Security Versus Capacity Tradeoffs in Large Wireless Networks Using Keyless Secrecy

Transcription

1 Security Versus Capacity Tradeoffs in Large Wireless Networks Using Keyless Secrecy Abstract We investigate the scalability of a class of algorithms that exploit dynamics of wireless fading channels to achieve secret unicast communication between multiple source-destination pairs in the presence of eavesdroppers. We first describe a construction that allows artificial noise generation to suppress eavesdroppers, yet yields a pernode throughput of Ωn / in a wireless network of n randomly located nodes, where denotes the average number of neighbors of a node. We then derive scaling laws for the number of eavesdroppers that can be tolerated in the network for both independent and collaborating eavesdropper models. In particular, we show that for some constant c such that 0 < c <, o c and o eavesdroppers can be tolerated under the independent and collaborating eavesdropper models respectively, while guaranteeing that the aggregate rate at which eavesdroppers intercept packets goes to 0. Thus, achievable security in number of eavesdroppers can be traded off against the per-node throughput by varying the average number of neighbors of a node or equivalently, by varying the transmit power. Introduction The idea of keyless secrecy also called, physical layer secrecy has attracted considerable attention recently as a method for key distribution in systems based on traditional computational cryptography, or as an alternative that, in contrast to the computational approach, assumes that eavesdroppers have infinite computational power. In wireless systems, for instance, a number of recent proposals exploit differences in the received signal across different nodes to achieve secret communication. Examples of such schemes include [8, 9, 0], which exploit common information in fading channel characteristics, and, [] which exploits independent packet loss of Eve and Bob. Other schemes [5, 3, 9] use artificial noise generation to ensure that the eavesdroppers receives a degraded version of the signal received by a legitimate receiver so as to ensure a positive rate of secure bits []. The scheme proposed in [5, 3], for instance, uses beamforming and interference cancellation, while the scheme in [9] exploits multi-user diversity to generate noise and confuse the eavesdroppers. In this paper, we study the scalability of keyless secrecy in large wireless networks, as opposed to the two hop setting generally considered in prior literature. Similar to [9], we use artificial noise generation to ensure secrecy of packets. In particular, for each packet hop, we select nearby nodes to generate noise that ensures maximal ambiguity at the eavesdroppers, while allowing the packet to be decoded by a legitimate receiver. Intuitively, the greater the number of nodes generating noise, the greater the number of eavesdroppers that can be tolerated at the cost of reducing the source-destination throughput. Thus, there is an inherent tradeoff between the achievable throughput of a node and the security of a wireless system. We therefore pose the fundamental question: for a given source-destination throughput, what is the maximum number of eavesdroppers that can be tolerated, while ensuring that with high probability w.h.p the aggregate rate at which eavesdroppers intercept packets henceforth referred to as the combined eavesdropper throughput goes to 0? In other words, we are interested in studying the security versus capacity tradeoffs for a class of algorithms that artificially generate noise to ensure message secrecy. The results obtained in this paper can be employed for network design in wireless systems that wish to guarantee a given secrecy rate per session w.h.p, hence addressing the difficult problem of secrecy rate selection when eavesdropper locations are unknown, or underneath network-level approaches that, for example, split a message into several smaller messages each of which is encoded and transmitted along different routes to avoid their capture. The efficient use of artificial noise generation for secrecy has been considered in two-hop relay networks in [5, 3, 9] and for arbitrary but known finite network topologies in [6]. Only a few works have considered secrecy in large networks. Reference [0] considers the density of eavesdroppers which can be tolerated when operating at the multiple unicast per-session throughput of [], and [7] considers the connectivity graph of large wireless networks when insecure links are removed. However, the results of [0] and [7] are for networks where eavesdropper locations are known to system nodes, and, hence, the network is able to route around eavesdroppers. The security of large mobile networks without delay or buffer constraints, where the mobility

