Ntwork Doupling for Sur Communitions in Wirlss Snsor Ntworks Wnjun Gu, Xiol Bi, Srirm Chllppn n Dong Xun Dprtmnt of Computr Sin n Enginring Th Ohio-Stt Univrsity, Columus, Ohio 43210 1277 Emil: gu, ixi, hllpp, xun @s.ohio-stt.u Astrt Sur ommunitions r highly mn y mny wirlss snsor ntwork (WSN) pplitions. Th rnom ky pr-istriution ( ) shm hs om wll pt to hiv sur ommunitions in WSNs. Howvr, u to its rnomnss in ky istriution n strong onstrint in ky pth onstrution, th shm n only ppli in highly ns ntworks, whih r not lwys fsil in prti. In this ppr, w propos mthoology ll ntwork oupling to solv this prolm. With this mthoology, wirlss snsor ntwork is oupl into logil ky-shring ntwork n physil nighorhoo ntwork, whih signifintly rlss th onstrint in ky pth onstrution of shm. W sign sur nighor stlishmnt protool (ll - ) s wll s st of link n pth pnny limintion ruls in oupl wirlss snsor ntworks. Our nlytil n simultion t monstrt th prformn nhnmnt of our solution n its ppliility in non-highly ns wirlss snsor ntworks. I. INTRODUCTION Wirlss Snsor Ntworks (WSNs) r gining wi ptn toy. A host of nw pplitions r ing rliz tht involv mny tiny wirlss snsors prforming snsing n ommunition tsks. Mny of ths pplitions r in hostil nvironmnts, n thir suss is ontingnt on prvnting th WSNs informtion from ing ssil to xtrnl mliious ttkrs. In this ppr, w rss th issu of proviing sur ommunitions in WSNs. Motivtion: A host of ky istriution thniqus hv n propos to hiv sur ommunitions in tritionl wir ntworks n wirlss ho ntworks. Howvr, thy nnot ppli in WSNs u to th uniqu hrtristis of WSNs lik ntwork sl, s of no ptur, physil onstrints in nrgy n mmory, t. For instn, th tritionl puli ky ryptogrphy [1], [2] is too nrgy onsuming to rri out y nrgy onstrin snsors. Th ky istriution ntr s shm [3] is ntrliz n not sll whn ntwork siz inrss. Othr thniqus lik using singl mstr ky for ll ommunition or stlishing uniqu pirwis kys twn h pir of nos r ithr too vulnrl unr ttk or my rquir too muh mmory, whih r ll unsuitl in WSNs. In orr to rss th ov onrns, th sminl shm s on Rnom Ky Pr-istriution ( in short) ws first propos in [4]. Eh snsor is initilly pr-istriut with smll numr of istint kys rnomly hosn from lrgr ky pool of kys. Two nos within ommunition rng of h othr (ll physil nighors) n irtly stlish pir-wis ky twn thm if thy shr t lst on pr-istriut ky. Altrntivly, two snsors n stlish pir-wis ky inirtly through ky pth trvrsing through othr snsors, with th onstrint tht ny two physilly nighoring snsors on this pth shr t lst on pr-istriut ky. For th rst of th ppr, physil nighors tht hv stlish pir-wir ky r ll sur nighors. Th shm is wily pt in WSNs u to its simpliity, low ovrh, slility n nrgy ffiiny. As suh, it hs srv s fountion for host of ky mngmnt protools in WSNs tht im towrs improving th proility of pir-wis ky stlishmnt, nhning th rsilin to no ptur, or rsing storg ovrh [5], [6], [7], [8], [9], t. Howvr, ll th s shms hv n inhrnt limittion. Th prformn of is stisftory only in highly ns snsor ntworks, whr th vrg numr of physil nighors pr no (i.., vrg physil no gr) [4], [5], [6]. As w know, suh high nsity is not lwys fsil in prti. In ft, u to th rnomnss in ky istriution n strong onstrint in ky pth onstrution, it oftn hppns tht mny physil nighors nnot om sur nighors, i.. th sur no gr is vry low, in non-highly ns ntworks. Consquntly, thy will hv low sur onntivity n r vry likly to prtition. Fig. 1 illustrts this. Th originl ntwork is shown in Fig. 1 (). Thr is n g twn two nos if thy r physil nighors. Th vrg physil no gr is st s. Th orrsponing sur ntwork gnrt y is shown in Fig. 1 () whr n g xists twn two nos if thy r sur nighors. Th vrg sur no gr in Fig. 1 () is only "!. It is muh smllr ompr to th vrg physil no gr. As n sn, th ntwork in Fig. 1 () is prtition into mny onnt omponnts. Two nos nnot ommunit surly if thy rsi in iffrnt onnt omponnts. Th s shms whn ppli to non-highly ns ntworks hv poor prformn. Our Contriutions: In this ppr, w im to solv th ov prolm. Our ontriutions r thr-fol. # Ntwork Doupling: W propos mthoology ll ntwork oupling for sur ommunitions in wir-
