Deecing Nework Inrusions vi Smpling : A Gme Theoreic Approch Murli Kodilm T. V. Lkshmn Bell Lborories Lucen Technologies 101 Crwfords Corner Rod Holmdel, NJ 07733, USA {murlik, lkshmn}@bell-lbs.com Absrc In his pper, we consider he problem of deecing n inruding pcke in communicion nework. Deecion is ccomplished by smpling porion of he pckes rnsiing seleced nework links (or rouer inerfces). Since smpling enils incurring nework coss for rel-ime pcke smpling nd pcke exminion hrdwre, we would like o develop nework pcke smpling sregy o effecively deec nework inrusions while no exceeding given ol smpling budge. We consider his problem in gme heoreic frmework, where he inruder picks phs (or he nework ingress poin if only shores ph rouing is possible) o minimize chnces of deecion nd where he nework operor chooses smpling sregy o mximize he chnces of deecion. We formule he gme heoreic problem, nd develop smpling schemes h re opiml in his gme heoreic seing. I. INTRODUCTION In his pper, we consider he problem of deecing inrusions in communicion nework. There is growing lierure on providing securiy in communicion neworks. Two key res of ineres in securiy re inrusion deecion nd inrusion prevenion. In his pper, we del wih he problem of inrusion deecion. Inrusion in neworks kes mny forms including denil of service cks, viruses inroduced ino he neworks, ec. Typiclly, in n inrusion problem, he inruder emps o gin ccess o priculr file server or websie in he nework. In his pper, we consider sylized inrusion problem. In his problem, he inruder emps o send mlicious pcke o given node in he nework. The nework emps o deec his inrusion. The deecion mechnism is pcke smpling nd exminion in he nework. The ide in smpling is h some porion of pckes rversing designed links (or rouer inerfces) re smpled nd exmined in deil o deermine wheher he pcke is n inruder pcke. This pcke exminion my be simple (limied o specific pcke heder fields s in pcke filering) or my involve more deiled exminion of he pcke. To preven pcke mis-ordering or reducion of link hroughpu his exminion hs o be done preferbly line res. Pcke smpling hs been previously proposed for vriey of neworking purposes. For insnce, he SRED scheme in [6] uses pcke smpling o esime he number of cive TCP flows in order o sbilize nework buffer occupncy for TCP rffic. Only pcke heders need be exmined for his scheme. The scheme proposed in [7], lso uses pcke smpling nd i is used for fir link-bndwidh llocion. Smpling hs lso been proposed o infer nework rffic nd rouing chrcerisics [3]. Wheres, hese pplicions require only smpling bsed on pcke heder comprisons, inrusion deecion my enil more horough exminion of smpled pckes. Also, unlike some of he smpling pplicions menioned bove, smpling for inrusion deecion requires ner line-speed pcke exminion since copying smpled pckes or pcke-heders for off-line nlysis is no sufficien o preven inruding pckes from geing hrough. Hence, in he design of n inrusion deecion scheme i is imperive o keep he smpling coss in mind. We sudy his inrusion deecion vi smpling problem in gme heoreic seing. Gme heory hs been used exensively o model differen neworking problems. This work includes he work of Shenker for modeling service disciplines [10], Akell e. l. for TCP performnce [2], nd Korilis, Lzr nd Ord [5] for modeling rouing problems. To he bes of our knowledge, his is he firs emp o model inrusion deecion vi smpling in communicion neworks using gme-heoreic frmework. This work is closely reled o drug inerdicion models. In priculr he work of Wshburn nd Wood [11] who considered drug inerdicion in gme heoreic frmework. This work differs from he drug inerdicion models in wo wys. Firs, in he drug inerdicion models he objecive is o deploy gens which is discree llocion problem. In our cse, he deecion is by mens of smpling. Therefore he gme heoreic resuls re much more nurl hn he discree llocion models. Secondly, in our cse, he gme heoreic problem nurlly leds o rouing problem (o