2 simplifies the problem by reducing it to a two-phase construction as in [5], has been considered in []. Hence, we are unaware of any prior work that has considered the achievable security of asympototically large static networks where eavesdropper locations are unknown.. Main Results We consider a large wireless network, in which nodes are distributed according to a D Poisson point process of unit intensity over a square of area n. Each node has on average Θ neighboring nodes i.e. nodes it communicates with directly, where = on / and goes to as n. Further, we consider eavesdroppers placed inside the square according to a Poisson process with intensity λ e n. In this setting, we obtain the following results:. We propose a construction in which nodes artificially transmit noise to ensure secrecy of a message at each hop. Under this scheme, we show using percolation theory that each node can achieve a throughput of Ω n / w.h.p. For example, when = n, each node achieves a throughput of Ω nn /, which corresponds to the classic Gupta-Kumar result [6]. Thus, our result shows that each node can achieve any desired throughput Λn such that on / Λn ω/n, by varying or equivalently, its transmit power.. When eavesdroppers operate independently i.e. do not collude, we show that for some constant c such that 0 < c 0 <, o c 0 eavesdroppers can be tolerated in the network while ensuring that the combined eavesdropper throughput, Λ E n, goes to 0 as n. Thus, the result clearly shows that by increasing, more eavesdroppers can be overcome at the cost of achievable throughput. We derive an identical scaling law even when eavesdroppers have multiple but fixed number of antennas, but for a different constant c,0 < c < c 0 <. 3. When the eavesdroppers collaborate with each other to decode the received signals, we show that o eavesdroppers can be tolerated while ensuring that Λ E n goes to 0.. Finally, we derive sufficient conditions on the number of eavesdroppers to break the secrecy of the system. In particular, we show that O c independent eavesdroppers and O c 3 collaborating eavesdroppers are sufficient to achieve a constant Λ E n, for some constants c > and 0 < c 3 <. Thus, our results imply that on c 0 independent eavesdroppers and o n collaborating eavesdroppers can be overcome while achieving the Gupta-Kumar pernode throughput of Ω n /. Higher throughput can be achieved while reducing the allowable number of gn A funtion gn = ohn if lim = 0. Similarly, gn = hn gn ωhn if lim hn = eavesdroppers. At the other extreme, we can tolerate up to o n eavesdroppers while achieving only a per-node throughput of ωn. Finally, we note that the scaling laws for the sufficient number of eavesdroppers to achieve a constant Λ E n show that the bounds for allowable number of eavesdroppers are tight up to a polynomial factor. The rest of the paper is organized as follows. Section presents the system assumptions. In Section 3, we describe a construction to achieve a per-node throughput of Ω n /. In Section, we analyze the number of independent eavesdroppers that can be tolerated in the network. In Section 5, we carry out a similar analysis for the case of collaborating eavesdroppers. In Section 6, we discuss the validity of our model and the applicability of our results for other fading models. Finally, we conclude in Section 7 and present future directions. System Model and Assumptions. We consider a collection of static nodes placed according to a Poisson point process of unit intensity over a square B n = [0, n] [0, n].. Also present in the D plane is a set E of passive eavesdroppers distributed according to a Poisson process of intensity λ e n. 3. We choose uniformly at random a matching of source-destination pairs, so that each node is the destination of exactly one source.. All links experience frequency-nonselective Rayleigh fading. The fading is assumed to be quasi-static, which is defined here as being constant over the transmission of a single secure message. The fading across different links and between transmissions on the same link are assumed independent. 5. Let T be the subset of nodes transmitting at a given time instant. Let d i denote the distance between an arbitrary pair of nodes R i and R. Then, a transmission from node R i is successfully received by node R only if the signal to interference plus noise ratio SINR at node R denoted as S exceeds a fixed decoding threshold γ i.e. P h S i α d i = N 0 + P h k α d γ, γ > k R k T where P denotes the transmit power and h i denotes the fading gain, a non-negative random variable that models fading on link R i R. N 0 is the ambient noise power at the receiver. Note that a single decoding threshold γ is employed for all transmissions for ease of exposition. However, if guaranteeing a given secrecy rate is the goal, separate thresholds for each of the eavesdropper and receiver can be employed, and identical scaling results can be obtained.