() Originl Ntwork () Ntwork with RKP () Ntwork with RKP-DE Fig. 1. Avrg sur no gr omprison twn $&%' n $&%' -( ) whn t most on proxy is us on h ky pth. Our $&%' -( ) hivs 40% improvmnt in vrg sur no gr. Th ntwork is of siz *,+-+-+/. * *,+-+-+/., whr 0-+-+ nos r ploy uniformly t rnom. All nos hv ommunition rng of *,1-1/. n th vrg physil no gr is 243 56*. W st %87*,+-+-+-+ n 9:7<;-+. lss snsor ntworks. In rnom ky pr-istriut snsor ntworks, thr xist two typs of rltionships twn ny two nos. On is logil (shring pristriut kys), n th othr is physil (within ommunition rng). In ntwork oupling, w oupl ths two rltionships in th snsor ntworks. As suh, for ny two nos onnt logilly, w n inpnntly fin pth for thm physilly n vi vrs. Th flxiility offr y oupling grtly nls fining mor logil n physil pths, thry nhning th hns of pir-wis ky stlishmnt twn physil nighors in th ntwork. # Protool Dsign: Bs on ntwork oupling mthoology, w sign nw protool for sur nighor stlishmnt twn physil nighors in th oupl ntwork. W ll our protool s th -=?> protool, whr logil ky pths r onstrut s on ky shring informtion. Thn, orrsponing physil ky pths r onstrut s on no nighorhoo informtion in our protool. # Dpnny Elimintion: Our thir ontriution is proposing novl pnny limintion ruls in our protool to tt n limint ky pnnis t link n pth lvl without ompromising xisting rsilin. In ky stlishmnt, whn multipl ky pths r onstrut, thr is possiility of som links (or pths) ing pnnt on othr links (or pths). Suh pnnis introu unnssry ovrh in trms of ommunition n omputtion. W point out tht suh pnnis xist in ll xisting s protools, whr multipl ky pths r us [5], [6], [9]. Our pnny limintion ruls n ppli to thm to minimiz thir ovrh s wll. To illustrt prformn improvmnt of our -=?> protool, in Fig. 1 (), w show th sur ntwork gnrt protool. Th vrg sur no gr y our -=@> in Fig. 1 () hs now inrs to A!"B, CED improvmnt ovr tht in Fig. 1 (). As our nlysis shows in Stion IV, th vrg sur no gr improvmnt of -=?> ovr is rouneed whn on proxy is us on h ky pth. With inrs in vrg sur no gr, th qulity of sur ommunitions nturlly inrss, monstrting th nfits of ntwork oupling, s lso shown in our prformn vlutions in Stion V. W wish to point out tht th mthoology of oupling in itslf is not nw in ntworking. In [10], onntion stlishmnt is oupl from QoS rsrvtion to nhn th ffiiny of frqunt short liv Intrnt onntions. Th nfits of oupling poliy from mhnisms in Intrnt routing hv n monstrt in [11]. In [12], n pproh is propos tht oupls ontrol from t in TCP ongstion ontrol. Anothr work is [13], whr pth nming is oupl from th tul pth to nl ttr t livry in ns snsor ntworks. Howvr, to th st of our knowlg, our work is th first on tht pplis this mthoology for sur ommunitions in wirlss snsor ntworks. Th rmining of our ppr is orgniz s follows. W isuss rnom ky pr-istriution n othr rlt works in Stion II. Th mthoology of ntwork oupling is introu in Stion III, n our sur nighor stlishmnt protool is til in Stion IV. In Stion V, w prsnt prformn vlutions, n finlly w onlu our ppr in Stion VI. II. THE RANDOM KEY PRE-DISTRIBUTION PROTOCOL IN WIRELESS SENSOR NETWORKS In sminl work in [4], th i of rnom ky pristriution ( ) ws first propos to stlish pir-wis kys in WSNs. Bfor nos r ploy rnomly in th ntwork, h no is pr-istriut with istint kys rnomly hosn from lrg ky pool of siz. Th st of kys pr-istriut in no F is ll th ky hin of no F, not y HGJIKFL. In Fig. 2, nos M, N, O n P r four physil nighors of no F. Eh no is pr-istriut with thr kys, whih r list si th orrsponing no. A soli lin xists twn two nos if thy r physil
Fig. 2. Communition Rng {k 1, k 4, k 5 } {k 4, k 6, k 7 } {k 1, k 2, k 3 } {k 5, k 8, k 9 } {k 6, k 8, k 9 } Pir-wis ky stlishmnt in $&%' protool. nighors (within ommunition rng), n sh lin xists twn two nos if thy r logil nighors (shr t lst on pr-istriut ky). Aftr ploymnt, h no sns mssg to its physil nighors, ontining its no ID n th ky IDs of its pristriut kys. If no F shrs t lst on pr-istriut ky with physil nighor, pir-wis ky twn thm n stlish irtly, suh s nos F n M in Fig. 2. To o so, no F n sn rnomly gnrt pir-wis ky to no M with th pir-wis ky nrypt using thir shr ky EQ. If no F os not shr ny ky with physil nighor, suh s no N, no F will ttmpt to stlish pir-wis ky inirtly using othr nos s proxis. Hr, ky pth is ttmpt to onstrut omprising of on or multipl proxis, whr ny two sussiv nos on th ky pth r physil nighors n shr t lst on pr-istriut ky. Th pir-wis ky gnrt y no F is snt to its physil nighor on th ky pth, with th onstrint tht th pirwis ky is nrypt/rypt in h hop till it rhs th stintion. Tht is, th logil (shring pr-istriut kys) onstrint n th physil (within ommunition rng) onstrint r oupl togthr uring ky pth onstrution. In Fig. 2, no M n th proxy twn nos F n N. Th pir-wis ky twn nos F n N is first gnrt y no F, thn it is snt to no M nrypt y ky RQ. No M will rypt th pir-wis ky, nrypt it y ky CS n sn to no N. Finlly no N rypts th pir-wis ky, n uss it to nrypt futur irt ommunition with no F. Th stnr ttk mol us in nlyzing sur ommunitions is on whr th ttkr os not ttmpt to isrupt ntwork oprtion; rthr it ttmpts to iphr s muh informtion s possil from snsor ommunitions [4], [5], [7]. As suh, th ttkr will typilly lunh two typs of ttks: link monitor ttk n no ptur ttk. In th link monitor ttk, th ttkr monitors n rors ll th wirlss ommunitions in th ntwork immitly ftr no ploymnt. In