A Graph-theoretic Framework for Identifying Trigger Nodes against Probabilistic Reactive Jamming Attacks

Abstract—During the last decade, Reactive Jamming Attack has emerged as a greatest security threat to wireless sensor networks, due to its mass destruction to legitimate sensor communications and difficulty to be disclosed and defended. Considering
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  JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 1 A Graph-theoretic Frameworkfor Identifying Trigger Nodes againstProbabilistic Reactive Jamming Attacks Ying Xuan, Yilin Shen, Incheol Shin, My T. Thai Abstract  —During the last decade, Reactive  Jamming Attack has emerged as a greatest security threat to wireless sensor networks,due to its mass destruction to legitimate sensor communications and difficulty to be disclosed and defended. Considering thespecific characteristics of reactive jammer nodes, a new scheme to deactivate them by efficiently identifying all trigger nodes  , whosetransmissions invoke the jammer nodes, has been proposed and developed. Such a trigger identification procedure could serve as aleverage subroutine for a jamming-resistent routing scheme and exhibit great potentials in enhancing the defense efficiency. By modelingthisprocedureasagraphoptimizationproblem,inthispaper,ontheonehand,weemployedanadvanced randomizederror-tolerantnon- adaptive group testing  technique and a classic clique-independent set  problem to further speed up the identification process, comparedto our previous work. On the other hand, by investigating two sophisticated jamming behavior models, we proposed an efficient algorithmwhich limits the identification false rates to desirable low levels. The theoretical analysis and simulation results illustrate the robustnessand efficiency of the proposed solution. Index Terms  —Trigger Identification, Clique-Independent Set, Error-tolerant Nonadaptive Group Testing, Graph Theory, Optimization,NP-Hardness. ! 1 I NTRODUCTION S INCE the last decade, the security of wireless sensornetworks (WSNs) has attracted numerous attentions, dueto its wide applications in various monitoring systems andinvulnerability toward sophisticated wireless attacks. Amongthese attacks, jamming attack where a jammer node disruptsthe message delivery of its neighboring sensor nodes withinterference signals, has become the most critical threat toWSNs. Thanks to the efforts of researchers toward this issue,as summarized in [11], various efficient defense strategies havebeen proposed and developed. However, a reactive variant of this attack, where jammer nodes stay quite until an ongoinglegitimate transmission (even has a single bit) is sensed overthe channel, emerged recently and called for stronger defendingsystem and more efficient detection schemes.Existing countermeasures against Reactive Jamming attacksconsist of jamming (signal) detection and jamming mitigation.On the one hand, detection of interference signals from jammernodes is non-trivial due to the discrimination between normalnoises and adversarial signals over unstable wireless channels.Numerous attempts to this end monitored critical communica- • Y. Xuan, Y. Shen, I. Shin and My T. Thai are with the Department of Computer Information Science and Engineering. E-mail: {  yxuan, yshen, ishin, mythai } @cise.ufl.eduThis is an extended version of “Y. Xuan, Y. Shen, I. Shin, M. T. Thai, ”OnTrigger Detection Against Reactive Jamming Attacks: A Clique-Independent Set Based Approach”, IPCCC, Phoenix, Arizona, 2009.” tion related objects, such as Receiver Signal Strength (RSS), Carrier Sensing Time (CST), Packet Delivery Ratio (PDR),compared the results with specific thresholds, which wereestablished from basic statistical methods and multi-modalstrategies [8][11]. By such schemes, jamming signals couldbe discovered, however, how to locate and catch the jammernodes based on these signals is much more complicated andhas not been settled. On the other hand, in order to mitigatethese attacks, two strategies were adopted at sensor nodes toescape from the detected interferences, namely, channel surfingand spatial retreats [11]. The former one employs frequencyhopping techniques at both communication ends [5][7][9], inwhich case jammer nodes are unable to find the current