Network Coding: An Instant Primer
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- Piers Cummings
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1 Network Coding: n Instnt Primer LC-REPORT Christin Frgouli EPFL - IC christin.frgouli@epfl.ch Jen-Yves Le oudec EPFL - IC jen-yves.leoudec@epfl.ch Jörg Widmer DoCoMo Ls widmer@docomoleuro.com STRCT Network coding is new reserch re tht my hve interesting pplictions in prcticl networking systems. With network coding, intermedite nodes my send out pckets tht re liner comintions of previously received informtion. There re two min enefits of this pproch: potentil throughput improvements nd high degree of roustness. Roustness trnsltes into loss resilience nd fcilittes the design of simple distriuted lgorithms tht perform well, even if decisions re sed only on prtil informtion. This pper is n instnt primer on network coding: we explin wht network coding does nd how it does it. We lso discuss the implictions of theoreticl results on network coding for relistic settings nd show how network coding cn e used in prctice. Ctegories nd Suject Descriptors H.4 [Informtion Systems pplictions]: Miscellneous Generl Terms Network Coding Keywords Network Coding 1. INTRODUCTION Communiction networks tody shre the sme fundmentl principle of opertion. Whether it is pckets over the Internet, or signls in phone network, informtion is trnsported in the sme wy s crs shre highwy or fluids shre pipes. Tht is, independent dt strems my shre network resources, ut the informtion itself is seprte. Routing, dt storge, error control, nd generlly ll network functions re sed on this ssumption. Network coding [2] is recent field in informtion theory tht reks with this ssumption. Insted of simply forwrding dt, nodes my recomine severl input pckets into one or severl output pckets. simple exmple in wireless context is three node topology, s shown in Figure 1. Liner network coding, in generl, is similr to this exmple, with the difference tht the xor opertion is replced y liner comintion of the dt, interpreted s numers over some finite field. This llows for much ville online from infoscience.epfl.ch Trditionl Method S S S S Network Coding S S xor S xor Figure 1: simple network coding exmple. Nodes nd wnt to exchnge pckets vi n intermedite node S (wireless se sttion). [resp. ] sends pcket [resp. ] to, which then rodcsts xor insted of nd in sequence. oth nd cn recover the pcket of interest, while the numer of trnsmissions is reduced. lrger degree of flexiility in the wy pckets cn e comined. In ddition to the throughput enefits evidenced in this exmple, network coding is lso very well suited for for environments where only prtil or uncertin informtion is ville for decision mking. Similr to ersure coding, successful reception of informtion does not depend on receiving specific pcket content ut rther on receiving sufficient numer of independent pckets. Liner comining requires enhnced computtionl cpilities t the nodes of the network. However, ccording to Moore s lw, processing is ecoming less nd less expensive. The ottleneck hs shifted to network ndwith to support the ever-growing demnd in pplictions nd QoS gurntees over lrge unrelile networks. Network coding utilizes chep computtionl power to increse network efficcy. The gol of this pper is to mke the sic concepts of network coding ville to the networking community. 1 Section 2 explins wht it is nd how it cn e implemented. Section 3 discusses existing results on the performnce enefits of network coding. Section 4 reviews proposls to use network coding in informtion nd networking systems. 2. WHT IS NETWORK CODING? 2.1 Liner Network Coding Consider system tht cts s informtion rely, such s router, node in n d-hoc network, or node in peer 1 For n exhustive list of literture on network coding see
2 to peer distriution network. Trditionlly, when forwrding n informtion pcket destined to some other node, it simply repets it. With network coding, we llow the node to comine numer of pckets it hs received or creted into one or severl outgoing pckets. ssume tht ech pcket consists of L its. When the pckets to e comined do not hve the sme size, the shorter ones re pdded with triling 0s. We cn interpret s consecutive its of pcket s symol over the field F 2 s, with ech pcket consisting of vector of L/s symols. With liner network coding, outgoing pckets re liner comintions of the originl pckets, where ddition nd multipliction re performed over the field F 2 s (see Section 2.5 for wht this mens). The reson for choosing liner frmework is tht the lgorithms for coding nd decoding re well understood. Liner comintion is not conctention: if we linerly comine pckets of length L, the resulting encoded pcket lso hs size L. In contrst to conctention, ech encoded pcket contins only frction of the informtion contined in originl pckets. One cn think of liner network coding s form of informtion spreding. s we explin in Section 3, this hs enefit in mny cses (s in Figure 1). 