On Secure Network Coding with Unequal Link Capacities and Restricted Wiretapping Sets

Transcription

1 On Secure Network Coing with Unequal Link Capacitie an Retricte Wiretapping Set Tao Cui an Tracey Ho Department of Electrical Engineering California Intitute of Technology Paaena, CA 9115, USA {taocui, Jörg Kliewer Klipch School of Electrical an Computer Engineering New Mexico State Univerity La Cruce, NM 8800, USA Abtract We are ecure network coing over network with unequal link capacitie in the preence of a wiretapper who ha only acce to a retricte number of k link in the network. Previou reult how that for the cae of equal link capacitie an unretricte wiretapping et, the ecrecy capacity i given by the cut-et boun, whether or not the location of the wiretappe link i known. The cut-et boun can be achieve by injecting k ranom key at the ource which are ecoe at the ink along with the meage. In contrat, for the cae where the wiretapping et i retricte, or where link capacitie are not equal, we how that the cut-et boun i not achievable in general. Finally, it i hown that etermining the ecrecy capacity i a NP-har problem. I. INTRODUCTION Information-theoretically ecure communication ue coing to enure that an averary eaveropping on a ubet of network link obtain no information about the ecure meage. A theoretical bai for information-theoretic ecurity wa given in the eminal paper by Wyner [1] uing Shannon notion of perfect ecrecy [], where a coet coing cheme bae on a linear maximum itance eparable coe wa ue to achieve ecurity for a wiretap channel. More recently, information-theoretic ecurity ha been tuie in network with general topologie. The ecure network coing problem wa introuce in [] for multicat wireline network where each link ha equal capacity, an a wiretapper can oberve an unknown et of up to k network link. For thi problem, contruction of information-theoretically ecure linear network coe are propoe in e.g. [] [5], where trae-off between ecurity, coe alphabet ize, an multicat rate of ecure linear network coe are coniere in [4]. In [6], [7] the work in [] i extene to multiple ource, where ranom key can now be generate at an arbitrarily given ubet of noe. Further, in [8], ecure communication i coniere for wirele eraure network. In thi paper, we conier ecure communication over wireline network with unequal link capacitie an retricte wiretapping et. In the cae of throughput optimization without ecurity requirement, the aumption that all link have Thi work ha been upporte in part by ubcontract # iue by BAE Sytem National Security Solution, Inc. an upporte by the Defene Avance Reearch Project Agency (DARPA) an the Space an Naval Warfare Sytem Center (SPAWARSYSCEN), San Diego uner Contract No. N C-01 an W911NF , by NSF grant CNS an CCF , an by Caltech Lee Center for Avance Networking. unit capacity i mae without lo of generality, ince link of larger capacity can be moele a multiple unit capacity link in parallel. However, in the ecure communication problem, uch an aumption cannot be mae without lo of generality. Inee, we how in thi paper that there are ignificant ifference between the equal capacity an unequal capacity cae. For the cae of equal link capacitie, the ecrecy capacity i given by the cut-et boun, whether or not the location of the k wiretappe link i known. The cut-et boun can be achieve by injecting k ranom key at the ource which are ecoe at the ink along with the meage []. However, we how that if the network ha unequal link capacitie an the wiretapper ha acce to an unretricte et of link in the network the cut-et boun i not achievable in general by any linear or nonlinear coing cheme. We further how that thi alo hol in the cae of an unretricte wiretapping et an equal unit link capacitie in the network. Finally, we are the complexity of etermining the ecrecy capacity if the location of the wiretapper i unknown. We how that thi problem, which i cloely relate to network interiction, i NP-har. II. NETWORK MODEL AND PROBLEM FORMULATION In thi paper we focu on acyclic graph for implicity; we expect that our reult can be generalize to cyclic network uing the approach in [9], [10] of working over fiel of rational function in an ineterminate elay variable. We moel a wireline network by a irecte acyclic graph G = (V, E), where V i the vertex et an E i the irecte ege et. There i a ource noe V an a ink noe V. Each link (i, j) E ha capacity c i,j. An eaveropper can wiretap a et A of link choen from a known collection W of poible wiretap et. Without lo of generality we can retrict our attention to maximal wiretap et, i.e. no et in W i a ubet of another. The choice of wiretap et A i unknown to the communicating noe, except where otherwie pecifie in thi paper. The ecrecy capacity i the highet poible ource-ink communication rate uch that the meage communicate i information theoretically ecret regarle of the choice of A, i.e. ha zero mutual information with the wiretapper obervation. In Section III an IV, we how that the cut-et boun i unachievable an that fining the ecrecy capacity i NP har,