mximize he service provider s chnces of deecing inruding pckes) which is bsen in he drug inerdicion problem. The soluion o he gme heoreic formulion is mximum flow problem nd he rouing problem cn be formuled s mulicommodiy flow problem. We lso consider vrious exensions nd vrins o he bsic models. Ack Node Mlicious Pcke II. PROBLEM DEFINITION The problem se-up is oulined in hree seps. Firs, we describe he nework, hen we define he dversries in he gmeheoreic frmework, nd finlly we describe he objecive of he gme h is plyed beween he dversries. Fig. 1. Nework Inrusion Gme Trge Node A. Nework Se-Up We consider nework G =(N,E) where N is he se of nodes nd E is he se of unidirecionl links in he nework. We ssume h here re n nodes nd m links in he nework. We ssume h he cpciy of link e E is denoed by c e nd he moun of rffic flowing on link e is denoed by f e.given wo nodes u nd v in he nework, le Pu v represen he se of phs from u o v in G. Given n m-vecor w, weusem uv (w) o denoe he mximum flow h cn be sen from node u o node v using w s he link cpciies. We use he prmeer w explicily when we define M uv () o indice h dependence of he mximum flow on he link cpciies. Corresponding o his mximum flow beween nodes u nd v, here is minimum cu comprising of se of links in he nework. This se of links in his minimum cu will be represened by Cu(c). v B. Nework Inrusion Gme The nework inrusion deecion gme is plyed on he nework beween wo plyers: he Service Provider nd he Inruder. The objecive of he inruder is o injec mlicious pcke from some ck node N wih he inenion of cking rge node N. We ssume h n inrusion is successful when he mlicious pcke reches he desired rge node wihou deecion. In order o deec nd preven he inrusion, he service provider is llowed o smple pckes in he nework. We ssume h smpling kes plce on he links in he nework. I is esy o modify he model o consider he cse, where he smpling is done he nodes in he nework. If during he course of smpling, he service provider smples he mlicious pcke hen he inrusion is ssumed o be deeced nd hwred. The gme is picorilly illusred in Figure 1. C. The Objecive nd he Consrins of he Gme If here is no bound on he moun of smpling h cn be done by he service provider, hen he service provider cn poenilly inspec every pcke h flows hrough he nework nd hence deec he mlicious pcke. Smpling he pckes flowing on link involves seing up he pproprie smpling filers nd exmining he pckes. These cn be firly expensive operions o perform in rel ime. Therefore, we ssume h he service provider hs smpling bound of B pckes per second over he enire nework. This smpling effor cn be disribued rbirrily over he links in he nework. One wy of implemening he smpling scheme is for ech link o pick some frcion of he pckes flowing hrough i nd send i o cenrl inrusion deecion node in he nework which exmines he pcke in more deil. The smpling bound cn be viewed s he mximum re which he inrusion deecion node cn process pckes in rel ime. If link e h hs rffic of f e flowing on i, is smpled re s e hen he probbiliy of deecing mlicious pcke on his link is given by p e = s e /f e.the smpling budge consrin implies h e E s e B. Weformule he gme heoreic problems in erms of p e. We ssume h boh he plyers hve complee informion bou he opology of he nework nd ll he link flows in he nework. The service provider cn hve ccess o his informion eiher from link-se rouing proocols wih rffic engineering exensions h disribue flow informion hroughou nework re or by explici link polling from mngemen sysems. We ssume h he dversry injecing inruding pckes hs his informion vilble s well since his mkes he service provider s deecion problem more difficul. Similrly, we lso ssume h he inruder is cpble of picking phs in he nework so s o mke he deecion problem for he service provider more difficul. However, in Secion V-A, we lso consider he cse where only shores ph rouing is llowed in he nework. 