2.2 Encoding ssume tht numer of originl pckets M 1,..., M n re generted y one or severl sources. In liner network coding, ech pcket through the network is ssocited with sequence of coefficients g 1,..., g n in F 2 s nd is equl to X = P n i=1 gim i. The summtion hs to occur for every symol position, i.e., X k = P n i=1 gim i k,wherem i k nd X k is the kth symol of M i nd X respectively. In the exmple of Figure 1, the field is F 2 = {0, 1}, symol is it, nd the liner comintion sent y S fter receiving M 1 = nd M 2 = is M 1 + M 2 (the + sign here is ddition in F 2, i.e., itwise xor). For simplicity, we ssume tht pcket contins oth the coefficients g =(g 1,..., g n), clled encoding vector, ndthe P n encoded dt X = i=1 gim i, clled informtion vector [7]. The encoding vector is used y recipients to decode the dt, s explined lter. For exmple, the encoding vector e i =(0,..., 0, 1, 0,...0), where the 1 is t the ith position, mens tht the informtion vector is equl to M i (i.e., is not encoded). Encoding cn e performed recursively, nmely, to lredy encoded pckets. Consider node tht hs received nd stored set (g 1,X 1 ),..., (g m,x m ) of encoded pckets, where g j [resp. X j ] is the encoding [resp. informtion] vector of the jth pcket. This node my generte new encoded pcket (g,x ) y picking set of coefficients P h 1,..., h m nd computing the liner comintion X m = j=1 hjxj. The corresponding encoding vector g is not simply equl to h, since the coefficients re with respect to the originl pckets M 1,..., M n ; in contrst, strightforwrd lger shows P tht it is given y g i m = j=1 hjgj i. This opertion my e repeted t severl nodes in the network. 2.3 Decoding ssume node hs received the set (g 1,X 1 ),..., (g m,x m ). In order to retrieve the originl pckets, it needs to solve the system {X j = P n i=1 gj i M i } (where the unknowns re M i ). This is liner system with m equtions nd n unknowns. We need m n to hve chnce of recovering ll dt, i.e. the numer of received pckets needs to e t lest s lrge s the numer of originl pckets. Conversely, the condition m n is not sufficient, s some of the comintions might e linerly dependent. However, nd this is mjor ppel of network coding, this is esy, s we discuss next. 2.4 How to Select the Liner Comintions The prolem of network code design is to select wht liner comintions ech node of the network performs. simple lgorithm is to hve ech node in the network select uniformly t rndom the coefficients over the field F 2 s,in completely independent nd decentrlized mnner [26]. With rndom network coding there is certin proility of selecting linerly dependent comintions [26]. This proility is relted to the field size 2 s. Simultion results indicte tht even for smll field sizes (for exmple, s =8) this proility ecomes negligile [29]. lterntively, we cn use deterministic lgorithms to design network codes. The polynomil-time lgorithm for multicsting in [21], sequentilly exmines ech node of the network, nd decides wht liner comintions ech node performs. Since ech node uses fixed liner coefficients, the pckets only need to crry the informtion vector. There lso exist deterministic decentrlized lgorithms tht pply to restricted fmilies of network configurtions [4]. 2.5 Prcticl Considertions Decoding: Decoding requires solving set of liner equtions. In prctice, this cn e done s follows. node stores the encoded vectors it receives s well s its own originl pckets, row y row, in so-clled decoding mtrix. Initilly, it contins only the non-encoded pckets issued y this node with the corresponding encoding vectors (if ny, else it is empty). When n encoded pcket is received, it is inserted s lst row into the decoding mtrix. The mtrix is trnsformed to tringulr mtrix 2 using Gussin elimintion. received pcket is clled innovtive if it increses the rnk of the mtrix. If pcket is non-innovtive, it is reduced to row of 0s y Gussin elimintion nd is ignored. s soon s the mtrix contins row of the form e i, this node knows tht x is equl to the originl pcket M i. This occurs t the ltest when n linerly independent encoded vectors re received. Note tht decoding does not need to e performed t ll nodes of the network, ut only t the receivers. Genertions: For ll prcticl purposes, the size of the mtrices with which network coding opertes hs to e limited. This is strightforwrd to chieve for deterministic network codes ut more difficult with rndom network coding. For the ltter, pckets re usully grouped into soclled genertions, nd only pckets of the sme genertion cn e comined [7]. Size nd composition of genertions my hve significnt impct on the performnce of network coding [11]. Similr considertions hold for the size of the finite field. oth prmeters llow to trde off performnce for lower memory requirements nd reduced computtionl complexity. Dely: The fct tht pckets need to e decoded hs minor impct on dely. It is usully not necessry to receive ll encoded pckets efore some of the pckets cn 2 The mtrix cn e trnsformed to reduced row echelon form, where within ech row, the leding term is 1, nd the position of the leding terms moves to the right.