3 6. Throughout this paper, we assume channel reciprocity i.e. h i = h i, for any two nodes R i and R. Further, the Rayleigh fading assumption implies that the fading gain h i follows an exponential distribution for all node pairs i,. Without loss of generality, we take E[ h i ] =. 7. Each node is assumed to have an infinite buffer for storing packets destined to other nodes. Further, the transmission and routing strategy for all nodes is agreed to in advance. While the routing strategy is determined in advance, the set of nodes on a path from a source to a destination may vary from one message to the next.. Problem Definition We define the per-node throughput Λn as the number of bits per second that w.h.p. all nodes can simultaneously transmit to their intended destinations. Let Λ E n be the rate at which eavesdropper E intercepts packets from legitimate nodes. Then, for a desired Λn, our goal is to determine the maximum intensity of eavesdropper process λ e n such that the combined eavesdropper throughput, Λ E n, defined as Λ E n = E E Λ E n goes to 0 both when the eavesdroppers operate independently, and the eavesdroppers collaborate with each other to decode packets 3 Achievable Per-Node Throughput We first describe a construction in which each node has Θ neighbors and which yields a per-node throughput of Ω n /, where represents the average number of neighbors of a node. As described in Section, satisfies the following two conditions: P = on /, and P lim = Similar to [, ], our construction uses percolation theory [] to show the achievability result. However, our construction differs from those in [, ] in two crucial aspects: Our result for the per-node achievable throughput allows system nodes to artificially generate noise to ensure secrecy of messages transported across the network. Hence, our transmission scheme needs to ensure successful delivery of messages to legitimate nodes in the presence of this added interference. Further, the fact that links experience time varying fading in our set up necessitates a substantially different approach towards relay selection from those described in [, ]. Our result implies that as grows, the per-node throughput decreases. However, we will see later that this results in a greater number eavesdroppers being allowed. Expressing the per-node throughput in terms of therefore provides us a better understanding of the tradeoffs between security and the achievable per-node throughput. Our capacity construction involves three maor steps:. Construction of a highway system,. Specification of a routing protocol, and 3. Specification of a transmission scheduling scheme Figure. Construction of the bond percolation model. The figure on the left hand side corresponds to division of the unit square into smaller cells. The figure on the right hand side is obtained by associating an edge with each cell, traversing it diagonally. 3. Highway Construction As shown in Figure, we divide the square B n into cells of side cn = /, where satisfies conditions P and P defined earlier. It is easy to see that the number of nodes inside a given cell is a Poisson random variable with parameter. As shown in Figure, for each cell, we define eight regions H H 8, each having an area /. As will become apparent in Section 3., these regions help in choosing a next hop relay node sufficiently far away from nodes transmitting artificial noise. We first show that each Figure. Regions H H 8 associated with each cell ĉ i. cell has a sufficient number of nodes in each of its regions H H 8. To establish this result, we use a Chernoff bound for a Poisson random variable [3][pp.97-98]. THEOREM 3.. Let X be a Poisson random variable with parameter λ.. If x > λ, then PX x e λ eλ x /x x. If x < λ, then PX x e λ eλ x /x x We are now ready to state and prove Lemma 3.. LEMMA 3.. Let N i be the number of nodes in region H, =,...,8 of a given cell ĉ i. Let E i denote the event { / Ni, }. For all i, / pn PE i > 6 e PROOF. The lemma follows from a straighforward application of Theorem 3. and the union bound. We declare a cell ĉ i open, if E i holds, and closed otherwise. From Lemma 3., it is easy to see that the probability pn that a given cell is open goes to as n. Similar to [], we map the constructed lattice to a bond percolation model. As shown on the right hand side of Figure, we draw a horizontal edge traversing a cell diagonally across half of the cells and a vertical edge across

