Secure Network Coding with a Cost Criterion

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1 Secure Network Coding with a Cost Criterion Jianong Tan, Murie Médard Laboratory for Information and Decision Systems Massachusetts Institute of Technoogy Cambridge, MA 0239, USA E-mai: {jianong, Abstract Network cost and network security are two of many network parameters that are important to network users. Whie these two parameters have been separatey considered in coded networks (networks that empoy network coding), a joint investigation of them both has not been done yet to our knowedge, thus providing the motivation for this work. In this paper, we consider the situation where a set of messages is to be muticasted across the network, and a known subset of these messages is of interest to a wiretapping adversary. The probem that we attempt to sove is to find a network coding scheme that has both a ow network cost and a ow probabiity of the wiretapper being abe to retrieve a the messages of interest. We make use of random inear codes in anticipation for decentraized impementation of the scheme, and focus on the probem of finding the muticast subgraph. As an exact agorithmic soution is difficut, we propose two heuristic soutions, and compare their performances to traditiona routing through a simuation study. Our resuts suggest that network coding can be more effective than routing for this ow cost and secure data muticast probem, especiay when the inks are not easiy tapped.. INTRODUCTION For any communication network or protoco, there are many performance parameters that are of importance to the network users, such as throughput rates and network robustness. When the concept of network coding was introduced by Ahswede et. a. [], the authors aready demonstrated that network coding can achieve higher throughput rates than traditiona routing. One important performance parameter is the network cost incurred for a given set of connections, and the compexity associated with the computation of the subgraphs needed to provide the connections. Whie the minimum cost muticast probem in routed networks requires the finding of a directed Steiner tree, which is NP-hard, the same probem in coded networks can be soved by a inear program in poynomia time [2], and aso be impemented in a decentraized manner [3]. In addition, simuation resuts have shown that network coding can provide the muticast connections at a ower cost than traditiona routing [3], [4]. Another important performance parameter is the security of the network. Cai and Yeung [5] considered the probem of using network coding to achieve perfect information security against a wiretapper who can eavesdrop on a imited number of network inks, and presented the construction of a secure inear network code for this purpose. Fedman et. a. [6] showed that the finding of a matrix for the construction of an optima secure network code is equivaent to finding a inear code with certain generaized distance properties. In a different setting, Ho et. a. [7] showed that randomized network coding is usefu in detecting Byzantine modification of data packets, thus providing data security against Byzantine attackers who arbitrariy modify data packets in the network. Whie both network cost and security have been separatey investigated in the network coding iterature, they have not been jointy investigated yet to our knowedge, thus providing the motivation for this paper. Here, we consider the situation where a set of messages is to be muticasted across the network, of which a subset is of interest to a wiretapper. The probem that we want to sove is to muticast the messages at a ow cost, whie keeping the network vunerabiity defined as the probabiity that the wiretapper is abe to retrieve a the messages of interest ow. Whie network security is not imited to the resiience of the network against wiretapping, the other notions of security are beyond the scope of this paper. In genera, we expect a trade-off between network cost and network vunerabiity. For instance, in routed networks, a cheapest cost approach to a unicast connection usuay seects a singe path. However, when the connection is to be resiient against wiretapping, mutipe disjoint paths may be used, which may increase the network cost [8], [9], [0], []. To iustrate further this trade-off and to show that network coding has the potentia of achieving a ower network vunerabiity than traditiona routing, consider the network shown in Figure (a) where each network ink has unit capacity and unit cost. Two random processes (denoted X and X 2 ) are to be muticasted from the source nodes s and to the sink nodes t and t 2, against a wiretapper who is interested in obtaining the random process X. The probabiity that any one particuar ink is tapped is 0.0, and edges are assumed to be tapped independenty of one another. Figures (b) to (e) show four different methods of achieving the muticast, and Tabe shows the corresponding network costs and vunerabiities. In Figures (b) and (c), each process is transmitted at unit rate, whie in Figures (d) and (e), each process X i is transmitted at a rate of two by spitting it into two processes X i and X i2. Figure (b) shows the singe path routing soution that minimizes the network cost, whie Figure (c) shows the non-trivia singe path network coding soution (note that the trivia soution is identica to the routing soution). Figure (d) shows the mutipath routing soution and Figure (e) shows the mutipath network coding soution.

