Detecting Wormhole Attacks in Wireless Networks Using Connectivity Information Ritesh Maheshwari, Jie Gao and Samir R Das Department of Computer Science, Stony Brook University Stony Brook, NY 11794-4400, USA {ritesh, jgao, samir}@cs.sunysb.edu Abstract—We propose a novel algorithm for detecting wormhole attacks in wireless multi-hop networks. The algorithm uses only connectivity information to look for forbidden substructures in the connectivity graph. The proposed approach is completely localized and, unlike many techniques proposed in literature, does not use any special hardware artifact or location information, making the technique universally applicable. The algorithm is independent of wireless communication models. However, knowledge of the model and node distribution helps estimate a parameter used in the algorithm. We present simulation results for three different communication models and two different node distributions, and show that the algorithm is able to detect wormhole attacks with a 100% detection and 0% false alarm probabilities whenever the network is connected with high probability. Even for very low density networks where chances of disconnection is very high, the detection probability remains very high. I. I NTRODUCTION Wireless ad hoc and sensor networks are typically used out in an open, uncontrolled environment, often in hostile territories. In particular, several important applications for such networks come from military and defence arenas. Use of wireless medium and inherent collaborative nature of the network protocols make such network vulnerable to various forms of attacks. In this paper our focus is on a particularly devastating form of attack, called wormhole attack [1]–[3]. Here, the adversary connects two distant points in the network using a direct low-latency link called the wormhole link. The wormhole link can be established by a variety of means, e.g., by using a network cable and any form of “wired” link technology or a long-range wireless transmission in a different band. The end-points of this link (wormhole nodes) are equipped with radio transceivers compatible with the ad hoc or sensor network to be attacked. Once the wormhole link is established, the adversary captures wireless transmissions on one end, sends them through the wormhole link and replays them at the other end. An example is shown in Figure 1. Here X and Y are the two end-points of the wormhole link. As the signals received on one end of the wormhole link are repeated at the other end, any transmission generated by a node in the neighborhood of X will also be heard by any node in the neighborhood of Y and vice versa. The net effect is that all the nodes in region A assume that nodes in region B are their neighbors and vice versa. For example, traffic between nodes like a and e can now take a one-hop path via the wormhole instead of a multi-hop B e Y d a X c Wormhole Link b A Fig. 1. Demonstration of a wormhole attack. X and Y denote the wormhole nodes connected through a long wormhole link. As a result of the attack, nodes in Area A consider nodes in Area B their neighbors and vice versa. path. If the wormhole is placed carefully by the attacker and is long enough, it is easy to see that this link can attract a lot of routes. Note that if the wormhole link is short, it may not attract much traffic, and hence will not be of much use to the adversary. Thus, throughout the paper we consider only such attacks in which the wormhole link is long enough so that regions A and B do not overlap. A. Significance of Wormhole Attack While wormhole could be a useful networking service as this simply presents a long network link to the link layer and up, the attacker may use this link to its advantage. After the attacker attracts a lot of data traffic through the wormhole, it can disrupt the data flow by selectively dropping or modifying data packets, generating unnecessary routing activities by turning off the wormhole link periodically, etc. The attacker can also simply record the traffic for later analysis. Using wormholes an attacker can also break any protocol that directly or indirectly relies on geographic proximity. For example, target tracking applications in sensor networks can be easily confused in the presence of wormholes. Similarly, wormholes will affect connectivity-based localization algorithms, as two neighboring nodes are localized nearby and the wormhole links essentially ‘fold’ the entire network. This can have a major impact as location is a useful service in many protocols and application, and often out-of-band location systems such as GPS are considered expensive or unusable because of the environment. A wormhole attack is considered dangerous as it is independent of MAC layer protocols and immune to cryptographic techniques. Strictly speaking, the attacker does not need to understand the MAC protocol or be able to decode encrypted packets to be able to replay them. In its most sophisticated form, the wormhole can be launched at the bit level or at the physical layer [4]. In the former, the replay is done bit-by-bit even before the entire packet is received (similar to cut-through routing [5]). In the latter, the actual physical layer signal is replayed (similar to a physical layer relay [6]). These forms of wormholes are even harder to detect. This is because such replays can happen quite fast and thus they cannot be detected easily by timing analysis. To distinguish these attacks from the simpler form of attack, where the wormhole nodes copy the entire packet before transmittal through the wormhole link, we will refer to this simpler form of attack as store-and-forward attack following