# On Secure Communication over Wireless Erasure Networks

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2 by ɛ ij. We assume that each receiver obtains all of the symbols along its incoming edges without interference. The capacity of this non wire-tapped network model was first given in [1] (and its achievablility alternatively demonstrated by a random linear network coding scheme in in [3]), and is given by the expression: C = min lg q 1 (1) S i S j S C ɛ ij where S is a vertex-cut. Specifically, S is any subset of the nodes which contains the source and not the destination. This paper extends the work on this notion of lossy wireless networks to demonstrate its secrecy capacity. node i, and γ ij be independent Bernoulli random variables with P [γ ij = 0] = ɛ ij. Then, for the network displayed in Figure 2, the matrix Y 2,s γ s2 0 [ ] Y 2,1 = 0 γ 12 Xs X Y d,1 0 γ 1 1d defines G A for the cut A = {(s, 2), (1, 2), (1, d)}. Note that when A is chosen to be the set of edges crossed by the vertex min-cut, then the expected value of G A corresponds precisely to the value of the min-cut capacity given in Equation (1). ε s1 1 ε1d s ε 12 d Dest. Source εs2 ε 2d 2 Fig. 1. System Model - Dotted Edges Represent an Example Edge Cut-set. cut Along the lines of the model in [8], the eavesdropper has access to any k edges of this network, in the sense that it can observe the outputs of each edge in the network. Specifically, if the packet transmitted along any wire-tapped edge is erased (or dropped), the wire-tapper as well the receiving node fail to receive it. Our objective is to ensure perfect secrecy of the message in the network from the wire-tapper, defined in the sense of [6]. Specifically, we wish to determine the highest rate possible such that the wire-tapper gains no information about the message being communicated, and that the mutual information between the wire-tapper s information and the source s intended message is zero. Let s V denote the source and d V the destination. This paper defines an edge cut as any set T E such that there does not exist a path from s to d in E \ T. We assume a path from s to d exists in E, so a cut must be nonempty. Also, note that E itself defines a cut in this network. The definition of a cut in this paper is distinct from that used in [1], which is a vertex-based definition of a cut in the network. Letting A E be a set of edges of the graph, we define a size m n incidence matrix. The number of columns n will be equal to the number of distinct parent vertexes of the edges in A. That is, n = I A, where I A = {i (i, j) A}. The number of rows m in a network with no interference will be equal to the number of edges in A, that is m = A. For each time slot t, the incidence matrix G A (t) will contain a 1 in each row corresponding to an edge whose symbol was not dropped, in the column corresponding to the corresponding parent vertex. Let X i denote the symbol transmitted from node i, Y j,i denote the symbol received at node j which had been transmitted from Fig. 2. Example Network and Edge Cut Set III. UPPER BOUND ON SECRECY CAPACITY Example 1: Consider for a moment a simple unicast scenario shown in Fig. 3. Assume that the source is directly Fig. 3. At the source node, k information symbols are encoded into h coded symbols, which are then simultaneously transmitted to the destination. connected to the destination through n network links where link i, 1 i n has erasure rate ɛ i, and that any k of these edges can be accessed by a wiretapper. If we assume that all erasure rates are equal, that is ɛ ɛ 1 =... ɛ n, then this network unicast is equivalent to the Wyner wiretap channel model in which the intended user observes the output of an erasure channel with the erasure rate ɛ n, and the wiretapper observes the output of a degraded channel with the erasure rate ɛ k. Outer Bound: Let S be any set of k edges in the network. Let T : S T E be a cut in the network. Then, the secrecy capacity of the network is bounded by: C lg q (E[rank(H T )] E[rank(G S )]) (2) where H T is the incidence matrix of the cut T and G S the incidence matrix of S.