1) Sregies for he Two Plyers: In he cse of he inruder, pure sregy would be o pick ph from P P for he mlicious pcke o rverse from s o. The inruder, in generl, cn use mixed sregy. In he cse of mixed sregy, he inruder hs probbiliy disribuion q over he se of phs in P such h P P q(p )=1.LeV = {q : P P q(p )=1} represen he se of fesible probbiliy llocions over he se
of phs beween nd. The inruder hen picks ph P P wih probbiliy q(p ). The sregy for he service provider is o deermine se of links on which smpling hs o be done. The sregy for he service provider is o choose he smpling re s e on link e such h e E s e B. If he mlicious pcke rverses link e wih smpling re of s e on link wih flow f e resuls in he mlicious pcke being deeced wih probbiliy p e = s e /f e.leu = {p : e E p ef e B} represen he se of deecion probbiliy vecors p h sisfy he smpling budge consrin. (Noe h p is n m- vecor.) Insed of viewing he service provider s picking he smpling res he links, we view he service provider s picking se of deecion probbiliies he links which belongs in he se U. Figures 2 nd 3 depic he inruder s nd he service provider s cions. Ack Node Fig. 2. Inruders Sregy: Pick ph from o Ack Node Fig. 3. Defenders Sregy: Pick he smpling res he links Inruder s Problem Service Provider s Problem Trge Node Trge Node Smpling on rcs belonging o - mincu 2) Pyoff Mrix: Assume h he inruder nd he service provider ech hve chosen sregy. This implies h he inruder hs picked probbiliy disribuion q over he se of phs in P nd he service provider hs picked se of deecion probbiliies p he links. The pyoff h we consider, is he expeced number of imes he mlicious pcke is deeced s i goes from o. For given ph P P,he expeced number of imes h pcke is deeced is given by e P p e. The probbiliy h his ph P is picked by he inruder is given by q(p ). Therefore he expeced number of imes pcke is deeced s i goes from he source o he desinion for fixed sregy from boh dversries is given by [ ] q(p ) p e. P P e P Inerchnging he order of summion, we ge [ ] q(p ) p e = p e q(p ). e P e E P P :P e P P This cn be equivlenly wrien in mrix form s q T Mp where M is n m P ph-rc incidence mrix. Ech row in M represens link in he nework nd ech column of M represens ph beween nodes nd. The enry correspondingorowe nd column P is se o one if e P nd o zero oherwise. A more nurl pyoff, is he probbiliy of deecion of he mlicious pcke s opposed o he expeced number of imes he mlicious pcke is deeced. In his cse for fixed ph P P, he probbiliy of he mlicious pcke being deeced is given by 1 e P (1 p e). This objecive is non-liner in p e which mkes he gme heoreic problem inrcble. However, he wo pyoffs h we oulined bove coincide if he opiml soluion for he service provider is o smple mos one link on ny ph P P wih q(p ) > 0. We cll his sregy miniml smpling sregy. We show ler h for ll he problems we consider, he opiml soluion is miniml smpling sregy. 3) Objecive of he Adversries: The inruder fers h if his sregy is known o he service provider hen service provider will choose sregy h mx p U P P q(p ) [ e P p e]. Therefore he objecive of he inruder is o pick disribuion q() h minimizes his mximum vlue. In oher words, he objecive of he inruder is o [ ] min mx q(p ) p e. q V p U P P e P The objecive of he service provider, using similr rgumen is [ ] mx min q(p ) p e. p U p V P P e P This is clssicl wo person zero-sum gme nd he following minmx resul is well known.