channelthat is used for the communication, so that the attack efficiencyis greatly decreased. The latter one requires sensor nodes toretreat from the possible jammed areas, then no sensor nodeswill be effected by the jamming signals [10][12]. However,owing to the limited power and spectral diversity [8] of wirelesssensors, these mitigation schemes are inefficient due to theirconsiderable computation and communication overheads.Instead of discovering the jammed areas, which may beinaccurate and unnecessarily large, we proposed a new solutionto mitigate the attacks by identifying the trigger nodes in [6],whose transmissions invoke the jammer nodes, and preventingthese trigger nodes from transmitting messages. Specifically,we provided a novel jamming-resistent routing scheme withregulating all identified trigger nodes as terminals, thereforeno messages will be transmitted from the trigger nodes and  JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 2 all jammer nodes will stay quiet. The motivation of studyingthis trigger identification problem is not limited in this reactive jamming scenario, but to provide a solution framework to themitigations against the reactive variant of a general scope of attacks.Since the performance of the trigger identification is criti-cal for the routing scheme and other applications that makebenefit from it, in this paper, we develop a solution for realsophisticated attack scenarios. Specifically, as many attackersplay tricks to evade detections, the feedbacks of jammer nodestoward sensed message transmissions can be non-deterministicor along with randomized time delays. To handle these unsurefactors, we introduce a novel randomized error-tolerant grouptesting scheme , which is combined with a clique-independentset model, and speeds up the identification procedure with lowerror rates under unreliable enviroments. The basic idea of oursolution is to partition the victim nodes (which are interferedby jamming signals) into multiple testing teams , and thenconduct group testing based on the constructed randomized ( d,z ) -disjunct matrix, by letting each victim node broadcasttest messages simultaneously and some leader  nodes gatherthe feedbacks (interference signals) from the jammers. Bygenerating test outcomes from these gathered feedbacks, allthe trigger nodes are identified via a prompt decoding process.Compared with our previous work [6], more sophisticated jammer behaviors are considered and handled in this paper bythe new group testing scheme, with lower time and communi-cation complexity, as well as accuracy guarantees. Moreover,we model the partitioning phase of victim nodes as a clique-independent set  problem, whose NP-Hardness on UDG (unitdisk graph) is shown.In the remainder of this paper, we first present the problemdefinition in Section 2, where the network model, victimmodel and attacker models are included. Then we introducetwo kernel techniques for our scheme, clique-independent set  and randomized error-tolerant non-adaptive group testing in Section 3. The core of this paper: trigger identification procedure and its error-tolerant extension toward sophisticated jammer behaviors are presented respectively in Section 4 and5. A series of simulation results for evaluating the systemperformance and validating the theoretical results are includedin Section 6. We also present some related works in Section 7and summarize the whole paper in Section 8. 2 P ROBLEM M ODELS AND D EFINITION 2.1 Network Model We consider a wireless sensor network consisting of  n sensornodes and one base station (larger networks with multiple basestations can be split into small ones to satisfy the model).Each sensor node has a uniform transmission radius r and isequipped with m radios for in total k channels throughout thenetwork, where k > m . The network can abstracted as a unit disk graph (UDG) G = ( V,E  ) , where any node pair i,j isconnected iff the Euclidean distance between i,j : δ  ( i,j ) ≤ r . 