2 even for the following pecial cae: 1) Scenario 1 i a wireline network with equal link capacitie, where the wiretapper can wiretap an unknown ubet of k link from a known collection of vulnerable network link. ) Scenario i a wireline network with unequal link capacitie, where the wiretapper can wiretap an unknown ubet of k link from the entire network. It i convenient to how thee reult for Scenario 1 firt, an then how the correponing reult for Scenario, by converting the Scenario 1 network coniere into correponing Scenario network for which the ame reult hol. a b c 1 4 e f g III. UNACHIEVABILITY OF CUT-SET BOUND Let S c enote the et complement of a et S. A cut for x, y V i a partition of V into two et V x an V c x uch that x V x an y V c x. For the x y cut given by V x, the cut-et [V x, V c x ] i the et of ege going from V x to V c x, i.e., [V x, V c x] = {(u, v) (u, v) E, u V x, v V c x}. (1) We can tate the following cut-et upper boun which applie to the general moel of Section II incluing both cenario 1 an : Theorem 1. Conier a network of point-to-point link, where link (i, j) ha capacity c i,j. The ecrecy capacity R i upper boune by min min c i,j. () {V : V i an cut} A W (i,j) [V,V c] Ac Thi upper boun applie whether or not the communicating noe have knowlege of the choen wiretap et A. Proof: Conier any ource-ink cut V an any wiretap et A W. Denote by X the tranmitte ignal from noe in V over link in [V, V] c an enote by Y an Z the oberve ignal from link in [V, V c] an in [V, V c ] A, repectively. We conier block coing with block length n. By the perfect ecrecy requirement H(M Z n ) = H(M) we have nr H(M Z n ) (a) H(M Z n ) H(M Y n ) + nǫ n =H(M Z n ) H(M Y n,z n ) + nǫ n =I(M;Y n Z n ) + nǫ n (b) I(X n ;Y n Z n ) + nǫ n (c) n n H(Y i Z i ) H(Y i X i, Z i ) + nǫ n, i=1 i=1 =ni(x;y Z) + nǫ n, =n (H(X Z) H(X Z,Y)) + nǫ n, =n (H(X Z) H(X Y)) + nǫ n, n max (I(X;Y) I(X;Z)) + nǫ n p(x) =n c i,j + nǫ n, (i,j) [V,V c] Ac () 5 Fig. 1. An example to how that the ecrecy rate without knowlege of wiretapping et i maller than that with uch knowlege. The wiretapper can wiretap any three of the five link in the mile layer. where ǫ n 0 a n + an (a) i ue to Fano inequality; (b) i ue to the ata proceing inequality an the fact that M X n Y n Z n form a Markov chain; (c) i ue to the efinition of the mutual information. Note that Theorem 1 i a generalization of the boun in [6] to arbitrary link capacitie. If the choice of wiretap et A i known to the communicating noe, the cut-et boun () i achievable uing a network coe that oe not en any flow on link in A. In the cae of unretricte wiretapping et an unit link capacitie, the ecrecy capacity i equal to the cut-et boun []. In contrat, we now how that the cut-et boun i not achievable in general when the wiretap et A i unknown, by coniering the example in Fig. 1, where the et of wiretappable link i retricte (Scenario 1). We ue the program Information Theoretic Inequalitie Prover (Xitip) [11] to how that the ecrecy capacity i boune away from the cut-et boun. We then convert the example into one with unequal link capacitie (Scenario ), an how the unachievability of the cut-et boun for thi cae alo. A. Retricte Wiretap Set (Scenario 1) In Fig. 1, let the mile layer link be 1-5 (from top to bottom) an the lat layer link be 6-8 (from top to bottom). All link have unit capacity. Let the ignal carrie by link i be calle ignal i, or S i. Let the ource information be enote X. For thi example, the ecrecy rate i two if any three of the five link in the mile layer are elete, i.e., the number of wiretappe link i three. The contraint require are that the ource information i a function of the ignal on the ink incoming link, an that there i zero mutual information between the ource information an the ignal on the link in each averarial ubet.

