficient than the BD protocol. That is to say, over time the energy savings afforded by each group Input: Occupancy sets = Oj j K , group { } O { } ∈ G G = g1, . . . , gt , hop distance h(gi, gj) between all pairs of clients, and a common PRF φ(). Output: Group key sj0 ,u for session with unique identifier u. ← O j0 index of largest occupancy set in ; ← ← C Oj0 , l 1; ∈ if gi Oj0 then ← compute the group key sj0 ,u φ (kj0 , u); end while C = G do ← ∈ O (il, jl) index of an occupancy set Ojl ∩ ƒ ∅ satisfying Ojl C = and a transmitter ∈ ∩ il Ojl C that maximizes the number of new clients that will obtain sj0 ,u per hop: key agreement in our protocol outstrip the overhead incurred for dynamic...
Protocol Discussion. We compared the performance of the protocols described by Algorithms 1 and 2 in a n = 100 client network using a random master key distribution with R = 100 and β = 0.2. To induce a random geometric graph topology, the clients were placed randomly in a unit square and a transmission radius of s to a destination set D. Algorithm 2 extends Algorithm 1 so as to minimize the sum of the depths of the multicast trees used r(n) = 3 . log n