th no ptur ttk, th ttkr will physilly ptur rtin numr of snsors ftr no ploymnt. On no is ptur, its pr-istriut kys r islos to th ttkr. Comining th pr-istriut kys islos n th mssgs ror, th ttkr will l to infr th pir-wis kys twn som nighoring nos, vn if th nos thmslvs r not ptur. Th pir-wis kys infrr y th ttkr r NUTWVYXZ[TWV]\_^4PWO, s is th orrsponing sur ommunitions twn thos nighoring nos. To vlut th prformn of protool, two typs of mtris r onsir. Th first is onntivity, whih inlus lol onntivity n glol onntivity. Lol onntivity is fin s th proility tht two physilly nighoring nos r l to stlish pir-wis ky in twn. Glol onntivity is fin s ithr th proility tht th whol sur ntwork (n xmpl is shown in Fig. 1 () or ()) is onnt, or th prnt of nos in th lrgst onnt omponnt of th sur ntwork. Th othr prformn mtri is rsilin, whih is fin s th proility tht pir-wis ky twn two nos is not ompromis givn tht thos two nos r not ptur. Th ovrll gol lrly is to mk onntivity n rsilin s high s possil. Th protool [4] hs riv wi ptn in WSNs u to its simpliity, low ovrh, slility n nrgy ffiiny. It hs srv s fountion for mny othr works s on rnom ky pr-istriution, iming to improv prformn or lowr ovrh [5], [6], [7], [8], t. In [5], th prformn of th si protool is nhn y onstruting multipl ky pths using proxis for pir-wis ky stlishmnt twn physilly nighoring nos. With multipl ky pths, s long s t lst on ky pth is unompromis, th pir-wis ky is sur. Similrly, [6] uss multipl two hop ky pths to nhn rsilin furthr unr slightly wkr ttk mol. W point out tht in oth works, vry high ntwork nsity (vrg physil no gr twn " n "A" ) is ssum to hiv stisftory prformn. Svrl othr works orthogonlly improv th si protool y xtning th ky strutur, xploiting rtin ntwork proprtis to nhn prformn, or rsing ovrh. In [7] n [8], th uthors inpnntly xtn th si protool y pr-istriuting ky struturs (ithr polynomils or vtors) inst of kys to stlish pir-wis kys. Whn th numr of ptur nos is smll, this protool hs muh ttr rsilin ompr to th si protool. Othr works lik [14], [15], [16] us powr ontrol, hnnl ivrsity or ntwork hirrhy to nhn prformn unr ssumptions on snsor hrwr, ntwork topology t. Rntly, som works hv us ploymnt knowlg to hiv omprl prformn with fwr numr of kys pr-istriut [17], [18], [19]. Ths works rly on th ssumption tht positions of nighoring nos in th ntwork r prtilly known priori, hlping in rsing th numr of kys pr-istriut to hiv omprl prformn. W point out tht our mthoology of ntwork oupling is orthogonl to ll th works ov, n n omplmnt thm to hiv furthr prformn improvmnt n ovrh rution.
III. NETWORK DECOUPLING IN RANDOM KEY PRE-DISTRIBUTED SENSOR NETWORKS A. Ntwork Doupling In rnom ky pr-istriut snsor ntworks, thr xist two typs of rltionships twn ny two nos. On is logil (shring pr-istriut kys), n th othr is physil (within ommunition rng). W n sprt ths two typs of rltionships y oupling rnom ky pristriut snsor ntwork into two grphs: logil on n physil on. Two nos in th logil grph hv n g twn thm if thy shr t lst on pr-istriut ky. Similrly two nos in th physil grph hv n g twn thm if thy r within ommunition rng of h othr. In th xmpl of Fig. 3 (), no F shrs ky with no M. Consquntly, nos F n M will hv n g twn thm in th logil grph. Bsis, no F is within th ommunition rng of th othr four nos. Consquntly in th physil grph, thr will n g from no F to th othr four nos. For th xmpl in Fig. 3 (), its oupl logil n physil grphs r shown in Fig. 3 () n () rsptivly. Dtil sription on how nos onstrut ths grphs is prsnt in Stion IV. In rnom ky pr-istriut snsor ntworks, w fin sur ommunition s th ommunition twn two nos whr ll mssgs trnsmitt (possily vi multihops) r nrypt. Now w will show how ntwork oupling hlps hiv sur ommunition. Thr r two ss possil, whr two nos in th ntwork n ommunit surly. Th first s is whr th two ommuniting nos shr pr-istriut ky (i.., thy r irtly onnt in th logil grph) n th nos r lso onnt (vi on or multipl hops) in th physil grph. In this s, th sour no n nrypt th mssgs using th shr pr-istriut kys, n h intrmit no in th physil grph n simply forwr th mssgs towrs th stintion, whih will rypt th mssgs using th shr pr-istriut kys. Th son s is on whr th two ommuniting nos o not shr pr-istriut ky (i.., thy r not irtly onnt in th logil grph), ut r onnt inirtly in th logil grph, n th two nos for h logil hop r onnt (irtly or inirtly) in th physil grph. In this s, nryption ours t h intrmit no in th logil grph, whil h intrmit no in th physil grph simply forwrs th mssgs. W point out tht in orr to pply oupling, h snsor ns to know oth th ky shring n no nighorhoo informtion mong its physil nighors. Not tht it will inur signifint ommunition ovrh to otin suh informtion on glol sl. Hn our ntwork oupling is purly loliz