3 In thi example, the cut-et boun i. To provie intuition, we firt how that ecrecy rate cannot be achieve by uing linear coing. Thi argument can be converte to an information theoretic proof that ecrecy rate cannot be achieve uing any coing cheme [1]. Suppoe ecrecy rate i achievable with a linear network coe. Firt note that the ource cannot inject more than unit amount of ranom key, otherwie the firt layer cannot carry two unit of ource ata. Let the ranom key injecte by the ource be enote K. For the cae when the ource inject a unit amount of ecret key, we firt have the following obervation. Signal 6 mut be a function of ignal 1, otherwie if the averary ee the ignal -4 then he know ignal 6-7. Alo, ignal 8 mut be a function of ignal 5, otherwie if the averary ee ignal 1, an 4, then he know ignal 7-8. Similarly we can how that ignal 8 mut be a function of ignal 1, an ignal 7 mut be a function of ignal. We conier the following two cae. Cae 1: ignal 5 i a linear combination of ignal preent at the ource noe. To achieve the full key rank conition on link 1, an 5, the top econ layer noe (a) mut put inepenent local key k 1 an k on link 1 an repectively. Link 7, whoe other input i inepenent of k, i then a function of k. Similarly, Link 8 i a function of k 1. Thi mean that the lat layer ha two inepenent local key on it. Cae : ignal 5 i a linear combination of ignal preent at the ource noe a well a a local key k injecte by the bottom econ layer noe (c). Cae a: k i alo preent in ignal 1. Then k i preent in ignal 6, an i inepenent of the key preent in ignal 7. Cae b: k i not preent in ignal 1. Then k i preent in ignal 8, an i inepenent of the key preent in ignal 7. From Cae 1, a, an b, we conclue that the ecrecy rate without knowlege of the wiretapping et by uing only linear network coing i le than two. We can alo how that the ecrecy rate i boune away from by uing the framework for linear information inequalitie [1]. Let X be the meage ent from the ource an Z i, i = 1,..., be the ignal on the link ajacent to the ource. We want to check whether H(X) ω i implie by (1) H(Z i ) 1, H(S j ) 1, i = 1,...,, j = 1,...,8, () H(X S 6, S 7, S 8 ) = 0, () I(X, Z 1, Z, Z, S 4, S 5, S 7, S 8 ; S 6 S 1, S, S ) = 0, (4) I(X, Z 1, Z, Z, S 1, S, S 5, S 6, S 8 ; S 7 S, S 4 ) = 0, (5) I(X, Z 1, Z, Z, S, S, S 6, S 7 ; S 8 S 1, S 4, S 5 ) = 0, (6) I(X; S 1, S, S ) = 0, I(X; S 1, S, S 4 ) = 0, (7) I(X; S 1, S, S 5 ) = 0, I(X; S 1, S, S 4 ) = 0, (8) I(X; S 1, S, S 5 ) = 0, I(X; S 1, S 4, S 5 ) = 0, (9) I(X; S, S, S 4 ) = 0, I(X; S, S, S 5 ) = 0, (10) I(X; S, S 4, S 5 ) = 0, I(X; S, S 4, S 5 ) = 0, (11) I(S 1 ; Z Z 1, Z ) = 0, I(S ; Z, Z Z 1 ) = 0, (1) I(S ; Z Z 1, Z ) = 0, I(S 4 ; Z 1, Z Z ) = 0, (1) I(S 5 ; Z 1, Z Z ) = 0, I(S 1 ; S 4 Z 1, Z, Z ) = 0, (14) I(S ; S 4, S 5 Z 1, Z, Z ) = 0, I(S ; S 5 Z 1, Z, Z ) = 0, (15) I(S 4 ; S 1, S, S 5 Z 1, Z, Z ) = 0, I(S 5 ; S, S, S 4 Z 1, Z, Z ) = 0, (16) I(S 1, S, S, S 4, S 5 ; X Z 1, Z, Z ) = 0, where the firt inequality i the capacity contraint, the econ contraint how that the ink can ecoe X, contraint () to (5) mean that the ignal in the lat layer are inepenent of other ignal given the incoming ignal from the mile layer, contraint (6) to (10) repreent the ecrecy contraint when any three link in the mile layer are wiretappe, an contraint (11) to (16) repreent the conitional inepenence between the ignal in the firt layer an thoe in the mile layer. In particular, (16) how that X (Z 1, Z, Z ) (S 1,..., S 5 ) form a Markov chain. Note that contraint () to (5) an (11) to (16) implicitly allow ome ranomne to be injecte at the correponing noe. We ue the Xitip program [11], which relie on the framework in [1], to how that H(X) 5/ i implie by the et of equalitie (4). Therefore, 5/ i an upper boun on the ecrecy rate when the location of wiretapper i unknown, which i le than the ecrecy rate achievable when uch information i known. B. Unequal Link Capacitie (Scenario ) We next