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ensure that two nodes share a key with a higher probability, thereby leading to greater chances of ensuring connectivity On the other hand, this could also increase the susceptibility of the network since, by compromising fewer nodes, the adversary would be able to eavesdrop successfully on a larger number of communications amongst the noncompromised nodes in the network For example, consider one extreme when the size of the key pool is 1 In this case, this scheme reduces to the simple deterministic case of a single key being used in the entire network While this scheme ensures high connectivity, it suffers from no resilience against node compromise since capture of a single node compromises the communication in the entire network Having a very large key pool causes the opposite behavior in terms of connectivity and resilience In such a case connectivity will be very low but resilience will be very high Several proposals have been developed based on this idea of random key predistribution The various proposals in this family of probabilistic key sharing approach differ in terms of the structure of the key pool, the number of common keys required, their applicability to a dynamic deployment scenario and the method of determining the common keys These schemes offer network resilience against capture of nodes since a node has very few keys deployed on it Thus, there is no need to consider tamper resistant nodes (though note that nontamper-resistant nodes might not be acceptable in some situations such as military deployments) This is an advantage given the problems such as cost and complexity associated with tamper-resistant nodes In addition this approach is more scalable for large-scale, dense deployments of sensor networks The drawbacks of these schemes are that the key setup is based on probabilistic arguments Additionally, setting up of a secure channel between nodes which do not share common keys might require communication over multiple hops This increases the workload and the latency associated with setting up secure links We next look at several solutions based on this idea of probabilistic key sharing In [39] the authors focus on very large, dense sensor networks The authors propose three random key predistribution schemes for such networks They are:
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the q-composite random key predistribution scheme two nodes need to share more than one common key in order to set up a secure link; the multipath key reinforcement scheme An attacker has to compromise many more nodes to achieve a high probability of compromising any communication the random-pairwise key scheme enables node-to-node mutual authentication whereby any node can determine the identity of the other node it is communicating with
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We rst consider the q-composite scheme This scheme is very similar to the one in [17] The difference, though, is that, in order to form a secure link, this scheme requires at least q common keys, q 1, between any two nodes, as opposed to the scheme in [17], where a single common key is needed for any two nodes to communicate securely By requiring more common keys between any two nodes, the connectivity of the network is impacted This is due to the fact that it is now more dif cult for two neighboring nodes to set up a secure link between themselves At the same time, the resilience of the network against node capture is increased This is because increasing the common key threshold makes it harder for an attacker with a given set of keys to break a link between two noncompromised nodes
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Now in order to make it possible for two nodes to establish a secure link with some probability, it is necessary to reduce the size of the key pool for a given size of the key ring on each node, but this makes it possible for the adversary to compromise a larger percentage of keys in the key pool by controlling nodes beyond a certain threshold number This implies that the network is more vulnerable when the number of nodes that have been compromised is larger than this threshold Thus, these opposing factors result in better security for small number of compromised nodes and worse security for large number of compromised nodes as compared with the basic scheme in [17] Thus q-composite scheme ensures that small-scale attacks are less productive as compared with large-scale attacks This is advantageous given that it might be easier to detect large-scale attacks which are also more expensive to launch In the q-composite scheme, during the initialization phase, a certain number of keys are selected randomly from a key pool and installed on a node During key-setup, a node must discover all the common keys it possesses with each of its neighboring nodes This can be achieved by using options as described earlier with the basic scheme Once a node determines a neighboring node with which it shares at least q keys, a secure link can then be formed between the two The actual key used to secure the link can be a hash of all the shared keys Secure links cannot be formed between nodes that share fewer than q keys Other operation details are similar to the scheme in [17] Determining the right size of the key pool for a given size of the key ring is critical here This is because, if the key pool size is too large, then the probability of any two nodes sharing at least q keys might be less than desired On the other hand, if the size of key pool is small, then it will result in easier compromise of the security of the entire network as the adversary has to determine a smaller number of keys The second scheme proposed by the authors is a multipath reinforcement scheme This is a method to strengthen the security of a secure link by establishing a new key to protect this link The new link key is set up through multiple disjoint paths It is assumed that the initial key setup is completed Thus, there exist secure links between the various nodes Consider two nodes A and B that do not need to be neighbors Assume that they have a secure link between themselves This secure link between A and B would be compromised if nodes elsewhere that have a superset of the keys shared between A and B were captured by the adversary One way to address this is to derive a new key between A and B once an initial secure link is formed If the new key is derived by sending the key fragments over multiple independent paths, then the updated key will be secure unless the adversary is able to eavesdrop successfully on all the paths The nal key can then be derived from these fragments such as by XORing all the fragments This is the basic idea behind the multipath reinforcement scheme The more paths are used to update the key, the more secure it is On the other hand, the longer the path is, the less secure it might be This is due to the fact that, if any one link on the path is insecure, the entire path is insecure and longer paths provide more compromise opportunities to the adversary Note that, if the key update is done only over a single path, then it might be possible for the adversary to record all traf c used to set up the keys and decrypt the key update message after it compromises the necessary keys As a result, the new updated key would be vulnerable It is clear that the communication overheads associated with this scheme can be signi cant due to the need to transmit the key fragments over multiple disjoint paths The authors show that this scheme has approximately 10 times more communication overhead over the basic scheme in [17], but then it provides an approximately 150 times
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