# Local Connectivity Tests to Identify Wormholes in Wireless Networks

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4 u v Figure 2. A legal network structure such as a bridge connecting two nodes on the boundary of a hole could also be identified as a wormhole in our definition. However, the same grah structure can be generated by also lacing wormhole antennas near u and v. Thus it is imossible to eliminate this case from our definition. 3. Removing all wormhole edges W W 1 increases the distance between W and W 1 to be at least k, but does not disconnect any art of the network. The set W is said to maximal in the sense that no node can be added to it while keeing true to the conditions above. This definition imlies that the diameter of W is at most 2. Sometimes we write a wormhole set simly as a k-wormhole to mean that τ is not relevant, or equivalently, τ =1. We remark that in certain cases, legal links can be identified as a wormhole set. Consider a network with a bridge connecting two nodes that are otherwise far aart in the network. Such a bridge or bridge like structure falls in our definition. See Figure 2. But such bridges could also be the result from a wormhole attack and there is no way to distinguish them from a real wormhole attack based on grah connectivity only. Thus, our tests will be on the aggressive side and also identify such structures, and reort them for further investigation. Finding a comlete biartite subgrah can be done in the centralized setting when the entire network toology is available. Estein [5] shows an algorithm that lists all comlete biartite subgrahs in a network with constant degree. The running time of the algorithm is linear in the size of the grah and exonential in the node degree. We will use this algorithm on local neighborhoods in the final stage of our algorithm to test that the wormholes detected have τ nodes on each side. 2.3 Local Connectivity Test The idea in our test is to observe that a wormhole attack connects two sets of nodes that are otherwise far away in the grah, while the wormhole set itself is contained in a very small neighborhood. As a node near a wormhole exands its neighborhood, the neighborhood grows on two sides of the wormhole edges. Removing a small region around the node removes the wormhole and disconnects the neighborhood into two comonents. Thus our local connectivity test is to check whether a neighborhood of a roer size will fall into multile connected comonents. Since wireless communication has a lot of local satial variations, checking the 1-ho neighborhood does not give reliable results. Thus we consider neighborhoods of different sizes. To be recise, we will introduce the following definitions. Definition 2.2. α-balland [α, β]-ring. An α-ball centered at node, written as B α() is the set of all nodes with distance at most α- hos from. All the nodes that are within β hos from but are more than α-hos away from are called the [α, β]-ring N [α,β] (). In symbols : N [α,β] () =B β () \ B α(). α, β are integers satisfying β>α 1. To test for a wormhole, we first introduce a basic [α, β]-ringconnectivity test, where α, β are integers satisfying β>α 1. Definition 2.3. [α, β]-ring-connectivity test for node. Consider the set of nodes N [α,β] () =B β () \ B α(), and the subgrah in u 1 u 2 v 1 v 2 W Figure 3. The thick circles reresent the nodes within the wormhole range, those on two sides corresond to W and W 1 resectively. The hysical wormhole link is not shown since it is not visible in the network connectivity. The darkly shaded region denotes the ball B 1(), which includes all nodes in W 1. Thus removing B 1() also removes all wormhole edges. The lightly shaded region denotes the ring N [1,2] (). It has two comonents, one near W and one near W 1. G induced by it. If this subgrah contains more than one connected comonents, the test returns true, and we say is a k-wormhole candidate for all k>2β. See Figure 3 for an examle. Guaranteed Detection of Wormhole Sets. We show that if there is a wormhole, the [α, β]-ring-connectivity test always detects it successfully. For now we consider the case that the network has just a single wormhole set. First we show that the connectivity test will surely label the nodes in a wormhole set. Theorem 2.4 (Guarantee of detection). Given a (k, τ)-wormhole set W, all the nodes in W will surely be detected by the [α, β]-ringconnectivity test, given that k>2β, β>α 1. PROOF. Consider a (k, τ)-wormhole set W. Without loss of generality, we take reresentative nodes W and argue that it must be labeled as wormhole candidate. Assume otherwise, then the subgrah induced by N [α,β] () remain as a single connected comonent, after we remove the α-ball of. Recall that all the nodes in W 1 are neighbors of, thus removing α-ball of with α 1 will surely remove all nodes in W 1. Thus all the wormhole edges are removed as a result. Intuitively the nodes in N [α,β] () were originally reached from through either the wormhole edges or not using any wormhole edges. After the wormhole edges are removed, these two sets naturally form disconnected comonents. We make this intuition rigorous in