Relay Secrecy in Wireless Networks with Eavesdropper

Transcription

1 Relay Secrecy n Wreless Networks wth Eavesdropper Parvathnathan Venktasubramanam, Tng He and Lang Tong School of Electrcal and Computer Engneerng Cornell Unversty, Ithaca, NY Emal : {pv45, th255, lt35}@cornell.edu Abstract Anonymous montorng of transmssons n a wreless network by eavesdroppers can provde crtcal nformaton about the data flows n the network. It s, therefore, necessary to desgn network protocols that mantan secrecy of routes from eavesdroppers. In ths work, we present a mathematcal formulaton of route secrecy when eavesdroppers observe transmsson epochs of nodes. We propose a schedulng technque to provde complete secrecy of routes, and based on that, characterze achevable rate regons for two hop source destnaton pars wth a common relay under two PHY models : transmtter drected and recever drected spread spectrum sgnalng. The results are also extended to the case when an addtonal constrant on packet loss s mposed. Index Terms - Network Securty, Secrecy, Multple Access, Packet Loss. I. INTRODUCTION Wreless networks are prone to anonymous montorng by eavesdroppers, who wsh to gan valuable network nformaton, namely source-destnaton pars and data flows. Equpped wth ths knowledge, t s possble for malcous adversares to then target specfc routes for ntruson or jammng. Actve ntruson attacks can be countered by sophstcated ntruson detecton mechansms. On the other hand, passve montorng does not affect the network operaton, and s not possble to detect. It s, therefore, necessary to modfy the network protocols, so that nformaton about data flows or source-destnaton pars are not traceable by eavesdroppers montorng node transmssons. The nference of routng nformaton from montored transmssons, known as traffc analyss attack, s done n a varety of ways. The eavesdropper can dentfy a flow of traffc by correlatng packet contents, packet lengths or transmsson epochs across multple nodes. Encryptng and random paddng of bts are some measures adopted to remove the correlaton of contents and lengths of packets across nodes. In Ths work s supported n part by TRUST (The Team for Research n Ubqutous Secure Technology), whch receves support from the Natonal Scence Foundaton (NSF award number CCF ) and the followng organzatons: Csco, ESCHER, HP, IBM, Intel, Mcrosoft, ORNL, Qualcomm, Prell, Sun and Symantec, and the U. S. Army Research Laboratory under the Collaboratve Technology Allance Program, Cooperatve Agreement DAAD The U. S. Government s authorzed to reproduce and dstrbute reprnts for Government purposes notwthstandng any copyrght notaton thereon. Informaton Flows Montored Nodes Fg. 1: Wreless Network wth Eavesdropper ths work, we are nterested n the desgn of secure transmsson schedules to prevent nference of routes by traffc analyss. In the absence of an eavesdropper, the transmsson schedule of relayng nodes are dependent on the arrval of packets, subject to the delay requrements. When transmssons are montored, however, t may be necessary to decouple the transmsson schedule of the nodes from the actual traffc flow to prevent flow correlaton. For delay senstve traffc, ths may not be possble wthout reducng network performance. In partcular, the desgn of such schedules would requre transmsson of dummy packets and could also result n packet drops. It s, therefore, necessary to characterze the achevable network performance under secrecy constrants. Sources Destnatons Fg. 2: m 1 Relay In ths work, we consder the problem of secrecy for a one-hop multplex relay as shown n Fgure 2. In partcular, we characterze the set of achevable relay rates, when packets are subjected to a strct delay crteron under two PHY models : transmtter drected and recever drected spread spectrum sgnalng. We provde transmsson schemes to prevent flow correlaton and show that as the delay ncreases, the achevable rate regon converges to the optmal regon. Furthermore, we also present achevable rate