hvior, whr h no otins lol informtion n onstruts its lol logil n physil grphs in istriut wy. B. Anlysis In this stion, w will monstrt th nfit of ntwork oupling quntittivly y nlysis. Spifilly, w will {k 1, k 4, k 5 } Fig. 3. {k 4, k 6, k 7 } {k 1, k 2, k 3 } {k 5, k 8, k 9 } () Smpl snsor ntwork {k 6, k 8, k 9 } oupl () Logil grph () Physil grph Doupl snsor ntwork into logil grph n physil grph. riv th proility for th s whr two physilly nighoring nos r l to ommunit surly. As mttr of ft, this proility is lso th proility tht two physilly nighoring nos r l to stlish pir-wis ky vi sur ommunition. Du to sp limittions, w only prsnt th nlysis for th s whr t most on proxy is us on ky pth. Intrst rrs r rfrr to [20] for th nlysis in gnrl s. For two physilly nighoring nos to ommunit surly, thr xist thr possil situtions: (1) Th two nos shr pr-istriut kys (irtly onnt in th logil grph), suh s nos F n M in Fig. 3 (). Clrly, thy n hiv sur ommunition irtly. W not th proility tht this sitution hppns s Q. (2) Th two nos o not shr pr-istriut ky (not irtly onnt in th logil grph), ut thy hv ommon physil nighor tht shrs pr-istriut kys with oth of thm. In Fig. 3 (), nos F n N o not shr pr-istriut ky, ut hv ommon physil nighor no M tht shrs on pr-istriut ky with oth of thm. Sur ommunition twn nos F n N n now hiv vi th hlp of no M ting s proxy. W not th proility tht this sitution hppns s `. (3) Th two nos o not shr pristriut ky (not irtly onnt in th logil grph), n thy nnot fin proxy stisfying th son sitution ov. But thr xists proxy tht shrs pr-istriut kys with oth of thos two nos, n is physil nighor of only on of thm. In Fig. 3 (), nos F n O o not shr pr-istriut ky, ut no M shrs on pr-istriut ky with oth of thm, n no M is physil nighor of only no F. Sur ommunition twn nos F n O n hiv vi th hlp of no M ting s proxy. W not th proility tht this sitution hppns s. Lt us fin oupl ntwork s th ntwork in whih th logil onstrint n th physil onstrint r lwys stisfi simultnously for h hop on sur ommunition pth. Thrfor two nos in oupl ntwork n hiv sur ommunition if n only if ithr of th first two situtions hppns. In th thir sitution, sur ommunition is not possil in oupl ntwork. On th othr hn, in oupl ntwork, sur ommunition is possil if ny of th thr situtions hppns. W not th proility tht two nos n hiv sur ommunition
{ { { { { ` using t most on proxy on h ky pth in oupl n oupl ntwork y f,gihujlk n &m-k,f,gihujlk rsptivly. Sin th ov thr situtions r isjoint, th xprssions of f,gihujlk n m-k_f,gihujlk r simply givn y, &fngih-jok Qqpr&`s (1) &m-k,f,gihujlk:tqqpuvẁpr& (2) Clrly, m-k,f,gihujlk x fngih-jok. This monstrts tht ntwork oupling nhns th hn for two nighoring nos to ommunit surly. In th following, w will riv th xprssions for Q, v` n &. Rll tht Q is th proility tht two nos shr t lst on pr-istriut ky. It is givn y, Q y:z C} 8 C : (3) If = h nots th vrg physil no gr, th vrg numr of nos in th ovrlpp ommunition rngs of two physilly nighoring nos is RABC!CA"= h [5]. Th proility tht &Q nos in th ovrlpp ommunition rngs of oth nos shr pr-istriut kys with on of thos two nos is ƒ i ˆ Šn vœ CŽ I Q6L Ž I_&z@Q-L Ž 6 nˆ Š n Œ. Th proility tht t lst on of th ov Q nos shrs pr-istriut kys with th othr no is z I, z Q L EŽ. Thrfor, ` is givn y, 6 nˆ Šn Œ ` yi, z Q L Žf Q CŽ I_:z Q L 6 nˆ Š n vœ RABC!CA= h vq Ž IK Q L CŽ I_:z8I_šzH Q L L, (4) For two physilly nighoring nos, th vrg numr of nos in th ommunition rng of on no ut outsi th ommunition rng of th othr no is RI =YhYz RAB"!EA= hwlqr BCE["= h. Similrly, & is givn y, 6 ˆ ` vœ všœi_:z QiL I_:z vẁl R BCE[C= h CŽ Qž vq Ž IK Q L I,:zH Q L i ˆ ` Ž Ž / Œ[ I_:z8I_:z Q L L_ (5) W point out tht f,gihujlk n m-k_fngihujlk (in ( ) n ( )) r lso th proilitis tht two physilly nighoring nos r l to stlish pir-wis ky twn thm (vi on proxy t most) in oupl n oupl ntwork rsptivly. Th rivtions of thm will us ltr in th nlysis in Stion IV. IV. SECURE NEIGHBOR ESTABLISHMENT PROTOCOL IN A. Ovrviw DECOUPLED NETWORKS In this stion, w isuss th sign of our nw protool for stlishing sur nighors in oupl rnom ky pristriut snsor ntworks. W ll our protool s - =?> protool. Th protool hs four mjor omponnts in its xution: ) onstruting lol logil n physil grphs in th oupl ntwork for h no, ) stlishing multipl physil ky pths twn physilly nighoring nos, Ÿ ) liminting pnnis mong th multipl ky pths, n ) stlishing pir-wis kys twn physilly nighoring nos. Th -=@> protool is istriut in its xution lik th tritionl protool. Similr to th mol in th tritionl shm in [4], th ntwork mol w onsir is on whr st of snsors r ploy rnomly. Eh snsor is pr-istriut with istint kys rnomly hosn from ky pool of siz. Th mjor iffrns twn our -=?