the following. Consider the nodes in N [α,β] (). We define the set N 1 to be the nodes whose shortest aths to go through nodes in W 1,andthe set N to be the nodes whose shortest aths to do not go through nodes in W 1. We argue that the two sets are disjoint, and form disconnected comonents. If the subgrah induced by N [α,β] () has only one connected comonent, take a node x / W in this subgrah. Since x N [α,β] (), x is within β hos from. There are also two shortest aths that connect from to x, one through the nodes in W 1 (denoted as P 1) and one not through the nodes in W 1 (denoted as P ). These two aths, concatenated, form a cycle of length at most 2β +1. See Figure 4 for an examle. We now argue that on ath P 1 there can only be one node q from W 1, that is, the node immediately after on P 1. Clearly, if there is another node q W 1 further down the ath P 1, then one can shortcut the ath P 1 as and q are also neighbors. This will contradict with the fact that P 1 is a shortest ath. Thus, removing the edges between W and W 1 willstillleavea ath connecting and q with totallength2β. This contradicts with the definition of a k-wormhole, where k > 2β. The arameters α, β can be varied. Our tests are aggressive, in W 1 q

5 W P x Figure 4. If N [α,β] () has only one connected comonent, then there is a ath connecting two nodes W, q W 1 not using any wormhole edges with total length at most 2β. the sense that a single wormhole attack will surely be identified for suitably small values of α and β. Thus detection is always guaranteed. Different arameters may introduce different tye of false ositives. For examle, a small β is likely to introduce false ositives that is, certain nodes in sarse regions may be wrongly identified as a candidate because their small neighborhoods are naturally disconnected. But using a large β will show that it is actually not a real wormhole node, since the neighborhoods are connected by a slightly longer ath. In our final algorithm we run multile tests with different arameters and outut the nodes that are labeled in all tests. We start with smaller values of α, β, and erform additional tests with larger values only on the nodes that are labelled as susicious as wormhole candidates by the earlier tests. We take to be wormhole set the nodes that that are detected by all the tests u to a suitable value. Once a set of candidates are detected, we can remove the links connecting the candidates. 2.4 The Wormhole Algorithm Based on the ring-connectivity test, we describe a simle distributed algorithm that identifies neighborhoods in a network as wormholes. Our goal is to detect wormholes of length k and greater. Since k must be greater than 2β, andβ is at least 2, the minimum ermissible value of k is 5. Let us denote by C [α,β] the set of nodes detected to be wormhole candidate by the [α, β]-ring-connectivity test erformed at each node in the network. Algorithm: Connectivity Metric Test. The algorithm consists of erforming the test on increasing values of (α, β) in lexicograhic order, and erforming subsequent tests only at nodes that are labelled candidates by all revious tests. More recisely, we select α =1, 2,..., (k 3)/2. And for each α, we erform the test for β = α +1,α+2,... (k 1)/2. Clearly, the result of the algorithm is a set of candidates (k 3)/2 α=1 P 1 (k 1)/2 C [α,β] β=α+1 q. What we have covered until now addresses the detection of some subgrahs whose resence have a large effect on the metric the basic symtom of a wormhole. Condition 2 in our definition of wormholes requires that each side of a wormhole have a size τ. We now describe how to check for this threshold. For this, we make use of the algorithm in [5] that finds the maximal comlete biartite subgrahs in any grah. Note that this entire hase can be ignored for τ =1. Algorithm: Test for τ Partitions. We take connected comonents of the subgrahs induced by the nodes detected as wormhole candidates after the connectivity metric test above. Let C be one such connected subgrah. W 1 On the subgrah C, we aly the algorithm of [5]. Let B be the set of maximal comlete biartite subgrahs generated by the algorithm. We write as a air (W,W 1) an element in B, where W and W 1 are the two artitions of the biartite grah. On each such biartite subgrah, we erform the following test. We consider a neighboring subgrah N that consists of nodes that are at a distance at most (k 1)/2 from all nodes in W = W W 1, but not the nodes in W itself. Let N,N 1,... be the connected comonents of N. For any edge (a, b) W W 1, if nodes a and b are neighbors to nodes of N, we check that these are in different comonents of N. For a grah that satisfies this condition, we check that W, W 1 τ. If there is a comlete biartite subgrah that satisfies all these conditions, we have detected a wormhole W = W W 1. Removal of Wormholes. One of the goals of detecting a wormhole is to be able to nullify it unobtrusively. We would like to retain the wireless nodes in action (thus keeing the sensing or comutational caabilities of the nodes), but eliminate the high volume of traffic assing through the wormhole link that creates the wormhole