3 e decoded (i.e., whenever Gussin elimintion leds to row in the form (e i,m i)). Together with reduction in the numer of required trnsmissions, the overll end-toend dely with network coding is usully not lrger thn the norml end-to-end dely in relistic settings. Finite field opertions: Network coding requires opertions in F 2 s, i.e., opertions on strings of s its. ddition is the stndrd itwise xor. For multipliction, one interprets sequence 0,..., s 1 of s its s the polynomil 0+ 1X s 1X s 1. Then one picks polynomil of degree s tht is irreducile over F 2 (there re severl of them, nd ech gives different representtion of F 2 s; for exmple, Rijndel s representtion of F 2 8 uses 1+X +X 3 +X 4 +X 8 ). Multipliction is otined y first computing the usul product of two polynomils (which gives polynomil of degree possily lrger thn s 1), nd then computing the reminder modulo the chosen irreducile polynomil. Division is computed y the Euclidin lgorithm. oth multipliction nd division cn e implemented efficiently with s shifts nd dditions [27]. If s is smll (e.g., s = 8), fster lterntive is to use discrete logrithms. In finite field there exists t lest one specil element α, clled genertor (for exmple, α = 0x03 = 1 + X is genertor in Rijndel s representtion of F 2 8), ny non-zero x cn e written in unique wy x = α l(x) ; l(x) is clled the logrithm [18]. Since l(xy) = l(x) + l(y), multipliction nd division cn e implemented y looking up the two tles tht mp x to l(x) nd vicevers. For s = 8 these re two tles of size 255 ytes. 3. WHT RE THE ENEFITS OF NET- WORK CODING? Theoreticlly proven results out network coding minly concern performnce improvements in sttic settings. We review these first nd then discuss rndom distriuted settings. 3.1 Throughput Gin in Sttic Environment primry result tht sprked the interest in network coding is tht it cn increse the cpcity of network for multicst flows. More specificlly, consider network tht cn e represented s directed grph (typiclly, this is wired network). The vertices of the grph correspond to terminls, nd the edges of the grph corresponds to chnnels. ssume tht we hve M sources, ech sending informtion t some given rte, nd N receivers. ll receivers re interested in receiving ll sources. Theorem 1. [23, 24] ssume tht the source rtes re such tht, without network coding, the network cn support ech receiver in isoltion (i.e. ech receiver cn decode ll sources when it is the only receiver in the network). With n pproprite choice of liner coding coefficients, the network cn support ll receivers simultneously. In other words, when the N receivers shre the network resources, ech of them cn receive the mximum rte it could hope to receive, even if it were using ll the network resources y itself. Thus, network coding cn help to etter shre the ville network resources (Figure 2). Network coding my offer throughput enefits not only for multicst flows, ut lso for other trffic ptterns, such s unicst. Consider gin Figure 2 ut ssume now tht x Trditionl Method Network Coding S 1 S 2 S 1 S 2 x 1 x 2 x 1 x 2 C x 2 x 1 C x 2 1 x 1 x 1 + x 2 E x 1 x 1 F E F R 1 D R 2 R 1 D R 2 Figure 2: (utterfly Network) S 1 nd S 2 multicst to oth R 1 nd R 2. ll links hve cpcity 1. With network coding (y xoring the dt on link CD), the chievle rtes re 2 for ech source, the sme s if every destintion were using the network for its sole use. Without network coding, the chievle rtes re less (for exmple if oth rtes re equl, the mximum rte is 1.5). source S 1 trnsmits to destintion R 2 nd S 2 to R 1. With network coding we cn send rte 1 to ech receiver, while without, we cn only send rte 1/2 to ech receiver. There exist directed grphs where the throughput gins of network coding for multicsting cn e very significnt [21, 6]. However, in undirected grphs (e.g., wired network where ll links re hlf-duplex) the throughput gin is t most fctor of two [17]. Experimentl results in [29] over the network grphs of six Internet service providers showed smll throughput gin in this cse. multicommodity flow prolem ). n interesting point is tht network coding llows to chieve the optiml throughput when multicsting using polynomil time lgorithms. In contrst, chieving the optiml throughput with routing is NP-complete: this is the prolem of pcking Steiner trees in CS theory. Thus, even when the expected throughput enefits of network coding re not lrge, we expect to e le to chieve them using simpler lgorithms. We expnd on this point in the following. 