the terminology used in [4]. B. Limitations of Prior Work and Our Contributions The current solutions for wormhole are limited particularly in connection with large sensor networks, where sensor nodes carry low-cost, relatively unsophisticated hardware and scalability is an important design goal. This rules out use of additional hardware artifact that several reported techniques use – such as directional antennas [7], GPS [2], ultrasound [8], guard nodes with correct location [9]. This also rules out fine grain timing analysis used in several techniques [2], [4]. Also, physical-layer attacks may be immune to timing analysis [4]. Finally, the scalability requirements rule out global clock synchronization [2] or any form of global computations [10]. In the current work, we develop a localized algorithm for detecting wormhole attacks that is purely based on local connectivity information. Such information is often collected any way by various upper layer protocols such as routing, thus may not present any additional overhead. No additional hardware artifact is needed making the approach universally applicable. No timing analysis is done ensuring that we can detect even physical layer attacks. Our technique does not use location information and is able to detect attacks that are launched even before the network is set up, that may influence localization. We expect that our technique is particularly useful for sensor networks as the existing techniques are quite limited there. Also, connectivity is not expected to change frequently in sensor networks, making our connectivity-based approach quite practical. The detection algorithm essentially looks for forbidden substructures in the connectivity graphs that should not be present in a legal connectivity graph. Understanding of the wireless communication model (i.e., a model that describes with some given confidence whether a link between two nodes should exist) helps the detection algorithm substantially, but is not strictly required. The models we require can be very general and we will demonstrate the capability of the detection using several realistic models such as quasi-unit disk graphs [11] and link models for Berkeley motes as modeled in the TOSSIM simulator [12]. II. R ELATED W ORK Several papers in literature have developed countermeasures for wormhole attacks. We discuss them in two categories. A. Approaches that Bound Distance or Time In [2] authors have considered packet leashes – geographic and temporal. In geographic leashes, node location information is used to bound the distance a packet can traverse. Since wormhole attacks can affect localization, the location information must be obtained via an out-of-band mechanism such as GPS. Further, the “legal” distance a packet can traverse is not always easy to determine. In temporal leashes, extremely accurate globally synchronized clocks are used to bound the propagation time of packets that could be hard to obtain particularly in low-cost sensor hardware. Even when available, such timing analysis may not be able to detect cut-through or physical layer wormhole attacks. In [13], an authenticated distance bounding technique called MAD is used. The approach is similar to packet leashes at a high level, but does not require location information or clock synchronization. But it still suffers from other limitations of the packet leashes technique. In the Echo protocol [8], ultrasound is used to bound the distance for a secure location verification. Use of ultrasound instead of RF signals as before helps in relaxing the timing requirements; but needs an additional hardware. In a recent work [4], authors have focused on practical methods of detecting wormholes. This technique uses timing constraints and authentication to verify whether a node is a true neighbor. The authors develop a protocol that can be implemented in 802.11 capable hardware with minor modifications. Still it remains unclear how realistic such timing analysis could be in low-cost sensor hardware. B. Graph Theoretic and Geometric Approaches LiteWorp [14] uses a combination of one-time authenticated neighbor discovery and use of guard nodes that attest the source of each transmission. The neighbor discovery process, however, can be vulnerable to wormhole attacks, if the attack is launched prior to such discovery. A followup paper from the same authors attempts to remove this inefficiency [15], however assumes availability of location information. As mentioned before, this itself could be suspect. In [9] a graphtheoretic framework is used to prevent wormhole attacks. The protocol assumes the existence of special-purpose guard nodes that know their “correct” locations, have higher transmit power and have different antenna characteristics. Use of such specialpurpose guard nodes make this approach impractical. In one approach, directional antennas are used to prevent wormhole attacks [7]. The authors develop a cooperative protocol where nodes share directional information to prevent wormhole endpoints from masquerading as false neighbors. that needs to be certified free from wormhole attack. However, use of directional antennas limits use of such protocols. In another approach [10] somewhat related, distance estimates between sensors that hear each other is used to determine a “network layout” using multi-dimensional scaling (MDS) technique. The technique is similar to localization of the network nodes in a metric space. Without any wormhole the network layout should be relatively flat. But the layout could be warped in presence of wormholes. The technique is purely centralized and is considerably susceptible to distance estimation errors. Finally, purely