2 regons when an addtonal constrant on packet loss s mposed. A. Related Work A countermeasure to traffc analyss attacks was frst provded through the noton of Mx-net by Chaum[1]. A Mx s an ntermedate node that reencrypts and reorders packets from multple sources to prevent matchng of source and destnaton streams. The dea has been used effectvely n provdng anonymous communcaton for nternet applcatons[2], [3], [4], [5]. For wreless networks wthout delay constrants, ndependent random transmsson schedules were used n[6] to prevent flow correlaton. The use of randomzed routes as a countermeasure to traffc analyss attacks has been consdered n [7], [8]. For low latency networks, t has been shown n [9] that smple mxng technques are not effectve to prevent correlaton of transmsson epochs. Ther proposed soluton utlzed the dea of transmttng dummy packets to make departure epochs dentcal rrespectve of the flows. The dea of havng fxed transmsson schedules ndependent of routes has also been consdered n [10], where the authors gve bounds on the performance loss ncurred due to the secrecy constrants. The theoretcal framework for secrecy n ths work s motvated by the noton of equvocaton developed by Shannon n [11]. The secrecy constrant we consder s a specal case of Shannon s equvocaton, known as maxmum secrecy[12], wheren the observatons provde zero nformaton about the source. A. Defntons II. PROBLEM SETUP Let the network be represented by a graph G = (V, E), where V s the set of nodes and E s the set of lnks between pars of nodes. A lnk (A, B) belongng to E denotes that node B can lsten to the transmssons from A and vce-versa. Let Y A = {Y A (1), Y A (2), } denote the tme nstants (known as departure epochs) at whch A transmts packets. The transmsson rate T A of a node A s defned as the average number of packets per unt tme transmtted by A. In other words, n T A = lm n Y A (n). Packets from source nodes are routed through ntermedate relays before they reach the destnaton. In general, the tasks carred out by a relay can be mult varous; t can choose to decode and re-encode blocks of packets, t can relay unaltered packets after a random delay or t can re-order the packets before transmsson. Re-encrypton and packet paddng occur at every node/relay to prevent any content based correlaton. We are concerned wth the knd of traffc, wheren each packet needs to be relayed wthn a fxed delay constrant. We restrct the tasks of a relay to packet-reorderng and tmng perturbaton. Dependng on ts transmsson schedule, a relay pcks departure epochs for the arrvng packets such that the delay constrant s satsfed. A packet that s not relayed wthn tme unts after arrval s dropped. A formal defnton of a relay functon s gven as follows. Let Y A = {Y A (1), Y A (2),, Y A (n)} represent the departure epochs of packets from A and Y B = {Y B (1), Y B (2),, Y B (n)} represent the departure epochs of packets from B. A 1 1 relay map s an algorthm that pcks a subsequence YA s of Y A and an equal length subsequence YB s of Y B such that, 0 YB s() Y A s (). If Y A = n and YA s = k(n), then the relay rate λ(m) of the 1 1 relay map M s gven by k(n) λ = lm n YA s(k(n)). The rate of a relay map s dependent on the transmsson rates of the nodes. A sngle node can serve as a relay to multple sources. An m 1 relay map s an algorthm that pcks subsequences YA s 1, YA s 2,, YA s m from departure epochs of m nodes A 1,, A m and a subsequence YB s from the departure epoch of the relay node B such that 1) YB s = m =1 Ys A. 2) Let Y s be the sequence formed by the concatenatng YA s 1,, YA s m and orderng the epochs n ascendng order. Then, Y s, 0 Y s B() Y s (). An m 1 relay map s assocated wth a relay rate vector λ(m) = (λ 1,, λ m ) whch s gven by where k (n) = Y s A. k (n) λ = lm n YA s (k (n)), B. Medum Access Constrants Nodes n a wreless network share a common channel and transmssons are susceptble to fadng and nterference. Dependng on the PHY model, the rates of transmsson are subjected to some medum access constrants specfed by a regon of rate vectors C. If the transmsson rates of the nodes belong to C, the packets are receved successfully at the recevng node. To ths extent, we consder two dfferent spread spectrum sgnalng models : transmtter drected and recever drected spreadng sequences. Transmtter Drected Spreadng Sequences : Each transmttng node n a shared channel uses an orthogonal spreadng code to transmt ts packets. The constrants on transmsson rates for the nodes are therefore ndependent. In other words, for a set of

