Many basic key distribution schemes specifically tuned to wireless sensor networks have been proposed in the literature. Recently, several researchers have proposed schemes in which they have used group-based deployment models and assumed predeployment knowledge of the expected locations of nodes. They have shown that these schemes achieve better performance than the basic schemes, in terms of connectivity, resilience against node capture and storage requirements. But in many situations expected locations of nodes are not available. In this paper we propose a solution which uses the basic scheme, but does not use group-based deployment model and predeployment knowledge of the locations of nodes, and yet performs better than schemes which make the aforementioned assumptions. In our scheme, groups are formed after deployment of sensor nodes, on the basis of their physical locations, and the nodes sample keys from disjoint key pools. Compromise of a node affects secure links with other nodes that are part of its group only. Because of this reason, our scheme performs better than the basic schemes and the schemes using predeployment knowledge, in terms of connectivity, storage requirement, and security. Moreover, the post-deployment key generation process completes sooner than in schemes like LEAP+.
Further, Anjum @cite_0 removed the assumption of group-based deployment model and also removed the assumption of knowledge of expected locations of nodes. He showed that the scheme performs better than the basic scheme; but the scheme requires nodes which can transmit at different power levels. In this scheme there are some special nodes which generate different random numbers (nonces) and transmit them at different power levels. Nodes receiving the same nonce can communicate, provided they are neighbors. Our scheme is different from this scheme, since we do not require the presence of nodes which can transmit at different levels. Instead of using different power levels, our scheme uses TTL scoping. In TTL scoping, after the deployment phase, some nodes transmit a broadcast packet containing the TTL (Time to Live) field, similar to that of IP packets in data networks.
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We study wireless ad hoc networks with a large number of nodes. We first focus on a network of n immobile nodes, each with a destination node chosen in random. We develop a scheme under which, in the absence of fading, the network can provide each node with a traffic rate spl lambda sub 1 (n)=K sub 1 (nlog n) sup -1 2 . This result was first shown in J. Hightower and G. Borriello (2001) under a similar setting, however the proof presented here is shorter and uses only basic probability tools. We then proceed to show that, under a general model of fading, each node can send data to its destination with a rate spl lambda sub 2 (n)=K sub 2 n sup -1 sup 2 (log n) sup -3 2 . Next, we extend our formulation to study the effects of node mobility. We first develop a simple scheme under which each of the a mobile nodes can send data to a randomly chosen destination node with a rate spl lambda sub 3 (n)=K sub 3 n sup -1 2 (log n) sup -3 2 , and with a fixed upper bound on the packet delay d sub max that does not depend on n. We subsequently develop a scheme under which each of the nodes can send data to its destination with a rate spl lambda sub 4 (n)=K sub 4 n sup (d-1) 2 (log n) sup -5 2 provided that nodes are willing to tolerate packet delays smaller than d sub max (n)<K sub 5 n sup d , where 0<d<1. With both schemes, a general model of fading is assumed. In addition, nodes require no global topology or routing information, and only need to coordinate locally. The above results hold for an appropriate choice of values for the constants K sub i , and with probability approaching 1 as the number of nodes n approaches infinity. We consider n wireless ad hoc network nodes with one antenna each and equidistantly placed on a line. The transmission power of each node is just large enough to reach its next neighbor. For this setting we show that a message can be broadcasted to all nodes in time O(log n) without increasing each node's transmission power. Our algorithm needs O(log n) messages and consumes a total energy which is only a constant factor larger than the standard approach where nodes sequentially transmit the broadcast message to their next neighbors. We obtain this by synchronizing the nodes on the fly and using MIMO (multiple input multiple output) techniques. To achieve this goal we analyze the communication capacity of multiple antennas positioned on a line and use a communication model which is based on electromagnetic fields in free space. We extend existing communication models which either reflect only the sender power or neglect the locations by concentrating only on the channel matrix. Here, we compute the scalar channel matrix from the locations of the antennas and thereby only consider line-of-sight-communication without obstacles, reflections, diffractions or scattering. First, we show that this communication model reduces to the SINR power model if the antennas are uncoordinated. We show that n coordinated antennas can send a signal which is n times more powerful than the sum of their transmission powers. Alternatively, the power can be reduced to an arbitrarily small polynomial with respect to the distance. For coordinated antennas we show how the well-known power gain for MISO (multiple input single output) and SIMO (single input multiple output) can be described in this model. Furthermore, we analyze the channel matrix and prove that in the free space model no diversity gain can be expected for MIMO. Finally, we present the logarithmic time broadcast algorithm which takes advantage of the MISO power gain by self-coordinating wireless nodes. Consider n nodes located in a sphere of volume V m sup 3 , each capable of transmitting at a data rate of W bits sec. Under a protocol based model for successful receptions, the entire network can carry only spl Theta (WV1 3n2 3) bit-meters sec, where 1 bit carried a distance of 1 meter is counted as 1 bit-meter. This is the best possible performance even assuming the node locations, traffic patterns, and the range power timing of each transmission, are all optimally chosen. If the node locations and their destinations are randomly chosen, and all transmissions employ the same power range, then each node only obtains a throughput of spl Theta (w (nlog sup 2 n)1 3) bits sec, if the network is optimally operated. Similar results hold under an alternate physical model where a minimum signal-to-interference ratio is specified for successful receptions. The proofs of these results require determination of the VC-dimensions of certain geometric sets, which may be of independent interest. When n identical randomly located nodes, each capable of transmitting at W bits per second and using a fixed range, form a wireless network, the throughput spl lambda (n) obtainable by each node for a randomly chosen destination is spl Theta (W spl radic (nlogn)) bits per second under a noninterference protocol. If the nodes are optimally placed in a disk of unit area, traffic patterns are optimally assigned, and each transmission's range is optimally chosen, the bit-distance product that can be transported by the network per second is spl Theta (W spl radic An) bit-meters per second. Thus even under optimal circumstances, the throughput is only spl Theta (W spl radic n) bits per second for each node for a destination nonvanishingly far away. Similar results also hold under an alternate physical model where a required signal-to-interference ratio is specified for successful receptions. Fundamentally, it is the need for every node all over the domain to share whatever portion of the channel it is utilizing with nodes in its local neighborhood that is the reason for the constriction in capacity. Splitting the channel into several subchannels does not change any of the results. Some implications may be worth considering by designers. Since the throughput furnished to each user diminishes to zero as the number of users is increased, perhaps networks connecting smaller numbers of users, or featuring connections mostly with nearby neighbors, may be more likely to be find acceptance.