how that the unachievability of the cut-et boun alo hol for the ecure network coing problem with unequal link capacitie (Scenario ). We convert the example of Fig. 1 by partitioning each non-mile layer link into 1 ǫ parallel mall link each of which ha capacity ǫ. Any three link can be wiretappe in the tranforme graph. For the cae where the location of the wiretap link i known, eleting any k (k ) non-mile layer link reuce the max flow by at mot k ǫ. When k = 0, the mincut i. When k 1 or at mot mile layer link are wiretappe, the min-cut between the ource an the ink i at leat after eleting thee wiretappe mile layer link, an the min-cut i at leat k ǫ ǫ after further eleting the k 1 non-mile layer link. Therefore, the cut-et boun i at leat ǫ. For the cae where the location of the wiretap link i unknown, we prove the unachievability of the cut-et boun in the tranforme network. Firt, conier the tranforme network with the retriction that the wiretapper can only wiretap any link in the mile layer. The optimal olution i exactly the ame a for the original network of the previou ubection, an achieve ecrecy rate at mot 5/. Now, conier the tranforme network without the retriction on wiretapping et, i.e., the wiretapper can wiretap any link in the entire network. A wiretapping only the mile layer link i a ubet of all poible trategie that the wiretapper can have, the ecrecy rate in the tranforme network i le than or equal to that in the former cae, which i trictly maller than the cut-et boun for ǫ trictly maller than 1 9. Therefore, the cut-et boun i till unachievable when the wiretap link are unretricte in the tranforme graph. (4)

4 1 a c x 4 ia ib j1 t 1 1 b y (a) Original Graph H ic ix j j i a iy j4 10 i b j 1 i c i x i y j j j 4 (b) Tranforme Graph G H Fig.. Example of NP-harne proof for the cae with knowlege of the wiretapping et. IV. NP-HARDNESS We how in the following that etermining the ecrecy capacity i NP-har by reuction from the clique problem, which etermine whether a graph contain a clique 1 of at leat a given ize r. From Section III, fining the ecrecy capacity when the location of the wiretap link i known to the communicating noe i the ame a the NP-har network interiction problem [14], which i to minimize the maximum flow of the network when a given number of link in the network i remove. To how that the cae where the location of the wiretap link i unknown i NP-har, we ue the contruction in [14] howing that for any clique problem on a given graph H, there exit a correponing network G H whoe ecrecy capacity i r when the location of the wiretap link i known if an only if H contain a clique of ize r. We then how that for all uch network G H, the ecrecy capacity for the cae when the location of the wiretap link i unknown i equal to that for the cae when uch information i known. We briefly ecribe the approach in [14] in the following. Given an unirecte graph H = (V h, E h ), we will efine a capacitate irecte network ĜH uch that there exit a et of link  in ĜH containing le than or equal to E h ( r link uch that Ĝ H  ha a maximum flow of r if an only if H contain a clique of ize r. For a given unirecte graph H = (V h, E h ) without parallel ege an elf loop, we create a capacitate, irecte graph G H = (N, A) a follow: For each ege e E h create a noe i e in a noe et N 1 an for each vertex v V h create a noe j v in a noe et N. In aition, create ource noe an etination noe. For each ege e E h, irect an arc in G H from to i e 1 A clique in a graph i a et of pairwie ajacent vertice, or in other wor, an inuce ubgraph which i a complete graph. ) Fig.. Illutration of Strategy 1. In thi figure, k = an only the 5 link in the firt layer can be wiretappe. with capacity an call thi et of arc A 1. For each ege e = (u, v) E h, irect two arc in G H from i e to j v an j u with capacity 1, repectively an call thi et of arc A. For each vertex v V h, irect an arc with capacity 1 from j v to. Let thi be