Modeling Pairwise Key Establishment for Random Key Predistribution in Large-scale Sensor Networks

Transcription

1 Modeling Pirwise Key Estlisment for Rndom Key Predistriution in Lrge-scle Sensor Networs Dijing Hung, Memer, IEEE, Mnis Met, Memer, IEEE, Appie vn de Liefvoort, Memer, IEEE, Deep Medi, Senior Memer, IEEE, Astrct Sensor networs re composed of lrge numer of low power sensor devices. For secure communiction mong sensors, secret eys re required to e estlised etween tem. Considering te storge limittions nd te lc of postdeployment configurtion informtion of sensors, Rndom Key Predistriution scemes ve een proposed. Due to limited numer of eys, sensors cn only sre eys wit suset of te neigoring sensors. Sensors ten use tese neigors to estlis pirwise eys wit te remining neigors. In order to study te communiction overed incurred due to pirwise ey estlisment, we derive proility models to design nd nlyze pirwise ey estlisment scemes for lrge-scle sensor networs. Our model pplies te inomil distriution nd modified inomil distriution nd nlyzes te ey pt lengt in op-y-op fsion. We lso vlidte our models troug systemtic vlidtion procedure. We ten sow te roustness of our results nd illustrte ow our models cn e used for ddressing sensor networ design prolems. I. INTRODUCTION Lrge-scle sensor networs re composed of lrge numer of low-powered sensor devices. According to [], te numer of sensor nodes deployed to study penomenon my e on te order of undreds or tousnds; depending on te ppliction, te numer my rec n extreme vlue of millions. Typiclly, tese networs re instlled to collect sensed dt from sensors deployed in lrge re. Witin networ, sensors communicte mong temselves to excnge dt nd routing informtion. Becuse of te wireless nture of te communiction mong sensors, tese networs re vulnerle to vrious ctive nd pssive ttcs on te communiction protocols nd devices. Tis demnds secure communiction mong sensors. Due to inerent storge constrints, it is infesile for sensor device to store sred ey vlue for every oter sensor in te system. Moreover, ecuse of te lc of postdeployment geogrpic configurtion informtion of sensors, eys cnnot e selectively stored in sensor devices. Altoug nïve solution would e to use common ey etween every pir of sensors to overcome te storge constrints, it offers we security. Mnuscript received Mrc 5; revised Ferury 6, April 6. D. Hung is wit te Deprtment of Computer Science & Engineering, Arizon Stte University, Tempe, AZ, USA (e-mil: dijing@su.edu. M. Met is wit Tumleweed Communictions. (emil: mnis.met@tumleweed.com. A. vn de Liefvoort nd D. Medi re wit te Deprtment of Computer Science nd Electricl Engineering, University of Missouri Knss City, USA (e-mil: ppie@umc.edu, dmedi@umc.edu. Rndom Key Predistriution (RKP scemes ([], [6], [5] nd [8] ve een proposed to provide flexiility for te designers of sensor networs to tilor te networ deployment to te ville storge nd te security requirements. Te RKP scemes propose to rndomly select smll numer of eys from fixed ey pool for ec sensor. Sensors ten sre eys wit ec oter wit proility proportionl to te numer of eys stored in ec sensor. Since te RKP scemes necessitte only limited numer of eys to e preinstlled in sensors, sensor my not sre eys wit ll of its neigor nodes. In tis cse, Pirwise Key Estlisment (PKE sceme is required to set up sred eys wit required frction of neigor nodes. Te PKE scemes require sensors to set up pirwise eys vi te nodes tt sre eys wit eiter or ot te sensors. Tis PKE pse involves communiction overed for finding te sortest pt to neigor node nd for setting up te pirwise ey troug tt pt. Te lesser te numer of eys preinstlled in ec sensor, te lower te proility tt sensor sres ey wit given neigor node. Consequently, te sensor requires more overed in te PKE pse wit te remining neigor nodes. Studies in [5] sow tt te energy consumption due to communiction in sensors is severl orders iger tn tt due to computtion overed. Te constrints suc s scrce ttery power nd limited storge necessitte reference model to study te trdeoff etween storge nd communiction overed involved during te PKE pse in RKP scemes. It my e noted tt te memory limittion of sensors restricts te numer of eys tt cn e preinstlled in ec sensor to smll numer. For exmple, te cpilities of sensor nodes for lrge-scle sensor networs cn e s limited s tose of Smrt Dust sensors [], [] tt ve only 8K of progrm nd 5 ytes for dt memory. Moreover, studies in [6] nd [8] sow tt smll ey pool size increses security vulnerilities. Tus, for lrge-scle sensor networs, smll numer of eys preinstlled in ec sensor nd lrge ey pool size result in smll vlue of proility (p tt two sensors sre eys (see ( in Section II-B.. Our studies sow tt te smller te vlue of p, te iger te numer of ops required to set up pirwise eys (A detiled nlysis is given in Section V. Anlyses presented in [6] nd [8] provide communiction overed in te PKE pse for up to 3 ops. Due to te restrictions mentioned ove, generl mtemticl model to study te communiction overed for te PKE pse is required. In tis pper, we propose proility model to nlyze

2 communiction overed requirements for te PKE pse in RKP scemes. Unlie te PKE sceme proposed in [8], our model is sed on te PKE sceme were sensors set up pirwise eys using only teir neigor nodes. Tis design significntly reduces te communiction overed involved in te PKE pse. Similr to te recent scemes in [], [6], [5], [8], [] nd [4], our model is sed on networs wit uniformly distriuted sensors. Our model pplies te inomil proility distriution nd modified inomil proility distriution (presented in Section III-B in op-y-op fsion. Tere re tree input prmeters to our model: proility of two sensors sring eys, verge numer of neigors of sensor, nd 3 proility tt two neigors of node sre eys nd re locted witin ec oter s communiction rnge. Our model cn e used for evluting te frction of neigors of sensor to wic it cn communicte securely. Furtermore, te model derives te proility tt for given sensor networ configurtion, every sensor cn securely communicte wit ll its neigors. We ten vlidte our model troug our proposed vlidtion procedure. Finlly, we use our model to nlyze te communiction overed in te PKE pse. Te rest of te pper is orgnized s follows: In section II, we provide