Theorem 1: There exiss n opiml soluion o he inrusion deecion gme where [ ] θ = min mx q(p ) p e = q V p U mx min p U q V P P P P e P [ ] q(p ) p e, e P where θ is he vlue of he gme. In he res of his pper, we show how his minmx opiml soluion cn be compued for he inrusion deecion gme nd use h insigh o roue flows in he nework. III. SOLUTION OF THE GAME We now consider he soluion of he minmx problem formuled in he ls secion. The ide is o ge some insigh ino he srucure of he problem which will enble us o exend he soluion o more complex cses. Consider he inruders problem. min mx p e q(p ). q V p U e E P P :P e For fixed q V he inner mximizion problem is he following: mx q(p ) p e e E P P :P e f e p e B e E p e 0 Associing dul vrible λ wih he budge consrin, we obin he following dul opimizion problem. f e λ λ 0 min Bλ q(p ) e E P P :P e Subsiuing his opimizion problem in he inruders minmx formulion mkes i he following minimizion problem. min Bλ q(p ) f e λ e E P P :P e q(p ) = 1 P P λ 0 Inerpreing q(p ) s flow on ph P, he consrin q(p ) f e λ P P :P e resrics he flow on link e o be f e λ. Therefore f e λ cn be inerpreed s he cpciy of link e. The consrin P P q(p )=1enforces one uni of flow o be sen from u v he source o he desinion. Assume h f e is he cpciy of link e in he nework. The objecive hen is o deermine he smlles scling fcor λ, on he links in he nework so h flow of one uni cn be sen from he source o he desinion. This cn be done s follows: Assume h link e hs cpciy f e nd deermine he mximum flow, M (f) from he o using hese cpciies. Se λ = M (f) 1. By scling he cpciies by λ, noe h flow of one uni is sen from o. The vlue of he gme θ = BM (f) 1. Any mximum flow from o cn be decomposed ino se of flows on phs from o using sndrd flow decomposiion echniques. From nework flow duliy, noe h corresponding o he mximum flow vlue here is minimum cu. The sble opering poin for he inruder nd he service provider re he following: Inruders Sregy: Solve he mximum flow M (f), from o using cpciy of f e on link e. Using sndrd flow decomposiion echniques, decompose he mximum flow ino flow on phs P 1,P 2,...,P l from o. wih flows of m 1,m 2,...,m l respecively. (Noe h l i=1 m i = M (f).) The inruder inroduces he mlicious pcke long he ph P i wih probbiliy m i M (f) 1. Service Providers Sregy: The service provider compues he mximum flow from o using f e s he cpciy of link e. Lee 1,e 2,...,e r denoe he rcs in he corresponding minimum cu wih flows f 1,f 2,...,f r.from duliy r i=1 f i = M (f). The service provider smples link e i re Bf i M (f) 1. We now illusre he bove resuls on he exmple shown in Figure 4. The numbers nex o he links re he flows on he links. How hese flows re genered is discussed in deil in subsequen secion. For now ssume h he flows on he links re given. Assume h here is smpling budge B of 5 unis. nd =1nd =5re he ck nd rge nodes respecively. The links (1, 2), (4, 5) belonging o he minimum cu re shown in hick lines. The minimum cu (nd hence he mximum flow) hs vlue of 11.5 unis. The inruder s sregy is he following: Inroduce he mlicious pcke long he ph 1-2-5 wih probbiliy 7.0/11.5 Inroduce he mlicious pcke long he ph 1-2-6-5 wih probbiliy 0.5/11.5