2.2 Victim Model Victim nodes refer to those sensor nodes whose transmissionsare disturbed by jamming signals, i.e., node v is a victim node iff  δ  ( J,v ) ≤ R for some activated  jammer J  . In this paper,we assume that each sensor can identify received jammingsignals and justify whether itself is a victim node. Furthermore,the results of these self-identifications are reported to thebase station by means of the existing message forwardingschemes periodically, therefore the set of victim nodes ismaintained at the base station. Since the detection of jammingsignals have been well developed with multi-modal statisticalmethods, the above assumptions are feasible even in unreliableenvironments.As a subset of the victim nodes, trigger nodes refer toa subset of victim nodes, whose transmissions activate the jammer nodes. In another word, node v is a trigger node iff  δ  ( J,v ) ≤ r for some activated jammer J  . Therefore theproblem studied in this paper is to identify all the trigger nodesfrom a given set of victim nodes. 2.3 Attacker Model We consider both a basic attacker model and several advancedattacker models in this paper. In the next sections, we willfirst illustrate our framework solution toward the basic attackermodel, and then validate its performance toward multipleadvanced attacker models theoretically and experimentally. 2.3.1 Basic Attacker Model  The basic attacker model is defined as follows: there existsat most J  ≪ n reactive jammer nodes in the network,whose transmission radiuses are R = αr with α > 1 .These jammer nodes keep idle until they sense any ongoinglegitimate transmissions and broadcast interference signals to jam all the sensors in distance R on this specific channel. Themaximum damages caused by the jammer nodes are limitedto the interferences toward specific sensor nodes on specifictransmission channels for a short period, instead of long-termdisabling the sensors. The motivation behind this assumptionarises from the basic goal of reactive jamming: disrupt themessage delivery with minimum energy cost. As soon as thesensors detect any jamming signals, the transmissions will beterminated, or continue on some other channels. Thus it isunnecessary for the jammer nodes to keep sending interferencesignals on this channel for a long time, or either disrupt allthe channels with large energy overheads as an active jammerdoes. Moreover, from the standpoint of the attacker, it willbe a waste to deploy two jammer nodes too close to eachother, thus we assume that for any two jammer nodes J  1 and J  2 , δ  ( J  1 ,J  2 ) ≥ 2 R − R ′ with a small overlap R ′ such that R ′ ≤ R − r (see Fig. 3 in the next section). 2.3.2 Advanced Attacker Models  Considering possible adjustments at the jammer nodes toevade the detection, we take into account two probabilistic  JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 3 attacker models: probabilistic attack  and variant response timedelay . In the first one, the jammer responds each sensedtransmission with a probability η independently. Practically, η is approaching 1 , to guarantee the attack efficiency. However,in order to validate the accuracy of our solution toward extremecases, we also consider small η in the theoretical analysis andsimulations. In the other model, the jammer delays each of its jamming signals with an independently randomized timeinterval. Similarly, too large delays do not make sense forpractical attacks, but our solution is also satisfiable under theseextreme cases.It is evident that most tricks in reactive jamming attack can be abstracted into either of these two models. Therefore,showing the efficiency of our identification toward such modelssuffices validating its applicability to practical defense systemsand unreliable WSN environments. 3 T WO K ERNEL T ECHNIQUES This section includes two advanced techniques which benefitour identification procedure. We first provide the NP-hardnessproof of the Clique-Independent Set  problem along with a sim-ple approximation algorithm, then introduce the randomized error-tolerant group testing by providing our novel design forrandomized ( d,z ) -disjunct matrix. 3.1 Clique-Independent Set Cliques-Independent Set is the problem to find a set of max-imum number of pairwise vertex-disjoint maximal cliques,which is referred as a maximum clique-independent set  (MCIS)[4]. Since this problem serves as the abstracted model of the grouping phase of our identification, its hardness is of greatinterest in this scope. To our best knowledge, it has alreadybeen proved to be NP-hard for cocomparability, planar, lineand total graphs, however its hardness on UDG is still an openissue. 