necessry cground nd relted wor in te re of RKP scemes. Section III descries te proposed proility model for pirwise ey estlisment nd ey-grp connectivity. Section IV presents te vlidtion metodology nd results of our proposed proility model. Communiction overed nlysis of using our proposed model is given is Section V. Section VI summrizes te wor. II. BACKGROUND OF RANDOM KEY PREDISTRIBUTION SCHEMES Te gol [] of RKP scemes is to reduce te numer of preinstlled eys in ec sensor considering unnown postdeployment geogrpic configurtion. Te preinstlled eys cn elp te sensor to set up pirwise eys wit its neigors. Sensors locted witin sensor s communiction rnge re clled neigors of tt sensor. In tis pper, we use terms sensor nd node intercngely. In tis section, we first list pses for pirwise ey setup in RKP scemes. We ten provide generl mtemticl cground of RKP scemes in literture. Finlly, we present relted wor. A. Te Pses In Rndom Key Predistriution Scemes Four min pses for ey setup in RKP scemes re presented s follows:. Key predistriution pse: A centrlized ey server genertes lrge ey pool offline. Te ey server is ssumed to e protected nd no dversry cn re into te server to revel te eys. Te procedure for offline ey distriution is s follows: Generte lrge ey pool of size P. Rndomly select m different eys for ec sensor from te ey pool to form ey ring. c Lod te ey ring into te memory of te sensor. d Assign unique node identifier or ey ring identifer to ec sensor.. Sensor deployment pse: Sensors re rndomly piced nd uniformly distriuted in lrge re. Typiclly, te verge numer of neigors of sensor (n is muc smller tn te totl numer of deployed sensors (n. 3. Key discovery pse: Two steps re involved in te ey discovery pse. In te first step, ec sensor ttempts to discover sred ey(s wit ec of its neigors. To ccomplis tis, te sensor cn rodcst its ey ring identifier to its neigors. Te sensor cn lso use te secret discovery protocol specified in [], [7], [7] to discover te sred ey witout using clertext rodcst. After te first step of te ey discovery pse, te sensor nows ll its neigors. Te set of ll neigors of sensor i is represented y W i nd W i = n. Te set of neigors of sensor i wo sre t lest one ey wit te sensor i is represented y Q i. Te set of neigors of sensor i wo do not sre ny ey wit sensor i is represented y R i. Tus, we ve W i = Q i R i nd Q i + R i = n. In te second step, every sensor i rodcsts its set Q i. Using te sets received from neigors, sensor cn uild ey grp (see Definition sed on te ey-sre reltions mong neigors. 4. Pirwise ey estlisment pse: If sensor i sres t lest one ey wit given neigor ( neigor in set Q i, te sred ey(s cn e used s teir pirwise ey(s. For ec of te neigors in set R i, sensor i uses te ey grp uilt during ey discovery pse to find ey pt (see Definition vi te neigors in set Q i to set up pirwise ey. Once pirwise ey is set up wit neigor in set R i, te neigor is included in set Q i nd deleted from set R i. Te ove PKE procedure cn e cieve y using source-routing sed pirwise ey PKE protocol [3]. Te gol of te PKE pse for sensor i is to set up pirwise eys wit its neigors in set R i nd stisfy f = Qi W i c, were c is te frction of te totl numer of neigors of sensor i tt re required to e reced. Definition (Key grp: A ey grp mintined y node i is defined s G i = (V i, E i were, V i = {j j W i j = i}, E i = {e j j, V i W j j W js}, S is reltion defined etween two nodes if tey sre t lest one ey fter te ey discovery pse. Definition (Key pt: A ey pt etween node A nd B is defined s sequence of nodes A, N, N,..., N j, B, suc tt, ec pir of nodes (A, N, (N, N,..., (N j, N j, (N j, B s t lest one sred ey fter te ey discovery pse. Te lengt of te ey pt is te numer of pirs of nodes in it. B. Mtemticl Foundtions of Rndom Key Predistriution Sceme An importnt proility (p is te proility tt two nodes sre t lest one ey fter te ey predistriution pse []. We first stte tis result ere for completeness. We ten

3 3 TABLE I COMPARISON OF EXISTING RKP SCHEMES Bsic sceme[] q-composite[6] Grid-sed[5] s-rkp[8], [5] -pt[6], [] Key pool (P unstructured unstructured structured structured unstructured Key selection (m RWR RWR restricted rndom restricted rndom RWR Sred-ey discovery CB/PSD CB/PSD CB/PSD CB/PSD CB/PSD Numer of ey pts one-pt one-pt one-pt one-pt -pt Communiction overed nlysis n/ prtil n/ prtil n/ RWR: rndom witout replcement. : Tey only present te nlysis for ey pt lengt nd 3. CB/PSD: cler-text rodcsting or privte sred-ey discovery. give two pproces proposed in te literture tt re used for computing te ey grp connectivity. Te proility tt two nodes sre t lest one ey (p : Given ey pool of size P nd tt ec sensor is loded wit rndomly selected m different eys from te ey pool, te proility tt two sensors sre t lest one ey is given s (for proof, see []: ( P P m p = m( m ((P m! = (P m!p!, ( ( P m (were P > m. Rndom grp pproc (RGA : An pproximtion metod to compute ey grp connectivity is proposed y Escenuer nd Gligor []. Teir metod utilizes te rndom grp teory []. Given desired proility of te grp connectivity P c = e e c, p l is te glol two-node connectivity proility tt lin exists etween ny two nodes: p l = ln(n n + c n, were n is te totl numer of sensors in te system. Using p l, we cn derive te locl two-node connectivity proility (p y mplifying p l wit fctor (n /n. Here, p is te estimted proility tt node sres ey wit ny of its neigors, were n is te verge size of its neigorood. p = n n p l. Note tt in [8], te utors define te P c s te proility for glol ey grp connectivity nd te p s te proility for te locl ey grp connectivity. Te RGA uses te proility of glol ey grp connectivity to estimte te locl ey grp connectivity. Tis pproc does not provide te ey pt lengt informtion of ey grp tt cn e useful in design of pirwise ey estlisment scemes. Moreover it estimtes te ey grp connectivity nd produces inconsistent results wen te neigorood size n is reltively smll (see our vlidtion results in Section IV-E. 3 Are covering pproc: Du et l. [8] uses te re covering pproc to nlyze te proility tt node cn set up pirwise eys wit ny of its neigors. Tey clculte te two-node connectivity proility s function of te overlpped rnge sred y sensor wit its neigors. During te PKE pse, te intermedite nodes my not necessrily e locted witin te source