Inroduce he mlicious pcke long he ph 1-3-4-5 wih probbiliy 4.0/11.5 The minmx sregy for he service provider is he following: Smple link 1-2 re 5/11.5 giving ol smpling re of (5 7.5)/11.5 on h link. Smple link 4-5 re 5/11.5 giving ol smpling re of (5 4.0)/11.5 on h link. Noe h θ =5/11.5 is he vlue of he gme. 1 Fig. 4. 7.5 7.5 2 3 5.5 Exmple of Nework 6.0 7.0 7.0 6 5 7.0 4 4.0 Minimum Cu The following observions cn be mde bou he minmx opiml soluion: The opiml sregy for he service provider is o smple pckes on he mincu wih respec o he rffic flows. This implies h long ny ph h he inruder would choose, he mlicious pcke will be smpled mos on one link. Therefore his is miniml smpling scheme. If B M (f) hen noe h he mlicious pcke will lwys be deeced. If B<M (f) hen here is non-null probbiliy h he mlicious pcke will no be deeced. IV. ROUTING TO IMPROVE THE VALUE OF THE GAME In he ls secion, we showed h he vlue of he nework inrusion gme is given by BM (f) 1. All long, we ssumed h he flow f on he links is fixed. The flows on he links re resul of rouing he demnds (ggrege rffic beween node pirs) in he nework. In his secion, we explore he cse where he service provider djuss he flows in he nework in order o mximize he vlue of he gme. Corresponding o ech pir of nodes in he nework, here could poenilly be demnds h hve o be roued from he firs node in his pir o he second node. Ech node pir beween which here is some demnd h hs o be roued is ermed source-desinion pir or commodiy. We ssume h here re K source-desinion demnd pirs (commodiies) in he nework. The source node Source Des. Demnds Pir 1-3 5.0 1-4 3.0 1-5 7.0 2-3 1.0 2-5 10.0 6-5 1.0 TABLE I SOURCE-DESTINATION PAIRS AND DEMANDS Source Phs Flow Des. Pir 1-3 1-3 5.0 1-4 1-2-3-4 0.5 1-3-4 2.5 1-5 1-2-6-5 6.0 1-2-3-4-5 1.0 2-3 2-3 1.0 2-5 2-5 7.0 2-3-5 3.0 6-5 6-5 1.0 TABLE II FLOWS FOR BASE CASE for commodiy k will be represened by s(k), he desinion node by d(k) nd he moun of demnd (bndwidh) h hs o be roued for his source-desinion pir is b(k). The service provider hs o roue hese flows in he nework respecing he link cpciy consrins. For he exmple shown in Figure 4, ech link is ssumed o hve cpciy of 10 unis. The differen source-desinion pirs nd he corresponding demnds re shown in Tble I. These demnds hve o be roued in he nework such h he link cpciy consrins re respeced. There re severl wys of rouing hese demnds. One commonly used mehod is o roue he demnds such h he mximum link uilizion in he nework is minimized. (This cn be solved s mximum concurren flow problem.) The link flows shown in Figure 4 re resul of rouing he demnds in order o minimize he mximum uilizion in he nework. The rouing is given in Tble II. We now explore he cse where he service provider roues hese flows such h he vlue of he nework inrusion gme is mximized. In oher words, he service provider roues he source-desinion demnds such h he mximum probbiliy of deecion of he mlicious pcke is incresed. We firs
formule his problem nd hen explore differen heurisics o solve he problem. Recll h P d(k) s(k) represens he se of phs beween he source node s(k) nd he desinion node d(k) for commodiy k. For noionl simpliciy, we refer o P d(k) s(k) s P k. Noe h P k represens he se of vlid phs o roue commodiy k. Le X = {x(p ) : P P k x(p) = d(k) k, k P P k :e P x(p ) c e e E}. Noe h X denoes n llocion of flow on phs in he nework which mees he demnd for ech commodiy while sisfying he cpciy consrins on he links in he nework. Given fesible rouing vecor, x X, he flow on link e is given by f e = k P P k :e P x(p ). From Secion III, he vlue of he gme is given by B/M (f). The objecive of he service provider hen is o roue he source desinion demnds such h he resuling vlue of M (f) is s smll s possible. Therefore he objecive of he service provider is o solve he following opimizion problem. min M ( x(p )). x X k P P k :e P This cn be wrien more explicily s min y(p ) P P v u :e P y(p ) x X P P y(p ) 0. k P P v u :e P x(p ) Unforunely his problem cnno be solved s liner progrmming problem. I is possible o reformule his problem s non-convex opimizion problem bu i is no cler if here is soluion echnique o solve his problem. We herefore develop wo differen heurisics o ge good soluions o his opimizion problem. Insed of minimizing he lef hnd side of he inequliy, we minimize he upper bound represened by he righ hnd side of he inequliy. Since c is fixed, his is equivlen o mximizing M(c k P P k :e P x(p )) subjec o he consrin h x X. We wrie his more formlly s: mx y(p ) k P P k :e P y(p ) P Pu v :e P P P v u x(p ) c e e E k P P k :e P x(p ) x(p ) = d(k) k P P k x(p ) 0 P P k j k y(p ) 0 P Pu v j k e E I is esy o view his s muli-commodiy flow problem wih K +1commodiies. There re he originl K commodiies nd n ddiionl commodiy beween nd. The size of he demnds for he firs K commodiies re known. We perform bisecion serch o deermine he lrges vlue of he commodiy K +1h sill resuls in fesible rouing for he firs K commodiies. In order o develop n efficien lgorihm i is beer o formule he problem s mximum concurren flow problem nd perform he bisecion serch for his problem insed. We do no give he deils of he soluion procedure. In he cse of he flow flushing lgorihm, he link flows for he exmple in Figure 4 re shown in Figure 5 nd he corresponding rouing is shown in Tble III. 6.6 6 A. Flow Flushing Algorihm Le he m-vecors c nd f represen he link cpciy nd he flow on he link respecively. The flow on he links is resul of rouing he differen source- desinion demnds in he nework. I is esy o see h M (f)+m (c f) M (c). This is rue since he se of flows in he wo erms on he lef hnd side of he inequliy is fesible flow for he righ hnd side of he inequliy. Therefore M (f) M (c) M (c f). If f is he resul of rouing he source-desinion demnds hen M ( x(p )) M(c) M(c x(p )). k P P k :e P k P P k :e P 1 Fig. 5. 7.76 7.23 2 3 3.95 Flow Flushing Algorihm 5.19 8.21 5 4 7.6 2.19 Minimum Cu 9.95 The mximum flow M (f) on his nework is 9.95 unis. The vlue of he gme θ =5/9.95. We now ouline noher
Source Phs Flow Des. Pir 1-3 1-3 5.00 1-4 1-2-3-4 0.76 1-3-4 2.23 1-5 1-2-5 7.00 2-5 2-5 1.21 2-6-5 6.60 2-3-4-5 2.19 2-3 2-3 1.00 6-5 6-5 1.00 TABLE III Cu rcs FLOWS FOR BASE CASE s heurisic h cn be used by he service provider o improve he probbiliy of deecion of he mlicious pcke. B. Cu Surion Algorihm This lgorihm relies on he fc h he mximum flow beween nd is upper bounded by he size of ny cu. Le C represen he se of links in some cu. Given ny link e E, leα(e) nd β(e) represen he sr nd end nodes of h link. The cu surion lgorihm picks some cu nd ries o direc flow wy from his cu. Once he sourcedesinion demnds re roued, his cu will be smll nd hence will limi he mximum flow. This is done s follows: Inroduce wo new nodes s nd. Inroduce n rc beween node s nd ll nodes α(e) for ll e C. Similrly inroduce links beween ech node β(e) for ech e C nd he node.the objecive now is o deermine he highes flow h cn be sen from s o while minining he fesibiliy of rouing he source-desinion demnds. The modificion of he nework is shown in Figure 6. The only links shown in he nework re he cu links. This problem cn be solved lmos ideniclly o he Flow Flushing Algorihm, excep h he K +1commodiy flows go beween nodes s nd. One wy of choosing he cu h is o be sured is s follows: Assume h we currenly hve rouing of he source-desinion demnds resuling in flow of f(e) on link e. Deermine minimum cu (using hese flows f s he cpciies). Tke his cu o be C nd now emp o sure his cu. Coninuing he exmple in Figure 4, ssume h he cu h we sure comprises of he links (1, 2) nd (4, 5). The links flows re shown in Figure 7 nd he corresponding flows re shown in Tble IV. The mximum flow M (f) on his nework is 8.0 unis. The vlue of he gme θ =5/8. Therefore, in his exmple, he cu Fig. 6. 