physical layer mechanisms can prevent wormhole attacks such as those involving authentication in packet modulation and demodulation [2]. Such techniques require special RF hardware. III. W ORMHOLE D ETECTION A LGORITHM The placement of wormhole influences the network connectivity by creating long links between two sets of nodes located potentially far away. The resulting connectivity graph thus deviates from the true connectivity graph. Our detection algorithm essentially looks for forbidden substructures in the connectivity graph that should not be present in a legal connectivity graph. Knowledge of the wireless communication model between the nodes helps our detection algorithm. This is because a communication model can help define what substructures observed in the connectivity graph could be forbidden. However, our approach is still applicable when the communication model is unknown. In this case we need to run an extra search procedure to determine a critical parameter for the detection algorithm. This parameter will be made clear later in this section. We first develop our wormhole detection algorithm, starting from the unit disk graph model and then general (known or unknown) communication models, and finally discuss how to automatically remove links created by wormhole once a wormhole is detected. A. Unit Disk Graph Model In unit disk graphs (UDG) each node is modeled as a disk of unit radius in the plane, modeling the communication range of the node with omni-directional antenna. Each node is a neighbor of all nodes located within its disk. UDGs have long been used to create an idealized model of multi-hop wireless networks. We start with this model and formulate our approach of wormhole detection. 1) Hardness of wormhole detection: We first note that under the UDG model, the problem of detecting wormhole attacks with connectivity information is NP-hard. This is observed from the equivalence of wormhole detection with UDG embedding. If the observed connectivity graph has no valid UDG embedding in the plane, it can be deduced that there must be a wormhole present in the network. This can happen when wormhole attack creates long-distance links (longer than unity) which should not exist in a UDG. Conversely, if the observed connectivity graph does admit a valid UDG embedding, then any algorithm based on connectivity information only will have to output ‘no wormhole’. In such a case, wormhole link, even present, is not distinguishable from a valid link in the embedded UDG. In the absence of any other information, this embedding has to be taken as the ground truth. This can happen, for example, when wormhole links are short and thus appear no different than a link in UDG. This can also happen when the link is indeed long, but lack of sufficient node density prevents detection. This issue will be clearer as we move forward in the paper. In such cases, wormhole detection has to use information other than the connectivity graph. It is known that finding a UDG embedding in 2D is a NP-hard problem [16]. Thus, it is equally hard to detect a wormhole attack using connectivity information alone. A similar relationship between wormhole detection and network localization is also exploited in [10]. The basic idea in our detection algorithm is to look for graph substructures that do not allow a unit disk graph embedding, thus can not be present in a legal connectivity graph. Due to the hardness result mentioned above, our algorithm will not guarantee the detection of wormhole in all cases. Rather, we aim to design a simple localized algorithm that provides a sufficiently high detection probability in connected networks. We will demonstrate the performance of the algorithm empirically in the next section. 2) Disk packing: The key notion we exploit is a packing argument – inside a fixed region, one cannot pack too many nodes without having edges in between. The forbidden substructures we look for are actually those that violate this packing argument. To be rigorous, we start with some definitions. Denote by p(S, r) the packing number, which is the maximum number of points inside a region S such that every pair of points is strictly more than distance r away from each other. We assume that no two network nodes are located at the same point. Denote by DR (u) a disk of radius R centered at u. D denotes just a unit disk to simplify notations. As a well-known fact [17], in a unit disk there can be at most 5 nodes whose pair-wise distances are strictly more than 1. Thus p(D, 1) = 5. Given two disks of radius R centered at u, v with distance r away, define by lune the intersection of the two disks, L(r, R) = DR (u) ∩ DR (v). When R = r = 1, we sometimes omit the radii and denote by L the lune of unit disks set at unit distance apart. Lemma 3.1. p(L, 1) = 2. Proof: Refer to Figure 2 for an illustration of a lune L. The line segment uv divides the lune into two parts, the upper and lower ones. The two intersections of the two unit circles centered at u, v are denoted p, q respectively. Denote by w the √ midpoint of segment uv. |pw| = 3/2 < 1. It is not hard to see that inside the upper half of the lune one can not place two nodes with their distance strictly larger than 1. Indeed, for any node x in the upper half of L, |xv| ≤ 1, |xu| ≤ 1, |xp| ≤ 1. Thus there can only be two nodes inside L with inter distance larger than 1. We can generalize the result for packing of disks of radius β, with the proof appearing in the appendix. p u v w q Fig. 2. One can only pack at most two nodes inside a lune with interdistance more than 1. Lemma 3.2. p(L(r, R), β) arccos(r/(2R + β)) − r ≤ 2R. ≤ 4r πβ 2 π8 (R/β + 1/2)2 · (R + β/2)2 − r2 /4 for Proof: See the appendix. Remark. Lemma 3.2 only gives a loose bound for p(L, β). When β = 1, Lemma 3.2 gives p(L, 1) ≤ 5, which is worse than the bound in Lemma 3.1. This motivates us to find a practical bound for p(L, β) by other techniques as will be shown later. 