node s communiction rnge. Tus, te sensor cnnot determine ey pts to set up pirwise eys wit its neigors. In oter words, te ey setuequests must e flooded insted of using cosen pts. Consequently, te communiction overed invoed y te pirwise ey requests is proiitively ig. Moreover, te re covering pproc is sed on te nlysis of geogrpicl loctions of nodes on ll possile ey pts. For te pt lengt greter tn 3, te nlyses of te node loctions nd te grpicl representtions re very complicted. In teir wor in [8], te utors only present te nlyses for ey pts wit lengts nd 3. C. Relted Wor Te first RKP sceme ws proposed y Escenuer nd Gligor [], nd we cll it te sic sceme. Te proposls tt followed re ll sed on te sic sceme nd tey propose improvements in terms of security. Te proposed improvements focus on tree spects: ey pool structure [8] [5], ey selection tresold [6], nd pt-ey estlisment protocol [6] []. Tle I sows five recent proposls nd te sic sceme. Cn et l. [6] proposed te q-composite sceme. In tis sceme, te ey selection tresold is set to q. To form secure lin, te sceme requires t lest q sred eys etween two nodes. Te structured ey pool sceme (s-rkp [8] proposed y Du et l. nd Grid-sed sceme [5] proposed y Liu nd Ning cnge te unstructured ey pool to structured ey pool. Te structured ey pool is formed y multiple ey spces. Witin ec ey spce, te ey spce structure uses te group ey sceme proposed y Blom [] nd furter developed y Blundo et l. [3]. Bot q-composite sceme nd -pt sceme [] use ey pts ( to setup pirwise ey. Te -pt sceme uses secret sring sceme to setup pirwise ey. In ll presented scemes, during te ey discovery pse, ot te cler-text rodcst discovery nd te privte sre-ey discovery sceme re specified. Clerly, te privte sre-ey discovery pproc involves more communiction overed. It my e noted tt te ey-grp connectivity prolem we ve considered ere s some similrity to glol networ connectivity prolems, i.e., te connectivity of n entire In teir proposed sceme, pirwise ey cn e derived y exclusive-or opertions on secret sres received from pts. Specified in [], using privte sre-ey discovery, for every ey on ey ring, ec node could rodcst of list α, E Ki (α, i =,...,, were α is cllenge. Te encryption of E Ki (α wit te proper ey y recipient would revel te cllenge α nd estlis sred ey wit te rodcsting node.

4 4 TABLE II NOTATIONS i numer of nodes selected on op i n Totl numer of sensors in te networ n Te verge numer of sensors witin sensor s rnge p Proility tt two sensors sre t lest one ey efore te PKE pse. p p = p p c Te proility tt ny two neigors of sensor re witin ec oter s rnge p p = p p c p p = p Te proility tt sensor cn set up pirwise ey wit given neigor wit exctly ops ( Te proility tt sensor cn set up pirwise ey wit given neigor witin ops p con Te proility tt sensor cn set up pirwise eys wit ll of its neigors witin ops networ; compreensive study on grp connectivity cn e found in [8]. Te recent wor y Xue nd Kumr [] ddresses te connectivity of wireless networs y inspecting te miniml numer of neigors in order to cieve glol connectivity. Note tt ot tese wors cieve networ connectivity of te entire grp. In our cse, te ey-grp connectivity is from te point of view of ec individul sensor (node wen glol nowledge/connectivity is not nown/possile due to limited storge nd communiction ility t ec sensor. In our sensor networ model, two neigoring sensors must e pysiclly visile vi direct wireless lin in order to set up direct ey in teir ey grps; in oter words, te wireless lins outside of sensor s communiction rnge re considered to e invisile for tt sensor. Consequently, given te storge nd communiction restrictions, we study te following prolem: te proility tt node cn estlis ey wit one or ll of neigors witin -op visile ey pt(s. III. PROBABILITY MODEL FOR PAIRWISE KEY ESTABLISHMENT In tis section, we nlyze te PKE pse of RKP sceme for lrge numer of sensors uniformly distriuted witin vst -dimensionl re. Te uniform distriution of sensors ws introduced in [] nd extensively utilized in mny proposls, suc s [6], [5] nd [8]. We derive ere te proility tt node cn rec ny of its neigors wit exctly ops nd te proility tt node cn rec ll of its neigors witin ops. Nottions used for tis wor re summrized in Tle II. A. Computing p c nd p In order to model te proility tt sensor cn set up pirwise ey wit ny of its neigors wit exctly ops, we first determine te proility tt ny two neigors, sy j nd, of sensor, sy i, re witin ec oter s rnge tis proility is denoted y p c. Our nlyticl pproc requires result on expected re of overlegion, A(x, given in [6]. In order to review tis result, consider Fig. ; we cn drw two circles wit centers s sensor node i nd j nd ec wit communiction rdius r resemling te rnge of sensors. Te Fig.. i r x c Overlpped Region Between Two Sensor Nodes. distnce etween i nd j is x. Te cumultive distriution function for te distnce etween node nd one of its neigors is given y F(x = Pr(distnce x = x /r. Tus, te proility density function is f(x = F (x = x/r. Te re of te overlpped region cd is: A cd (x = r cos ( x r x r x 4. Te expected re of te overlpped region is given y [6]: A(x = r j A cd (xf(x = (π r =.5865πr. As sown in Fig., node must e locted witin te sded region cd in order to e in node i s nd node j s rnge simultneously. On n verge, p c is te rtio of te sded re to te wole re of te circle. Using te result for A(x, we cn ten determine p c s d p c =.5865 πr /(πr = ( Furtermore, given node i nd its rnge, te proility tt ny two nodes witin te rnge of node i sre ey nd re in ec oter rnge is given s: p = p c p. (3 In our cse, p =.5865 p due to (. It my e noted tt p is computed sed on te ssumption tt every sensor s circulr communiction rnge wit equl rdius. Oter mecnisms cn e devised to find p for different configurtions. However, ny different mecnism will not ffect our proility model introduced in Section III-B nd Section III-C. B. Computing Te sic ide eind our pproc is te proility of node selection on ec op. Te selection follows te inomil distriution or modified inomil distriution. Te inomil proility mss function for given n nd p is represented s: f(n, p = ( n p ( p n, =,...,n. Te modified inomil proility mss function is represented s follows: ( n f(n, p, p = p ( p n, =,...,n (4 were proilities p nd p need not e te sme.