4 the remaining half. In this way, we obtain a grid of horizontal and vertical edges, each edge being open with probability pn, independently of all other edges. For convenience of exposition, we introduce a quantity m that denotes the number of horizontal or vertical edges composing the side length of the box B n. Then, m = n/ cn. Note that m as n. Thus, in terms of number of edges, the square B n of area n, has dimensions m m. We divide the box B n into rectangular slabs of dimensions m κm, where κ > 0. Similar to [], we show that each slab contains a large number of edge-disoint left-to-right crossing paths i.e. paths consisting of open edges connecting the shortest sides of the rectangular slab. Let R i m denote the i-th rectangular slab of B n of dimensions m κm ε m, where i m κm ε m. We choose ε m > 0 as the smallest value such that the number of m rectangular slabs κm ε m is an integer. As noted in [], ε m = o as m. Let Cm i denote the maximal number of edge-disoint left-to-right crossing paths of rectangle R i m and let N m = min i Cm. i We then have the following theorem which shows the existence of a large number of crossing paths in each of the rectangular slabs of B m. THEOREM 3.3 NUMBER OF CROSSING PATHS. For all κ > 0 and 5/6 < pn <, there exists a δ satisfying 0 < δ < κ such that lim m PN m δm = 0 PROOF. The proof is identical to Theorem 5 in [] except that the probability, pn, that a cell is open is a function of n. Noting that pn > 5/6 for large enough n and taking limits as n, the inequality 6 in [] yields the condition 0 < δ < κ. Thus, Theorem 3.3 shows not only that each rectangular slab has a large number of crossing paths but also that each node is at most κm away from a crossing path. Similarly, by dividing B n into vertical rectangular slabs of sides κm ε m m, we can show that there exist δm top-to-bottom crossing paths inside each rectangular slab. Using the union bound, we conclude that there exist Ωm left-to-right and top-to-bottom crossing paths of B n w.h.p. These paths form a backbone that we call the highway system. 3. Routing Protocol We next describe our routing protocol used for carrying packets from various sources to their respective destination. Our routing protocol involves three phases:. The draining phase in which a source transmits a packet to a nearby entry point situated on a horizontal crossing path.. The highway phase in which the packet is moved first along a horizontal crossing path and then along a vertical crossing path until it arrives at an exit point, which is suitably close to the destination node. Figure 3. Relay selection for each of cell containing horizontal edge and vertical edge. For each cell type, the packet can only be forwarded to one of at most six possible adacent cells indicated by the arrows. Only nodes inside the shaded area are available for relay selection. 3. The delivery phase in which the packet is transmitted from the exit point to the destination node via an intermediate relay node. In contrast to the routing strategy in [, ] where the nodes on a routing path are fixed over the entire set up, we opportunistically choose a relay node at each hop with a sufficiently large fading gain. As a result, for the same source-destination pair, the routing paths may vary from one packet transmission to the next. 3.. Highway Phase In the highway phase, a packet is forwarded from a node in an open cell to another node in an adacent open cell. The node to forward the packet to is chosen using a relay selection mechanism described next. Unlike [], we do not assume apriori knowledge of fading coefficients. Let cell ĉ i denote the current location of the packet and let adacent open cell ĉ denote its next hop. The choice of the adacent cell ĉ depends on what type of edge i.e. horizontal or vertical each of the cells ĉ i and ĉ have. The relay in cell ĉ is then chosen as follows:. A transmitter R i broadcasts a pilot signal and announces the co-ordinates of its cell ĉ i as well as those of its adacent cell ĉ to which the packet is next destined.. Depending on ĉ i and ĉ s co-ordinates, only nodes within the shaded area of ĉ, as shown in Figure 3, participate in relay selection. Lemma 3. implies that the number of eligible relay nodes,, N e, lies between / and. Each eligible node R then measures its fading gain, h i, from R i s pilot. 3. Let ˆn =. Then, a node R announces itself the relay if h i > ˆn.. In case no relay is chosen, in keeping with the quasistatic Rayleigh fading assumption, the transmitter R i once again broadcasts a pilot signal and steps -3 are repeated again. The probability that exactly one node annouces itself as the relay is therefore p s = N e ˆn Ne ˆn ˆn ˆn 8 ˆn = ˆn e 8 > 0