2 (b) Singe path routing s X X2 X t 2 t 2 (d) Mutipath routing X (a) A simpe network s t 2 t 2 X2 X2 X22 s X X2 X2 X22 t 2 t 2 X22 X22 (c) Singe path network coding X X2 s X+X2 X+X2 X X2 t 2 t 2 X X2 X X2 (e) Mutipath network coding X X2 X22 s X X2+X22 X2+X22 X2 X22 t 2 t 2 X2 X22 Figure : A simpe network exampe. X2 X22 Tabe : Network costs and vunerabiities for the simpe network. Average cost Network (per bit) Vunerabiity Singe path routing Singe path coding Mutipath routing Mutipath coding From this simpe exampe, we see that, whie the singe path routing soution offers the owest network cost, it resuts in the highest network vunerabiity. Whie the network vunerabiity can be reduced by empoying network coding or mutipath routing, the network cost inevitaby increases. It shoud aso be noted that the mutipath network coding soution returns the owest network vunerabiity, at a cost equa to that of mutipath routing. The rest of this paper is structured as foows. Section 2 describes the probem in greater detai. Section 3 describes the genera soution to the probem, which is difficut in genera, and suggests two heuristic methods that can be used to sove the probem. Simuation resuts using the two proposed heuristics, together with a discussion of the resuts, are presented in Section PROBLEM FORMULATION A communication network is represented by a directed graph G = (V, E), where V is the set of vertices (or nodes) and E is the set of edges (or inks). To each edge E, we associate two non-negative numbers c and d, which are the cost per unit fow and the ink capacity, respectivey. For simpicity, the edges are assumed to be free of deays, but the resuts sti appy for networks with deays [3]. In this network, a set of r discrete independent random processes W = {W,..., W r } is to be transmitted to a fixed set of sink nodes, T = {t,..., t T } V. Each random process W i is generated at the source node s Wi, and is assumed to have a constant integer entropy rate of ρ. Thus, it can be decomposed into ρ independent random processes, each with unit entropy rate. We denote the j-th such component of W i as the random process X ρ(i )+j, and we denote the source node of process X i as s Xi. In the network, there exists an adversary who is interested in knowing a given subset of W, denoted by W interest. Without oss of generaity, we assume that the adversary is interested in the first k random processes (i.e. W interest = {W,..., W k }, k r). In terms of the unit entropy processes, we say that the adversary is interested in the set X interest = {X,..., X kρ }. This adversary is abe to tap network inks and retrieve a the messages that are being transmitted aong them. To mode this wiretapping behavior, we associate with each edge, a number p [0, ] to denote the probabiity that the wiretapper wi be abe to retrieve the messages sent aong it. Edges are assumed to be tapped independenty of one another, and we et Y tapped be the (random) set of messages transmitted aong the tapped inks. We define the network vunerabiity ν to be the probabiity that the conditiona entropy of W interest, given Y tapped, is zero: ν := P [H (W interest Y tapped ) = 0]. This definition of the network vunerabiity is ogica because in order for the wiretapper to figure out the identities of a the messages in W interest, he wi need to make a guess out of 2 H(Winterest Y tapped) possibiities. Assuming that the muticast probem is feasibe given the network topoogy, the probem to be soved here is the design of a network coding scheme that has both a ow overa network cost and a ow network vunerabiity. To define the probem more concretey, we denote the overa network cost by the variabe µ = { E} c z, where z is the amount of fow transmitted on the ink. The probem is then to minimize some function F (µ, ν), which is an increasing function of both µ and ν. Once again, as we expect a trade-off between network cost and vunerabiity, it is often inadequate to minimize ony µ or ν. 3. PROBLEM SOLUTION We separate this network coding probem into two parts. The first part deas with the finding of the coding subgraph, incuding the amount of information fow to be put onto each network ink. The second part invoves the actua network code to be impemented in the subgraph. The network code describes how data packets in the network interact with one another, and is important to ensure that the origina messages can be decoded at the sink nodes. In anticipation of a distributed impementation of the soution, we make use of random inear network codes, which are sufficient for muticast [2]. The random inear coding mode 2