5 5 stge A stge C stge B su-stge B stge A stge C su-stge B - - stge C = - - n'- n'- nodes n'- - n'- nodes n'- nodes ( = ( = (c 3 Fig.. Node selection for computing. Fig. gives grpicl view of our pproc. Node is sensor tt wnts to set up pirwise ey wit one of its neigors, sy node. s rnge contins n nodes including te node. is te proility tt node cn set up pirwise ey wit given neigor wit exctly ops. As sown in Fig., our eqution is derived in tree stges. Stge C lwys represents te finl stge tt node cn rec node ; stge B represents ll te intermedite ops wen 3; nd stge A represents te first op wen. For =, s sown in Fig. (, tere is no intermedite node. Terefore, p is te proility tt node sres t lest one ey wit node. If node nd node sre ey(s, te sme ey(s cn e used s pirwise ey(s nd no dditionl ey setup is required. Oterwise, tey cnnot set up pirwise ey(s directly nd must go troug te PKE pse to set up pirwise ey(s. Tus, we ve: ( = p. (5 Te cse for = is sown in Fig. (. In stge A, we select nodes from n nodes s te first op nodes wo sre t lest one ey wit node, were =,...,n. Since our gol is to derive te proility tt node is recle wit exctly ops, once node is fixed, we cn only select t most n nodes s te first op nodes. We ( n (p ( p n for n now ve te eqution p = stge A. Tis eqution cn e interpreted s follows: out of n nodes re selected for te first op; te proility tt nodes sre eys wit node is (p ; te proility tt n nodes do not sre eys wit node is ( p n ; lso, in ec selection, we ve te condition tt node does not sre eys wit node (represented y p = p. Following stge A, is te stge C in te second op. Any selected node in stge A my sre eys wit node. Tus te proility tt t lest one of te selected nodes in stge A sres ey wit node is given s: p, in wic p mens tt ll selected nodes in stge A do not sre ey wit node. It my e noted tt te proility tt node selected in stge A sres eys wit node is p, s derived in Section III-A. Tus, we ve, n ( n ( = p (p ( p n ( p. (6 = For = 3, s sown in Fig. (c, tere exists tree stges. In stge A, te mtemticl expression is similr to te expression we presented for =. One difference ( n (p p ( p n is n is tt eqution p = cumultive modified inomil distriution function multiplied y p (using (4, p = p p, p = p. Te expression (p p stnds for te condition tt none of te nodes selected for te first op sre eys wit node nd node simultneously. Since tere must e t lest one node ville on op, te mximum numer of nodes tt cn e selected from n cndidte nodes is n. In stge B (only sustge B exist wen = 3, te formul is cumultive inomil distriution function tt nodes re selected from te remining nodes (n nodes fter te selection for te first op nd ec of te nodes sres t lest one ey wit t lest one of te nodes. Te expression for stge B is ( n ( p ( p n. given s: n = All te nodes selected for tis op re eligile to connect to node. Te expression of stge C is p, wic mens tt t lest one of te nodes sres ey wit node. Using te expressions of ll te stges, we derive te following eqution for = 3: (3 = n p = n = ( n (p p ( p n. ( n ( p ( p n ( p. (7 For 4, tere exists tree stges; see Fig. (c. Te nlysis of stge A is te sme s tt for = 3. We ve te expression p n + ( n = (p p ( p n for stge A. Tere re two su-stges in stge B, we denote tem s B nd B. Te expression of sustge B represents te formuls from op to op. It represents itertive process tt is used on ec op. In oter words, on op i, i nodes re selected from te left over nodes of previous i ops. For exmple, on op, we select nodes from n nodes. We now sow induction s follows: on op i, we cn select i nodes from n i j= j nodes. Following (4, p = ( p i p sows te proility of selection of

6 6 stge C stge A stge C stge A stge B stge C - - n' nodes ( = n' nodes ( = n' nodes (c 3 Fig. 3. Node selection for computing p con. node tt sres t lest one ey wit t lest one of te i nodes in te op i nd tis selected node does not sre ey(s wit node ; p = p i represents te proility tt node does not sre ny ey wit i nodes in te op i. For ec op i, te vlue of i s to e less tn n + i i j= j to gurntee tt t lest one node is ville t ec of te ops from i + to. Te su-stge B represents te op. As discussed in te nlysis of (3, te nodes selected for op re ll eligile to sre ey wit node. Te nlysis of stge C is te sme s tt of stge C for = 3. Tus, we rrive t: ( 4 + n = p = i= ( n (p p ( p n i X n + i j i= j= i X n i j= j [( p i ] i ( p p i n X n j j= = X n j= j ( ( n p p ix j j= X j j= ( p. (8 Te proility tt node cn rec ny of its neigors witin ops is represented s (. Te formul is ten given s: ( = (i, (9 i= were (i is s derived for different vlues of i erlier in tis section. Note tt (9 is not dependent on rdius r nor on te totl numer of sensors n; it is dependent on n, p, nd p. C. Key Grp Connectivity In tis susection, we study te ey grp connectivity of given sensor networ. A grp is sid to e connected if ny two of its vertices cn e joined y pt, disconnected oterwise [4]. We sy tt ey grp, G i, is -op-connected t vertex i if vertex i cn rec ny oter vertex of te ey grp wit pt no more tn ops in lengt. We now derive te proility tt ey grp, G, mintined y node is -op-connected. Our eqution derivtion includes tree stges s sown in Fig. 3. Tere re n nodes witin s rnge. p con is te proility tt te ey grp G is -op-connected. Stge C lwys represents te finl op in wic node cn rec ll its neigors; stge B represents ll te intermedite ops wen 3; nd stge A represents te first op wen. We first present te formul for = (see Fig. 3(. It is esy to see tt p con ( is: p con ( = (p n. ( For =, s sown in Fig. 3(, tere re two stges: te first stge A nd te finl stge C. In stge A, out of n nodes ( n re selected for te first op. Tis is inomil proility mss function wit proility p. Tus, we ve te inomil proility distriution n ( n = p p (n to represent te first op. Unlie in expressions for (, te vlue of for p con ( cn e n. In stge C, te proility tt ec of n nodes sres t lest one ey wit t lest one of te nodes is given y p. Ten, te proility of connecting to ll n nodes on op is ( p n. Te p con ( is given s follows: p con ( = n = ( n p ( p n ( p n. ( For 3, s sown in Fig. 3(c, tere re tree stges. Te nlysis nd mtemticl expression of stge A re te sme s tt for =. Tere re ops in te stge B. For op, we select nodes from n nodes tt sre eys wit nodes in op. Now, we cn do induction s follows: in op i, we cn select i nodes from n i j= j nodes; it is inomil proility mss function wit te proility p i tt ec of i op nodes sres t lest one ey wit t lest one of te i op nodes nd n i j= j nodes do not sre ey(s wit i op nodes. Ten, we

