Figure 3 illustrates the probability of establishing

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Figure 3 illustrates the probability of establishing a direct key between two neighbor sensor nodes for different standard deviation σ of the deploy- ment of sensor nodes. We can clearly see the high probability of establishing a direct key between two neighbor sensor nodes for a small value of σ . This indicates that our framework greatly improves the key pre-distribution as long as the sensor nodes in the same group are indeed close to each other after deployment. However, the result also shows that the probability of establishing di- rect keys for our schemes decreases when the sensor deployment deviation σ increases. In particular, the benefits introduced by our group-based ap- proaches as compared with existing solutions will diminish when the sensor nodes in the same group are not close to each other. This is also con- firmed by our later study on the probability of establishing indirect keys and the security in hostile environments. This problem, however, is not surprising since our study is motivated by the location-based grouping of sensor nodes. When the group-based deployment model turns out to be ineffective, our approaches cannot do much to improve the performance. Nevertheless, our study indicates an interesting research direction where the performance of key management is improved by investing more efforts on the sensor deployment. To better show the effectiveness of our framework, we also compare them with the existing key pre-distribution techniques such as the random pairwise keys scheme, 8 the random subset assignment scheme, 10 and the grid-based scheme. 10 In addition, we will also compare our schemes with Copyright © 2010. World Scientific Publishing Company. All rights reserved. May not be reproduced in any form without permission from the publisher, except fair uses permitted under U.S. or applicable copyright law. EBSCO Publishing : eBook Collection (EBSCOhost) - printed on 2/16/2016 3:46 AM via CGC-GROUP OF COLLEGES (GHARUAN) AN: 340572 ; Beyah, Raheem, Corbett, Cherita, McNair, Janise.; Security in Ad Hoc and Sensor Networks Account: ns224671
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86 D. Liu, P. Ning and W. Du 0 0.2 0.4 0.6 0.8 1 1.2 0 50 100 150 200 Standard deployment deviation Probability of having direct keys between neighbors RS RK Grid Our schemes CPKS Fig. 3. Probability of having direct keys between neighbor nodes. “RS” represents the random subset assignment scheme. “RK” represents the random pairwise keys scheme. “Grid” represents the grid-based scheme. “CPKS” represents the closest pairwise keys scheme. the closest pairwise keys scheme (the extended version), 13 which exploits the knowledge of the expected locations of sensor nodes to facilitate key pre-distribution. For the random subset assignment scheme and the grid-based scheme, we assume the same number of bivariate polynomials in the system and the same number of polynomial shares stored on each sensor node as the polynomial-based instantiation. Thus, there are m + n = 200 bivariate polynomials in the polynomial pools for the random subset assignment scheme and the grid-based scheme. The random subset assignment scheme assigns the polynomial shares of 2 randomly selected polynomials from the pool to each sensor node. The grid-based scheme arranges 200 polynomials
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