3 that we invoke henceforth foows that described in the origina paper by Ho et. a. [3]. In particuar, we et the coding subgraph be represented by G = (V, E ), where every edge E has unit capacity and carries the random process Y ( ). The finite fied used is denoted by F 2 u, and the set of a transmitted processes is denoted by Y : Y = {Y ( )}. { E } Since a the random processes in W have to be muticasted to T, we transate the probem into a singe muticast probem by introducing into G, a pseudo-source node α that generates a the processes in W and transmits them to the actua source nodes s W,..., s Wr via pseudo-edges (α, s Wi ). Noting that the probem of finding the minimum cost subgraph for a singe muticast can be cast into a inear programming probem [2], we structure the subgraph finding probem as the foowing constrained optimization: where: minimize F (µ, ν) subject to z = ρ, s.t. tai() = α, z x (t), E, t T, {: tai()=v} x (t) σ (t) v = {: head()=v} x (t) = σ (t) v, () v V, t T, d x (t) 0, E, t T, rρ if v = α, rρ if v = t, 0 otherwise. One difficuty of this optimization is that, whie it is easy to compute µ given x (t) and z, the same is not true for ν, as it depends on the actua processes sent aong the inks (i.e. the network code). As an exampe, consider the network shown in Figure 2, with ρ =, W = {W, W 2 } = {X, X 2 }, W interest = {W } = {X }. Figure 2(a) shows the vaues of x (t) and z, whie Figure 2(b) shows the network code. Consider the inks = (s, ) and 2 = (2, t 2 ), which have the same x (t) and z vaues, but are carrying different processes. Note that whie the tapping of wi enabe the retrieva of X, the soe tapping of 2 gives no information about X. From this exampe, we see that ν depends not just on x (t) and z, but aso on the actua network code. Another difficuty is that ν and F (µ, ν) may not be convex functions of x (t) and z. For instance, consider the network shown in Figure (e), where we now assume that every edge has a capacity of two. We make a singe change to the network code, whereby for the edge (, ), in addition to transmitting X 2, we aso transmit X 22 in an uncoded manner. As the knowedge of X 22 does not hep the wiretapper in knowing the identity of either X or X 2, this increase in z for the edge (, ) does not affect ν. As a resut of these difficuties, we sha ook at some heuristic methods to sove the probem. (a) Vaues of (z, x (t), x (t2) ) (,,0) s (,0,) (,,0) (,,) 2 (,,0) (,0,) t t 2 (,0,) (b) Actua network code s X X2 X X+X2 X2 2 X+X2 X+X2 Figure 2: Network vunerabiity depends on the network code. t 3. FIRST HEURISTIC Consider the scenario where the foowing statements are true. Whie these statements are not true in genera, they can be reasonabe approximations to the actua network code when the processes are we-mixed throughout the network. First, we have: Y ( ) = Y ( 2) = 2,, 2 E. Hence, the tapping of any n inks in E by the wiretapper wi provide him with a set of at most min(rρ, n) ineary independent equations in terms of the input processes X i. Second, we have: where: Y ( ) = rρ i= ζ,i F 2 u\{0}, ζ,ix i, E, t 2 E, i rρ. Hence, in order to retrieve a the messages in W interest, the wiretapper wi need to have a set of exacty rρ ineary independent equations in terms of the X i s [4]. Under these conditions, ν is then upper bounded by the probabiity that the wiretapper manages to tap at east rρ inks in E, and this probabiity, denoted as ν, can be obtained from the vaues of z and x (t) in the foowing manner. Consider a ink E that is carrying a fow amount z R + 0. Since each ink in E has unit capacity, the number of unit capacity inks in E that correspond to the ink is approximatey [z ] the nearest integer to z. With this, we et L be the discrete random variabe denoting the amount of information fow successfuy tapped on ink. As p is the probabiity of tapping ink, we then have the z-transform of L as (to avoid confusion, the transform variabe z is repaced by w): L (w) = E[w L ] = p w [z ] + ( p ). Now, et L tota be the random variabe denoting the tota amount of information fow successfuy tapped in the network. Since edges are assumed to be tapped independenty of one another, we have: L tota (w) = L (w) = P (L tota = i)w i { E} i 3