3 nodes A 1,, A n, the medum access regon s gven by A 1 D 2 C = {(T A1,, T An : T A C A, = 1,, n}. A 2 B D 1 Recever Drected Spreadng Sequences : The nodes transmttng to a common node/relay use the same spreadng sequence. Each recevng node has an ndependent rate constrant. The sum-rate of nodes transmttng to a sngle recever s therefore bounded by a maxmum value. If nodes A 1,, A n transmt packets to node B, the regon C s gven by C. Secrecy C = {(T A1,, T An : T A C B }. As mentoned earler, by correlatng transmsson epochs from multple nodes, the eavesdropper can obtan nformaton about routes wthn the network. The goal s, therefore, to schedule transmssons so as to maxmze the secrecy of the routes wth respect to the eavesdropper. Secrecy can be formally defned as follows. Let A = {A 1, A 2,, A k } be a set of nodes and F 2 A denote the set of all ordered node-pars n A ( F = A ( A 1)). Snce transmssons from nodes not physcally connected can be correlated to nfer a flow, t s necessary to consder all possble node-pars. Durng a gven sesson, the set of node-pars n F that requre non-zero relay rate s denoted by the flow vector F 2 F. We defne A to have complete relay secrecy f the transmsson epochs of the nodes n A and F are ndependent. In other words, the condtonal dstrbuton p(y A1, Y A2,, Y Ak F ) = p(y A1, Y A2,, Y Ak ). (1) If for any flow vector F, the jont dstrbuton of transmsson epochs s unaltered, then t s mpossble to nfer the flow to any degree of accuracy. D. Achevable Rates A rate vector R = (R 1,, R m ) for a set of node-pars wth common relay {(A 1, B), (A 2, B),, (A m, B)} s an achevable rate vector, f there exsts a condtonal dstrbuton p(y A1, Y A2,, Y Am F ) and an m 1 relay map such that followng condtons are satsfed 1) The transmsson rate {T A1, T A2,, T Am, T B } satsfy the medum access constrants (4). 2) For every realzaton (Y A1,, Y Am ), λ (M) R, = 1,, m. 3) {A 1,, A m, B} have complete relay secrecy. In the followng secton, we present an achevable rate regon for the specal case of provdng relay secrecy for an m 1 multplex relay (Fg. 3), where a sngle node relays packets from m nodes. The results A m D m Fg. 3: m 1 Relay are presented for the two PHY models dscussed n Secton II-B. III. RATE REGION In the absence of an eavesdropper, the flow-rates achevable n a network can be obtaned purely from the topology and medum access restrctons. In the presence of eavesdropper, however, the secrecy condton mposes addtonal constrants whch can lower the achevable rates. The secrecy condton n (1) ndcates that the dstrbuton of transmsson epochs are ndependent of the flows. A specal case of ths condton s when, the transmsson schedule of each node s drawn from an ndependent dstrbuton and the margnal dstrbutons are not dependent on the flows. In other words, for a set of nodes A 1,, A k p(y A1,, Y Ak F ) = p(y A1 )p(y A2 ) p(y Ak ). Statstcal ndependence of departure epochs s a suffcent condton to ensure relay secrecy. In general, t may be possble to desgn schedules such that the transmsson epochs are not ndependent and yet guarantee relay secrecy. The noton of flow ndependent schedules has also been consdered n [10], [7], wheren the transmsson schedules were fxed apror rrespectve of the data flows. We assume that the sources generate packets at Posson tme ponts whch determne the schedules of the source nodes. In order to satsfy the secrecy condton, the relay nodes generate departure epochs from ndependent Posson processes. To an eavesdropper montorng the nodes, t s mpossble to decpher the actual flows by observng tme ponts, snce at all tmes, the schedules are statstcally ndependent. However, due to the delay constrant, the secrecy condton leads to a reduced rate regon, whch s characterzed n the followng sectons. A. Recever Drected Sgnalng As mentoned earler, n order to ensure complete relay secrecy for an m 1 multplex relay, all the nodes nvolved have statstcally ndependent transmsson schedules. The schedules for the source nodes s determned by the source packet generaton process, whle the relay generates an ndependent Posson pont process. When the spreadng sequences are recever