7 7.9.9 Accumulte Poisson Distriution Accumulted Norml Distriution Numer of neigors (n Communiction rdius (r Fig. 4. Cumultive Poisson Distriution, r =. Te verge numer of neigors of sensor is n = ρπr = 5. Fig. 5. δ =. Cumultive Norml Distriution, r =. δ =.5r =.5 nd conclude tt te stge B is sequence of cumultive inomil distriution functions nd ec successive function depends on te previous op nodes selection. Te nlysis of stge C is te sme s tt for =. Ten, we ve te following expression for 3, p con ( 3 n ( n = p ( p n = i= i X n j j= i= ( p i n n i X j= ix j= i j ( p i i j ( X n j p j=. ( In order to deploy sensor system using RKP scemes, we first select different vlues of p nd plug tem into (-( to find out suitle vlue of p to cieve te required vlue of connectivity proility p con. Once p is found, we cn pply ( to select te proper P (ey pool size nd m (te numer of eys to e preinstlled in sensor. IV. VALIDATION METHODOLOGY AND RESULTS In (3 (, we ve te following ssumptions: ec sensor cn communicte directly wit n neigors, were n is te verge numer of neigors of sensor, ec sensor s te sme communiction rdius r, nd 3 p is fixed vlue. Here, troug systemtic pproc, we sow tt our mtemticl derivtions sed on ove ssumptions re sound pproximtions. We consider following two prmeters in our vlidtion: te numer of neigors of sensor, nd te communiction rdius of sensor. We lso consider two different distriutions: sensors re uniformly distriuted witin vst re, nd sensor s communiction rdius follows norml distriution wit men r nd stndrd devition δ or uniform distriution witin te intervl [r δ, r + δ]. A. Numer of Neigors of Sensor We consider A to e lrge sensor deployment re, i.e., A π(r were r is te verge trnsmission rnge of sensor nd A = π(r. We define ρ = n/a is te sensor deployment density. Te proility tt node is plced witin re A is p = A /A. Proility, p(x, tt x of n nodes re plced in te re A is: ( n p(x = n = n p n ( p n n. (3 Wen n nd A A, we cn pproximte tis solution wit Poisson distriution [9]: p(x = n (npn n e np! (n A A n A n n e A! (ρa n n! e ρa (ρπr n n e ρπr. (4! Te cumultive poisson distriution function P(X is: P(X = e ρπr X n = Te verge numer of neigors of sensor is: (ρπr n n. (5! n = ρπr. (6 In order to ssign numer of neigors of sensor, we cn simply mp uniform distriution in te rnge of [, ] to te cumultive poisson distriution. For exmple, for ec sensor, we rndomly select numer witin te rnge [, ] (y-xis in Fig. 4; sed on te istogrm sown Fig. 4, we cn find its corresponding x-coordinte, wic is te numer of neigors ssigned to te sensor.

8 8 c r r r j x i c r r r i x j d i r r r x c r j d d r r, r r r r r, r r Fig. 6. Coverge re wit different communiction rdius. B. Generting p We ssume tt sensors ve verge communiction rdius r wen tey re sipped out of fctories. We consider ot norml distriution nd uniform distriution to model te communiction rdius of sensor. We use men r nd stndrd devition δ =.5r. We cn esily derive te proportion of distriution tt is elow given numer of stndrd devitions from te men. It cn e sown tt only.3% of te popultion will e less tn or equl to vlue two stndrd devitions elow te men. Similrly, te sme portion tt is ove te men. Using te similr mpping tecnique, we cn drw te cumultive norml distriution function for sensor s communiction rdius. For exmple, sown in Figure 5, we cn rndomly select vlue from te y- xis (witin te intervl of dsed lines; ten we cn ssign te rdius of sensor from te x-xis to sensor. Note tt te vlue selected for sensor is witin δ from te men r. In te cse of uniform distriution, we rndomly select rdius witin te rnge [r δ, r + δ] for ec sensor. In rel world, te communiction rdii of sensors re of vrying lengt. Fig. 6 sows te coverge re of sensors in tree different scenrios. Sensor i (wit communiction rdius r s two neigors j nd wit communiction rdii r nd r, respectively: were r r, r, r r r (te nlysis of te scenrio r r r is te sme, nd 3 r, r r. In ll our following nlysis, we ssume r r. In order to set up pirwise eys, two sensors must e locted witin ec oter s communiction rnge. In Fig. 6, te sde re is te intersected coverge re etween two sensors wit smller communiction rdii. It is esy to prove tt, in ll tree scenrios, te intersected coverge re used to compute p c is te intersected re of two sensors wit less communiction rdii. We note tt, in te scenrio r, r r sown in Fig. 6, if is locted in te re etween te dsed circle (centered y node i wit rdius r nd te circle centered y i (wit rdius r, node will not consider i s its neigor. Tus node must locted witin te sde re. Te re cd sown in Fig. 6 is: A cd (x = cos ( γ + x γ xγ + cos ( γ + x γ xγ γ γ S(S x(s γ (S γ, (7 were S = (x+γ +γ /, γ nd γ represent two of te less communiction rdii of r, r, nd r (we ssume γ γ. In Section III-A, we presented te cumultive distriution function for te distnce etween te node nd one of its neigors wit F(x = Pr(distnce x = x /r, were we ssume ec node s te sme communiction rdius r. Te proility density function is f(x = df(x/dx = x/r. For te sensor networ wit different communiction rdii, we ve te proility density function f(x = df(x/dx = x/γ. Tus, te expected coverge re cd is given y: A(x = γ Ten, we cn compute p c s follows: nd, p is computed using (3. C. Vlidtion Procedure A cd (xf(xdx. (8 p c = A(x/πr (9 Our im ere is to vlidte te soundness of proility models nd p con derived erlier in Section III. We use te following procedure to estimte nd p con to compre ginst te models presented in Section III:. Select n to e verge numer of neigors for sensor.. For te verge communiction rdius of sensors, r, use te exponentil mpping metod presented in Section IV-A to find te numer of neigors of sensor. 3. Select distriution for ec neigor, nd use te metod presented in Section IV-B to ssign rdius for ec sensor. 4. Use (7 (9 to compute p c for ec pir of neigors, nd ten use (3 to derive te proility p for te sme. Bsed on te derived vlue of p, ssign pirwise eys for ec pir of neigors. 5. Compute nd p con sed on te steps to 4 descried ove. Te ove procedure is run 3, times for ec distriution selected in step-3, nd ten te verge vlue for nd p con is otined over te 3, runs; note tt we consider uniform nd norml distriution seprtely for tis step.