> protool n th tritionl protool r u to th first thr omponnts. In th tritionl protool, ky pths r stlish in ntwork whr th physil n logil grphs r oupl. On th othr hn in our -=@> protool, th physil n logil grphs r sprt/oupl. Th first omponnt of our -=?> protool is h no onstruting ths two lol grphs oupl from h othr. Th lol logil grph is onstrut s on ky shring informtion n th lol physil grph is onstrut s on no nighorhoo informtion, following th mthoology of ntwork oupling isuss rlir in Stion III. Th protool is to stlish logil ky pths twn two physilly nighoring nos s on th logil grph, n for ths logil ky pths, orrsponing physil ky pths r stlish s on th physil grph. Th oupling ftur nls mor ky pths (oth logil n physil) to onstrut whn ompr to th tritionl protool. Not tht whn multipl ky pths (h with multipl links/hops) r onstrut, thr is possiility of som links (or pths) ing pnnt on othr links (or pths). Suh pnnis introu unnssry ovrh in trms of ommunition n omputtion. Th son omponnt in our -=@> protool proposs novl pnny limintion ruls to tt n limint suh pnnis without ompromising th xisting rsilin. Eh omponnt in our -=@> protool is sri in til low. thir omponnt in our -=@> B. Lol Grphs Constrution Aftr no ploymnt, h no otins th ky shring n no nighorhoo informtion within its ommunition rng y lol ommunition with its physil nighors. W ssum tht from lol ommunition, h no n trmin whthr ny two of its physil nighors r physil nighors or not. This n sily on y xhnging nighor informtion uring initil ommunition. With this informtion, h no onstruts lol logil grph ( j ) n lol physil grph (h ). In th lol logil grph, two nos r onnt if thy shr t lst on pr-istriut ky, whil in th lol physil grph, two nos r onnt if thy r within ommunition rng of h othr. Not tht our protool ns only lol informtion xhng n is purly istriut. In this ppr, w ssum h no otins th lol informtion within its ommunition rng (on-hop). Informtion ross multipl
hops n otin y furthr informtion xhng, ut will inur mor ommunition ovrh. C. Ky Pths Constrution Algorithm 1 shows th psuoo of ky pths onstrution xut y h no in th ntwork. In Algorithm 1, nots n ritrry no, whil j I L n h IK L r its lol logil n physil grphs rsptivly. Initilly th logil ky pth tr of no ( g ) is mpty. Th ky pths onstrution is xut in two stps s shown in Algorithm 1. First, g is onstrut y no s on its lol logil grphj_i L (lins to ). This logil ky pth tr g ontins ll th logil ky pths twn n ll its sur nighors. Thn, no onstruts orrsponing physil ky pths s on oth g n its lol physil grphhi L (lins B to 6Ÿ ). Th pnny hking in lin Ÿ n " will isuss in th nxt sustion. Logil ky pth tr onstrution: Th protool onstruts logil ky pth tr (lins to ) using vrint of th stnr pth-first-srh lgorithm, in whih no oul hosn multipl tims (on iffrnt pths). Hr rik L nots th st of physil nighors of no. Fig. 4 shows th rsultnt logil ky pth tr for no F in th xmpl of Fig. 3 (). By xuting th lgorithm just on on its lol logil grph in Fig. 3 (), no F is l to otin ll logil ky pths to ll its nighors. Tking no P s n xmpl, no F otins two logil ky pths twn no F n no P, tht r Fªs M[snN[s P n F«s/MWsnO«snP. Physil ky pths onstrution: Aftr otining th logil ky pth tr ( g ), no gins to onstrut physil ky pths for its nighors (lins B to 6Ÿ ). For h physil nighor, no first otins st of logil ky pths twn n ( gw ) from g. Out of ll suh ky pths in gw, som of thm will limint s on pnny hking (s isuss in th nxt sustion). Th st of pths tht pss th pnny hking is not s gw. Finlly, for ll logil ky pths in g4, orrsponing physil ky pths ( gw ) r otin. In Fig. 3 (), th logil ky pth Fªs/MWsnO«snP ontins logil hop ±MWsnOu twn two non-nighoring nos. From Fig. 3 (), w s tht physil pth 8MWsnFªsnO? n rpl th ov logil hop. Thrfor, for logil ky pth 8Fªs MWs Os P, its orrsponing physil ky pth is Fªs/MWsnFªsnO«snP, in whih h hop is twn two physilly nighoring nos. Mssg nryption/ryption ours for h logil hop, whil mssg trnsmission ours for h physil hop. Hr, w slt th physil pth with fwst hops to rpl logil hop twn non-nighoring nos. Othr poliis n hosn if nrgy onsumption, lo lning, t. r to onsir. D. Dpnny Elimintion W now isuss limintion of link n pth pnnis in stps Ÿ n " of Algorithm 1. Gnrlly, if mor ky pths r us, rsilin is nhn. This is us whn multipl ky pths xist twn two nos, th ttkr ns to ompromis ll ky pths in orr to ompromis th Algorithm 1 Psuoo of Ky Pths Constrution 1: Log Ky Pth Tr Construt(, j I L, g ) 2: for h ³² rik L 3: if µq\ =?P X«P4 OEP6 NU G ¹«P4N6}\ º IK «sn vsn g L» ¼½q½s thn 4: ¾C ^6P4ZW UI &s, ªs, g L-À 5: µát4º P4 FE,¹ šzwpwp GTW ^6 fz4 N- UIK «s j,i L-sn g}l-à 6: n if 7: n for 8: Phy Ky Pths Construt(, h IK L, g ) 9: for h ³² rik L 10: TM- _FC\ H,¹PJ^4P4 wtâhf}ãkãã T4ºE\fNUF}ÃP4 XªFC,¹ª^YM-P4 fäšpwp6 FE O? HI gw LªZ[TWVÅ g À 11: gw t FE,¹ =@PnXªP4 OCP4 N- G ¹PWNi\Æ º I gw L-À 12: TM- _FC\ H,¹PYNUTWZWZ[P[^,XªTW O"\ ºÇ^6P4 :TÂ]Xª¹Ŗ^i\fNUF}ÃP4 X«FE,¹«^ gw ªZ[TWVÅ gw À 13: n for 14: Insrt(,, g ) 15: ¾C ^4P4Z4 T[OEP <\ _T g F^ F<Ni¹\ à O?T T[OCP & Fig. 4. Logil Ky Pth Tr of No È pir-wis ky stlish twn thm. Howvr, this is not lwys tru. Existing links (or pths) my hv pnnis mong thm suh tht th ompromis of som links (or pths) utomtilly ls to th ompromis of othr pnnt links (or pths). Clrly, th prsn of suh pnny os not nhn rsilin. Thy only inrs ovrh in trms of oth storg n nrgy onsumption (u to ommunition n omputtion). In this sustion, w propos two novl pnny limintion ruls to rs suh ovrhs without ffting th rsilin of th pir-wis kys stlish. 