4 and the expression for ν is: ν = P (L tota rρ) = rρ i=0 P (L tota = i), which is an increasing function with respect to each z, but may not be convex in nature. Agorithm beow summarizes how to obtain ν from the vector z = (z ). In the agorithm, k is the vector of poynomia coefficients of L (w) and k tota is the vector of poynomia coefficients of L tota (w), which is obtained by convoving a the k vectors. Note that the symbo is used to denote the convoution operation. Agorithm : Given z, cacuate ν. z [z ] E; for each E do k zero vector of ength (z + ); if z = 0 then k [] ; ese k [] p ; k [z + ] p ; end end k tota k k 2 k 3... ; h ength(k tota ); h ν k tota [j]; j=h rρ+ With this, we repace the objective function in () by the function F (µ, ν ), which is a function in terms of z ony. For our simuations, we considered a specific form of the objective function, given by: F (µ, ν ) = µ + ων, where ω is a non-negative weighting variabe that represents the reative importance of the network vunerabiity with respect to the overa network cost. 3.2 SECOND HEURISTIC One major drawback of the first heuristic is that the cacuation of ν can be computationay intensive, as it requires the cacuation of the poynomia coefficients of L tota (w). Because of this compexity, a different heuristic is proposed here. As mentioned in the introduction, we first note that in routed networks, the use of disjoint data paths can decrease the network vunerabiity. Returning to our probem, consider the spreading of information fow across the network inks. If a the fows are concentrated aong a singe muticast tree, then P (L tota rρ) can be quite high, as each used ink carries a arge amount of fow, and one ony needs to tap a few of them to have L tota rρ. However, if the fows are spread out across network inks instead, one wi need to tap more network inks to have L tota rρ. Hence, we expect the network vunerabiity to decrease as information fow is spread across the network. A straightforward method to spread information fow across network inks is to introduce to each ink E, a stricty convex and increasing cost function in terms of z. For our simuations, we considered the addition of a quadratic cost function to each ink. Specificay, the objective function in () is repaced by the foowing function: where µ + ω µ quad, µ quad = { E} p (z /ρ) 2, and ω is a non-negative weighting variabe that represents the reative importance of the network vunerabiity with respect to the overa network cost. Here, it is important to note that ω ω in genera, because ν, whie µ quad r 2. As it is much easier to compute µ quad than ν, this heuristic is easier to impement than the previous one. 4. SIMULATION RESULTS AND DISCUSSION Our simuations are done using the network topoogies made avaiabe by the Rocketfue project [5]. For our simuations, we make the foowing simpifying assumptions. First, we assume that a network inks have infinite capacities (d E). Next, we assume that a the network inks have the same probabiity of being tapped (p = p E). For each scenario, we first decide on the network parameters, incuding the number of input random processes to be transmitted (r), the throughput rate for each process (ρ), the number of processes of interest to the wiretapper (k) and the number of sink nodes ( T ). We aso decide on the network to be used, which gives us the c vaues. We then proceed to run simuations with the above network parameters, and cacuate the mean network cost and network vunerabiity. For each simuation, each source node in S is randomy and uniformy chosen from V with repacement, before each sink node in T is randomy and uniformy chosen from V S without repacement. During this process, we aso ensure that the muticast probem is feasibe, by requiring the presence of at east one path from each source node to each sink node. Foowing this, we et p vary from 0 to 0., and for each vaue of p, we proceed to sove for five different muticast subgraphs. We first consider the singe path routing and the mutipath routing soutions, before going on to consider the singe path network coding soution by having the objective function in () as µ. Finay, we consider the two heuristics proposed in this paper. For each subgraph, we estimate the network vunerabiity in the foowing way. We randomy generate 2000 sets of tapped inks based on p, and decide for each set, if it enabes 4