9 9 Rdius r follows Uniform distriution rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =,p =..3 rdius= rdius=3. rdius=5 teoreticl rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =3,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =3,p =..6 rdius=.4 rdius=3 rdius=5 teoreticl rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =,p =.3 Numer of ops (n =3,p =.3 Numer of ops (n =5,p =.3 ( n = ( n = 3 (c n = 5 Rdius r follows Norml distriution rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =,p =..3 rdius= rdius=3. rdius=5 teoreticl rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =3,p =. rdius= rdius=3 rdius=5 teoreticl Numer of ops (n =3,p =..6 rdius=.4 rdius=3 rdius=5 teoreticl rdius= rdius=3 rdius=5 teoreticl p r Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl p r Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl p r Numer of ops (n =,p =.3 Numer of ops (n =3,p =.3 Numer of ops (n =5,p =.3 (d n = (e n = 3 (f n = 5 Rdius r follows Uniform distriution.5 x rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =,p =. rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =,p =. Numer of ops (n =,p =.3 (g n = rdius= rdius=3 rdius=5 teoreticl p con rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =3,p =. rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =3,p =. Numer of ops (n =3,p =.3 n = 3 rdius= rdius=3 rdius=5 teoreticl p con rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p =.3 (i n = 5 Rdius r follows Norml distriution.5 x 3.5 rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =,p = rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =3,p =...5 rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p = rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =,p =. Numer of ops (n =,p =.3 rdius= rdius=3 rdius=5 teoreticl p con rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =3,p =. Numer of ops (n =3,p =.3 rdius= rdius=3 rdius=5 teoreticl p con rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p =. rdius= rdius=3 rdius=5 teoreticl p con Numer of ops (n =5,p =.3 (j n = ( n = 3 (l n = 5 Fig. 7. n =, 3, 5, p =.,.,.3, rdius =,, 3. (-(f: vlidte ; (g-(l: vlidte p con.

10 TABLE III THE CONNECTIVITY PROBABILITY FOR ey grp: ( METHOD VS. D. Vlidtion Results p con METHOD VS. RGA METHOD ( p con RGA n (p (p (p : = 5 nd ( 5 >.99999, : = 5 nd p con (5 >.99999, : P c =.99999, n =. In Fig. 7, we plot computed verge vlue from te vlidtion procedure nd te teoreticl result derived from our proposed proility models (see su-figures ( (f nd p con (see su-figures (g (l. In Fig. 7, ec sufigure sows tree scenrios wit p =.,.,.3 for different vlues of n =, 3, 5 were ec plot considers tree vlues or rdius t, 3, 5 to compre ginst te teoreticl result. We ssume tt sensor s communiction rdius follows eiter uniform distriution wit te rnge r δ r + δ, or norml distriution wit men r nd stndrd devition δ in our vlidtion process. Our results sow tt te proposed proility models fit te metod used for vlidtion. Reders migt notice te gps etween te simultion results nd teoreticl results in figures (g (l. Note owever tt te evluted proilities (see y-xis re very smll; tus, tis difference s less prcticl significnce, nd our nlyticl model cn e considered to e very ccurte. E. Comprison of ey grp connectivity metods Next, we compre ey grp connectivity. We cn find te ey grp connectivity proility y using ny of te following tree metods: Using te proility tt node connects to ny of its neigors witin ops (see (9, we cn derive te frction of neigors of node for wic te pirwise eys cn e set up witin ops. Tis pproc only provides te numer of neigors (cn e derived y n tt connect to node witin ops. Wen, node cn set up pirwise eys wit prcticlly ll its neigors, ten we cn sy te ey grp is connected. Te second metod is s descried in Section III-C (see (-(. To strt wit, tis metod only considers directly recle nodes nd pirwise-ey sring reltions mong sensors witout considering te geogrpicl loction of ec sensor, wic significntly reduces te nlyticl complexity. In ddition, it provides te ey pt lengt informtion wic is vlule for sensor networ designers to evlute or design sensor system. Te tird evlution metod, RGA, is s descried in Section II-B. wic is y []. In Tle III we compre te tree pproces. Note tt te RGA metod requires te totl numer of sensors (n wile our pproc does not. We find tt te RGA metod my produce inconsistent result, especilly wen te neigorood size is reltively smll, for exmple, p > wen n. In ddition, compred to p con metod, te RGA metod requires iger p vlue wen n < 4 nd lower p vlue wen n > 4. Since our model s een vlidted vi simultion (see Section IV-D, our comprtive results in Tle III sow tt te RGA metod will result in inccurte prmeter settings in sensors. F. Using Connectivity or ( to Deploy Sensor Networs Our model llows to nswer questions suc s Cn our proposed nlyticl model elp sensor networ designers to deploy sensor networs nd evlute rndom ey predistriution (RKP sceme wit given storge constrints nd networ configurtions?. We riefly illustrte tis spect. For instnce, we migt ve te following requirements: Deploy uniformly-deployed sensor networ suc tt ec node cn estlis pirwise eys, witin 5 ops, to % of ll its neigoring sensors. Tis requirement cn e trnslted to te following prmeters used in our model, i.e., p con = nd = 5. Once p con is fixed, we need to determine te importnt RKP prmeter, p. Recll tt te selection of p is determined y evluting Eqution (. Since p con nd re nown, tere re tree vriles in Eqution (; tey re n, p, nd p. From Eqution (3, we now tt p is function of p. Tus, te prolem trnsltes to solving Eqution ( in order to determine unnowns n nd p. Reltions etween different vlues of nd p so determined cn e found in Tle III. For exmple, if we coose n = 3, ten p =.649. In fct, we cn crete multiple similr tles wit different connectivity requirements. Te next step is to determine RPK sceme prmeters. We coose tree RKP scemes for our nlysis. Tey re: sic sceme, q-composite sceme, nd s-rkp sceme (see Tle I. Te mtemticl representtions of p for ec RKP sceme re given in Section II-B. nd in Appendices A nd B. For dditionl informtion out tese scemes, refer to te pulictions [], [6], nd ([5], [8]. Tus, we cn use equtions (, (, nd ( to determine te numer of eys to e instlled in ec sensor nd te size of te ey pool. Bsed on te ove discussion, it is esy to see tt our nlyticl model provides nice mtemticl representtion for ey-grp connectivity for given numer of ops. Tis model cn elp networ designers to evlute te communiction overed (restricted y te numer of ops, i.e., wit te considertions of storge cpility of sensors (i.e., te vlue of m. n