the et of arc A. Thi complete the contruction of G H = (N, A) = ({} {} N 1 N, A 1 A A ). In Fig., we give an example of the graph tranformation, where H = ({1,,, 4}, {a, b, c, x, y}). We replicate [14, Lemma ] a follow. Lemma 1. Let G H be contructe from H a above. Then, there exit a et of arc A 1 A 1 with A 1 = E h ( ) r uch that the maximum flow from to in G H A 1 i r if an only if H contain a clique of ize r. After obtaining G H, we generate ĜH by replacing each arc (i e, j v ) with E h parallel arc each with capacity 1/ E h an call thi arc et Â. We carry out the ame proceure for arc (j v, ) an call thi arc et Â. Then ĜH = (N, A) = ({} {} N 1 N, A 1  Â). For the cae when the location of wiretap link i known, it i hown in [14] that the wort cae wiretapping et  mut be a ubet of A 1. By uing Lemma 1, thi cae i NP-har. Now, we conier the cae where the wiretapping et i unknown, an how that the ecrecy capacity of ĜH when the wiretapper accee any unknown ubet of k = E h ( ) r link i r if an only if H contain a clique of ize r. We ue the following achievability reult from [15], which ue a trategy where ranom key injecte by the ource are either cancele at intermeiate noe or ecoe by the ink: Strategy 1 achievable rate: Connect each ubet of link A W in the network G to a virtual noe t A, an connect both t A an the actual ink to a virtual ink A. Let R A be the minimum cut capacity between an t A. The virtual link between t A an A ha capacity R A, an the virtual link between the actual ink an A ha capacity R. Thi i illutrate in Fig.. If the min-cut between the ource an each virtual receiver A i at leat R + R A, the ecrecy rate R i achievable. From Lemma 1, the conition that H contain a clique of ize r i equivalent to the conition that the max-flow to the ink in G H after removing any k link from A 1 i r. We now how that the latter conition i equivalent to the conition t 10

5 that the ecrecy capacity of G H when the wiretapper accee any unknown ubet of k link from A 1 i r. We create a virtual ink connecting each ubet of k link from A 1 an the actual ink. A the wiretappe link are connecte to the ource irectly, the min-cut between each virtual ink an the ource i at leat k + r. Since r i the cut-et upper boun on the ecrecy rate, by uing Strategy 1 the ecrecy rate r i achievable, which i equal to the ecrecy rate when the location of wiretap link i known. Finally, we how that the ecrecy capacity of G H when any k link of A 1 are wiretappe (cenario 1) i equal to the ecrecy capacity of ĜH when any k link are wiretappe (cenario ). Since each econ layer link ha a ingle firt layer link a it only input, wiretapping a econ layer link yiel no more information to the wiretapper than wiretapping a firt layer link. When ome link in the thir layer are wiretappe, let the wiretapping et be  =  1  where  1 an  1 k 1. Thu A 1  1 contain at leat ( r ) +1 arc. We create noe t   with their correponing incient link a ecribe in Strategy 1. A removing link in A 1 i equivalent to removing link in H, after removing link in H correponing to  1, H contain a ubgraph H 1 containing ( r ) ege plu at leat an ege e = (u, v). Cae 1: H 1 i a clique of ize r. In thi cae, the number of vertice with egree greater than 0 in H 1 e i r +. Cae : H 1 i not a clique. H 1 contain at leat r+1 vertice with egree greater than 0. Accoring to [14, Lemma 1], the max-flow in G H i equal to the number of vertice in H with egree greater than 0. In both cae, the max-flow of G H after removing link in  1 i at leat r + 1. Let R  be the max-flow capacity from the ource to  in G H  1. We can ue a variant of the For-Fulkeron (augmenting path) algorithm, e.g., [16], a follow to contruct a maxflow ubgraph D from to  in GH  1 atifying the property that after removing D from G H  1, the min-cut between an i at leat r + 1 R  r + 1  / E h r + 1 ( E h 1)/ E h > r, (5) where we have ue  E h 1. Coniering the network G H  1 with all link irection revere, we