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Secure connectivity of wireless sensor networks Ayalvadi Ganesh University of Bristol Joint work with Santhana Krishnan and D. Manjunath

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Problem statement N nodes uniformly distributed on unit square Pool of P cryptographic keys Each node is assigned K keys at random Two nodes can communicate if they are within distance r of each other and share a key Q: For what values of N, K, P and r is the communication graph fully connected?

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Background: Random key graphs Eschenauer and Gligor (2002): Key distribution scheme for wireless sensor networks Yagan and Makowski (2012): Analysis of the full visibility case Theorem: Suppose P= (N). Let K 2 /P = (log N+ N )/N Then, P(connected) 1 if N + and P(connected) 0 if N

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Heuristic explanation Probability of an edge between two nodes is approximately K 2 /P Mean degree of a node is approximately NK 2 /P = log N + N Edges are not independent but, if they were: – key graph would be an Erdos-Renyi random graph – has connectivity threshold at mean node degree of log N

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Background: Random geometric graphs N nodes uniformly distributed on unit square Edge probability g(x/r N ) for node pairs at distance x from each other Boolean model: g(x) = 1(x<1) Penrose: Let Nr N 2 = log N + N P(connected) 1 if N , and 0 if N

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Generalisations Mao and Anderson: – Similar model but with Poisson process of nodes on infinite plane. – Same scaling of r N – Under suitable conditions on g, show a threshold between having isolated nodes in unit square, and no components of finite order in unit square

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Results for geometric key graphs Mean node degree r 2 K 2 /P If r 2 K 2 /P = log N + c, then P(graph is disconnected) > e c /4 If r 2 K 2 /P = c log N and c>1, then P(graph is connected) 1

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Upper bound on connection probability Graph is disconnected if there is an isolated node P(node j is isolated) (1 r 2 K 2 /P) N exp( r 2 NK 2 /P) e c /N Bonferroni inequality: P(there is an isolated node) ≥ i P(i is isolated) i

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Isolation of pairs of nodes

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Lower bound on connection probability Approach for ER graphs – Compute probability that there is a connected component of m nodes isolated from other n m – Take union bound over all ways of choosing m nodes out of n, and over all m between 1 and n/2

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Approach for geometric key graphs Tesselate unit square with overlapping squares of side r/ 2

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Approach for geometric key graphs Are there disconnected components of different sizes in the unit square? Are there “locally” disconnected components of different sizes within the small squares of side r/ 2, considering only nodes within that square?

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Big picture of proof There are no small – size O(1) – components in the unit square disconnected from rest There are no large – size > 6 – locally disconnected components in any small square Can also bound the number of nodes in small components within a small square : very few of them So how might the graph be disconnected?

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Notation N: number of nodes in unit square r: communication radius of a node P: size of key pool K: number of keys assigned to each node n= r 2 N: expected number of nodes within communication range p=K 2 /P: approximate probability that two nodes share a key

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Assumptions N,K,P ,K 2 /P 0 nK 2 /P c log N for some c>1 K > 2 log N Corollary: – Number of nodes in each small square is (log N) – concentrates near its mean value of n/(2 ) – uniformly over all squares

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Within a small square n/(2 ) nodes, full visibility Mean degree is nK 2 /(2 P) = c/(2 ) log N Even if edges were independent, expect to see local components of size up to 2 , somewhere in the unit square Show there are no bigger components, taking edge dependence into account

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Within a small square Say there is a connected component of size m isolated from the rest Say these m nodes have mK j keys between them Then – j ≥ m 1 – None of the other n m nodes in the square has one of these mK j keys

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Number of keys among m nodes Assign K distinct keys to first node Assign subsequent keys randomly with replacement P(collision at (i+1) th step) i/P, independent of the past P(j collisions) P(≥j collisions) ?

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Collision probability bounds X 1, X 2, …, X n independent Bernoulli random variables X i ~ Bern(p i ) Y = X 1 +…+X n Z is Poisson with the same mean as Y Hoeffding (1956): Z dominates Y in the convex stochastic order

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Within the big square Say nodes 1,2,…,m form a connected component isolated from the rest. Then – for some permutation of 1,2,…,m there is an edge between each node and the next – they hold mK j keys between them, for some j – there is no edge between the remaining N m nodes and these m

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Putting the pieces together Most nodes belong to a giant component Each small square may contain some nodes that are locally isolated or within small components These must either be connected to the giant component in a neighbouring cell, or to another small component Latter is unlikely, doesn’t percolate

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