3.2 Roustness nd dptility The most compelling enefits of network coding might e in terms of roustness nd dptility. Intuitively, we cn think tht network coding, similrly to trditionl coding, tkes informtion pckets nd produces encoded pckets, where ech encoded pcket is eqully importnt. Provided we receive sufficient numer of encoded pckets, no mtter which, we re le to decode. The new twist tht network coding rings, is tht the liner comining is performed opportunisticlly over the network, not only t the source node, nd thus it is well suited for the (typicl) cses where nodes only hve incomplete informtion out the glol network stte. Consider gin Figure 1 nd ssume tht nd my go into sleep mode (or my move out of rnge) t rndom nd without notifying the se sttion S. If the se sttion S rodcsts (or ), the trnsmission might e completely wsted, since the intended destintion might not e le to receive. However, if the se sttion rodcsts xor, or more generlly, rndom liner comintions of the informtion pckets, the trnsmission will ring new informtion to ll ctive nodes. This rgument is the sis for the significnt performnce improvements reported in Section 4.2.
4 Coupon Collector Prolem This is generic prolem for which theoreticl result is ville, nd which explins the performnce gins reported in Section 4.1. Consider network with n nodes nd O(n) messges 3. ll nodes would like to receive ll messges. centrlized gossip-sed protocol llows to disseminte the messges in Θ(n) rounds, where in ech round Θ(n) pirs of nodes exchnge one messge. To perform the sme tsk in decentrlized mnner, we need Θ(n log n) rounds. This is ecuse, for receiver specific messge my ecome rre, i.e., hrd to collect. llowing codes to use network coding nd propgte rndom liner comintions of their messges insted, mkes ll pckets equl : it is sufficient for node to collect ny n messges. This pproch llows to disseminte ll messges in Θ(n) rounds [8]. Thus, network coding llows to chieve the optiml performnce using simple decentrlized lgorithm. Pcket Ersure Networks We then consider networks where pckets my e dropped. We rgue tht for some pplictions the enefits of network coding outweigh those of the pproches employed tody. Forwrd error correcting (FEC) schemes t pcket level offer n lterntive to utomtic repet request (RQ), for pplictions with high lod nd low dely requirements. Recently, we hve witnessed the emergence of Fountin codes, set of rte-less codes prticulrly suited for such pplictions. Fountin codes re still end-to-end: pckets re encoded t the source nd decoded t the destintion, while intermedite nodes re only llowed to replicte nd forwrd pckets [25]. pplying ides from network coding in this context, i.e., llowing intermedite nodes to lso form liner comintions, cn led to significnt improvements. For exmple, consider source tht would like to trnsmit informtion to destintion C. On the pth from to C there exists router tht cn perform network coding opertions. ssume tht node sends encoded pckets, tht re dropped on pths nd C with proility ɛ nd ɛ C respectively. Using n end-to-end FEC scheme, i.e., hving the destintion C decode the pckets it receives, restricts the rte to R 1 (1 ɛ )(1 ɛ C). If we llow the router to perfectly decode nd re-encode, we will chieve the optiml min-cut rte R 2 min{(1 ɛ ), (1 ɛ C)}, ut t the cost of dditionl dely: we hve to wit t node to receive sufficient encoded pckets to e le to decode nd re-encode the informtion. Using n RQ scheme will gin llow to chieve rte R 2, ut gin t the cost of incresed dely. lterntively, node cn t ech time instnce form nd send rndom liner comintions of the encoded pckets it hs received up to tht time, without witing for ll encoded pckets. We cn then chieve the optiml rte R 1 without n dditionl dely. Moreover, it is sufficient to perform xor opertions. This scheme in its full generlity cn e pplied over n ritrry network topology, nd with diverse trffic lod (multicsting, unicsting, rodcsting, etc.) [19, 20]. 