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The PBGA is a genetic algorithm and follows the basic scheme of this type of algorithm, performing crossover and mutation over a selected pool of individuals that represent ANNs. The working cycle of the PBGA is very standard: 1. Creation of a random population of N individuals using the representation mentioned earlier. 2. Fitness calculation over the whole population. 3. Selection of 2N individuals using a tournament selection operator. 4. Panmitic crossover with a probability Pc over the 2N population. The crossover operator is applied twice over the same parents and the offspring with highest tness is selected. 5. After crossover, an N individual offspring population is produced. 6. Mutation with probabilities Psm, Ppm over the offspring population. 7. Fitness calculation over the offspring population. 8. Elitism that substitutes the worst individuals of the offspring population with the best individuals of the original population. 9. Return to step 3 for n generations. Six parameters must be established by the user in the PBGA: (1) maximum number of neurons of the ANNs, (2) population size, (3) crossover probability, (4) structural mutation probability, (5) parametric mutation probability, and (6) number of generations of evolution. All of these are problem dependent but, as we will show in the next section, their values are intuitively easy to set up.

(See Section i.l.)

As one of the most fundamental security services, pairwise key establishment enables the sensor nodes to communicate securely with each other using cryptographic techniques such as encryption and authentication. However, due to the resource constraints on sensor nodes, it is undesirable for them to use traditional pairwise key establishment techniques such as public key cryptography and key distribution center (KDC). Instead of the traditional approaches to key establishment, a promising alternative is key predistribution, where keying materials are predistributed to sensor nodes before deployment. As two extreme cases, one may set up a global key among the network so that two sensor nodes can establish a key based on this global key, or assign each sensor node a unique random key with each of the other nodes. However, the former is vulnerable to the compromise of a single node, and the latter introduces substantial storage overhead on sensor nodes. There have been a number of recent advances in key predistribution for sensor networks, starting from the probabilistic key predistribution scheme proposed by Eschenauer and Gligor [3]. In this section, we describe several of these advances. 5.3.1 Random Key Predistribution Based on Key Pools

If no such W exists or if there is more than one such, an error is declared. (We may assume that the receiver outputs a dummy index such as 0 in this case.) 7. There is a decoding error if W = W . Let E be the event {W = W }. Analysis of the probability of error Outline: We rst outline the analysis. Instead of calculating the probability of error for a single code, we calculate the average over all codes generated at random according to the distribution (7.62). By the symmetry of the code construction, the average probability of error does not depend

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