11 TABLE IV COMMUNICATION OVERHEAD ANALYSIS FOR ( AND ( 8.5 (i Weigt (i Weigt (i Weigt (i Weigt Hop (i n = (% n = 3 (% n = 5 (% n = 7 (% ( ( V. COMMUNICATION OVERHEAD ANALYSIS In tis section, we use our proposed proility model to nlyze te communiction overed involved in te PKE pse. We nlyze te required numer of ops to set up pirwise ey nd te communiction overed for PKE distriuted on ec op. Since sensors estlis pirwise eys vi only te neigors nodes in our model, our nlysis is sed on te verge numer of neigors (n nd is independent of te size of te sensor networ (n. Te numer of neigors (n of sensor is usully less tn. For our computtions, wen (.99999, we ssume tt sensor cn set up pirwise eys wit ll its neigors witin ops. We refer to te condition ( p s p-frction -connected condition. A specil cse wen ( is referred to s strong -connected condition. In Tle IV, we sow proility under te conditions strong 4-connected nd.5-frction 8- connected. It my e noted tt te communiction overed during pirwise ey estlisment for strong -connected sensor networs is minly distriuted witin first 3 ops. As sown in Fig. 8, wit increses in te neigorood size, te pirwise ey estlisment communiction overed sifts from op to op nd drops drmticlly on op 3. On te oter nd, te pirwise ey estlisment communiction overed for.5-frction -connected sensor networ is distriuted in more tn 3 ops. Fig. 9 sows tt te proility curves ecome fltter wit increse in te neigorood size nd te proility pes sift to iger op numers. We define te communiction weigt on op i s (i/ (. We notice tt te igest communiction weigt for ec curve lso sifts to iger op numer wit increse in te neigorood size. Tus, we summrize our findings s follows: For strong -connected ( (.99999: te pirwise ey estlisment communiction overed is minly distriuted witin first tree ops. te pirwise ey estlisment communiction overed sifts from te first op to te second op wen n increses ( n. Proility tt two nodes sre ey(s (p r Communiction weigt for ec op (% (<=3>= n = n =3 n =5 n =7 3 Numer of ops Numer of ops n = n =3 n =5 n =7 Fig. 8. Sensor networ ey estlisment communiction overed distriution for ( For p-frction -connected ( < (.99999: For given n nd mximum numer of ops, wen ( decreses, te pirwise ey estlisment communiction overed sifts from lower numer ops to iger numer ops, te vlue of te pe proility decreses nd sifts to iger numer op, te proility curve ecomes fltter, nd p (= ( decreses (see te cnges from Fig. 8 to Fig. 9. For given ( nd te mximum numer of ops, wen n increses, te pirwise ey estlisment communiction overed sifts from lower numer ops to iger numer ops, te vlue of te pe proility decreses, te proility curve ecomes fltter, nd te p (= ( decreses (see Fig From findings nd, we me following oservtions: for fixed n nd te mximum numer of op, te decreses of ( cuses decrese in p ; for fixed given (, te increses in n cuses decrese in p ; te increse in te mximum numer of llowed ops lso cuses decrese in p. Tese oservtions for

12 Proility tt two nodes sre ey(s (p r Communiction weigt for ec op (% (<=8>=.5 n = n =3 n =5 n =7 Numer of ops Numer of ops n = n =3 n =5 n =7 Fig. 9. Sensor networ ey estlisment communiction overed distriution for ( 8.5. te PKE pse led us to te study of trdeoff etween te communiction overed, wic is restricted y, nd te storge overed of sensor, wic is restricted y p. It my e noted from ( tt for fixed P, te smller te p te smller te m. On te oter nd, te smller te mximum numer of ops, te lesser te communiction overed involved. Te exct trdeoff study etween p nd for different scenrios requires dditionl nlysis wic is out of scope of tis pper. VI. SUMMARY In tis pper, we ve derived two nlyticl proility models for lrge-scle sensor networs to nlyze te PKE pse for RKP scemes. Troug vlidtion procedure, we sow te roustness of our mdoels. Our models cn elp designers to nlyze te PKE pse for RKP scemes in following wys: to study te ey grp connectivity, wic in turn elps to determine te numer of eys to e preinstlled in ec sensor nd te rnge of sensor, nd te pirwise ey estlisment pt lengt cn elp in determining te communiction overed during pirwise ey estlisment nd ten evlute if designed te PKE sceme fulfils te energy consumption requirements for sensor. Te softwre for te nlyticl models nd te vlidtion process re ville t ttp:// dmedi/softwre/. REFERENCES [] I. F. Ayildiz, W. Su, Y. Snrsurmnim, nd E. Cyirci, A survey on sensor networs, IEEE Communictions Mgzine, vol. 4, pp. 4, August. [] R. Blom, An optiml clss of symmetric ey genertion systems, in EUROCRYPT 84, ser. Lecture Notes in Computer Science, vol. 9. Pris, Frnce: Springer-Verlg, 985, pp [3] C. Blundo, A. D. Sntis, A. Herzerg, S. Kutten, U. Vccro, nd M. Yung, Perfectly-secure ey distriution for dynmic conferences, Informtion nd Computtion, vol. 46, no., pp. 3, 998. [4] B. Bolloás, Modern Grp Teory. Springer-Verlg, 998. [5] D. W. Crmn, P. S. Kruus, nd B. J. Mtt, Constrints nd pproces for distriuted sensor networ security, NAI L, Tec. Rep., Septemer. [6] H. Cn, A. Perrig, nd D. Song, Rndom ey predistriution scemes for sensor networs, in Proceedings of 3 Symposium on Security nd Privcy. Los Almitos, CA: IEEE Computer Society, 3, pp [7] R. Di Pietro, L. V. Mncini, nd A. Mei, Efficient nd resilient ey discovery sed on pseudo-rndom ey