4 the intermediate nodes is known to the receiver. In the rest of this document, we will utilize the linear network coding framework for coding at the intermediate nodes in [1]. Note that our proof for secrecy capacity can be modified to the case when random network coding is used [1]. Let n be a positive integer. Define m ne(rank(h T )) and l ne(rank(g S )). Then C = m l n lg q. Intuitively, we show that if m un-erased packets reach the legitimate destination and l un-erased packets reach the wire-tapper, then there exists a coding scheme that achieves C. The encoding scheme is as follows: A transmit vector of length m packets for some ɛ > 0 is constructed consisting of (m l nɛ) information (or data) packets and (l + nɛ) noise packets that are independent of the data source and chosen uniformly from GF (q). We refer to this as the vector x m, with a (m l nɛ) denoting data packets, b l+nɛ denoting noise packets, and x m = [a (m l nɛ) b (l+nɛ) ]. A linear encoding scheme is used at each node, which results in n m transfer matrices M n and F n for the legitimate destination and the wire-tapper respectively. By the weak law of large numbers (which results in the notion of strong typicality as in [16]), the received vector at the legitimate destination is at least an (m nɛ)-sized subset of the n-length vector M n x m with probability greater than 1 δ. Similarly, the wire-tapper observes at most an (l + nɛ)-ary subset of the vector F n x m with high probability. In [1, Section 7-A], the authors show that averaging over all matrices, the average probability of error in decoding can be made less than ɛ. Therefore, the information vector a m l nɛ is received successfully with high probability at the destination allowing for a rate arbitrarily close to C ɛ. Let the (l + nɛ)-ary subset of the vector F n x m received at the wire-tapper with high probability be denoted as: I(a m l nɛ ; z l+nɛ ) = H(z l+nɛ ) H(z l+nɛ a m l nɛ ) (a) = H(z l+nɛ ) H(ẑ l+nɛ a m l nɛ ) (b) = H(z l+nɛ ) H(ẑ l+nɛ ) (c) = H(z l+nɛ ) (l + nɛ) lg q (d) (l + nɛ) lg q (l + nɛ) lg q = 0 where: (a) follows from the definition of ẑ (l+nɛ) and the property of entropy. (b) follows from the independence between a (m l nɛ) and b (l+nɛ). (c) follows from the invertibility of ˆF (l+nɛ), and that b (l+nɛ) are i.i.d. and uniform. Or in essence, no information is recovered by the wire-tapper. This concludes the achievability proof. V. UPPER BOUND COUNTER EXAMPLE Counter to the authors intuitive expectations, the upper bound of Section III is not tight in general. This section provides a counter example demonstrating this fact. s ε=.4 r ε=.8 ε=.5 ε=1 d z l+nɛ F l+nɛ x m Fig. 4. Upper Bound Counter Example Define ẑ (l+nɛ) ˆF (l+nɛ) b (l+nɛ) where ˆF (l+nɛ) is a (l+nɛ) (l+nɛ) matrix of the last (l+nɛ) columns from F l+nɛ. A key step is for us to choose coefficients for the network (i.e., one particular M n and F n ) such that: a (m l nɛ) is decodable at the legitimate destination with probability of error arbitrarily small, and ˆFl+nɛ is invertible. Note that a random choice in coefficients ensures both of these simultaneously with high probability [1], [17], and therefore a particular set of M n and F n exist that satisfy these requirements. By Fano s inequality [16], the information recovered by the wire-tapper is upper bounded by: Consider the three node network shown in Figure 4. This network has two independent edges which connect the source to the destination. The wiretapper may choose any one edge to wiretap, but it is clear that the set consisting of the edge S = {sr} is the most favorable. Note that this set S does not lie in the minimum cut edge set of the original network T min = {rd, sd 1, sd 2 }, whose cut value is 1 (.8)(.5) =.6. We evaluate along the cut T = {sr, sd 1, sd 2 }, (E[rank(H T )] E[rank(G S )]) = (1 (.4)(.5)(.8)).6 =.24 to find the minimum of the outerbounds on capacity from Equation 2. However, the actual secrecy capacity of this network is equal to zero. The wiretapper receives the data transmitted by the source, with a loss of 40% of all the symbols. The destination will also receive the data transmitter by the source, again with a loss of 40% of the symbols. Therefore, whatever the destination d can decode, the wiretapper will also be able to decode, and there is no secrecy capacity.

5 VI. CONCLUSIONS This work investigates the secrecy capacity, or specifically the equivocation rate, of a wire-tapped wireless erasure network. Secrecy capacity for this network takes the same intuitive form as it does in many other network models: we desire to maximize the difference in the amount of data that the receiver can interpret and the amount of information from the source the wire-tapper could receive. In a simple, error and interference-free network, this result has the straightforward interpretation of subtracting the number of wire-tapped edges from the number of edges in the minimum cut. For the wireless erasure network, the subtraction is equivalent, given the modified cut-set bound. This paper shows that the upper-bound is achievable when the wire-tappers are chosen from a specific subset of the nodes, that is, along the minimum cut of the original network. The strategy is interesting because of its history in the secrecy context, and its application to a non-traditional secrecy model. When the wiretappers do not lie along the minimum-cut of the original network, we have shown that our upper-bound may not, in general, be tight. REFERENCES [1] A. F. Dana, R. Gowaikar R. Palanki, B. Hassibi, M. Effros, Capacity of wireless erasure networks, IEEE Trans. on Inform. Theory, Vol. 52, pp , March [2] D. Julian, Erasure Networks, Proc. IEEE ISIT,pp. 138, [3] D. S. Lun, M. Médard, and M. Effros, On coding for reliable communication over packet network,proc. 42nd Annual Allerton Conf. on Commun. Control, and Computing, September [4] M. Xiao, M. Médard, and T. Aulin, A Binary Coding Approach for Combination Networks and General Erasure Networks, Proc. IEEE ISIT, [5] I. Csiszár and J. Körner, Broadcast channels with confidential messages, IEEE Trans. Inform. Theory, 24(3):339348, May [6] A. D. Wyner, The wire-tap channel, Bell Syst. Tech. J., 54(8): , October [7] Y. Liang and H. V. Poor, Generalized multiple access channels with confidential messages, IEEE Trans. Inform. Theory, [8] N. Cai and R. W. Yeung, Secure Network Coding, ISIT [9] J. Feldman, T. Malkin, C. Stein, R. A. Servedio On the Capacity of Secure Network Coding, Proc. 42nd Annual Allerton Conference on Communication, Control, and Computing, September [10] S. E. Rouayheb and E. Soljanin, On wiretap networks II, in Proc International Symposium on Information Theory, (ISIT 07), Nice, France, June [11] K. Bhattad and K. R. Narayanan, Weakly secure network coding, Netcod 2005, Italy, April [12] K. Jain, Security based on network topology against the wiretapping attack, IEEE Wireless Communications, pp , Feb [13] B. Smith and S. Vishwanath, Unicast transmission over multiple access erasure networks: capacity and duality, IEEE Information Theory Workshop, Lake Tahoe, CA, Sep [14] S. Bhadra, P. Gupta, and S. Shakkottai, On network coding for interference networks, in Proc. IEEE ISIT 2006, Seattle, WA, Jul [15] A. Avestimehr, S. Diggavi, D. Tse, Wireless network information flow, Proc. 45th Annual Allerton Conference on Communication, Control, and Computing, September [16] R. W. Yeung, A First Course in Information Theory, Kluwer Academic Press. [17] T. Ho, R. Koetter, M. Medard, D. R. Karger and M. Effros, The benefits of coding over routing in a randomized setting, IEEE International Symposium on Information Theory (ISIT), [18] A. Shamir, How to Share a Secret, Communications, 1979.

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