3) Forbidden substructure for wormhole detection: The packing results are used to define forbidden substructures for unit disk graphs. The wormhole connects all nodes in region A with all the nodes in region B (Figure 1). Thus we can have two independent (i.e., non-neighbor) nodes in region A, say, a, b, that share three common neighbors c, d, e in region B that are independent. This constitutes a forbidden structure, since in any valid UDG embedding of the connectivity graph the three common neighbors must be within the intersection of disks centering a, b. Since they are independent, their pairwise distance must be more than 1. By Lemma 3.1 we know that this can not happen. Thus the discovery of this forbidden substructure reveals the existence of a wormhole. However, this technique of finding forbidden substructure cannot always guarantee detection of wormholes because the existence of nodes like c, d, e in region B is dependent on the density of nodes in the network. The technique will fail when region B has only 2 nodes, for example. For such low density cases, we need to go beyond 1-hop and look for similar forbidden substructures among k-hop neighbors. Here, we will look for fk common independent k-hop neighbors of two non-neighboring nodes. fk is a parameter to be discussed momentarily. To summarize, the forbidden substructures we will use in our algorithm are the following. • • 3 independent common 1-hop neighbors: Two nonneighboring nodes having 3 independent common neighbors; In general, we have fk independent common k-hop neighbors: Two nonneighboring nodes having fk independent common k-hop neighbors. We call fk the forbidden parameter of the wormhole detection algorithm. fk must be more than the packing number for unit distance inside the lune of two disks of radii k (modeling the k-hop neighborhood) placed at distance 1 (modeling the lower bound for the distance between non-neighbors). Thus, fk = p(L(1, k), 1)+1, with p(L(1, k), 1) as the corresponding packing number to be determined by Lemma 3.2 or other methods. Also, from Lemma 3.1, for k =1, f1 = 3. For a communication model that is not unit disk graph, the determination of fk will be discussed in subsection III-C. If a network has one of these forbidden substructures, we know for sure that there is a wormhole. For a given node density, if there is wormhole present, the possibility of finding it improves with increasing k. This is because larger neighborhoods simply provide more nodes to work with, thus increasing the possibility of finding forbidden substructures. Our evaluations in the next section show that testing for 1hop is often sufficient to provide a very high detection rate requiring 2-hops only for very sparse, disconnected or irregular networks. This makes the approach quite practical. B. Algorithm Description Recall that the wormhole detection algorithm is to search by each node a forbidden structure in its neighborhood. The algorithm is localized and distributed. Each node searches for forbidden structures in its k-hop neighborhood. We will explain the algorithm for the general k-hop detection. In our empirical studies k ≤ 2 was found sufficient for most of the cases. Each node u maintains the list of 2k-hop neighbors N2k (u). Node u finds a non-neighboring node, v, from N2k (u) and checks their k-hop neighbor lists to compute their common k-hop neighbors Ck (u, v). Note that to find a non-empty Ck (u, v) set, node u need not look for v beyond 2k hops. We now need to look for the existence of the forbidden substructure (i.e., fk independent nodes) in Ck (u, v). One way to do this would be to compute the maximum independent set among Ck (u, v) and comparing the size of this set with fk . But computing the maximum independent set is a NPhard problem, even for unit disk graphs [18], [19]. Thus we relax the detection rule by finding a maximal independent set (a set of independent nodes such that no other node can be included), which can be done by a simple greedy algorithm: we start from an empty set, pick an arbitrary node and include it in the independent set, remove its neighbors, and continue until we run out of nodes in Ck (u, v). The resulting set is a maximal independent set. We compare the size of the maximal independent set thus obtained with the forbidden parameter fk . If it is equal or larger than fk , then we output ‘wormhole detected’. The outline of the algorithm is as follows. 1) In a preprocessing stage, find the forbidden parameter fk , based on the node distribution and communication model. (For UDGs, the bound on fk can be derived from Lemmas 3.1 and 3.2. We discuss other techniques of finding fk in practice in the next subsection, which also generalize to non-UDGs.) 2) Each node u determines its 2k-hop neighbor list, N2k (u), and executes the following steps for each nonneighboring node v in N2k (u). 3) Node u determines the set of common k-hop neighbors with v from their k-hop neighbor lists. This is Ck (u, v) = Nk (u) ∩ Nk (v). This can be determined by simply exchanging neighbor lists. 4) Node u determines the maximal independent set of the sub-graph on vertices Ck (u, v), by using the greedy algorithm presented above. 