pre-deployment, in Proceedings of 8t Interntionl Prllel nd Distriuted Processing Symposium (IPDPS 4, April 4. [8] W. Du, J. Deng, Y. S. Hn, nd P. K. Vrsney, A pirwise ey pre-distriution sceme for wireless sensor networs, in Proceedings of t ACM Conference on Computer nd Communictions Security (CCS 3, Octoer 3, pp [9] W. Du, J. Deng, Y. S. Hn, P. Vrsney, J. Ktz, nd A. Klili, A pirwise ey pre-distriution sceme for wireless sensor networs, ccepted y te ACM Trnsctions on Informtion nd System Security, 5. [] L. Escenuer nd V. D. Gligor, A ey-mngement sceme for distriuted sensor networs, in Proceedings of 9t ACM Conference on Computer nd Communiction Security (CCS-, Novemer, pp [] V. D. Gligor nd P. Donescu, Fst encryption nd utentiction: Xcc encryption nd xec utentiction modes, in Proceedings of nd NIST Worsop on AES Modes of Opertion, August. [] J. Hill, R. Szewczy, A. Woo, S. Hollr, D. E. Culler, nd K. S. J. Pister, System rcitecture directions for networed sensors, in Proceedings of Arcitecturl Support for Progrmming Lnguges nd Operting Systems,, pp [3] D. Hung, M. Met, nd D. Medi, Source routing sed pirwise ey estlisment protocol for sensor networs, in Proceedings of 4t IEEE Interntionl Performnce Computing nd Communictions Conference, 5, pp [4] D. Hung, M. Met, D. Medi, nd H. Lein, Loction-wre ey mngement sceme for wireless sensor networs, in Proceedings of ACM Worsop on Security of Ad Hoc nd Sensor Networs (SASN 4, Octoer 4, pp [5] D. Liu nd P. Ning, Estlising pirwise eys in distriuted sensor networs, in Proceedings of t ACM Conference on Computer nd Communictions Security (CCS 3, Octoer 3, pp [6] D. Liu, P. Ning, nd R. Li, Estlising pirwise eys in distriuted sensor networs, ACM Trnsctions on Informtion nd System Security, vol. 8, no., pp. 4 77, 5. [7] M. Met, D. Hung, nd L. Hrn, RINK-RKP: A sceme for ey predistriution nd sred-ey discovery in sensor networs, in Proceedings of 4t IEEE Interntionl Performnce Computing nd Communictions Conference, 5. [8] M. D. Penrose, On -connectivity for geometric rndom grp, Rndom Structures nd Algoritms, vol. 5, no., pp , Septemer 999. [9] P. E. Pfeiffer nd D. A. Scum, Introduction to Applied Proility. New Yor: Acdemic Press, 973. [] J. H. Spencer, Te Strnge Logic of Rndom Grps (Algoritms nd Comintorics, ser.. Springer Verlg,. [] F. Xue nd P. R. Kumr, Te numer of neigors needed for connectivity of wireless networs, Wireless Networs, vol., pp. 69 8, 4. [] S. Zu, S. Xu, S. Seti, nd S. Jjodi, Estlising pir-wise eys for secure communiction in d oc networs: A proilistic pproc, in Proceedings of t IEEE Interntionl Conference on Networ Protocols (ICNP, Novemer 3. APPENDIX We riefly review ere te mtemticl cground for two RKP scemes: q-composite sceme [6] nd s-rkp sceme [5], [6], [8], [9]. A. q-composite sceme According to [6], te proility to set up secure lin requires t lest q eys. Te proility tt two nodes cn set up q eys is denote s p(q. Tus, p(q is given s: p(q = ( P q ( ( P q (m q (m q m q ( P m.

13 3 Te proility tt two nodes cn set up secure lin is: B. s-rkp sceme p (q = p(q + p(q p(m. ( Te s-rkp sceme s een independently proposed y Du t l. [8], nd Liu nd Ning [5]. In tis sceme, te ey pool P of RKP scemes is constructed y ω ey spces nd ec ey spce (i.e., ey mtrix is te structure of N su-ey-spce (i.e., n rry of eys, te size of ec ey spce is λ +. For structured ey pool, we cn determine te numer of eys (m tt pre-instlled in sensor is given y m = τ(λ +, were τ is numer of ey spces re selected for ec sensor wit τ < ω, nd λ + is numer of eys instlled in ec sensor from ec of selected ey spces. Te proility p tt two sensor nodes sre t lese one ey is: p ( ω ω τ = τ( τ = ( ω τ ((ω τ! (ω τ!ω!. ( Deep Medi is Professor of Computer Networing, Computer Science nd Electricl Engineering Deprtment t te University of Missouri-Knss City, USA. He received B.Sc. (Hons in Mtemtics from Cotton College/Guti University, Indi, M.Sc. in Mtemtics from te University of Deli, Indi, nd P.D. in Computer Sciences from te University of Wisconsin-Mdison, USA. Prior to joining UMKC in 989, e ws memer of tecnicl stff t AT&T Bell Lortories. He ws n invited visiting professor t te Tecnicl University of Denmr nd visiting reserc fellow t Lund Institute of Tecnology, Sweden. He is Fulrigt senior specilist. His reserc interests re resilient multi-lyer networ design, networ routing nd design, sensor networs. He s pulised over seventy ppers, nd is co-utor of te oo Routing, Flow, nd Cpcity Design in Communiction nd Computer Networs (4, nd te fortcoming oo Networ Routing: Algoritms, Protocols, nd Arcitectures, ot pulised y Morgn Kufmnn Pulisers. Dijing Hung (M /ACM received is B.S. degree from Beijing University of Posts & Telecommunictions, Cin 995. He received is M.S., nd P.D. degrees from te University of Missouri Knss City, in nd 4, respectively. He is n Assistnt Professor in te Computer Science & Engineering Deprtment t te Arizon Stte University. His current reserc interests re computer networing, security, nd privcy. Mnis Met is currently Senior Softwre Engineer t Tumleweed Communictions, Redwood City, CA. He erned is P.D. in Computer Science in 6 nd M.S. in Computer Science in from University of Missouri Knss City (UMKC, USA. He erned is B.E. in Computer Engineering from Mumi University, Indi in 999. His reserc interests re in Cryptogrpy, Networ Security, nd sensor networs. Appie vn de Liefvoort is Professor nd Cir of te Deprtment of Computer Science Electricl Engineering t te University of Missouri-Knss City, were e s een since 987. Prior to joining UMKC, e ws fculty memer t te University of Knss. He received grdute degrees in Computer Science nd Mtemtics from te University of Ners Lincoln, nd from te Ktoliee Universiteit in Nijmegen, te Neterlnds, respectively. His reserc interests include Queueing Teory nd performnce modeling of computer- nd communiction networs, specilizing in liner lgeric queueing teory, te mtrixexponentil distriution, nd correlted mtrix-exponentil sequences.