3.1.1 NP-hardness  In this section, we prove the NP-hardness of this problemon UDG via a polynomial-time reduction from the Maximum Independent Set  problem on planar graph with maximum nodedegree 3 to it.From [20], the Maximum Independent Set  problem is NP-hard on planar graph with maximum degree 3, and from [21],any planar graph G with maximum degree 4 can be embeddedin the plane using O ( | V  | 2 ) area units such that its vertices areat integer coordinates and its edges consist of line segments of the form x = i or y = j , for any integers i and j . Theorem 3.1: Clique-Independent Set  problem is NP-hardon Unit Disk Graph. Proof: Given an instance G ′ = ( V  ′ ,E  ′ ) of such a MIS  problem, whose optimal value is denoted as MIS  ( G ′ ) , weconstruct an instance G = ( V,E  ) of the CIS  problem asfollows: • Embed G ′ in the plane in the way mentioned above [21]. • For each node v i ∈ V  ′ , attach two new nodes v i 1 and v i 2 to it and form a triangle N  i = { v i 1 ,v i 2 ,v i 3 } , where eachedge of this triangle N  i is of a unit length r = √  33 . • Since each nodes v i is incident to at most three edges,for all edges ( v i ,u ) , ··· , ( v i ,v ) , move their endpoint from v i to different v ij s, e.g., ( v 1 ,u ) changes to ( v 11 ,u ) and ( v 1 ,v ) to ( v 12 ,v ) . Afterwards, for each of such edges e = ( u,v ) , assume that it is of length t , we divide itinto t pieces and replace each piece with a concatenationof 2 triangles (not necessarily equilateral), as shown inFig. 1(b). Therefore, any edge e ij = ( v i ,v j ) ∈ E  ′ of length | e ij | becomes a concatenation of  2 | e ij | 3-cliques,denoted as { c 1 , 1 ij ,c 1 , 2 ij ,c 2 , 1 ij , ··· c | e ij | , 1 ij ,c | e ij | , 2 ij } . Because of the triangles N  i s, the two triangles at each corner of Fig.1(b) may need slight stenches, which can be done inpolynomial time. • The resulting graph G is then a unit disk graph with radius r = √  33 . 9 9 V1V3V4V2 (a) Instance G ′ = ( V   ′ ,E  ′ ) of MISproblem on planar graph with maximumdegree 3 N1N3N2 N4 (b) Instance G = ( V,E  ) with radius √  33 of Clique-Independent Set problem on UDG Fig. 1. Polynomial Time Reduction The reduction is as follows:( ⇒ ): if  G ′ has a maximum independent set M  , for each u i ∈ M  , we choose cliques of two kinds in the correspondinginstance G : (1) the clique N  i at u i ; (2) for each incident edge e ij = ( u i ,u j ) , choose cliques { c 1 , 2 ij ,c 2 , 2 ij ,c 3 , 2 ij , ··· ,c | e ij | , 2 ij } .Since the clique N  j at u j shares a vertex with c | e ij | , 2 ij , it cannotbe selected. For any edge e jk = ( u j ,u k ) where u j / ∈ M  and u k / ∈ M  , choose cliques { c 1 , 2 jk ,c 2 , 2 jk , ··· c | e jk | , 2 jk } . It is easy to  JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 4 verify that all the cliques selected are vertex-disjoint from eachother.Assume that after embedding G ′ into the plane, each node v i ∈ V  ′ has coordinate ( x i ,y i ) , then edge length | e ij | = ∥ v i ,v j ∥ 1 = | x i − x j | + | y i − y j | . Therefore if we have anindependent set of size | M  | = k for G ′ , we then have a cliqueindependent set of size k ′ = k + ∑ ( i,j ) ∈ E  ′ | e ij | .( ⇐ ): if  G has a clique independent set of size k ′ , since thelengths of the embedded edges are constant, then G ′ hasexactly an independent set of size k = k ′ − ∑ ( i,j ) ∈ E  ′ | e ij | .The proof is complete. 3.1.2 Algorithms  There have been numerous polynomial exact algorithms forsolving this problem on graphs with specific topology, e.g.,Helly circular-arc graph and strongly chordal graph [4], butnone of these algorithms gives the solution on UDG. In thispaper, we employ the scanning disk approach in [3] to find allmaximal cliques on UDG, and then find all the MCIS  usinga greedy algorithm. In fact, by abstracting this problem as a Set Packing problem, we can obtain a √  n -approximation algo-rithm, however, it exhibits worse performance than the greedyalgorithm proposed in our trigger identification procedure. 3.2 Error-tolerant Randomized Non-Adaptive GroupTesting Group Testing was proposed since WWII to speed up theidentification of affected blood samples from a large samplepopulation. This scheme has been developed with a completetheoretical system and widely applied to medical testing andmolecular biology during the past several decades [1]. Noticethat the nature of our work is to identify all triggers out of alarge pool of victim nodes, so this technique intuitively matchesour problem. 