1) Link Dpnny Elimintion: W illustrt link pnny with n xmpl in Fig. 5. No F otins logil ky to its physil nighor pth ÉF«s i6 snnsno«s i6 snpes/âªs i6 s M@ no M. If w not ÊI \/sæë}l s th st of shr pr-istriut kys us to nrypt th mssgs on th logil hop Ê\/s ËJ in logil ky pth, thr xists link pnny twn th hops ÌN[snOH n œpes  in tht ÊIKN[s OEL ÍÎuI PEs  L. Sin oth nos N n  shr kys RQ n C`, thr must xist nothr shortr logil ky pth 8Fªs 6i s N[s ªs ii s M, whih
{k 1, k 2 } {k 1, k 2, k 3 } f Fig. 5. Link Dpnny Exmpl {k 1, k 2, k 4 } {k 1, k 2, k 3 } {k 4, k 5, k 6 } {k 1, k 2 } {k 4 } Fig. 6. {k 2 } {k 4 } {k 2, k 4, k 7 } Pth Dpnny Exmpl hs ttr rsilin thn th originl on. This is us th ompromis of ny logil hop twn nos O n P will ompromis th originl ky pth finitly, whil it is possil tht th shortr ky pth is not ompromis. On th othr hn, th ompromis of th shortr ky pth will finitly ompromis th originl ky pth 1. Also, using shortr ky pth will sv ovrh. W formlly fin link pnny s follows. Link Dpnny: Givn two logil hops ÿ\ Q sæë Q n Ê\ ` s Ë ` in logil ky pth, thr xists link pnny twn ths two hops if ithr uik\ Q s Ë Q LHÍÅÊI \ ` s Ë ` L or ui \ ` sæë ` LwÍ8uIK\ Q sæë Q L. Our link pnny limintion rul is tht on suh link pnny is tt on logil ky pth, th protool will limint tht logil ky pth. In th ov xmpl, th logil ky pth ÏFªs ii snn[s Os ii snpes ªs i6 s M will limint sin shortr ky pth 8Fªs snns ªs s/m with i6 6i ttr rsilin xists. Th psuoo of link pnny hking is givn in Algorithm 2 (lins to ). In Algorithm 2, Z[T[TW nots th root no of th logil ky pth tr, FC,¹vI &s,z[t[tw,l nots th st of nos on th logil ky pth from to Z[T[TW, n ª XªFEZWP4 nots th prnt no of no on th tr. As w n s in Algorithm 2, link pnny will hk to output ¼½q½ or Ð ¼¾Cµ, whih is rturn in lin Ÿ of Algorithm 1. 2) Pth Dpnny Elimintion: Aprt from link pnny, nothr typ of pnny ll pth pnny my xist. In Fig. 6, thr r two logil ky pths twn nos F n M. Howvr, w n s tht th ompromis of th ky pth ÑF«s N[s M< (islosur of kys I Qs "`4L or I "SWL ) lwys ls to th ompromis of th othr ky pth FªsnO«s Mš, ut not vi vrs. Thrfor, givn tht ky pth ÏFªsnNs M xists, th othr ky pth ÒFªs Os/M oms runnt in trms of rsilin, n lso inurs ovrh. 1 W formlly prov tht th shortr ky pth improvs rsilin ovr th originl on in [20]. Algorithm 2 Psuoo of Dpnny Chking 1: Link Dpnny Chking(,, ) 2: if Ó no ÄѲ Pth(,Z[T[TW ), s.t. ÊI ªs, ª XªFEZWP4,L Í ÊI Ä snä XªFEZWP4,L, thn 3: ZWP4 f ªZW r𠼚¾Eµ À 4: ls if Ó no ı² Pth(,Z[T[TW ), s.t. ÊI Ä snä X«FEZ[P6,L ÍyuIK «sn «X«FEZ[P4,L, thn 5: ZWP4 f ªZW r𠼚¾Eµ À 6: ls ZWP4 f ªZW r ¼½q½ À 7: n if 8: Pth Dpnny Chking( g4 ) 9: š gw t g4 EÀ 10: whil Ó pths X n Ô<²Õ gw, s.t. X is wkr thn Ô OR Ô is wkr thn X, o 11: if X is wkr thn Ô, thn 12: š gw š gw :Ö X À 13: ls if Ô is wkr thn X, thn 14: š gw š gw Ö Ô}À 15: n if 16: n whil 17: Z[P4 f «ZW š gw À Dnoting th st of logil hops on logil ky pth X s µh, n not th st of shr pr-istriut kys us on logil hop ¹ s ÊI ¹ªL, pth pnny is formlly fin s follows. Pth Dpnny: Givn two logil ky pths X n Ô, thr xists pth pnny twn X n Ô if ithr of th following two onitions is stisfi. (1) rã T4ºE\fNUF}Ã:¹«T/Xu¹ ² µáø[s<ótf8ã T4ºE\fNUF}à ¹«T/X ¹«JI ¹«?²Ñµh"L-sÙ^C -:uiæ¹ ÚL ÍÛuIƹ«LUÀ (2) Ñà T4ºE\fNUF}Ã<¹T/XܹݲݵhsrÓÌFœÃKT4º}\ NiFEÃÙ¹«T/X±¹«ÞIƹ ² µáøil-s^c -«uiæ¹ ÚLwÍ8uI ¹ªL- If th first onition of pth pnny is stisfi, w ll pth XÇÄšPWFP4Z thn pth Ô. Similrly, pth Ô is ÄšPWFP4Z thn pth X if th son onition is stisfi. Our pth pnny limintion rul is tht ftr tting pth pnny twn two logil ky pths, our protool will limint th wkr on. In th ov xmpl, th logil ky pth will limint. In s two pths stisfy ßFªsnO«s MÎ oth onitions in th pth pnny, w n limint on of thm s on rtin poliis (.g., th pth with mor physil hops). Th psuoo of pth pnny hking is givn in Algorithm 2 (lins B to W ). E. Pir-wis Ky Estlishmnt On th physil ky pths r onstrut ftr pnny limintion, h snsor will gnrt istint ky shrs t rnom, n sn h ky shr on h physil ky pth for h sur nighor. Th mssgs r trnsmitt t h physil hop, whil thy r rypt/nrypt t h logil hop. Tk th logil ky pth 8F«s/MWsnO? in Fig. 3 () s n xmpl. Its orrsponing physil ky pth is ÑFªs MWs F«s Or. W ssum ky shr ªà â Q,á Wã is trnsmitt on this ky pth. W not ä s th
