Group is approximately m n 2 copyright 2010 world

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group is approximately m + n 2 . 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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Key Pre-Distribution for Sensor Networks Using Group Deployment Knowledge 81 4.3. Polynomial-Based Instantiation The idea presented in the previous subsection allows a sensor node to es- tablish a shared key with any other sensor node in the same group easily. In addition, it also provides the perfect security guarantee. However, the hash key-based instantiation has the following two problems. First, the storage overhead for the pre-distributed pairwise keys increases linearly with num- ber of groups ( n + m ). Although we can always allocate enough space for these keying materials, it is always desirable to allow the trade-off between security and storage overhead given resource constraints. Second, for the hash key-based scheme, once the groups are constructed, it is non-trivial to further extend the network and deploy new groups of sensor nodes. As a result, it is also desirable to seek other instantiations where the network can be easily extended. This subsection provides an instantiation that meets these requirements. In this instantiation, we use the basic polynomial-based pairwise key establishment in Ref. 23 as the basic building block of our group-based key pre-distribution. The resulting scheme is similar to the grid-based scheme. 10 However, it also differs from the grid-based scheme in that this instantiation exploits the node grouping idea to make sure that the sensor nodes sharing the same polynomial will be close to each other. This imme- diately provides some nice properties, as we will see later. Compared to the hash key-based scheme, this instantiation allows us to make trade-offs be- tween security and storage, while still provides an efficient way to establish a pairwise key between any two sensor nodes. Specifically, for any group G (either a deployment group or a cross group), we generate a unique symmetric t -degree bivariate polynomial f g ( x, y ) with the property of f g ( x, y ) = f g ( y, x ). Every sensor node i G gets pre-distributed a polynomial share f g ( i, x ) by evaluating the bivariate polynomial f g ( x, y ) at x = i . It is assumed that the polynomial f g ( x, y ) is only known by a trusted server. To establish a pairwise key with node j , node i only needs to compute f g ( i, j ), which equals f g ( j, i ), the value that can be computed by node j as well. There is no additional communication involved. For example, as shown in Figure 2, the deployment group G 1 includes nodes 1, 2 and 3. We generate a random 2-degree bivariate polynomial f G 1 ( x, y ) with the property of f G 1 ( x, y ) = f G 1 ( y, x ) and then pre-distribute f G 1 (1 , x ), f G 1 (2 , x ), and f G 1 (3 , x ) to node 1, node 2 and node 3, respec- tively.
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