4. WHERE CN NETWORK CODING E USED? 3 We use Knuth s nottion: f(n) = O(g(n)) mens tht there exists constnt c nd integer N such tht f(n) cg(n) for n > N; f(n) = Θ(g(n)) denotes tht f(n) = O(g(n)) s well s g(n) = O(f(n)). In the following, we list numer of pplictions of network coding nd discuss how the enefits mentioned in Section 3 improve performnce in concrete settings. 4.1 P2P File Distriution Proly the most widely known ppliction using network coding is vlnche [12, 1]. Generlly, in peer-topeer content distriution network, server splits lrge file into numer of locks. Peer nodes try to retrieve the originl file y downloding locks from the server ut lso distriuting downloded locks mong them. To this end, peers mintin connections to limited numer of neighoring peers (rndomly selected mong the set of peers) with which they exchnge locks. In vlnche, the locks sent out y the server re rndom liner comintions of ll originl locks. Similrly, peers send out rndom liner comintions of ll the locks ville to them. node cn either determine how mny innovtive locks it cn trnsmit to neighor y compring its own nd the neighor s mtrix of decoding coefficients, or it cn simply trnsmit coded locks until the neighor receives the first non-innovtive lock. The node then stops trnsmitting to this neighor until it receives further innovtive locks from other nodes. Coding coefficients re trnsmitted together with the locks, ut since locks usully hve size of hundreds of kiloytes, this overhed is negligile. Network coding helps in severl respects. 1) It minimizes downlod times; in such lrge scle distriuted peer-topeer system, optiml pcket scheduling is very complex, prticulrly if the prticipting hosts only hve very limited informtion out the underlying network topology. With network coding, the performnce of the system depends much less on the specific overly topology nd schedule. Consequently, very simple mechnisms tht construct rndom overly cn e used. The uthors show tht network coding outperforms trditionl forwrding or FECsed peer-to-peer systems y significnt mrgin. 2) Due to the diversity of the coded locks, network coding sed solution is much more roust in cse the server leves erly (efore ll peers hve finished their downlod) or in the fce of high churn rtes (where nodes only join for short period of time or leve immeditely fter finishing their downlod). 3) In contrst to forwrding sed protocols, their network coding protocol suffers only smll performnce penlty when incentive mechnisms to cooperte re implemented (e.g., tit-for-tt to prevent free-riding). 4.2 Wireless Networks idirectionl trffic in wireless network: s shown in Figure 1, network coding cn improve throughput when two wireless nodes communicte vi common se-sttion. This setting cn e extended to the cse of multi-hop routing in wireless network (or ny other network with physicl lyer rodcst) where the trffic etween the two end nodes is idirectionl [30] nd oth nodes hve similr numer of pckets to exchnge. Given schedule tht lterntes etween djcent routers, fter few initil steps ll intermedite routers hve pckets uffered for trnsmission in oth directions of the pth. Whenever trnsmission opportunity rises, router comines two pckets, one for ech direction, with simple xor nd rodcsts it to its neighors. oth receiving routers lredy know one of the pcket