¾ ã ã â â â ã ¾ ã ã mssg nrypt with ky. Th ky shr trnsmission is xut s follows: FYåæçMšè ÊF@ si 8FªsnO s6 ªà Q,á Wä Ž s M:åǽ F?èÏ 8O s ÊF? s6 ÊO s6 à Q_á ä/é s FYåæÛO?è ÊF? s6 ÊO s6 ªà Q_á Wä é In th mssg tht no F sns to no M, 8F? nots th sour no, n ÊFªs O nots th rmining physil ky pth. In th mssg tht M sns to F, th 8O without nryption nots th rmining physil ky pth sin F nnot rypt th mssg using ky. Similrly, no F n trnsmit nothr rnom ky shr ªà â ` á on nothr logil ky pth ÉF«s/MWsnNsnPEsnO to no O. No O my lso onstrut othr ky pths 2, n trnsmit its ky shrs to no F. Finlly, nos F n O n omput ommon pir-wis ky vi som simpl oprtion suh s it-wis XOR oprtion, s on ll th ky shrs thy oth gnrt. F. Anlysis In this stion, w will riv th xprssion for th vrg sur no gr in oth protool n -=?> protool, not y =]ê ë:ì n =?ê ë:ì qî rsptivly. Du to sp limittions, w only prsnt th rivtion for th s whr on proxy is us on h ky pth. Intrst rrs r rfrr to [20] for th gnrl s nlysis whr ritrry numr of proxis r us on h ky pth. In Stion III, w riv xprssions for fngih-jok n m-k_fngihujlk in (1) n (2) rsptivly, whih not th proility tht two physilly nighoring nos r l to onstrut ky pth in -=@> protool n protool with t most on proxy rsptivly. With this, w n riv = ê ë:ì n = ê ë:ì»î s, = ê ë:ì = h fnguhujlk[s (6) = ê ë:ì qî = h &m-k,f,gihujlkws (7) whr rll tht = h nots th vrg physil no gr. Th improvmnt of = ê ë:ì qî ovr = ê ë:ì, not y ¾, is thn givn y, = ê ë:ì qî z = ê ë:ì & = ê ë:ì Q pr ` (8) Rll tht Q, v` n & in (3), (4) n (5) rsptivly r funtions of ky pool siz, ky hin siz n vrg physil no gr = h. Unr iffrnt vlus of = h, w omput th vlus of = ê ë:ì, = ê ë:ì qî n th improvmnt ¾ in Tl I (ïð4"c", ÉA ). W n s tht ntwork oupling improvs th vrg sur no gr unr ll situtions. Th improvmnt in vrg sur no gr hlps to nhn th prformn of rnom ky pr-istriution in trms of onntivity n rsilin, whih will monstrt using simultions in th following stion. 2 Not shown in Fig. 3 () TABLE I IMPROVEMENT OF (:ñròó ôrõ ö OVER ( ñò&ó UNDER DIFFERENT (ø = h 5 10 15 20 25 = ê ë:ì 1.66 4.26 7.59 11.52 15.89 2.27 6.16 10.96 16.22 21.68 37% 45% 44% 41% 36% = ê ë:ì»î V. PERFORMANCE EVALUATION In this stion, w rport xprimntl t to monstrt th prformn of our -=?> protool ompr to th tritionl protool unr vrious ntwork n ttk prmtrs. Th mtris w stuy r onntivity (lol onntivity n glol onntivity) n rsilin. W lso stuy th ovrh of our -=@> protool ompr to th tritionl protool in trms of ommunition n omputtion. A. Simultion Environmnt Th snsor ntwork is squr rgion of siz 4"""V * 6C""V, in whih 6"C snsors r ploy uniformly t rnom. Th ommunition rng Z is th sm for ll snsors n is hosn s on th sir vrg physil no gr = h (ignor th ounry fft). Eh no is wr of th ky shring n no nighorhoo informtion within its ommunition rng. Th following r th fult vlus for th prmtrs unlss othrwis spifi: vrg physil no gr = h?ñ6, ky pool siz ù±4"c", ky hin siz ]ya". Th ttk mol is on whr th ttkr n monitor ll links in th ntwork, n n ptur up to ú nos. By fult, útòa. Eh simultion is run 6C tims with iffrnt rnom ss, n th t prsnt is th vrg ovr 6" runs. In oth protool n -=?> protool, h no tris to stlish pir-wis ky with h of its physil nighors using multipl ky pths s solly on its lol informtion. B. Snsitivity of Conntivity to = h 1) Lol Conntivity: In Fig. 7, w stuy th snsitivity of lol onntivity to vrg physil no gr =Jh. W osrv tht th lol onntivity in -=@> protool is onsistntly highr thn tht in th protool. Th improvmnt is in ft mor signifint (out ŸCAED improvmnt) for non-highly ns ntworks whr = h û". In protool, th onsirtion of oth physil n logil onstrints in ky pth onstrution limits th numr of ky pths twn physil nighors, spilly whn th ntwork is not ns. Howvr, th rlxtion/oupling of th onstrints s rsult of ntwork oupling nls th vilility of mny mor ky pths grtly nhning lol onntivity. 2) Glol Conntivity: Th finition of glol onntivity hr is th prnt of nos in th lrgst onnt omponnt of th sur ntwork (n xmpl is shown in Fig. 1 () or ()). In Fig. 8, w osrv tht th glol onntivity of our