5) If the maximal independent set size is equal or larger than fk , node u declares the presence of a wormhole. The way the algorithm is presented makes it appear as if some work is duplicated (nodes u and v are doing the same computation by symmetry). These can be easily resolved by using some priority rules based on node ids. The algorithm presented above depends only on the 2k and k-hop neighbor lists of each node. If the wormhole attacks are required to be detected as soon as they are in place, ideally our algorithm can be run everytime there is a change in topology. Since it is a local algorithm, only the nodes affected by the change in topology need to re-run it. In practice, the requirement to run it immediately after the attack is placed is not so strict. In such cases, the algorithm can be run periodically depending on the security requirements and the network condition. For example, in mobile networks it is probably more sensible to run it periodically, while in static networks, it should be triggered by changes in topology. The message and time complexity of the algorithm is dependent on k. As we mentioned, for all cases we considered in our simulations, including fairly low density cases, k ≤ 2 has been sufficient. In cases where the network in fact has enough density to be connected and is fairly uniform (like in most practical cases), k = 1 has been found to be sufficient. The computational cost for k = 1 is roughly O(d3 ), where d is the average degree of the nodes. Essentially a node checks each of O(d2 ) non-neighboring nodes in its 2-hop neighborhood, and pays a cost of O(d) for finding the maximal independent set size in the intersection list. For any practical network, d is typically a small constant. So the detection algorithm is quite efficient. C. Consideration of Node Distribution and General Communication Model Consideration of node distribution is important in the performance of our algorithm. The packing number fk − 1 used above, i.e., the maximum number of independent common khop neighbors of two independent nodes, is the theoretical worst case bound for an arbitrary distribution. If the sensors are deployed with a known distribution, then the forbidden parameter fk we use in the forbidden substructure can be much smaller than the theoretical worst case. For example, for the 2-hop detection case, p(L(1, 2), 1) ≤ 18 by lemma 3.2, providing f2 = 19. Unless the node density is very high, it is unlikely that we will be able to find that many common independent 2-hop neighbors between two non-neighboring nodes to be able to detect a wormhole attack. This observation prompts us to tune this critical parameter fk according to the specific node distribution and not relate it directly to the packing number that models an absolute bound. In general, the smaller fk is, the higher the detection rate. When fk is too small, we may have false positives as some legal configuration may be identified as wormhole. The second important consideration is the communication model. The unit disk graph model considered so far is an overly simplified model for wireless communications. Experiments show that packet reception range is not a perfect disk [20]. Our approach can be generalized to any communication model, and even to situations where communication model is unknown. The algorithm indeed remains the same. But the preprocessing step involving the determination of the forbidden parameter fk in the first step of the algorithm differs. In following we describe a number of techniques to obtain the forbidden parameter fk in practice. 1) Known models: For any practical node deployment we typically know the radio propagation characteristics for the specific hardware used subject to the deployment environment, as well as the spatial distribution of nodes. We could try to find fk directly using mathematical or geometrical constructs. For example, a quasi-unit disk graph model [11] assumes that two nodes have a link if their distance is within α ≤ 1 and do not have a link if their distance is larger than 1. If two non-neighboring nodes have f1 independent common neighbors, these nodes must be within the lune L(α, 1) and are pairwise distance α away. Thus the packing number is f1 = p(L(α, 1), α) + 1. In general, we have fk = p(L(α, k), α) + 1. For all communication models, it may not be always possible to evaluate such expressions, or even write such mathematical constructs. In such cases, we can run simulations with the targeted distribution to obtain an estimated connectivity graph, with which we can estimate the forbidden parameter fk . For example, for any pair of non-neighboring nodes we can find the maximal independent set among their common khop neighbors and take the maximum as fk −1. Our simulation results in this paper actually use this method and obtain tight bounds for fk . Notice that when the communication model is probabilistic, the maximum number of independent neighbors of two non-neighboring nodes, f1 − 1, is also probabilistic. Thus false positives are possible in theory under our detection algorithm. 