contruct augmenting path via epth firt earch from to, tarting firt by contructing augmenting path via link in Â, until we obtain a et of path correponing to a max flow of capacity R  between an Â. We a further augmenting path until we obtain a max flow (of capacity at leat r+1) between an, which may caue ome of the path travering link in  to be reefine while not changing their total capacity. The ubgraph D conit of the final et of path travering link in Â. Thu, the path remaining after removing D have a total capacity lower boune by (5). Therefore, the min-cut between the ource an in G H   1 D i at leat r, an the min-cut between the ource an in G H i at leat r + R   + R 1  = r + R Â. By uing Strategy 1, a ecure rate of r i achievable when  i wiretappe. Thu, the ecrecy rate for the cae when the location of the wiretap link i unknown i equal to that for the cae when uch information i known with an unretricte wiretapping et. We have thu prove the following theorem. Theorem. For a ingle-ource ingle-ink network coniting of point-to-point link an an unknown wiretapping et, computing the ecrecy capacity i NP-har. V. CONCLUSION We have coniere ecure network coing in the preence of a wiretapper. In contrat to previou reult for network with equal capacity link an unretricte wiretapping et for which the cut-et boun i ientical to the ecrecy capacity, we have hown for a imple example network that the ecrecy capacity i trictly maller than the cut-et boun if the network ha equal capacity link an the wiretapper ha only acce to a retricte wiretapping et. In aition, bae on thi reult we have alo hown that the cut-et boun i not achievable in general if the wiretapping et i unretricte but the network conit of link of unequal capacity. Finally, we have aree the complexity of etermining the ecrecy capacity if the location of the wiretapper i unknown. It i hown that thi problem, which i cloely relate to network interiction, i NP-har. REFERENCES [1] A. Wyner, The wire-tap channel, Bell Sytem Technical Journal, vol. 54, no. 8, pp , Oct [] C. E. Shannon, Communication theory of ecrecy ytem, Bell Syt.Tech. J., vol. 8, pp , [] N. Cai an R. Yeung, Secure network coing, in Proc. of IEEE ISIT, June 00, p.. [4] J. Felman, T. Malkin, R. Serveio, an C. Stein, On the capacity of ecure network coing, in Proc. of Allerton Conference on Communication, Control, an Computing, Sept [5] S. Y. El Rouayheb an E. Soljanin, On wiretap network II, in Proc. of IEEE ISIT, Nice, France, June 007, pp [6] N. Cai an R. W. Yeung, A ecurity conition for multi-ource linear network coing, in Proc. of IEEE ISIT, Nice, France, June 007, pp [7] Z. Zhang an R. Yeung, A general ecurity conition for multi-ource linear network coing, in Proc. of IEEE ISIT, Seoul, Korea, June 009, pp [8] A. Mill, B. Smith, T. Clancy, E. Soljanin, an S. Vihwanath, On ecure communication over wirele eraure network, in Proc. of IEEE ISIT, July 008, pp [9] R. Koetter an M. Méar, An algebraic approach to network coing, IEEE/ACM Tran. Networking, vol. 11, no. 5, pp , Oct. 00. [10] T. Ho, M. Méar, J. Shi, M. Effro, an D. R. Karger, On ranomize network coing, in Proc. of Allerton Conference on Communication, Control, an Computing, Sept. 00. [11] Xitip - information theoretic inequalitie prover, [1] T. Cui, Coing for wirele broacat an network ecrecy, Ph.D. iertation, California Intitute of Technology, 009. [1] R. W. Yeung, A framework for linear information inequalitie, IEEE Tran. Inform. Theory, vol. 4, no. 6, pp , Nov [14] R. K. Woo, Determinitic network interiction, Mathematical an Computer Moeling, vol. 17, no., pp. 1 18, 199. [15] T. Cui, T. Ho, an J. Kliewer, Achievable trategie for ecure network coing for general network, in Information Theory an Application Workhop, 010. [16] B. C. Dean, M. X. Goeman, an N. Immorlica, Finite termination of augmenting path algorithm in the preence of irrational problem ata, Lecture Note in Computer Science, pp , 006.