NP-Completeness

We show in the following that determining the secrecy capacity is NP-complete when the location of the wiretap links is known or unknown by reduction from the clique problem, which determines whether a graph contains a clique² of at least a given size r.

The case when the location of the wiretap links is known by the source and the wiretapper can choose the wiretapping set to minimize the secrecy rate is closely related to the network interdiction problem [79]. The network interdiction problem is to minimize the maximum flow of the network when a certain number of links in the network is removed, which is a special case of the secrecy communication problem when the location of the wiretap links is known and q_i,j = p_i,j, ∀(i, j) ∈ E. It is shown in [79] that the network interdiction problem is NP-complete. Therefore, the case where the location of the wiretap links is known is also NP-complete. To show that the case where the location of the wiretap links is unknown is NP-complete, we use the construction in [79] showing that for any clique problem on a given graph H, there exists a corresponding networkG^Hwhose secrecy capacity isrwhen the location of the wiretap links is known if and only if H contains a clique of size r. We then show that for all such networks the secrecy rate for the case when the location of the wiretap links is unknown is equal to that for the case when such information is known, which shows that there is a one-to-one correspondence between clique problem and the secrecy capacity problem.

We briefly describe the approach in [79] in the following. Given an undirected graphH= (Vh,Eh), we will define a capacitated directed network ˆG^H such that there exists a set of links ˆA^′ in ˆG^H containing less than or equal to |Eh| − ^r₂

links such that ˆG^H−Aˆ^′ has a maximum flow of r if and only if H contains a clique of size r.

2A clique in a graph is a set of pairwise adjacent vertices, or in other words, an induced subgraph which is a complete graph.

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4 1

a

y b

c

x

(a) Original Graph

s d

ia

iy

ix

i_c i_b

j1

j₄ j3

j2

(b) Transformed Graph

Figure 5.4: Example of NP-completeness proof for the case with knowledge of wire- tapping set

For a given undirected graph H = (V^h,E^h) without parallel edges and self loops, we create a capacitated, directed graph G^H = (N,A) as follows: For each edge e ∈ Eh

create a node ie in a node set N¹ and for each vertex v ∈ V^h create a node jv in a node set N2. In addition, create source node s and destination node d. For each edge e∈ E^h, direct an arc in G^H froms to ie with capacity 2 and call this set of arcs A1. For each edge e = (u, v) ∈ Eh, direct two arcs in G^H from ie to jv and ju with capacity 1, respectively and call this set of arcs A². For each vertex v ∈ V^h, direct an arc with capacity 1 from jv to d. Let this be the set of arcs A3. This completes the construction ofG^H = (N,A) = ({s} ∪ {d} ∪ N¹∪ N²,A¹∪ A²∪ A³). In Fig. 5.4, we give an example of the graph transformation, where H = ({1,2,3,4},{a, b, c, x, y}).

We replicate [79, Lemma 2] as follows.

Lemma 5.3 Let G^H be constructed fromH as above. Then, there exists a set of arcs A^′1 ⊆ A¹ with |A^′1|=|E^h| − ^r₂

such that the maximum flow from s to d in G^H− A^′1

is r if and only if H contains a clique of size r.

After obtainingG^H, we generate ˆG^Hby replacing each arc (ie, jv) with|Eh|parallel arcs each with capacity 1/|E^h| and call this arc set ˆA². We carry out the same procedure for arcs (jv, dl) and call this arc set ˆA3. Then ˆG^H = (N,A) = ({s} ∪ {d} ∪

N¹∪ N²,A¹∪Aˆ²∪Aˆ³). For the case when the location of wiretap links is known, it is shown in [79] that the worst case wiretapping set ˆA^′ must be a subset of A1. By using Lemma 5.3, this case is NP-complete.

Now, we consider the case where the wiretapping set is unknown, and show that the secrecy capacity of ˆG^H when the wiretapper accesses any unknown subset of k =|Eh| − ^r₂

links isr if and only ifH contains a clique of size r. From Lemma 5.3, the condition thatHcontains a clique of size ris equivalent to the condition that the max-flow to the sink in G^H after removing any k links from A1 is r. We now show that the latter condition is equivalent to the condition that the secrecy capacity of G^H when the wiretapper accesses any unknown subset of k links from A1 is r. We create a virtual sink connecting each subset of k links from A¹ and the actual sink.

As the wiretapped links are connected to the source directly, the min-cut between each virtual sink and the source is at least 2k+r. Since r is the cut-set upper bound on the secrecy rate, by using the key cancelation scheme (Strategy 1) in Section 5.4 with 2k global keys the secrecy rateris achievable, which is equal to the secrecy rate when the location of wiretap links is known.

Finally, we show that the secrecy capacity of G^H when any k links of A¹ are wiretapped is equal to the secrecy capacity of ˆG^H when any k links are wiretapped.

Since each second layer link has a single first layer link as its only input, wiretapping a second layer link yields no more information to the wiretapper than wiretapping a first layer link. When some links in the third layer are wiretapped, let the wiretapping set be ˆA^′ = ˆA^′1∪Aˆ^′3 where ˆA^′3 6= ∅ or |Aˆ^′3| ≥ 1 and |Aˆ^′1| ≤ k −1. Thus A1 −Aˆ^′1

contains at least ^r₂

+ 1 arcs. As in Section 5.4, we create a nodet^Aˆ′ by connecting all wiretapping links in ˆA^′ and a virtual sink d^Aˆ′. Connect the actual sink to d^Aˆ′ with a link of capacity r and connectt^Aˆ′ to d^Aˆ′ with a link of capacity R_s→Aˆ^′, where R_s→Aˆ^′

is the min-cut between the source and t^Aˆ′. As removing links in A1 is equivalent to removing links in H, after removing links in H corresponding to ˆA^′1, H contains a subgraph H1 containing ^r₂

edges plus at least an edge e = (u, v). We consider two cases.

Case 1: H1 is a clique of size r. In this case, the number of vertices with degree

greater than 0 in H¹∪e isr+ 2.

Case 2: H1 is not a clique. H1 contains at leastr+ 1 vertices with degree greater than 0.

According to [79, Lemma 1], the max-flow inG^His equal to the number of vertices in H with degree greater than 0. In both cases, the max-flow of G^H after removing links in ˆA^′1 is at least r+ 1. Let R_s→Aˆ^′3 be the max-flow capacity from the source to Aˆ^′3 in G^H−Aˆ^′1 and D a corresponding max-flow subgraph. After removing D from G^H −Aˆ^′1, the min-cut between the source and the actual sink is at least r + 1−

|Aˆ^′3|/|E^h| > r+ 1−(|E^h| −1)/|E^h| > r. Therefore, the min-cut between the source and d^Aˆ′ in G^H −Aˆ^′1 − D is r, and the min-cut between the source and d^Aˆ′ in G^H is r+R_s→Aˆ^′. On the other hand, the total amount of wiretapped data is at most 2|Aˆ^′1|+|Aˆ^′3|/|Eh| ≤2|Aˆ^′1|+(|Eh|−1)/|Eh| ≤2k−2+(|Eh|−1)/|Eh|<2k. By using the precoding scheme in Strategy 1 in Section 5.4, if the source sends r message symbols and 2k random keys, perfect secrecy is achieved when ˆA^′ is wiretapped and the size of finite field is large enough. Thus, the secrecy rate for the case when the location of the wiretap links is unknown is equal to that for the case when such information is known with unrestricted wiretapping set. We have the following theorem.

Theorem 5.4 Computing the secrecy capacity no matter whether the location of the wiretap links is known is NP-complete.