5 the rodcst is coded over, while the other pcket is new. Thus, ech rodcst llows two routers to receive new pcket, effectively douling the cpcity of the pth. The uthors in [30] lso discuss distriuted implementtion tht works when trnsmissions re not synchronized nd the wireless chnnel is lossy nd hs rndom dely. Overhering pcket of neighor tht is coded over informtion previously forwrded to the neighor serves s pssive cknowledgment. This llows to mke etter use of trnsmission opportunities t routers tht only hve new pckets uffered for single direction. In this cse, one of the new pckets is comined with n old pcket for the reverse direction for which no pssive cknowledgment hs een received. Residentil wireless mesh networks: Even limited form of network coding which only uses xor to comine pckets my significntly improve network performnce in wireless mesh networks [16]. ll trnsmissions re rodcst nd re overherd y the neighors. Pckets re nnotted with summry informtion out ll other pckets node lredy herd. This wy, informtion out which nodes hold which pckets is distriuted within the neighorhood. node cn xor multiple pckets for different neighors nd send them in single trnsmission, if ech neighor lredy hs the remining informtion to decode the pcket. In experiments with hrdwre, the uthors show tht their mechnism lmost doules network throughput. Mny-to-mny rodcst: Network-wide rodcst is used for numer of purposes in d-hoc networks (e.g., route discovery) nd cn e implemented much more efficiently with network coding [15, 11]. lredy simple distriuted lgorithm for rndom network coding reduces the numer of trnsmission y fctor of 2 or more, leding to significnt energy svings. In such setting, lrger trnsmit power directly trnsltes into reduction in the numer of required trnsmissions, which llows for interesting energy trdeoffs. Energy expenditure is either evenly distriuted mong the nodes or covered y only few nodes (mye with longer ttery life). There is lrger flexiility in the distriution the energy requirements compred to conventionl lgorithms. lgorithms for chllenging wireless networks which my e very sprse or highly moile re investigted in [28]. Usully, lgorithms for such networks employ some form of intelligent flooding to comt the dverse network conditions. With network coding, performnce improvements re much lrger thn the ones reported ove, indicting tht network coding is prticulrly suitle when roust opertion is of high importnce. The uthors present simple mechnism for network wide rodcst tht hs much lower overhed thn flooding nd lso discuss prcticl implementtion issues for network coding in wireless networks. Similr performnce enefits cn e expected when nodes spend most of their time in sleep mode to sve energy or use rndomized emforming with directionl ntenns. ll of these cses hve in common tht specific pir of nodes is not likely to e le to communicte t given time. 4.3 d-hoc Sensor Networks Untuned rdios in sensor networks: novel nd interesting ppliction for network coding is, to use it to cope with untuned trnsceivers [22]. For sensor nodes which should e s simple nd chep to mnufcture s possile, the qurtz oscilltor to tune the rdio to specific frequency mkes up significnt frction of hrdwre cost nd design complexity. The uthors propose to replce the nlog oscilltor y much simpler on-chip resontor. s consequence, the trnsceivers rdio frequencies depend to some degree on vritions in the mnufcturing process nd two such devices re not very likely to e le to communicte directly. However, in dense sensor networks with thousnds of tiny devices nd multiple rdios per device, multi-hop pth etween informtion source nd dt sink will most proly exist. With rndom network coding, it is possile to use these pths without hving to explicitly find them, nd without the excessive overhed of flooding. In their pper, the uthors do not propose specific protocol ut rther give theoreticl nlysis, tht, given their ssumptions, untuned rdios with network coding perform only fctor of 1/e worse thn perfectly tuned rdios without network coding. They conclude, tht operting networks in such rndomized fshion my e preferle to the more trditionl wy of controlled opertion, since it implies simpler rdio rchitecture. Dt gthering in sensor networks: n interesting dt gthering lgorithm for sensor networks is presented in [9]. Nodes hve storge spce for one single pcket. Overherd pckets from neighoring nodes re multiplied with rndom coefficient nd dded to the existing informtion. Sensor nodes re thus not le to decode, ut in sensor network the gol is only to mke the dt ville to (more cple) sink node. Sensor informtion is pro-ctively distriuted to smll numer of other nodes, so tht sink node cn reconstruct n dt pckets with high proility y contcting only n sensor nodes nywhere in the network. 