2) Unknown models: When nothing is known about the node distribution and/or communication model, it becomes harder to estimate fk . In this case, we run the detection algorithm with a standard parametric search for the unknown parameter fk . We start with a large initial value for fk , and run the algorithm as presented before. If no wormhole is detected, we halve fk and rerun the algorithm. Notice that when fk is small enough, false positives will show up. We choose fk to be the value when only a very small fraction of nodes report wormholes, or the minimum number of tolerable B e Y d a X Wormhole Link c b A Fig. 3. Example of second possible placement of the forbidden substructure. false positives. One good mechanism would be to run this parametric search in a safe part of the network, guaranteed to be free from wormhole, before deploying it in the entire network. We can then estimate the parameter such that there is no false positive detection in the safe part and apply the parameter for the entire network If no such safe part can be ascertained, the search must run in the network that has potentially been inflicted with wormholes already. In that case, a “threat level” must be assumed. The threat level is to be used as a guidance for what fraction of nodes must report wormholes before fk will not be reduced any further. D. Wormhole removal Once a forbidden structure is discovered, it is usually expected that user should manually intervene and remove the wormhole links. Here, we devise a simple approach that can be used to isolate all links possibly affected by wormhole without manual intervention. We outline the approach for the 1-hop detection case for UDGs. It can be easily extended for other cases. After a successful 1-hop detection in UDGs, we have two non-neighboring nodes a, b with 3 common independent neighbors c, d and e. Figure 1 shows one possible placement of these nodes to form the forbidden substructure, such that a and b are placed in one region (lets call it region A, without loss of generality) and c, d and e are placed in another region, B. Another possible placement is shown in Figure 3. Here, a, b are located in region A; d, e are located in region B; but c is located just outside A neighboring a and b. It can be verified that these are the only two placements possible. One can define two types of nodes neighboring the wormhole region – uncorrupted and corrupted nodes. An uncorrupted node is a node which is not in the transmission radius of the wormhole nodes. Thus they are the nodes just outside A and B. Corrupted nodes are the ones which do hear wormhole nodes and are thus inside regions A and B. Corrupted nodes have their neighbor lists corrupted due to the presence of the wormhole link. Our wormhole removal algorithm tries to identify, and blacklist, all nodes that are possibly corrupted nodes (the rest are surely uncorrupted nodes). Once identified, each corrupted node then purges its neighbor list by verifying it with the neighbor lists of its neighboring uncorrupted nodes. Note that even one link due to wormhole placement left out un-removed will still make a huge damage to the network. Thus our removal scheme allows error on the aggressive side and removal of legals links, as long as all the illegal links are definitely removed. Inferring from the two placements discussed above, one can say that nodes which satisfy any of these two conditions must include all corrupted nodes: • • The node is a neighbor of both a and b, or, The node is a neighbor of at least 2 nodes out of c, d and e. This identification method finds a set of suspicious nodes that will include all corrupted nodes and may also include some uncorrupted nodes. While on the other hand, all nodes not identified by this method, are definitely uncorrupted nodes that do not have fake links created by wormhole. To remove the fake links, each suspicious node, u, takes the intersection of its neighbor set, N (u) with the neighbor sets of its neighboring uncorrupted (non-suspicious) nodes. Any node v ∈ N (u) which is not part of any such intersections, is blacklisted by node u. All future transmissions from such nodes will be ignored by node u making the wormhole attack ineffective. When all suspicious nodes finish blacklisting nodes from their neighbor list, this completes wormhole removal. We note that the removal is a bit aggressive to guarantee that all illegal links due to wormhole will be removed, however, some legal links may be removed as well. This algorithm is not evaluated here for brevity. IV. S IMULATION R ESULTS In this section, we present simulation results demonstrating the effectiveness of our algorithm in detecting wormhole attacks. In particular, we evaluate the probability of successful detection for networks with various node distributions and connectivity models. We consider three different connectivity models in our simulations: a) unit disk graph, b) quasiunit disk graph and c) the model used in the TOSSIM simulator [12], which is based on real empirical data from a motes testbed. We evaluate the algorithm with two different node distributions: i) grid distribution with some perturbations (modeling a planned sensor deployment) and ii) random distribution. A. Details of Models and Evaluation Approach In the quasi-UDG model, if the transmission radius of the nodes in the network is R and the quasi-UDG factor is α (where, 0 ≤ α ≤ 1), then there exists a link between every pair of nodes within distance αR. If the distance is greater than R, then there is no link. If the distance d between a node pair is within [αR, R], we assume presence of a link with probability d R−αR . For all our quasi-UDG simulations, we used α = 0.75. In the TOSSIM model, the provided LossyBuilder tool is used to generate bit error probabilities (say, Pb ) between node pairs. In order to build the connectivity graph, it is assumed that the link exists with probability (1 − Pb ). Note 0.6 Detection Disconnected False +ve 0.4 10 Detection Disconnected False +ves f_1 f_2 0.6 0.4 8 6 4 0.2 0.2 2 0 0 0 0 2 4 6 8 10 12 14 16 0 3 Average Degree 6 9 16 1 14 0.8 Probability 0.8 Probability Probability 0.8 12 Detection Disconnected False +ve f_1 f_2 0.6 0.4 10 8 6 4 0.2 2 0 12 0 1 3 5 Average Degree (a) 12 Forbidden Parameter 1 Forbidden Parameter 1 7 9 11 13 15 Average Degree (b) (c) 1 1 12 0.8 0.8 10 1 0.6 0.4 6 4 0.2 0.2 2 0 0 0 1 3 5 7 9 11 Average Degree 13 15 3 5 7 9 11 13 15 Average Degree (d) Probability 0.4 Probability