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polynomial share of each node as f(I,y) where I is the unique ID corresponding to the node and stores these polynomial shares on the node I Now to establish a pairwise key, any two nodes will have to evaluate the polynomial at the ID of the other node using the polynomial shares present on the node As a result, any two nodes with IDs I and J compute the common key as f(I, J) f(J, I) Thus node I evaluates the stored polynomial at point J while node J evaluates the stored polynomial at point I Thus, each node will need to store the coef cients of a t-degree polynomial Blundo et al [42] proves that such a scheme is unconditionally secure Further a coalition of less than t 1 compromised nodes will not be able to determine a pairwise key used by any two noncompromised nodes A coalition of t 1 or more compromised nodes though can pool their shares together and can hence determine the t degree polynomial As a result, such a coalition of t 1 or more nodes can determine the pairwise keys used by any pairs of nodes Thus, we see that the basic scheme in [42] can tolerate at most t compromised nodes To strengthen this further, the authors in [18, 41] enhance this scheme using the concept of random key predistribution In this case the TA generates a pool of multiple distinct bivariate symmetric polynomials Each polynomial has a unique identity As earlier, the pairwise key establishment is performed in three phases namely key predistribution, shared key discovery and path key establishment The goal of each phase is the same as earlier except that we are now dealing with polynomials instead of numbers Thus, we see that this general framework encompasses the schemes in [42] and [17] as special cases Speci cally, when the polynomial pool has only one polynomial we
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obtain the scheme in [42] On the other hand when all the polynomials are of zero degree, we obtain the scheme of [17] The authors propose two methods for allocating the polynomial shares to the nodes in the setup phase In the rst method, the TA selects a random subset of polynomials in the pool and assigns the shares corresponding to these polynomials to the node After deployment, two neighboring nodes set up secure links between themselves if they have at least one common polynomial Thus, this scheme is similar to the basic probabilistic scheme in [17], where instead of randomly selecting keys from a large key pool and assigning them to nodes, polynomials are chosen from a pool and the polynomial shares are assigned to nodes However, the difference is that, while in [17] the same key will be shared by multiple nodes after the shared key discovery phase, in this scheme there will be a unique key between each pair of nodes after the shared key discovery phase Further, if fewer than t 1 shares of the same polynomial are compromised, then keys constructed using that polynomial will not be disclosed The rst method provides for more uncertainty in terms of key distribution as compared with the other probabilistic distribution schemes Hence, it can be expected to perform better than the basic scheme [17] for a larger number of compromised nodes, but once the number of compromised nodes exceeds a threshold, this scheme will degenerate at a faster rate than the other schemes with respect to the security of the network The results shown by the authors do indeed con rm this In fact the authors show that, when the fraction of compromised direct links is less than 60 percent, given the same storage constraint, the proposed scheme provides a signi cantly higher probability of ensuring secure communication between noncompromised sensors than the earlier methods Note also that unless the number of compromised nodes sharing a polynomial exceeds a threshold, compromise of nodes does not lead to the disclosure of keys established by noncompromised nodes The second method to distribute polynomial shares in the setup phase is the grid-based key predistribution We explain this method with the aid of Figure 310 Let the maximum
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