4 drected, the constrants on transmsson rates are ndependent for dfferent recevng nodes. When characterzng the achevable rates for an m 1 relay, we assume that the destnaton nodes for dfferent sources are dstnct. Therefore, the constrant on the rates of the relay node are ndependent for each destnaton node. A B D Fg. 4: 1 1 Relay To characterze the achevable rates for a 1 1 relay map, we use the BOUNDED-GREEDY-MATCH (BGM) algorthm proposed n [13] that optmally maps Pont processes wth the least packet drops. Snce epochs are generated accordng to ndependent Posson processes, the strct delay constrant makes t mpossble to relay all transmtted packets. The relay rate s, therefore, strctly less than the transmsson rates of the nodes. Let node A be the source node sendng packets to destnaton D through relay B (see Fg. 4). The algorthm s as follows; When a packet arrves at B, f there exsts a departure epoch wthn of the arrval nstant and has not been matched to any prevous arrval, t s assgned to the arrved packet. Otherwse, the packet s dropped. The transmsson schedule of A s obtaned from the generaton tmes of packets whle node B generates an ndependent Posson process of a fxed rate and uses the algorthm to map arrval epochs to the generated schedule. Theorem 1: If the maxmum transmsson rates allowed to nodes B and D are C B and C D respectvely, respectvely, the maxmum achevable relay rate R between (A, B), when Y A, Y B are ndependent Posson processes s obtaned when T A = C B, T B = C D and s gven by R = C B C D. (2) C 2 B 1+C B C B = C D C D(e C (C B C D ) 1) B C De (C B C D ) C B Proof: Refer to Appendx. As s evdent from the expresson n Theorem 1, as, the maxmum relay rate s gven by mn{c B, C D }. Smlarly as C B, the maxmum rate s C D for any fnte and vce-versa. Clearly, when s fnte, the transmsson rates T A, T B of the nodes are strctly greater than the achevable nformaton relay rate, thereby resultng n packet drops. Packet losses can, however, be countered f the source employs forward error correctng (FEC) schemes. Snce the PHY layer s a recever drected sgnalng scheme, for an m 1 relay, each outgong stream from the relay would be an ndependent Posson process whch depends on the transmsson rate allowed to the partcular destnaton node. If there s no delay constrant, =, then the achevable rate regon s dentcal to the rate regon wthout any secrecy constrant; the relay node can store ncomng packets and regenerate them ndependently usng a Posson scheduler [6]. The rate regon s then determned solely based on medum access constrants. The achevable relay rate regon when nodes A 1,, A m send packets through relay B to destnatons D 1,, D m respectvely s gven by R C B ; R C D. (3) We assume the relay node decodes the packet headers and can dstngush packets arrvng from multple sources. Hence, the relay can treat the ncomng packets from each source as a separate stream, although the streams have the same spreadng code. For each source-destnaton par, the relay node uses the BGM algorthm to map the packets n each arrval process to the correspondng outgong stream. The followng theorem characterzes the achevable rate regon of ths m 1 relay map. Theorem 2: Let C B be the maxmum Tx. rate allowed to the relay B and C D the maxmum allowed rate to destnaton node D. An achevable rate regon R r for the m 1 relay s gven as follows. (R 1,, R m ) R r f T A1,, T Am such that ( C D e (T A C D ) 1 ) R T A, T C D e (TA CD ) A C B T A Proof: Refer to Appendx. T A represents the transmsson rate from source node A to the relay. Snce the BGM algorthm has been proven to mnmze the packet loss[14], ths strategy provdes the best achevable