Detection Disconnected False +ves 13 Detection Disconnected False +ve f_1 f_2 0.6 0.4 11 9 7 0.2 5 0 Forbidden Parameter 0.6 8 Detection Disconnected False +ves f_1 f_2 Forbidden Parameter Probability 15 0.8 3 2 4 6 8 10 12 14 16 18 Average Degree (e) (f) Fig. 4. Probability of wormhole detection, graph disconnection and false positives for various configurations. The first three graphs are for a Perturbed Grid node distribution with p=0.2 for (a) UDG (b) Quasi-UDG and (c) TOSSIM connectivity models. The next three graphs are for Random node distribution with (d) UDG (e) Quasi-UDG and (f) TOSSIM connectivity models. that the TOSSIM model does not assume that the links are bidirectional. Our algorithm works irrespective of whether the links are directional or bi-directional. Each simulation is run with 144 nodes. Since our technique is localized (we use only 1-hop and 2-hop detections in our experiments) and the simulations so far concentrate on detecting only a single wormhole, simulating a very large networks is not required to determine the performance of our approach. For the grid-like topologies the nodes are placed in a 12 × 12 grid. Then their x and y coordinates are changed to a randomly chosen value between [x − px, x + px] and [y − py, y + py] respectively, where p is the perturbation parameter. Values of p from 0.0 to 0.5 have been used, but for brevity we only show results of p=0.2 here. For the random case, x and y coordinates are chosen randomly. As noted before node density is an important factor in our algorithm. Node density is varied in different experiments by changing the geographic area containing the nodes (for TOSSIM) or by changing the transmission radius of the nodes (for UDG and Quasi-UDG cases). After the topology is created, the nodes are connected using the given connectivity model. Once the connectivity graph is established, the following experiments are performed: • • Connectivity in the entire network is checked. The network is assumed disconnected if any two nodes do not have a path to each other.1 The wormhole detection algorithm is run to see whether there is a false positive. (At this time, there is no 1 While our technique is independent of whether the entire network is connected or not, connected networks are more useful from a practical standpoint. wormhole attack) A wormhole attack is established between two randomly chosen locations. The algorithm is run again to see whether it detects the wormhole. The algorithm was run with k ≤ 2 only. We will see momentarily that this already gives very good results for most practical scenarios. We have repeated each experiment for 10, 000 times with randomly generated topologies and attacks, but with the same node distribution model and connectivity model, and then reported various probabilities for different node densities. Three probabilities are computed: (i) probability of detection, (ii) probability of false positive and, (iii) probability of network disconnection. To avoid boundary effects, we have not considered the boundary nodes when calculating the degree, testing for disconnected networks, etc. • B. Results Figure 4 shows all our performance results for the three types of communication models and two types of node distribution models. Recall that the forbidden parameter fk is an input parameter to our algorithm and is evaluated separately in a pre-processing step as shown in subsection III-C. Figure 4 also shows fk values for different experiments. For UDG cases, it is observed that only 1-hop detection is enough for all cases except at very low densities (average degree ≤ 1), and f1 is constant at 3. Thus, we do not show the fk curves for UDG graphs. In general, the following observations can be made from the results: • Our algorithm provides very good results (no false alarms and 100% detection) when the network disconnection Probability 12 0.8 10 1 hop Detection 1+2 hop Detection f_1 f_2 0.6 0.4 8 6 4 0.2 2 0 Forbidden Parameter 1 0 3 5 7 9 11 13 15 Average Degree Fig. 5. Comparison of 1-hop vs 1 and 2-hop detection. 1 V. C ONCLUSION 0.8 Probability where the communication model and/or the node distribution are unknown. The given scenario uses k = 1, quasi-UDG model and nodes placed in grid with perturbation of 0.2 and average degree of 6. Both the false positive probability (in the absence of wormhole) and detection probability (in presence of wormhole) are shown for different values of f1 in Figure 6. There exist critical values of f1 (4 in the plot) where the detection probability is close to 100%, but the false positive probability is close to 0%. This demonstrates that if the parametric search is used in a safe network, a suitable value for f1 can be estimated by simply observing the false positive probabilities. When f1 is reduced from a large value, the detection of real wormholes shows up first before false positives. 