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compromised sensor nodes, assuming that the sensor nodes can communicate with each other Second, this scheme is sensitive to node compromises Even if some nodes are compromised, there is still a high probability that a pairwise key can be reestablished between two noncompromised nodes Third, a sensor node can directly determine whether it can establish a direct key with another node, and if it can, which polynomial should be used As a result, there is no communication overhead during polynomial share discovery 534 Improving Random Key Predistribution with Expected Deployment Locations Despite the recent advances, key management in distributed sensor networks is still not an entirely solved problem This is especially because the performance of these schemes, particularly the probability of establishing a common key between communicating sensors and the ability to tolerate compromised sensors, are highly dependent on the memory available on sensor nodes.

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Because of the resource constraints on sensor nodes and the need to lower the cost of sensor networks, it is always desirable to reduce the memory required by key management and allocate more resources for the sensor network applications In some applications, the sensors may have low mobility, and we may be able to predetermine the location of the sensors to a certain extent In this case, we can use the sensors location information to improve the performance of pairwise key predistribution In this subsection, we describe a simple location-aware deployment model as well as a pairwise key management scheme that can take advantage of the location information More details of random key predistribution using deployment knowledge can be found in refs [7] and [8] 534.

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1 A Location-Aware Deployment Model We assume that sensors are deployed in a two-dimensional area called the target eld, and two sensors can communicate with each other if they are within each other s signal range The location of a sensor can be represented by a coordinate in the target eld Each sensor has an expected location that can be predicted or predetermined After the deployment, a sensor is placed at an actual location that may be different from its expected location We call the difference between the expected location and the actual location of a sensor the deployment error for the sensor Thus, this model can be characterized by the following three parameters: 1 Signal Range dr A sensor can receive messages from another sensor if the former is located within the signal range of the latter.

Thus, for every f 1, with probability exceeding 1 1/f , group G(i) contains at most (1 + )E[X] = n/k + (3n ln f )/k stations Now, Theorem 52 guarantees that G(i) can be initialized, with probability exceeding 1 1/f , in 4(n/k + (3n ln f )/k) + 8 ln f + o(n + (3n ln f )/k + ln f ) = O(n/k + log f ) time slots Put differently, the probability that G(i) cannot be initialized in O[n/k + log( f k)] = O(n/k + log k + log f ) time slots is less than 1/f k By (93), all the groups can be initialized, with probability exceeding 1 1/f, in O(n/k + log k + log f ) time slots Thus, Step 2 terminates, with probability exceeding 1 1/f , in O(n/k + log k + log f ) time slots.

We model the signal range of a sensor as a circle centered at its actual location with the radius dr For simplicity, we assume that the radius dr de ning the signal range is a networkwide parameter, and denote the signal range by dr We say two sensors are neighbors if they are physically located within each other s signal range..

By Corollary 43, Step 3 terminates in 2 log k time slots Further, Step 4 can be completed in one time slot Thus, we have proved the following result Theorem 61 The task of initializing a single-hop, k channel, n station radio network with CD terminates, with probability exceeding 1 1/f , ( f 1) in O(n/k + log k + log f ) time slots To extend the result of Theorem 61 to radio networks with no-CD we note that in Step 2 we need to use an initialization protocol for radio networks with no-CD The complexity of all the other steps remains the same Recall that group G(i) has at most n/k + (3n ln f)/k stations with probability exceeding 1 1/f By Theorem 5.