effect. We do this by removing the edges W W 1 in the biartite grah. Test for network connectivity. Once a wormhole has been detected and removed, we flood from any one node in it and ensure that the flood reaches all other nodes. This is to guarantee that the network remains connected as required by our definition. Provable guarantee. Now we are ready to show our main result. The [α, β]-ring connectivity test is guaranteed to label all nodes in a real wormhole, but may label some legal nodes incorrectly. Together withγ-artition test, the removal and the connectivity test, the false ositives are removed so our detection recisely identifies a wormhole in our definition. Theorem 2.5. Any (k, τ) wormhole W = W W 1 is detected by our test. And, our detection is surely a (k, τ) wormhole. PROOF. To show the first claim that our test is effective, we simly need to show that in each of the succession of tests, a real wormhole set (W,W 1) is not eliminated. First, (W,W 1) is by definition a maximal biartite grah. Therefore, it will be one of the grahs detected by [5]. Next we need to show that if (a, b) W W 1,anda and b are neighbors to the neighbor set N, they are neighbors to different connected comonents of N. Suose to the contrary that they are neighbors to the same connected comonent. Then there is a node c N that is at a distance at most (k 1)/2 from both a and b. Thus, there is a ath of length k 1 from a to b not assing through W. This contradicts the definition of a (k, τ) wormhole. Finally, by definition, W, W 1 τ. Thus every legitimate wormhole is detected by the test. Now we show that our detections follow the wormhole definition. It is clear that our detection generates a biartite grah (W,W 1) satisfying that each side has at least τ nodes. By the test of τ-artition, we see that without edges in the biartite grah the nodes in W and W 1 can only be connected by aths of length at least k. By the wormhole removal and connectivity test, the removal of the edges in the biartite subgrah does not disconnect the network. Thus the detected structure recisely follows the definition of a wormhole. Scalability and Communication Costs. The detection method is naturally local and distributed. It is local in the sense that communication distances are bounded by a known arameter, and comletely indeendent of the size of the global network. Each node

6 only uses the connectivity information of the nodes within its β neighborhood, whose size just deends on the average network degree and not on any other roerty of the network. This makes the algorithm scalable to networks of any size. For the test for τ sized artition, we aggregate the data about the set C and the adjoining comonents of N to a single node, and conduct the comutation at that node. The algorithm from [5] can be comutation intensive in a dense network, since its cost is exonential in degree. But note that we do this only at a few small neighborhoods we consider very likely to contain a wormhole. The overall cost for the network is therefore tyically not large. Also, this ste can be ignored for τ =1, which is the value we use in simulations and get very good results. 2.5 Discussions on Parameters As shown in the revious section, our [α, β]-ring connectivity test algorithm surely labels the nodes in a wormhole set. If we use the τ-artition test and the wormhole removal and connectivity test we recisely identify a wormhole. It is nice to have such theoretical guarantee but in ractice one suggestion is to use the [α, β]-ring connectivity test only, for the reason of simlicity and low communication requirement. In this way we do not lose any detection ower but may identify some false alarms. In this section we discuss a few interesting cases and in articular how the arameters may influence the erformance of the algorithm. Effect of k. The user sulied arameter k essentially determines the sensitivity of the algorithm. A smaller value of k makes the algorithm more sensitive. It can detect smaller wormholes, but introduces a greater chance of false ositives. A larger value of k may miss some smaller wormholes, but rovides more reliable detection of the longer wormholes. Longer wormholes are more dangerous, since they introduce larger distortion to the grah metric and attract more traffic. Thus, in a sense the algorithm s accuracy automatically scales with the effect of the wormhole, or the danger osed by it. The Influence of Parameters α, β. Recall that in our detection algorithm there are arameters α, β satisfying k>2β, β>α 1. α is the size of the neighborhood around to be removed. α is at least one. β must be at least one greater than α to allow a nonemty ring between the α ho and β ho. While clearly a sufficiently long wormhole will surely be detected for many different combinations of these arameters, an intelligent choice of arameters can lead to fewer false alarms. Our final algorithm tries different sets of arameters and take the intersection of their candidate sets. Notice that our sufficiency roof guarantees that any real wormhole nodes will definitely ass all such tests so we will not miss any real wormholes. We discuss the