5 Tabe 2: Network vunerabiities for three different networks, with r = 4, ρ = 0, k =, T = 4, and p = 0.0 or 0.. Exodus Testra Tiscai p = 0.0 p = 0. p = 0.0 p = 0. p = 0.0 p = 0. Singe path routing Mutipath routing Singe path coding Heuristic (ω = 3 05 ) Heuristic 2 (ω 0 = 3000) the wiretapper to retrieve a processes in Winterest. The sampe mean is then taken to be the network vunerabiity (ν). From Tabe 2, we observe that when p = 0.0, not ony are the vaues of ν associated with the two heuristics comparabe to that of mutipath routing, but they are often the owest vaues out of the five. However, when p increases to 0., both heuristics fai to match up to mutipath routing in terms of the network vunerabiity. In fact, we observe that the second heuristic yieds the highest network vunerabiity. To expain these observations, we consider the coded network shown in Figure 2(b) where the wiretapper is interested in X. When p is very sma, we can approximate the situation as one where the wiretapper has negigibe probabiity of tapping two or more network inks. Thus, the ony way for him to retrieve X is by tapping one of the two inks that are carrying X in an uncoded fashion. For a simiar routing strategy on the network, at east four network inks must be used for the muticast of X, and the tapping of any one of these inks wi enabe the retrieva of X. Thus, routing resuts in a higher vaue of ν than network coding. In genera, when p is sma, better security is achieved by network coding as Ytapped is often sma, and the wiretapper is unabe to retrieve enough degrees of freedom to decode Winterest entirey. In routed networks, we first note that the tapping of inks carrying messages outside Winterest does not hep the wiretapper in the retrieva of the messages in Winterest. However, in coded networks, the tapping of such inks can potentiay aid the wiretapper. Consider the network in Figure 2(b), athough H(X X + X2 ) > 0 and H(X X2 ) > 0, we note that H(X {X +X2, X2 }) = 0. In genera, for coded networks, there are more ways for the wiretapper to decode the messages in Winterest from Y. This reduces the security advantage of transmitting encoded messages, and resuts in the higher vaues of ν seen in coded networks when p is arge. Figure 3 shows the pots of two possibe functions that F (µ, ν) can take, for the Exodus network. Figures 3(a) and 3(b) show the pots for F (µ, ν) = µ + ων, where ω = 3 05 (the vaue of ω used in the first heuristic). Figures 3(c) and 3(d) show the pots for F (µ, ν) = µν. From Figures 3(a) and 3(b), we see that the first heuristic performs the best out of the three coding strategies for a vaues of p, but the second heuristic performs quite we when p is sma. In addition, the routing cases perform worse than the coding cases for sma vaues of p, mainy because of the higher network cost incurred in routed networks. From Figures 3(c) and 3(d), we again observe that the coding schemes perform better than the routing schemes for sma vaues of p. Simuations done on the Testra and Tiscai networks show simiar resuts. In particuar, we see that for the case where F (µ, ν) = µν, both routing soutions returned higher vaues of µν than the coding soutions for a vaues of p between 0 and 0.. The corresponding pots for the Testra and Tiscai networks are shown in Figures 4(a) and (b), respectivey. Figure 5 shows the pot of network vunerabiity against p for the first heuristic, but with different vaues of r. The vaues of ω were appropriatey scaed to compensate for the higher vaues of µ with increasing r. Figure 3: Pot of two possibe reaizations of F (µ, ν) against p. (Exodus network, r = 4, ρ = 0, k =, T = 4). Figure 4: Pot of µν against p for the Testra and the Tiscai networks. (r = 4, ρ = 0, k =, T = 4). 5

6 Figure 5: Pot of ν against p for different vaues of r (first heuristic). (Exodus network, ρ = 0, k =, T = 4). From Figure 5, it is observed that when r increases from 2 to 6, ν decreases, but when r increases further to 2, ν does not decrease anymore. On the contrary, ν increases for sma vaues of p when r = 2. This interesting behavior can be expained as foows. When r increases, it affects the network vunerabiity in severa ways. First, the mean number of encoded processes X i in the transmitted processes Y ( ) grows as O(r). This decreases ν as the wiretapper wi, on average, need to retrieve more degrees of freedom from the network in order to decode a the messages in W interest. Second, the size of Y grows as O(2 r ), increasing the number of ways W interest can be decoded from the messages in Y, thus increasing ν. Finay, the average amount of fow on the network inks grows as O(r). Thus, for any fixed p, the