# Achievable Strategies for General Secure Network Coding

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3 while Y h is a degraded version of Z h. We consider block coding with block length n. We have nr s H(M Z n ) (a) H(M Z n ) H(M Y n ) + nǫ n (b) =H(M Z n d,z n h) H(M Y n d,y n h) + nǫ n (c) H(M Z n d,y n h) H(M Y n d,y n h) + nǫ n (d) H(M Z n d,yn h ) H(M Zn d,yn d,yn h ) + nǫ n =I(M;Y n d Z n d,y n h) + nǫ n (e) I(X n ;Yd n Z n d,yh) n + nǫ n, n n = H(Y d,i Z n d,yh) n H(Y d,i X n,z n d,yh) n (f) + nǫ n, n H(Y d,i Z d,i,y h,i ) + nǫ n, =ni(x;y d Z d,y h ) + nǫ n, n H(Y d,i X i,z d,i,y h,i ) =n (I(X;Y d,y h ) I(X;Z d,y h )) + nǫ n, n max (I(X;Y d,y h ) I(X;Z d,y h )) + nǫ n, p(x) (g) =n (1 p i,j ) (1 q i,j ) (i,j) [V s,vs c] d (1 p i,j ) + nǫ n, h =n (1 p i,j )+ (i,j) [V s,vs c] A max(q i,j p i,j, 0) + nǫ n, where ǫ n 0 as n + and (a) comes from Fano s inequality. (b) follows from the definition of Y d,y h,z d,z h. (c) comes from the fact that M X n (Z n d,zn h ) (Zn d,yn h ) forms a Markov chain. (d) follows from conditioning reduces entropy. (e) comes from the fact that M X n (Yd n,yn h ) (Zn d,yn h ) forms a Markov chain and A B C D I(A; C D) I(B; C D). To show this inequality, we have I(A; C D) I(B; C D) =I(A; C, D) I(A; D) I(B; C, D) + I(B; D) =I(B; D A) I(B; C A) = I(B; C A, D) 0. (10) (f) follows from the fact that conditioning reduces entropy and that Y d,i is independent of other variables given X i,z d,i,y h,i. (g) is because both I(X;Y d,y h ) and I(X;Z d,y h ) are maximized when the entries of X are i.i.d. Bernoulli(1/2). By decoupling the secrecy coding from the routing or network coding as in the achievability proof of Theorem 1, Theorem 1 can be readily extended to the multicast case. The proof is similar to the unicast case. We thus give the following theorem without proof. Theorem 2. Consider a multicast problem in a wireline erasure network G = (V, E) with a single source s V and a set of destinations D V. A secret message M is multicast from s to all nodes in D. There exists a wiretapper in the network that can wiretap at most k links, and the wiretapped messages are denoted as Z. Assug that the destination has complete knowledge of the erasure locations on each link of the network and the locations of the wiretapped links, the secrecy multicast capacity of the network is given by C s = where d D {V s: V s is an s d cut} {A A [V s,vs c ], A k} (1 p i,j )+ (i,j) [V s,vs c] A max(q i,j p i,j, 0), (11) H(M Z) = H(M). (12) B. Achievable Strategies for Unknown Wiretap Set In the case of unrestricted wiretapping sets, unit link capacities, and p i,j = q i,j on every edge (i, j), the secrecy capacity can be achieved using global keys generated at the source and decoded at the sink [3]. The source transmits r secret information symbols and k random key symbols, where r + k is equal to the -cut of the network. This scheme does not achieve capacity in general networks. Intuitively, this is because the total rate of random keys is limited by the cut from the source to the sink, whereas more random keys may be required to fully utilize large capacity cuts with large capacity links. In this case, capacity can be improved by using a combination of local and global random keys. A local key is injected at a non-source node and/or canceled at a non-sink node. However, it is complicated to optimize over all possible combinations of nodes at which keys are injected and canceled. Thus, we propose the following more tractable family of constructions, which we will use in subsequent sections. In the following, we focus on the case of a single source and a single sink. Let z i,j be the actual flow on link (i, j). For the case where q i,j p i,j, by using random linear z coding on each link, node i sends i,j () random linear combinations over link (i, j) to guarantee that j can decode