influence of the arameters in the following. When β is increased, the [α, β]-ring has more nodes in it. For an examle, take a look at Figure 6 (i). If α = 1,β = 2,the ring has two nodes that are not connected. But if we increase β to be 3, the ring has three nodes in one connected comonent. It is also clear that the newly included nodes are always connected to the nodes already in the ring, so there will not be any newly connected emerged comonents in the ring. Increasing β will always reduce the number of false alarms. The issue is that β cannot be increased arbitrarily due to uer bound of k and higher communication/comutation cost. The arameter α works in an interesting way regarding false alarms. First, when α is small, there can be many false ositives in a network that is not well connected. Take a look at Figure 6 (ii). In articular, when there are small dangling nodes, these nodes may lead to identifications of some false ositives. But increasing α can enclose all these dangling nodes inside the α ball and thus remove them. For a dangling comonent with deth of l, using an α l will include all dangling nodes inside the α ball and thus eliminate the false ositives created this way. On the other hand, making α too big may remove a bridge in the network and thus create falsely identified candidates. Take a look at Figure 6 (iii). Asmallα does not disconnect the bridge but a large α can fully remove the bridge and reort as a candidate (false alarm). Eliminate False Alarms with τ. So far in our discussion we focused on the length of a wormhole, denoted by the arameter k, as the minimum ho distance between nodes in W and W 1 once the wormhole edges are removed. Another arameter in a wormhole definition is the size of W and W 1. A wormhole antenna takes all the signal it hears and broadcasts to the other antenna. Thus all the nodes within direct communication range of a wormhole antenna will be affected by the attack. In a case when the node density has a lower bound τ (i.e., an antenna laced at any location can hear from at least τ nodes), then it is clear that W, W 1 τ. We can also use this roerty to eliminate the false alarms. This avoids identifying isolated edges that act as connection between otherwise distant arts of a sarse network. 2.6 Multile Wormhole Sets When the network has multile wormhole sets, our [α, β]-ringconnectivity test can also detect these wormhole sets if they are far away (and thus indeendently alter the network connectivity) or too close (thus removing the α neighborhood will remove all related wormhole edges). Theorem 2.6. When there are multile k-wormhole sets, the nodes in the wormhole sets are surely icked u by our [α, β]-ring-connectivity test, given that k>2β, β>α 1,andeither one of the following conditions holds for each air of wormhole sets W, W : 1. The minimum ho distance between any two nodes that belong to different wormhole sets W, W is greater than β There are two nodes W, W such that, are within α 1 hos of each other. PROOF. In the first case, the two wormhole sets W, W are far aart. Thus when we run the test at a node W, all edges involved are within β hos from. That means the existence of W does not affect the test we run around. Thus all nodes in W are still identified. In the second case, the two wormhole sets are close. Basically there is a node W and W,, are within α 1 hos of each other. Now if we do a test on any node x W,then all the nodes in W are within α hos from each other. Thus the wormhole tests running on will remove the wormhole edges of both W and W. Thus the test will also turn out to label as a candidate, since there cannot be any edges of W that affect the results (i.e., decrease the number of connected comonents). The test for size τ of wormholes can be carried out as usual. In the second case, the detection of the comlete biartite grahs can hel in identifying the fact that there are in fact two wormholes. The case when our detection algorithm fails with multile wormholes is when the multile wormholes are carefully laced at a roer distance from each other such that they interfere. An examle is shown in Figure 6. The removal of the α-ball around a node does not leave the nodes in the ring in different connected comonents as they can ossibly be connected through another wormhole. In fact, in this case any single wormhole itself does

8 False Positive Number False Positive Number Perturb Grid UDG α=2 α=4 α=6 α= False Positive Number False Positive Number Perturb Grid Quasi UDG α=2 α=4 α=6 α= False Positive Number False Positive Number Random Placement UDG α=2 1 α=4 α=6 5 α= (a) (b) (c) Random Placement Quasi UDG Random Placement UDG Random Placement Quasi UDG 1 α=2 6 β=5 6 β=5 α=4 β=7 β=7 α=6 4 β=9 4 β=9 5 α=8 2 2 (d) (e) (f) Figure 7. The number of false ositive nodes on a network with 5 nodes. In the first four figures, we vary α to be 2, 4, 6, 8 and take β = α +2. In the last two figures, we take α =3and take β as 5, 7, 9 resectively. (a) Perturbed grid with UDG model, erturbation ratio =.4. (b) Perturbed grid with quasi-udg model, =.4. quasi-udg radius r =,q =.5. (c) Random distribution with UDG model. (d) Random distribution with quasi-udg model, r =,q =.5. (e) Random distribution with UDG model.