5 where N e is the number of eligible relay nodes. Thus, the probability of a successful relay selection is a constant bounded away from 0. In other words, a relay R is chosen in a finite expected time equal to e 8. Further, the chosen relay is such that h i >. We note that the lower bound on p s can be significantly improved by employing more time slots for relay selection. However, these improvements do not change the asymptotic results obtained in this paper. 3.. Draining Phase In the draining phase, a source node chooses an entry point located on a horizontal crossing path to transmit its next packet to. Before choosing an entry point, each source node is mapped to a horizontal crossing path and a set of eligible entry points chosen as follows. w Slices Source Entry Point Figure. Entry Point Selection. Only nodes in cells with a horizontal edge and which are located at least κm and at most κm from the cell containing the source i.e. shaded cells are eligible for selection.. Division of B n into rectangular slabs: We divide the square B n into an integer number of rectangular slabs of size m κm ε m, where m = n/ cn. Figure shows a portion of a rectangular slab. We choose a κ such that there are at least δ m crossing paths inside each rectangular slab. These crossing paths are numbered,...,n m.. Mapping source node to a horizontal crossing path: We next divide each rectangular slab into δm smaller slices, each of dimensions m w. Figure depicts the case where w =. Each source node in the i-th slice is then mapped to the i-th horizontal crossing path and therefore, chooses an entry point located along the i-th path. Note that as depicted in Figure, a crossing path could potentially traverse multiple slices. Further, even though a source node may be located on a crossing path, the mapping scheme can potentially lead the source node to access an entry point on an entirely different crossing path. The main goal of our mapping scheme is to achieve load-balancing by distributing source nodes across different crossing paths in a rectangular slab. In fact, we next prove Lemma 3. which shows that each crossing path has a bounded number of source nodes accessing it. Lemma 3. uniformly bound the number of streams carried by a highway. We know that the number of streams carried by a highway is bounded by the number of nodes, N s, in a rectanglular slice s of dimensions m w. There are w n slices in total, each of which has an area m cn w cn = w n. Then, LEMMA 3.. lim PN s w n, s = PROOF. Using the Chernoff and union bounds, we get PN s w n, s w e w n n as n tends to infinity. 3. Shortlisting eligible entry points: Once the source nodes are mapped to crossing paths, the set of eligible entry points for each source are determined as follows: a Only nodes located inside open cells at a horizontal distance of at least κm and at most κm from the cell containing the source node are considered for selection. As will become clear in Section 3.3, the reason for choosing an entry point sufficiently far away from the source ensures that the entry point does not experience too much interference from the noisegenerating nodes. b Finally, the entry points are chosen from only those open cells containing a horizontal edge. As shown in Figure, only nodes in the shaded cells are eligible to function as entry points. Thus, the number of eligible cells i.e. shaded cells in Figure available for entry point selection is at least κm which occurs when the source is located at the boundary of a rectangular slab at most κm. From Lemma 3., we therefore conclude that there are at least κm and at most κm eligible entry points available for selection. In Section 3.3.3, we will describe a transmission scheme for the draining phase that schedules only those source nodes in a given time slot which are at least κm cells apart from each other. The reader can convince himself after a little thought that in each time slot we have a : mapping between the source nodes transmitting in that slot and the corresponding eligible entry points. Note that steps -3 are performed only once and not repeated every time a source needs to transmit a packet. Finally, we are ready to choose an entry point for a source node S. Entry point selection: The actual entry point P is chosen from the set of eligible points using the scheme described in Section 3.. by setting ˆn = κm/ in Step of the relay selection scheme. Once again, it can

6 be easily verified that the probability of successful entry point selection is at least /e 8 i.e. a constant bounded away from 0. Further, the chosen entry point P has a fading gain h SP > κm to the source S Delivery Phase The mapping of destinations to vertical crossing paths is analogous to the mapping described in Section 3... In this section, we therefore focus only on exit point selection and the subsequent delivery of packets to destination nodes. Exit Point Selection: A packet destined for a node D located in cell ĉ d is forwarded along a vertical crossing path, until it arrives at a node P located in cell ĉ p whose vertical distance to ĉ d is κm. Node P then becomes the exit point for destination D. For illustration, one can rotate Figure clockwise by 90 o and treat the node labeled Source as the destination node D instead. Assuming that the packet for D is moving from left to right in Figure or top-to-bottom in the rotated figure, the leftmost shaded cell in