4.4 Network Tomogrphy In [10], network coding is used to infer the loss rtes of links in n overly network. For conventionl ctive proing, pckets re usully multicst to severl receivers. The receivers experience the sme loss event which provides informtion out losses in the underlying multicst tree. fter sufficiently lrge numer of proe pckets, shred links nd their loss rtes cn e identified with resonle ccurcy. In such setting, network coding provides dditionl flexiility since pckets re not only duplicted t rnching points of the multicst tree, ut my lso e merged. If multiple senders unicst pckets to single receiver, nd these pckets re comined within the network, it llows to infer the topology in much the sme wy s multicsting from one sender to multiple receivers. Furthermore, if the network code (i.e., the specific wy in which pckets re comined t the nodes) is known in dvnce, the coding coefficients contined in the proe pckets provide dditionl informtion out the originl pckets tht were comined (nd consequently which pckets were lost in which prt of the tree). y exploiting these fetures, it is possile to significntly reduce the numer of ctive proes nd the link stress generted y these proes. Network coding hs lso een proposed for pssive network monitoring. s with ctive monitoring, rndom (ut fixed) network code llows receivers to determine which coefficients re expected under norml opertion. When the otined coefficients differ, the receiver cn drw from tht conclusions out the filure ptterns [13].
6 4.5 Network security In [5], the uthors investigte the prolem of designing secure network codes for wiretp networks, where certin links cn e ccessed y ttckers. They ssume tht it is known which links re tpped. The source comines the originl dt with rndom informtion nd designs network code in wy tht only the receivers re le to decode the originl pckets. Furthermore, the mutul informtion etween the pckets otined y the evesdroppers nd the originl pckets is zero (security in the informtion theoretic sense). The fct tht with network coding, nodes cn only decode pckets if they hve received sufficient numer of linerly independent informtion vectors llows for weker form of security [3]. Such codes re more efficient, ut n ttcker who hs n 1 outofn liner comintions only hs to guess the content of single pcket to e le to decode ll n pckets (hence the nme wek security ). Finlly, network coding simplifies the protection ginst modified pckets in network [14]. t norml network (nd no dditionl protection), n intermedite ttcker my mke ritrry modifictions to pcket to chieve certin rection t the ttcked destintion. However, in the cse of network coding, n ttcker cnnot control the outcome of the decoding process t the destintion, without knowing ll other coded pckets the destintion will receive. Given tht pckets re routed long mny different pths, this mkes controlled mn-in-the-middle-ttcks more difficult. 5. CONCLUSION Network coding my hve impct on the design of new networking nd informtion dissemintion protocols. y llowing to etter spred the informtion content in the network environment, it cn simplify distriuted lgorithms. Relile multicst is n exmple where the existing solutions need to e re-thought; we hve shown in our review tht emerging res such s d-hoc networks, overly infrstructures nd sensor networks re strting to enefit of network coding. More such pplictions re expected to emerge. 