0.6 Detection False +ve 0.4 0.2 0 1 3 5 7 9 11 Forbidden Parameter Fig. 6. Estimation of the forbidden parameter in a quasi-UDG model. In this paper we propose a practical algorithm for wormhole detection. The algorithm is simple, localized, and is universal to node distributions and communication models. Our simulation results have confirmed a near perfect detection performance whenever the network is connected with a high enough probability, for common connectivity and node distribution models. We expect that this algorithm will have a practical use in real-world deployments to enhance the robustness of wireless networks against wormhole attacks. ACKNOWLEDGMENT • • • probability is 0. This observation is independent of communication or node distribution model used. Detection probability does drop for low density cases; however, in such cases the likelihood that the network is disconnected also increases (hence the usefulness of the network also drops). Amongst all results, even with a 50% chance of the network being disconnected, our algorithm has detected the wormhole attack in 90% or more cases. For the same average node density, detection performance gets worse as the randomness of deployment (in terms of node distribution or communication model) increases. For example, the detection rate is better in UDG Perturbed Grid scenario (Figure 4a) than UDG Random scenario (Figure 4d) or Quasi-UDG Perturbed Grid scenario (Figure 4b) and so on. This phenomenon is expected because the estimation of fk is more accurate in less random scenarios. As mentioned earlier, 1-hop only detection does not perform well in non-UDG cases. Figure 5 compares 1-hop detection probability with the case when both 1 and 2-hop detection algorithm were used (2-hop detection runs only when 1hop fails), under the setup of Figure 4e with Random node distribution and Quasi-UDG connectivity model. Note that as the value of parameter f1 increases, the 1-hop detection fails to detect wormhole attacks in some cases, and hence 2-hop detection kicks in. Finally, we run a set of simulations demonstrating how the forbidden parameter fk can be estimated for a situation Ritesh Maheshwari and Samir Das’s research has been partially supported by the NSF grants CNS-0519734, OISE0423460, CNS-0308631 and a grant from the SensorCAT center. R EFERENCES [1] P. Papadimitratos and Z. J. Haas, “Secure routing for mobile ad hoc networks,” in SCS Communication Networks and Distributed Systems Modeling and Simulation Conference (CNDS 2002), 2002. [2] Y. C. Hu, A. Perrig, and D. Johnson, “Packet leashes: a defense against wormhole attacks in wireless networks,” in INFOCOM, 2003. [3] K. Sanzgiri, B. Dahill, B. Levine, and E. Belding-Royer, “A secure routing protocol for ad hoc networks,” in International Conference on Network Protocols (ICNP), November 2002. [4] J. Eriksson, S. Krishnamurthy, and M. Faloutsos, “Truelink: A practical countermeasure to the wormhole attack,” in ICNP, 2006. [5] L. M. Ni and P. K. McKinley, “A survey of wormhole routing techniques in direct networks,” Computer, vol. 26, no. 2, pp. 62–76, 1993. [6] A. Scaglione and Y. W. Hong, “Opportunistic large arrays: Cooperative transmission in wireless multihop ad hoc networks to reach far distances,” IEEE Transactions on Signal Processing, vol. 51, no. 8, 2003. [7] L. Hu and D. Evans, “Using directional antennas to prevent wormhole attacks,” in Network and Distributed System Security Symposium (NDSS), 2004. [8] N. Sastry, U. Shankar, and D. Wagner, “Secure verification of location claims,” in ACM Workshop on Wireless Security (WiSe 2003), September 2003. [9] R. Poovendran and L. Lazos, “A graph theoretic framework for preventing the wormhole attack in wireless ad hoc networks,” ACM Journal of Wireless Networks (WINET), 2005. [10] W. Wang and B. Bhargava, “Visualization of wormholes in sensor networks,” in WiSe ’04: Proceedings of the 2004 ACM workshop on Wireless security, New York, NY, USA, 2004, pp. 51–60. [11] F. Kuhn and A. Zollinger, “Ad-hoc networks beyond unit disk graphs,” in Proc. 2003 Joint Workshop on Foundations of mobile computing, 2003, pp. 69–78. [12] “TOSSIM: A simulator for tinyos networks,” User’s manual in TinyOS documentation. [13] S. Capkun, L. Buttyn, and J. P. Hubaux, “SECTOR: Secure tracking of node encounters in multi-hop wireless networks,” in 1st ACM Workshop on Security of Ad Hoc and Sensor Networks (SASN), October 2003. [14] I. Khalil, S. Bagchi, and N. B. Shroff, “LITEWORP: A Lightweight Countermeasure for the Wormhole attack in multihop wireless network,” in International Conference on Dependable Systems and Networks (DSN), 2005. [15] I. Khalil, S. Bagchi, and N. Shroff, “MOBIWORP: Mitigation of the wormhole attack in mobile multihop wireless networks,” in Second International Conference on Security and Privacy in Communication Networks (SecureComm 2006), 2006. [16] H. Breu and D. G. Kirkpatrick, “Unit disk graph recognition is NP-hard,” Computational Geometry. Theory and Applications, vol. 9, no. 1-2, pp. 3–24, 1998. [Online]. Available: citeseer.ist.psu.edu/breu93unit.html [17] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 2nd ed. New York, NY: Springer-Verlag, 1993. [18] M. R. Garey, R. L. Graham, and D. S. Johnson, “Some NP-complete geometric problems,” in Proc. 8th Annu. ACM Sympos. Theory Comput., 1976, pp. 10–22. [19] M. R. Garey and D. S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness. New York, NY: W. H. Freeman, 1979. [20] D. Ganesan, B. Krishnamachari, A. Woo, D. Culler, D. Estrin, and S. Wicker, “Complex behavior at scale: An experimental study of lowpower wireless sensor networks,” UCLA, Tech. Rep. UCLA/CSD-TR 02-0013, 2002. A PPENDIX Lemma 3.2: We use a packing argument. u r v R + α/2 Fig. 7. Packing in a lune L(r, R). Suppose we place a set of nodes P inside L(r, R) with their inter distances more than β. Thus we place disks of radius β/2 on each node in P . All the disks are disjoint. Further, all the disks are inside a slightly larger lune L(r, R + β/2), which has an area of 2(R + β/2)2 arccos(r/(2R + β)) − r (R + β/2)2 − r2 /4. Thus p(L, β) is no more than the maximum number of non-overlapping disks of radius β/2 packed inside the lune L(r, R + β/2). The total area of the disks centered on P , π(β/2)2 · |P | ≤ 2(R + β/2)2 arccos(r/(2R + β)) − r (R + β/2)2 − r2 /4. Thus p(L, β) ≤ |P | ≤ π8 (R/β + 1/2)2 arccos(r/(2R + β)) − 4r (R + β/2)2 − r2 /4 as claimed. πβ 2

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