3.3 Traditional Non-adaptive Group Testing The key idea of group testing is to test items in multipledesignated groups, instead of testing them one by one. Thetraditional method of grouping items is based on a designated 0 - 1 matrix M  t × n where the matrix rows represent the testinggroup and each column refers to an item, as Fig. 2 shows. M  [ i,j ] = 1 implies that the j th item appears in the i th testinggroup, and 0 otherwise. Therefore, the number of rows of thematrix denotes the number of groups tested in parallel and eachentry of the result vector V  refers to the test outcome of thecorresponding group (row), where 1 denotes positive outcomeand 0 denotes negative outcome.Given that there are at most d < n positive items amongin total n ones, all the d positive items can be efficiently andcorrectly identified on the condition that the testing matrix M  is d -disjunct: any single column is not contained  by the unionof any other d columns. Owing to this property, each negativeitem will appear in at least one row (group) where all the M  =  0 0 0 0 1 1 1 10 0 1 1 0 0 1 10 1 0 1 0 1 0 11 1 1 1 0 0 0 01 1 0 0 1 1 0 01 0 1 0 1 0 1 0  testing = ⇒ V  =  001111  Fig.2. Binarytestingmatrix M  andtestingoutcomevector V  . Assumed that item 1 ( 1 st column) and item 2 ( 2 nd column) are positive, then only the first two groups returnnegativeoutcomes,becausetheydonotcontainthesetwopositive items. On the contrary, all the other four groupsreturn positive outcomes. positive items do not show up, therefore, by filtering all theitems appearing in groups with negative outcomes, all the left ones are positive . Although providing such simple decodingmethod, d -disjunct matrix is non-trivial to construct [1][2]which may involve with complicated computations with highoverhead, e.g., calculation of irreducible polynomials on GaloisField. In order to alleviate this testing overhead, we advancedthe deterministic d -disjunct matrix used in [6] to randomizederror-tolerant d -disjunct matrix, i.e., a matrix with less rowsbut remains d -disjunct w.h.p. Moreover, by introducing thismatrix, our identification is able to handle test errors undersophisticated jamming environments. 3.4 Error-tolerant Randomized Designs In order to handle errors in the testing outcomes, the error-tolerant non-adaptive group testing has been developed using ( d,z ) -disjunct matrix, where in any d +1 columns, each columnhas 1 in at least z rows where all the other d columns are 0 .Therefore, a ( d, 1) -disjunct matrix is exactly d -disjunct. Byuse of  ( d,z ) -disjunct matrix, we can still correctly identify d positive items, even in the presence of at most z − 1 test errors.A prompt decoding scheme by differentiating positive andnegative items based on their number of appearances in groupswith negative outcomes is summarized in [1]: considering anysingle positive item i and negative item j . Suppose there are c negative groups containing i , then these c groups (tests)have errors, hence there are at most z − 1 − c other groupsturning negative outcomes to positive outcomes. Due to thedefinition of  ( d,z ) -disjunct matrix, column j appears in atleast z negative groups where none of the d positive itemsexist, so even z − 1 − c of these groups are turned into positiveones, the number of negative groups containing j is at least z − ( z − 1 − c ) = c + 1 > c . It is evident that by sorting allthe suspected items by their number of appearances in negativegroups, those d items with smallest number of appearances arepositive.In the literature, one the one hand, numerous deterministicdesigns for ( d,z ) -disjunct matrix have been provided [1],  JOURNAL OF L A TEX CLASS FILES, VOL. 6, NO. 1, JANUARY 2007 5 however, these constructions often suffer from high compu-tational complexity, thus are not efficient for practical useand distributed implementation. On the other hand, to ourbest knowledge, the only randomized construction for ( d,z ) -disjunct matrix dues to Cheng’s work via q  -nary matrix [19],which results in a ( d,z ) -disjunct matrix of size t 1 × n withprobability p ′ , where t 1 = 4 . 