expected size of Y tapped increases, and ν increases. As the increase in r can affect the network vunerabiity in these different ways, the net effect of an increase in r on ν is ambiguous. However, the simuation resuts suggest that when r is sma, an increase in r decreases ν, whie for arger vaues of r, an increase in r increases ν. This is probaby due to the different growths of the network parameters mentioned in the previous paragraph. As the size of Y grows as O(2 r ), when r is sma, the growth of Y is sma, and the net effect of an increase in r is a decrease in ν. However, when r gets bigger, the growth of Y becomes much arger, resuting in a net increase in ν. 5. CONCLUSION In this paper, we have considered the probem of providing a set of connections at both a ow cost and a high eve of security against wiretapping. As an exact soution is difficut, we have presented two heuristic methods of finding the coding subgraphs that can achieve this. Through our simuation resuts, we observe that a trade-off between network cost and network vunerabiity aso exists in coded networks. In addition, we observe that network coding can be more effective than traditiona routing for ow cost and secure data muticast, especiay when the inks are not too easiy tapped. In this study, we have focused on the probem of finding the coding subgraph, and conducted our simuations in a centraized manner. However, it shoud be noted that the finding of the coding subgraph can done in a decentraized manner for the second heuristic that we have proposed, since it has a stricty convex cost function for each network ink [3]. As we have assumed the use of a random inear network code, which itsef is impementabe in a distributed manner, we concude that our second heuristic soution to the probem can be impemented in an entirey decentraized manner. As we have ony considered the issue of resiience against wiretapping in this work, future research can be done to investigate the other security issues of network coding. Furthermore, as we have ony suggested two simpe heuristic approaches to the probem, aternate agorithmic approaches remain as cear avenues for future work. REFERENCES [] R. Ahswede, N. Cai, S.-Y. R. Li, and R. W. Yeung, Network information fow, IEEE Trans. Inform. Theory, vo. 46, no. 4, pp , Ju [2] D. S. Lun, M. Médard, T. Ho, and R. Koetter, Network coding with a cost criterion, IEEE Internationa Symposium on Information Theory and its Appications, [3] D. S. Lun, N. Ratnakar, R. Koetter, M. Médard, E. Ahmed, and H. Lee, Achieving minimum-cost muticast: A decentraized approach based on network coding, Proc. - IEEE INFOCOM, Mar [4] P. A. Chou, Y. Wu, and K. Jain, Practica network coding, Proc. 4st Annua Aerton Conference on Communication, Contro, and Computing, Oct [5] N. Cai and R. W. Yeung, Secure network coding, Proc. - IEEE Internationa Symposium on Information Theory, pp. 323, [6] J. Fedman, T. Makin, R. A. Servedio and C. Stein, On the capacity of secure network coding, Proc. - 42nd Annua Aerton Conference on Communication, Contro and Computing, September [7] T. Ho, B. Leong, R. Koetter, M. Médard, M. Effros, and D. R. Karger, Byzantine modification detection in muticast networks using randomized network coding, Proc. - IEEE Internationa Symposium on Information Theory, pp. 44, [8] J. Jeong, and J. Ma, Security anaysis of muti-path routing scheme in ad hoc networks, Lecture Notes in Computer Science, vo. 3320, pp , [9] W. Lou, W. Liu, and Y. Fang, SPREAD: Enhancing data confidentiaity in mobie ad hoc networks, Proc. - IEEE INFOCOM, vo. 4, pp , [0] J. Yang, and S. Papavassiiou, Improving network security by mutipath traffic dispersion, Proc. - IEEE MILCOM, vo., pp , 200. [] C. K.-L. Lee, X.-H. Lin, and Y.-K. Kwok, A mutipath ad-hoc routing approach to combat wireess ink insecurity, IEEE Internationa Conference on Communications, vo., pp , [2] S.-Y. R. Li, R. W. Yeung, and N. Cai, Linear network coding, IEEE Trans. Inform. Theory, vo. 49, no. 2, pp , Feb [3] T. Ho, M. Médard, J. Shi, M. Effros and D. R. Karger, On randomized network coding, Proc. - 4st Annua Aerton Conference on Communication Contro and Computing, Oct [4] R. Koetter, and M. Médard, Beyond routing: An agebraic approach to network coding, Proc. - IEEE INFOCOM, vo., pp , [5] N. Spring, R. Mahajan, and D. Wethera, Measuring ISP topoogies with Rocketfue, IEEE/ACM Trans. Networking, vo. 2, no., pp. 2-6, Feb

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