1 Fig. 7. Cu Surion Algorihm Nework Se-up 5.3 9.7 2 3 1.0 7.15 4.7 Cu Surion Algorihm 8.15 6 5 8.15 4 1.7 Minimum Cu 7.0 surion lgorihm gives beer soluion h he flow flushing lgorihm. V. VARIANTS AND EXTENSIONS We consider severl vrins of he problem oulined bove. The firs vrin h we consider is he cse where he inruder cn inroduce he mlicious pcke one of se of nodes A N. We ssume h / A. The second vrin h we consider is he cse where he objecive of he inruder is o rech ny one of of se of nodes T N. We ssume h A T =. Boh hese cses re esy o solve by inroducing super source node h is conneced o ll nodes in A nd connecing ll nodes in T o super sink node. The gme is now plyed beween he super source node nd he super sink node. Anoher vrin is he cse where he inruder cn inroduce he pcke ny one
Source Phs Flow Des. Pir 1-3 1-3 5.00 1-4 1-3-4 3.00 1-5 1-3-4-5 1.70 1-2-5 5.30 2-5 2-5 2.85 2-6-5 7.15 2-3 2-3 1.00 6-5 6-5 1.00 TABLE IV FLOWS FOR BASE CASE Noe h L(d) represens he mximum flow h cn be sen from ll he nodes in A o he desinion node d. Thevlueof he gme is B/L(d). VI. EXPERIMENTAL RESULTS In his secion we evlued he lgorihms developed on wo neworks. The firs nework is shown in Figure 8. Ech undireced link in he figure represens wo direced links ech hving cpciy of 10 unis. We performed he following experi- 2 5 12 11 of se of nodes A bu we ssume h he inruder does no hve conrol of he rouing in he nework. Insed, we ssume h he rouing in he nework is shores ph rouing like in OSPF or IS- IS. We erm his shores ph rouing gme. 1 3 6 13 A. Shores Ph Rouing Gme We now consider he problem where he rouing in he nework is long shores phs. We ssume h ech link hs lengh nd pckes re roued from he source o he desinion long shores phs ccording o his lengh meric. We ssume h ies re broken rbirrily. Therefore given ny wo nodes in he nework, here is unique ph from one node o he oher. Given rge node, ll pckes rriving his node rverse he shores ph ree. Shores ph rouing implies h here is unique ree rooed he desinion. A pcke inroduced ny node in he nework rverses he unique ph from h node o he desinion long he links in he shores ph ree. We use A o represen he se of nodes h he inruder cn inroduce mlicious pcke ino he nework. The objecive of he inruder is o deermine which node of his se A o inroduce he pcke ino nd he objecive for he service provider is o deermine he smpling re he links subjec o smpling budge of B. The min difference beween his problem nd he problem h we originlly sudied is he fc h i is esy o compue he mximum flow nd hence he minimum cu on ree. The lgorihm for solving his problem is he following: Elimine ll lef nodes in he rouing ree h do no belong o A. LeT represen his ree. Le P (i) represen he predecessor of node i on T. Se L(i) = for ll lef nodes. While here re no lef nodes do Pick lef node i. Lee be he edge connecing i o P (i). Se L(P (i)) L(P (i)) + min{l(i),f e }. Oupu L(). 4 Fig. 8. Experimenl Nework 1 8 7 mens: Single ck node nd single rge node. (3 problems). Muliple ck node nd single rge node. (1 problem). Muliple ck node nd muliple rge node. (1 problem). For ech of he cses, we rn hree differen lgorihms. 1) Rouing o minimize he highes uilized link wih f 1 represening he m-vecor of link flows s resul of his rouing lgorihm. 