28 d 2 log21 −  p ′ + 4 . 28 d 2 log n + 9 . 84 dz +3 . 92 z 2 ln2 n − 11 −  p ′ with time complexity O ( n 2 log n ) . Compared with this work,we advance a classic randomized construction for d -disjunctmatrix, namely, random incidence construction [1][2], to gen-erate a ( d,z ) -disjunct matrix which can not only generatecomparably smaller t × n matrix, but also handle the case where z is not known beforehand, instead, only the error probabilityof each test is bounded by some constant γ  . Although z can bequite loosely upperbounded by γt , yet t is not an input. Themotivation of this construction lies in the real test scenarios,the error probability of each test is unknown and asymmetric,hence it is impossible to evaluate z before knowing the numberof pools.We only show the performance of this new construction,namely, ETG algorithm in this section. For the review purpose,we include the details of the construction and proofs in theAppendix. Theorem 3.2: ETG algorithm produces a ( d,z ) -disjunct ma-trix with t = 2  ( d + 1) d +1 d d  z − 1 + ln(11 −  p ′ ) + ( d + 1)ln n  rows with probability p ′ for an arbitrarily large constant p ′ . Corollary 3.1: The ( d,z ) -disjunct matrix is asymptoticallysmaller than the one constructed by Cheng [19]. Corollary 3.2: The time complexity of  ETG algorithm isasymptotically smaller than that of Cheng’s algorithm, giventhat d < √  n . Corollary 3.3: Given that each test has an independent errorprobability γ  , ETG algorithm produces a ( d,z ) -disjunct matrixwith t = τ  ln n ( d +1) 2 − 2 τ  ( d +1)ln(1 −  p ′ )( τ  − γ  ( d +1)) 2 with probability p ′ ,where τ  = ( d/ ( d + 1)) d . 4 T RIGGER I DENTIFICATION P ROCEDURE FOR B ASIC A TTACKER M ODEL In this section, we present the trigger identificationprocedure for the basic attacker model, where the jammers deterministically and immediately broadcasts jamming signalson the particular channel which carries the sensed messagetransmissions between sensor nodes. Therefore, as longas some jamming signals are received, at least one of thebroadcasting victim nodes is a trigger. In the next section,we will further investigate the performance of our solutiontowards some sophisticated attack models, in order to showthe robustness of this scheme in real scenarios. 4.1 Identification Overview The trigger identification can be sketched as follows (Fig. 3):Assume that at the beginning of the identification phase, all jammer nodes are idle and all the victim nodes in grey andblue have been discovered beforehand. The set of victims aredivided into interference-free teams, where the transmissions of victim nodes within one team will not invoke a jammer node,whose interference signals will disrupt the communicationswithin another team, as shown in Fig. 3. We call these teams testing teams in the remainder of the paper. Base StationSensor    Field J 1 J 3 Sensor Nodes J 2 V 1 V 2 V 4 V 5 V 7 V 6 V 10 V 13 V 11 V 12 V 19 V 8 V 20 V 18 V 15 V 14 V 17 V 9   r R    J 2 R’ BS Fig. 3. Nodes in grey and blue are victim nodes around jammer nodes, where blue nodes are also trigger nodes,which invoke the jammer nodes. The identification of trigger nodes involves two paralleltesting types: (1) Denote the set of victim nodes within each testing team as W  , and the number of trigger nodes (tobe estimated) as d , then a group testing procedure will runsimultaneously over each testing team , to identify the d triggernodes from | W  | victim ones; (2) Victim nodes within each testing team will be divided into the multiple group s, accordingto a randomized ( d, 1) -disjunct matrix, as mentioned in Section3.2. Each group of victim nodes will be tested on a differentchannel, to avoid interference among groups.The testing procedure within each pool is two-fold: (1) Eachgroup i is corresponding to a row in the testing matrix M  , andassigned with a different channel frequency f  from that of other groups. Let a victim node j broadcast a single bit on f  , iff  M  [ i,j ] = 1 , to activate possible jammer nodes nearby.Assume M  has t rows and each sensor has m radios, thenonly m groups can be tested at a time, and all t groups canbe tested within ⌈ tm ⌉ rounds. This is because one victim node
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