(f) Random distribution with quasi-udg model, r =,q =.5. ositive nodes dros very fast. Effect of α and β. From Figure 7 shows that the increase of α and β reduces the number of false ositive nodes. This resonates with our design idea in which we test the (α, β) arameters in lexicograhic order, gradually removing false ositives. Notice that we take the candidates that ass all tests, the number of false ositives is very small. Wormhole Placement. Certain schemes roosed earlier are extremely sensitive to the ositions of wormholes. For examle, the WormCircle method [4] divides the wormhole ositions into different cases and under certain cases the detection rate is high, while in other cases, e.g. lacing wormhole antennas on network boundaries, the detection rate is much lower. Our method is not influenced much by the wormhole lacement. We show different scenarios in Figure 8. It shows that we can lace wormhole antennas near the network outer boundary, or near holes, the detection is always effective and accurate. 3D Wireless Networks. A wireless Network may be deloyed in 3D sace, say, under water or in a multi-floor building. Most revious results would fail in 3D networks. The method that uses forbidden substructure in [13] can be extended to 3D, but would need very high node densities and detailed radio models. The WormCircle method [4] strictly assumes the underlying geometry to be two dimensional, and does not generalize to 3D at all. Figure 8. Examle of wormhole lacement, Network size is 1, average degree is 6, α =1, β =3. Figure 9. Wormhole detection in a 3D network. Network toology is formed by using a 3D grid with erturbation. The network has 1 nodes. We use α =3, β =5. The wormhole transceivers are located near a air of diagonal corners and the nodes affected are accurately detected as highlighted in the figure. Our method oerates urely in terms of grah connectivity, with-

9 out any deendency on the dimension of the network. Therefore it works naturally in 3D. Figure 9 shows an examle of wormhole detection on 3-dimensional wireless network. The behavior of our method in a 3D network is similar to that on a 2D network Communication Cost Our detection mechanism requires all nodes to articiate initially, and the susicious nodes articiate more rounds using different arameters. For a test using arameter α, β, a node will need to gather the connectivity information for all nodes within β hos. While the nodes articiating in more detection rounds will introduce higher communication cost, the number of articiants is fairly small comared to the total number of nodes. Figure 1 shows the communication cost in terms of ackets transmitted for each node on average for the entire detection rocess. There are a few interesting observations. First, the communication cost is smaller for networks built by a erturbed grid model than networks of randomly distributed nodes. This is because there are fewer false ositives in a erturbed grid. Second, when the network density increases, obviously it would incur a higher cost to collect the connectivity in local neighborhood as there are more nodes. However, when the average degree increases, fewer nodes are marked susicious in the first round, which leads to a decrease of communication cost in later rounds. The combination of the two factors shows the interesting trend of first increasing and then decreasing for the case of a network of random node distribution. Packets Transmitted Per Node Grid α=4 Grid α=6 Rnd α=4 Rnd α= Figure 1. Communication cost in terms of ackets transmitted. Network has 5 nodes, β = α +2. Grid is erturbed grid with UDG, erturbation ratio =.4. Rnd is node random lacement with UDG. Figure 11. Multile Wormholes. Left: α = 1, β = 3; Right: α =2, β = Multile Wormholes When multile wormholes are laced simultaneously, they may interfere with each other, making the detection harder. The interference of two wormholes deends on the relative ositions of their antennas: as long as there exists at least one antenna which is far away from other antennas, those two wormholes will not affect each other in terms of detection. Figure 11 shows three scenarios. From to to bottom, in the first one the antennas of the two wormholes are far away from each other. In the middle, several wormholes share one antenna and the other antennas are far from each other. In both cases the wormhole nodes are well recognized. The last case is an interesting examle where the second wormhole reduces the length of a revious existing wormhole. The wormhole nodes are detected for smaller values of α, β (left). But they are not detected when we use a larger set of α, β arameters (right) Comarison with Wormcircle Figure 12. A wormhole detected by localized wormcircle at a regular node, in a quasi unit disk grah. The 3-ho ring has two comonents. Edges in dashed blue show the breadth first trees in the two cases. The red solid edge is detected as a cut edge, imlying a long cycle in one of the trees and a false detection.

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