2) Rouing wih flow flushing lgorihm wih f 2 represening he m-vecor of link flows s resul of his rouing lgorihm. 3) Rouing wih cu surion lgorihm wih f 3 represening he m-vecor of link flows s resul of his rouing lgorihm. Le M(f i ) for i =1, 2, 3 represen he mximum flow h cn be sen from node o using f i s he link cpciies. If B is he smpling budge, hen he vlue of he gme θ = B/M(). Tble V shows he vlues of M() insed of θ. The smller h vlue of M, he beer he chnces of deecion for given smpling budge. 9 10 15 14
Ack Trge M(f 1 ) M(f 2 ) M(f 3 ) Node(s) Node(s) 1 13 13.2 8.9 9.1 5 7 9.2 7.55 7.55 7 11 16.4 7.3 7.1 1,2,4,8 13 16.2 9.4 8.7 1,2,4,8 12,13,14 24.88 19.5 18.9 TABLE V COMPARISON OF DIFFERENT ROUTING ALGORITHMS From he ble, noe h he mximum flow vlue nd hence he vlue of he gme cn be chnged significnly by chnging he rouing in he nework. In mos of he exmples he performnce of he flow flusing lgorihm nd he cu surion lgorihm re quie similr, nd beer hn he simple minimizion of mximum link uilizion lgorihm A. Effec of Cpciy on he Vlue of he Gme As he moun of spre cpciy in nework increses, he opporuniy o reroue flows increses. This implies h he service provider cn improve he probbiliy of deecion by exploiing he spre cpciy o reroue flows. We illusre his in he following se of experimens, using second exmple nework, where he cpciy of he links in he exmple nework re fixed some consn vlue C. IfhevlueofC increses, hen he opporuniy o reroue flows goes up. We consider he inrusion deecion gme beween nodes =1nd =13.The demnds in he nework re uniformly disribued beween zero nd one. We firs run he lgorihm o roue he flow such h mximum uilizion of ny link is minimized. This mximum uilizion vlue versus he link cpciy C is shown in Figure 9. As he mximum uilizion becomes lower, he moun of spre cpciy o reroue flows increses in he nework. This implies h boh he flow flushing lgorihm s well s he cu surion lgorihm will hve more lerne phs. In Figure 10, we show he performnce of he flow flushing lgorihm s he vlue of C increses. The srigh line in he plo shows he performnce of he bse cse which is he rouing lgorihm h minimizes he mximum uilizion. The mximum flow is independen of he vlue of C. In he cse of he flow flushing lgorihm, he mximum flow vlue decreses wih incresing link cpciy. I sympoes bou 8.8. The sme kind of performnce ws observed in he cse of oher ck-rge pirs s well s he cse for muliple ck sies. VII. CONCLUDING REMARKS We considered he problem of deecing inruding pckes in nework by mens of nework pcke smpling. Since MAX UTILIZATION 0.5 0.6 0.7 0.8 0.9 5 6 7 8 9 10 LINK CAPACITY Fig. 9. Mx. Uilizion vs. Link Cpciy for flow rouing o minimize mximum link uilizion. 1-13 MAXIMUM FLOW 9 10 11 12 13 14 Fig. 10. MAXFLOW BEFORE FFA MAXFLOW AFTER FFA 5 6 7 8 9 10 LINK CAPACITY Performnce of flow flushing lgorihm for differen link cpciies pcke smpling nd exminion in rel-ime cn be expensive, he nework operor mus devise n effecive smpling scheme o deec inruding pckes injeced ino he nework by n dversry. We considered he scenrio where he dversry hs considerble informion bou he nework nd cn eiher pick phs o minimize chnces of deecion or cn pick suible nework ingress-poin if only shores ph rouing is llowed. The deecion vi smpling problem ws formuled in gme-heoreic frmework. The soluion o his gme-heoreic problem is mx-flow problem from which he sble opering poins re obined. We lso considered he nework op-
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