1 1 0.9 Lol Conntivity 0.8 0.6 0.4 0.2 Fig. 7. RKP RKP-DE 5 10 15 20 25 Avrg Physil No Dgr (D p ) Snsitivity of lol onntivity to (qø Glol Conntivity 0.8 0.6 0.4 0.2 0 Fig. 8. RKP RKP-DE 5 10 15 20 25 Avrg Physil No Dgr (D p ) Snsitivity of glol onntivity to (qø Rsilin 0.8 0.7 0.6 Fig. 9. RKP RKP-DE 5 10 15 20 25 Avrg Physil No Dgr (D p ) Snsitivity of rsilin to (qø 1 1 0.9 Lol Conntivity 0.8 0.6 0.4 0.2 RKP RKP-DE 40 50 60 70 80 Ky Chin Siz (k) Fig. 10. Snsitivity of lol onntivity to 9 Glol Conntivity 0.8 0.6 0.4 0.2 0 RKP RKP-DE 40 50 60 70 80 Ky Chin Siz (k) Fig. 11. Snsitivity of glol onntivity to 9 Rsilin 0.7 0.5 0.3 0.1 Fig. 12. RKP (x=50) RKP (x=100) RKP-DE (x=50) RKP-DE (x=100) 40 50 60 70 80 Ky Chin Siz (k) Snsitivity of rsilin to 9 n ü -=@> protool is highr thn tht of th protool in ll situtions. Th improvmnt is spilly signifint in non-highly ns ntworks (up to CED improvmnt). This improvmnt is rsult of th phs trnsition phnomnon in rnom grphs [21]. Aoring to this phnomnon, th lrgst onnt omponnt in rnom grph with nos jumps from ýyi þoÿ L to ýyik L whn th vrg no gr rhs yon rtin thrshol. With ntwork oupling in our -=@> protool, suh jump in glol onntivity ours whn = h is roun 6 ompr to th protool whn = h is roun WA. Anothr osrvtion is tht th glol onntivity whn = hx 4 in our -=?> protool is similr to th glol onntivity whn = h t in th protool. This monstrts tht w n otin similr lvls of glol onntivity with muh fwr nos ompr to th numr of nos n in th protool. C. Snsitivity of Rsilin to = h In Fig. 9, w stuy th snsitivity of rsilin to =Yh. W s tht th rsilin is highr in our -=?> protool ompr to tht of th tritionl protool in gnrl. Th improvmnt is onsistnt xpt whn th ntwork is vry sprs (= h ta ). Ntwork oupling not only inrss th numr of ky pths twn physilly nighoring nos, ut lso rss th numr of logil hops of mny ky pths, oth of whih hlp nhn th rsilin. Whn ntwork oms vry sprs, only singl ky pth n onstrut for most situtions, thus th improvmnt iminishs. D. Snsitivity of Conntivity n Rsilin to n ú In Fig. 10, 11 n 12, w stuy th snsitivity of onntivity n rsilin to n ú. In Fig. 10 n 11, w s similr pttrn in snsitivity of onntivity to s tht to = h. This is us th inrs in nhns th proility tht two nos shr pr-istriut kys, whih mks th lol logil grph mor ns. This n lso hiv y inrsing = h s wll. Ovrll, our -=@> protool hivs ttr prformn thn tht of protool, n th prformn improvmnt is spilly signifint in nonhighly ns ntwork. On th othr hn, givn th sm prformn rquirmnt, our -=?> protool n sv ompr with th storg ovrh ( ) up to roun Ÿ"}D protool. For xmpl, givn tbc in th protool, our -=?> protool n hiv similr prformn with roun (or smllr thn)!". In Fig. 12, w stuy th snsitivity of rsilin to ky hin siz unr iffrnt vlus for numr of ptur nos ú. W osrv tht th rsilin of our -=?> protool is ttr thn tht of th protool for ll ss. Th improvmnt is spilly mor pronoun for lrgr ú (i.., strongr ttks), whih furthr monstrts th fftivnss of our -=?> protool. Th vlu of ú os not impt onntivity, so w o not show th snsitivity of onntivity to ú. E. Ovrh An importnt nillry ftor juging th prformn of our protool is th inurr ovrh. Th storg ovrh ( ) in our -=?> protool is lss thn tht of th protool unr similr prformn, s isuss ov. Hr
` Œ w fous our isussion on ommunition n omputtion ovrh. 1) Communition Ovrh: In our protool, h snsor stlishs pir-wis kys with = sur nighors on vrg. In orr to stlish pir-wis ky with h sur nighor, th snsor ns to sn mssgs on h ky pth. If w not th vrg numr of ky pths twn pir of snsors s h, n not th vrg numr of hops of physil ky pth s ¹Rh, th vrg numr of mssgs snsor sns/forwrs is Œ. In prti, snsors my not n to us ll th ky pths vill if th rsilin rquirmnt n mt with fw short physil ky pths. Thrfor, th vlus for h n ¹ h will rltivly smll. Ovrll, th ommunition ovrh in our protool is similr to tht of th tritionl protool. 2) Computtion Ovrh: Th omputtion ovrh is omint y two mjor prts in our protool, whih r ky pths onstrution n ky shrs trnsmission. In ky pths onstrution, th vrint of pth-first-srh lgorithm w us is lightwight, spilly in non-ns ntwork whr = h is mort. In ky shrs trnsmission, th nryption/ryption oprtion opts lightwight symmtri lgorithm, n this oprtion ours only on logil hop sis. Ovrll, th omputtion ovrh is mil. VI. FINAL REMARKS In this ppr, w propos ntwork oupling to sprt th logil rltionship from th physil rltionship in rnom ky pr-istriut snsor ntworks. W sign sur nighor stlishmnt protool ( -=?> ) in oupl snsor ntworks, n lso sign st of pnny limintion ruls for liminting link n pth lvl ky pnnis mong th ky pths. W onut til nlysis s wll s xtnsiv simultions to vlut our propos solution. Our t show tht signifint prformn improvmnt n hiv using our solution in non-highly ns ntworks. Our futur work will onsist of prtilly implmnting our propos solution on th xisting snsor ntwork tst t OSU [22]. ACKNOWLEDGMENT W thnk th nonymous rviwrs for thir invlul fk. This work ws prtilly support y NSF unr grnts No. ACI-0329155 n CCF-0546668. REFERENCES [1] W. Diffi n M. E. Hllmn, Nw irtions in ryptogrphy, IEEE Trnstions on Informtion Thory, vol. IT-22, no. 6, pp. 644 654, Novmr 1976. [2] R. L. Rivst, A. Shmir, n L. M. 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