rates, when the transmsson schedules are drawn from ndependent Posson processes. It s easly seen from the theorem, that as, all rates that satsfy the medum access constrants (3) are achevable by ths technque. The achevable rate regon for a 2 1 relay wth recever drected sgnalng s shown n Fgure 5. B. Transmtter Drected Sgnalng When the sgnalng s transmtter drected, the constrant on the transmsson rates are ndependent for each source node and the relay. Moreover, snce the transmsson rate constrant for the relay s ndependent of the number of destnatons, the followng results hold even f multple source nodes share a common destnaton. If there s no delay constrant, =, then the achevable rate regon s dentcal to the rate regon wthout any secrecy constrant; the rate regon s then determned solely based on medum access constrants,.e.,, R C A ; R C B. (4)

5 Achevable Rate Regon : C D1 =3,C D2 =4,C B =5 R r = the packet from A s assgned that epoch. If Z = 0, then the packet that arrved earlest s assgned that epoch. By consderng all possble ndex assgnments and prorty values, the rate regon s obtaned. The algorthm for 2 nodes s formally stated n Table assumng A 1 has ndex 2 and prorty value α. 2.5 R R 1 Fg. 5: Achevable Regons for 2 1 relay wth Recever drected sgnalng : = 1 An achevable rate regon when s fnte can be obtaned through a drect extenson of the sngle source relay case consdered n the prevous secton. The relay node gnores the orgn of the packets and executes the BGM algorthm on the jont traffc from all the nodes. Ths strategy, whch we refer to as homogenous relay map results n an achevable rate regon R H gven by Theorem 3: (R 1,, R m ) belongs to R H ff T A [0, C A ], = 1,, m s.t C B (e ( j TA j CB) 1) R = T A C B e ( ( j TA j CB)) j T. A j Proof : Snce the relay gnores the source of the packets, t apples BGM algorthm on the jont arrval process of transmsson rate j T A j. The proof follows from Theorem 1. It s easly shown that as ncreases, the regon R H converges to the optmal rate regon gven by (3). Smlarly, as dscussed n the sngle relay case, as C B, t s possble to acheve all rate vectors satsfyng the medum access constrants. The regon n Theorem 2 can be sgnfcantly mproved f orgn of packets are taken nto consderaton. The algorthm we propose s the followng. The nodes transmttng to the relay are assgned unque ndces from 1 to m such that the node wth a hgher ndex s gven more prorty when n contenton. Every subset of nodes S 2 A s assgned a prorty value α(s) [0, 1]. As long as there s no contenton between packets from dfferent sources for a partcular departure epoch, the relay functons as a homogenous relay map. If packets from any subset of nodes S contend for the same departure epoch, the relay generates a Bernoull random varable Z B(α(S)). Let A be the node n S wth the hghest ndex. If Z = 1, then TABLE I: BOUNDED-PRIORITY-MATCH (α-bgm). BOUNDED-PRIORITY-MATCH(Y A1, Y A2, Y B, α, ): m 1 = m 2 = n = 1; whle (m 1 Y A1 or m 2 Y A1 ) and n Y B f Y B (n) Y A1 (m 1 ) < 0 and Y B (n) Y A2 (m 1 ) < 0 At Y B (n) Tx dummy packet; n = n + 1; else f Y B (n) Y A1 (m 1 ), Y B (n) Y A2 (m 2 ) > Y s A 1 = Y s A 1 Y A1 (m 1 ), Y s B = Ys B Y B(n) ; Drop Y A2 (m 2 ) ; Increment m 1, m 2, n by 1 ; else f Y B (n) Y A1 (m 1 ) >, Y B (n) Y A2 (m 2 ) Y s A 2 = Y s A 2 Y A2 (m 2 ), Y s B = Ys B Y B(n) ; Drop Y A1 (m 2 ) ; Increment m 1, m 2, n by 1 ; else f Y B (n) Y A1 (m 1 ), Y B (n) Y A2 (m 2 ) Generate random varable Z B(α) If Z = 0, = arg mn{y A1 (m 1 ), Y A2 (m 2 )} else = 1 Y s A = Y s A Y A (m ), Y s B = Ys B Y B(n) ; m = m + 1, n = n + 1 else Drop Y A1 (m 1 ), Y A2 (m 2 ); Increment m 1, m 2 by 1 ; end end end Let R P denote the set of all rate vectors achevable by usng the prorty relay map (all prorty assgnments). The followng theorem provdes bounds for R P. Theorem 4: Let R out = {(R 1, R 2 ) : R f(c A, C D ), R f( C A, C D )}, (5) where f(a, b) = a b(e (a b) 1) be ( (a b)) a. (6) Then, R H R P R out. Proof: Refer to Appendx The outer bound R out descrbed n the theorem s a tghter bound (than (3)) to the achevable rate regon when we restrct the epochs to be ndependent Posson transmssons. Fgure 6 plots an example of the dfferent regons for a 2 1 relay. As can be seen, the achevable rate regon of the prorty relay map R P nearly concdes wth the outer bound. As ncreases, the regons R H, R P and R out converge to the optmal regon gven by (4). C. Fxed Packet Loss As mentoned n Secton III-A, the fnte delay constrant mposed on the transmsson schedule results n packet loss. Hence, t s necessary for the source to use a forward error correcton scheme to ensure relable