6. REFERENCES [1] vlnche: File swrming with network coding. plo/vlnche.spx. [2] R. hlswede, N. Ci, S. R. Li, nd R. W. Yeung. Network informtion flow. IEEE Trnsctions on Informtion Theory, July [3] K. httd nd K. R. Nyynn. Wekly secure network coding. In Proc. First Workshop on Network Coding, Theory, nd pplictions (NetCod), pr [4] C. Frgouli nd E Soljnin. Decentrlized network coding. Informtion Theory Workshop, Octoer [5] N. Ci nd R. W. Yeung. Secure network coding. In Interntionl Symposium on Informtion Theory (ISIT), [6] C. Chekuri, C. Frgouli, nd E. Soljnin. On verge throughput nd lphet size in network coding. ccepted to IEEE Trns. Inform. Theory. [7] P.. Chou, Y. Wu, nd K. Jin. Prcticl network coding. In Proc. llerton, Oct [8] S. De nd M. Medrd. lgeric gossip: network coding pproch to optiml multiple rumor mongering. In Proc. llerton, Oct [9]. G. Dimkis, V. Prhkrn, nd K. Rmchndrn. Uiquitous ccess to distriuted dt in lrge-scle sensor networks through decentrlized ersure codes. In Symposium on Informtion Processing in Sensor Networks (IPSN), pr [10] C. Frgouli nd. Mrkopoulou. network coding pproch to overly network monitoring. In llerton Conference, Sept [11] C. Frgouli, J. Widmer, nd J.-Y. L. oudec. network coding pproch to energy efficient rodcsting: from theory to prctice. Technicl Report LC-REPORT , ccepted t Infocom 2006, EPFL, July [12] C. Gkntsidis nd P. Rodriguez. Network coding for lrge scle content distriution. In Proc. IEEE Infocom, Mimi, FL, Mr [13] T.Ho,.Leong,Y.Chng,Y.Wen,ndR.Koetter. Network monitoring in multicst networks using network coding. In Interntionl Symposium on Informtion Theory (ISIT), [14] T. Ho,. Leong, R. Koetter, M. Mdrd, M. Effros, nd D. R. Krger. yzntine modifiction detection in multicst networks using rndomized network coding. In Interntionl Symposium on Informtion Theory (ISIT), [15] J. Widmer, C. Frgouli, nd J. Y. Leoudec. Energy efficient rodcsting in wireless d hoc networks. First Workshop on Network Coding, Mrch [16] S. Ktti, D. Kti, W. Hu, H. Rhul, nd M. Medrd. The importnce of eing opportunistic: Prcticl network coding for wireless environments. In llerton, [17] Z. Li nd. Li. Network coding in undirected networks. Proceeding of CISS 2004, [18] C. H. Lim nd P. J. Lee. More flexile exponentition with precomputtion. In Proc. dvnces in Cryptology: 14th nnul Interntionl Cryptology Conference, ug [19] D. S. Lun, M. Medrd,, nd M. Effros. On coding for relile communiction over pcket networks. In llerton nnul Conference on Communictions, Control, nd Computing, [20] P. Pkzd, C. Frgouli, nd. Shokrollhi. On low complexity coding for line networks. IEEE Interntionl Symposium on Informtion Theory (ISIT), ustrli, Sept [21] P. Snders, S. Egner, nd L. Tolhuizen. Polynomil time lgorithms for network informtion flow. Proc. 15th CM Symposium on Prllel lgorithms nd rchitectures, [22] D. Petrovic, K. Rmchndrn, nd J. Rey. Overcoming untuned rdios in wireless networks with network coding. In Proc. First Workshop on Network Coding, Theory, nd pplictions (NetCod), Itly, pr [23] R. hlswede, N. Ci, S-Y. R. Li, nd R. W. Yeung. Network informtion flow. IEEE Trns. Inform. Theory, pges , July [24] S-Y. R. Li, R. W. Yeung, nd N. Ci. Liner network coding. IEEE Trns. Inform. Theory, 49: , Fe
7 [25]. Shokrollhi. Rptor codes. Sumitted to IEEE Trns. Informtion Theory, [26] T. Ho, R. Koetter, M. Medrd, D. R. Krger, nd M. Effros. The enefits of coding over routing in rndomized setting. Interntionl Symposium on Informtion Theory (ISIT), pge 442, July [27] N. R. Wgner. The Lws of Cryptogrphy with Jv Code. ville online t Nel Wgner s home pge, [28] J. Widmer nd J.-Y. L. oudec. Network coding for efficient communiction in extreme networks. In Workshop on dely tolernt networking nd relted networks (WDTN-05), Phildelphi, P, ug [29] Y. Wu, P.. Chou, nd K. Jin. comprison of network coding nd tree pcking. ISIT 2004, [30] Y. Wu, P.. Chou, nd S.-Y. Kung. Informtion exchnge in wireless networks with network coding nd physicl-lyer rodcst. Technicl Report MSR-TR , Microsoft Reserch, ug
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