6 1 seems to be useful if the wiretapped links are upstream of the -cut while Strategy 2 is useful if the wiretapped links are downstream of the -cut. In general, these two strategies can be combined to obtain a higher secrecy rate. We can partition the graph G = (V, E) into two graphs G 1 = (V, E 1 ) and G 2 = (V, E 2 ), where each edge e in V is partitioned into two parallel edges e (1) E 1 and e (2) E 2 with capacity c (1) e and c (2) e (c e = c (1) e +c (2) e ), respectively. We then apply strategy 1 on graph G 1 and strategy 2 on graph G 2. Combining (14) and (15), we get max R s (1) + R s (2) subject to f (1),A i,j R s (1) f (1),A j,i = + ( ) z 1 qi,j, 1, if i = s, R s (1) ( ) z 1 qi,j 0, f (1),A i,j z i,j c (1) i,j, (i, j) E, f (2),A i,j f (2),A j,i j j { = ( ) g(2) 1 qi,j R w,i, g (2) i,j g (2) j,i = j j R s (2) ( + R w,s, if i = s, R s + ) v V,v d R w,v, if i = d, R w,i, f (2),A i,j g (2) i,j, g(2) i,j c(2) i,j, (i, j) E, c (1) i,j + c(2) i,j = c i,j, (i, j) E. (16) Consider the example in Fig. 2, where all links have unit capacity and p i,j = q i,j for all edges (i, j). Let the middle layer links be 1-5 (from top to bottom) and the last layer links be 6-8 (from top to bottom). Any three of the five links in the middle layer can be wiretapped. We show that the optimal solution must inject random keys at the source and random keys at the second layer nodes and some random keys are canceled at the fifth layer. At the beginning of Strategy 2, we have shown that only global keys at the source is not sufficient. We next consider when random keys are only injected at the second layer nodes. It is clear that if the source does not inject any random key the second layer nodes must inject some random keys otherwise perfect secrecy cannot be achieved. As the maximum secrecy rate is two obtained by the cut-set bound, we can move some portion of random keys from the second layer to the source. Therefore, there are random keys injected at both the source and the second layer nodes. By solving the linear programs (14) and (15), we find that both strategy 1 and strategy 2 achieve a rate 1.2. For strategy 3, a rate is achievable. Even though strategy 3 cannot achieve the outer bound 5/3 given in [11], it outperforms both both strategy 1 and strategy 2. This also shows that strategy 3 is not simply a time sharing strategy between strategy 1 and strategy 2 over the whole network. For numerical computation of achievable rates in scenarios 1 and 2, we note that the number of possible wiretapping sets, and thus the size of the LPs, are exponential in the size k of each wiretap set, so they are useful for small k. IV. CONCLUSION We have considered the secrecy capacity of wireline networks with restricted wiretapping sets and for unequal capacity links. For such a scenario we have shown that inserting global keys at the source represents a suboptimal strategy. This is in contrast to the case of unrestricted wiretapping sets and equal capacity links where a global key strategy achieves the secrecy capacity. In particular, we have proposed achievable strategies where random keys are canceled at intermediate non-sink nodes, injected at intermediate non-source nodes, or where a combination of these strategies is considered. REFERENCES [1] A. Wyner, The wire-tap channel, Bell Systems Technical Journal, vol. 54, no. 8, pp , Oct [2] C. E. Shannon, Communication theory of secrecy systems, Bell Syst.Tech. J., vol. 28, pp , [3] N. Cai and R. Yeung, Secure network coding, in Proc. of IEEE ISIT, June 2002, p [4] J. Feldman, T. Malkin, R. Servedio, and C. Stein, On the capacity of secure network coding, in Proc. of Allerton Conference on Communication, Control, and Computing, Sept [5] S. Y. El Rouayheb and E. Soljanin, On wiretap networks II, in Proc. of IEEE ISIT, Nice, France, June 2007, pp [6] A. Mills, B. Smith, T. Clancy, E. Soljanin, and S. Vishwanath, On secure communication over wireless erasure networks, in Proc. of IEEE ISIT, July 2008, pp [7] R. Koetter and M. Médard, An algebraic approach to network coding, IEEE/ACM Trans. Networking, vol. 11, no. 5, pp , Oct [8] T. Ho, M. Médard, J. Shi, M. Effros, and D. R. Karger, On randomized network coding, in Proc. of Allerton Conference on Communication, Control, and Computing, Sept [9] T. Cover and J. Thomas, Elements of Information Theory, [10] R. W. Yeung, Information Theory and Network Coding. Springer, August [11] T. Cui, T. Ho, and J. Kliewer, On secure network coding over networks with unequal link capacities, Submitted to IEEE Transactions on Information Theory. Online: arxiv.org, arxiv: v1, 2009.

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