6 R Achevable Rate Regon : C A1 =3, C A2 =4,C B = R 1 Fg. 6: Achevable Regons for 2 1 relay wth Transmtter drected sgnalng : = 1 R P R out R H = recovery of packets at the destnaton. In practce, t may be necessary to desgn strateges for a fxed packet drop fracton ɛ dependng on the nature of data and avalablty of good codes. The followng theorem characterzes an achevable rate regon for the m 1 relay, such that the packet drop fracton s less than a fxed ɛ. Theorem 5: 1. The achevable relay rate regon R ɛ for the m 1 relay wth packet loss constrant ɛ for transmtter drected sgnalng s gven by R R H S ɛ, where { } S (R 1,, R m ) : R x(1 ɛ) and x s the soluton of C B x C B exp( (x C B )) x. (7) 2. The achevable relay rate regon R r ɛ for the m 1 relay wth packet loss constrant ɛ for recever drected sgnalng s gven by R r R r S ɛ, where S {(R 1,, R m ) : R x (1 ɛ), = 1,, m}, and x s the soluton of C D x. (8) C D exp( (x C D )) x Proof: Refer to Appendx The rate regon n Theorem 5 s obtaned by usng the homogenous relay map scheme descrbed n SectonIII-B coupled wth the constrant on sumtransmsson rate due to the packet loss fracton ɛ. IV. CONCLUSIONS In ths work, we formally defned the problem of hdng data flows from eavesdroppers observng transmsson epochs. We proposed a possble soluton for provdng complete secrecy and characterzed, achevable rates for a multplex relay n Posson traffc. Allowng relays to perform re-encodng s a worthwhle extenson to pursue. Although we have consdered only a sngle relay system, the basc deas are extendable to longer routes also. REFERENCES [1] D. Chaum, Untraceable electronc mal, return addresses and dgtal pseudonyms, Communcatons of the ACM, vol. 24, pp , February [2] Q. Sun, D. R. Smon, Y. Wang, W. Russell, V. N. Padmanabhan, and L. Qu, Statstcal dentfcaton of encrypted web browsng traffc, n Proceedngs of the 2002 IEEE Symposum on Securty and Prvacy, (Berkeley, Calforna), p. 19, May [3] C. Gulcu and G. Tsudk, Mxng e-mal wth babel, n Proceedngs of the Symposum on Network and Dstrbuted System Securty, pp. 2 19, February [4] G. Danezs, R. Dngledne, and N. Mathewson, Mxmnon: desgn of a type anonymous remaler protocol, n Proceedngs of 2003 Symposum on Securty and Prvacy, pp. 2 15, May [5] M. K. Reter and A. D. Rubn, Crowds: anonymty for Web transactons, ACM Transactons on Informaton and System Securty, vol. 1, no. 1, pp , [6] X. Hong, P. Wang, J. Kong, Q. Zheng, and J. Lu, Effectve Probablstc Approach Protectng Sensor Traffc, n Mltary Communcatons Conference, 2005, (Atlantc Cty, NJ), pp. 1 7, Oct [7] S. Jang, N. H. Vadya, and W. Zhao, A mx route algorthm for mx-net n wreless moble ad hoc networks, n Proceedngs of IEEE Moble Sensor and Ad-hoc and Sensor Systems, pp , October [8] J. Kong and X. Hong, Anodr: Anonymous on demand routng wth untraceable routes for moble ad-hoc networks, n ACM lntematonal Symposum on Moble Ad Hoc Nefworkng and Computng, (Annapols, MD), June [9] Y. Zhu, X. Fu, B. Graham, R.Bettat, and W. Zhao, On flow correlaton attacks and countermeasures n mx networks, n Proceedngs of Prvacy Enhancng Technologes workshop, May [10] B.Radosavljevc and B. Hajek, Hdng traffc flow n communcaton networks, n Mltary Communcatons Conference, [11] C. E. Shannon, Communcaton theory of secrecy systems, Bell System Techncal Journal, [12] A. Wyner, The wretap channel, Bell Syst. Tech. J., vol. 54, pp , [13] A. Blum, D. Song, and S. Venkataraman, Detecton of Interactve Steppng Stones: Algorthms and Confdence Bounds, n Conference of Recent Advance n Intruson Detecton (RAID), (Sopha Antpols, French Rvera, France), September [14] T. He and L. Tong, Detectng Steppng-stone Traffc n Chaff: Fundamental Lmts and Robust Algorthms, Tech. Rep. ACSP-TR , Cornell Unversty, June [15] D. Cox and H. Mller, The Theory of Stochastc Processes. New York: John Wley & Sons Inc., Proof of Theorem 1 APPENDIX To prove the theorem, we adopt the technque used n [14]. Consder the two pont processes Y A, Y B. If a packet n Y A, say at tme t s desgnated as dummy packet by the BGM algorthm, we nsert a vrtual packet at the t + n Y B. Smlarly, f a packet at tme t n Y B s desgnated as dummy packet, we nsert a vrtual packet at tme t n Y A. Now we consder the dfference process Z = {Y B () Y A ()} between

7 the two processes. At every occurrence of a dummy packet, the dfference process hts a reflectng barrer, ether at 0 or at. The net probablty of chaff s, therefore, the probablty of httng ether barrer. If the transmsson rates of node A and B are T A and T B respectvely, from the analyss n [15], we know that the probablty of httng s gven by Pr{Z() = } = 1 TA T B T B. TA e (TA TB) TA T B It s easy to see that the fracton of chaff n Y A s ɛ A = T B Pr{Z() = } T A (1 Pr{Z() = }) = Snce the rate of relayed packets ncreases wth the transmsson rates of ether nodes, the achevablty of the theorem s proved. In [13], the authors have shown that the BGM algorthm nserts the least chaff fracton for any par of pont processes. Hence, for any (T A, T B ), t s mpossble to obtan a hgher nformaton relay rate than (2). Proof of Theorem 2 Snce the nodes use recever drected sgnalng, the relay node generates an ndependent outgong Posson process for each source. If the source transmsson rate s T A and the maxmum allowed rate to destnaton D s C D, we know from theorem 1 that R = T A C D (e (TA CD ) 1) C D e ( TA ) T A. (9) Snce the maxmum allowed transmsson rate to the relay s C B, the sum-rate must satsfy T A C B. (10) Combnng (9) and (10), the theorem s proved. transmssons are ndependent Posson processes) s gven by ( ( ) R ) max = f C A, C B. (12) The best rate for A s obtaned when T = 0, j s zero. By replacng j C A j by C A n (12), we can obtan the remanng condtons that specfy R out. Proof of Theorem 5 1. We consder the homogenous relay map. From T B T Theorem 1, we know that for a set of transmsson A. T B e (TA TB) rates of sources(t A1,, T Am ) the least fracton of T A chaff n the ncomng stream s gven by C B ( T A ) C B exp( (( T A ) C B )) T A, when the relay transmts at the hghest rate. It s easly shown that ɛ s an ncreasng functon of T A. Hence, an upper bound on ɛ corresponds to an upper bound on the sum transmsson rate T A. Therefore, for any rate vector that satsfes T A x where x s gven by 7, the homogenous relay map guarantees that relay rates satsfy the packet loss constrant. 2. In the case of recever drected sgnalng, the outgong streams to destnatons are ndependent. The packet loss for any stream s dependent only on the transmsson rate of the source node and the rate of that partcular outgong stream. If the maxmum allowed rate to the destnaton D s C D and ɛ s the allowed packet loss, then from Theorem 1, we know that the maxmum allowed source transmsson rate x satsfes C D x. C D exp( (x C D )) x Combnng the above equaton wth the achevable rate regon R r, we get the result. Proof of Theorem 4 The nner bound s trvally shown as the homogenous map s a specal case of the prorty map when α(s) = 0, S. The outer bound s obtaned usng the optmalty of BGM algorthm. Let node A transmt at rates T. Then, the sum nformaton relay rate obtaned by usng the homogenous map s gven by: ( ) R = f T, C B. (11) Snce BGM nserts the least fracton of dummy packets[13], ths s the maxmum sum-rate achevable for the gven transmsson rates. It s easy to see that R n (11) s an ncreasng functon of T. Therefore, the maxmum sum-rate possble (when The vews and conclusons contaned n ths document are those of the authors and should not be nterpreted as representng the offcal polces, ether expressed or mpled, of the Army Research Laboratory or the U.S. Government.