Figure is the cell ĉ p. Eligible Relay Nodes: Once an exit point P is chosen, the packet is delivered to node D by P via an intermediate relay node from a set of eligible relay nodes. A node R located in cell ĉ r is eligible for relay selection if it satisfies the following conditions:. The cell ĉ r must lie between ĉ p and ĉ d i.e. its y- coordinate must lie between those of ĉ p and ĉ d.. The cell ĉ r must contain a vertical edge 3. The vertical distance between cells ĉ r and ĉ p must be at least κm, and. The vertical distance between cells ĉ r and ĉ d must also be at least κm. It is easy to see that there are κm cells satisfying the above criteria. From Lemma 3., we conclude that the number of eligible relay nodes N e satisfies κm N e κm. Once again, the reader can convince himself that by scheduling only those exit points which are at least κm cells apart in a given time slot, we obtain a : mapping between the exit points and eligible relay nodes. Relay Selection: Finally, the relay R is chosen as follows:. P transmits a pilot signal. Each eligible node R measures its fading gain, h RP to P.. Next, D transmits a pilot signal and once again, each eligible node R measures its fading gain h RD to node D. 3. Finally, a node R satisfying min h R P, h R D > ˆn/, where ˆn = κm, announces itself as the relay. Since the minimum of iid exponential random variables with mean is also an exponential random variable with mean /, it is easy to check that the probability that a node R announces itself as the relay is / ˆn. Proceeding as before, the probability of a successful relay selection Figure 5. Transmission Scheduling. Simultaneous transmissions by nodes in shaded cells are at least d cells apart. In the figure, d =. is p s = N e ˆn Ne ˆn ˆn ˆn ˆn ˆn e > 0 Once again, p s is a constant bounded away from 0. Further, the selected relay R satisfies: min h RP, h RD > κm ˆn/ =. 3.3 Transmission Scheduling Scheme Finally, we describe the last step in our construction viz. the transmission scheduling scheme, which is illustrated in Figure 5. The shaded cells represent the cells containing the transmitting nodes in a given time slot. Further, simultaneous transmissions are spaced d cells apart. The choice of d between the transmitting cells is different for the highway and the draining/delivery phases and will be derived separately for each phase. Each node in our construction transmits with a power P n d dcn α. Thus, unlike [, ], nodes in our set up transmit using different power settings in the highway and draining/delivery phases. We will explain later in this section why different power settings are necessary unlike [, ]. An important ingredient of our transmission scheduling scheme is a mechanism for choosing noise-generating nodes and is discussed next Artificial Noise Generation Unlike the constructions in [, ], the primary obective in our work is to achieve message secrecy in the presence of eavesdroppers. To guard against the message being succesfully decoded by an eavesdropper, we require a subset of system nodes to generate noise simultaneously with the message transmissions. Regardless of which phase a transmission occurs in, we observe that greater the number of noise-generating nodes the greater will be the number of eavesdroppers that are unable to decode a given transmission. At the same time, the chance that a legitimate node decodes the transmission is also reduced. Ideally, we want to maximize the amount of generated noise, while ensuring that a legitimate receiver decodes a transmission w.h.p. Consider a transmission from a node R i in cell ĉ i to a next hop node R. Regardless of the phase, the set of

7 noise-generating nodes are chosen according to the following rule: Nodes inside cell ĉ i within a radius rn = ν/ from R i transmit noise, where ν = / 3+5α/ πγ. The rationale behind the choice of rn will become clear, when we show that the receive SINR S at node R exceeds γ, for both the highway and the draining/delivery phases. Intuitively, choosing noise-generating nodes close to R i serves two purposes: i noise-generating nodes are far away from the receiving node R, and ii eavesdroppers close to R i receive sufficient interference to keep them from decoding R i s transmission Receive SINR in Highway Phase Consider a transmission from a node R i to another node R and let S be R s received SINR. Then, P n d h i /di α S = N 0 + P n di near + P n di f ar where P n d denotes R i s transmit power, I near denotes the near-cell interference at R caused by noisegenerating nodes in cell ĉ and I f ar denotes the interference from the rest of the network. From Section 3.., we know that h i >. Further, in the highway phase d i cn. Hence, P n d / cn α S N 0 + P n di near + P n di f ar I near is upper bounded as h k I near R k R cn/ α where R denotes the set of noise-generating nodes inside ĉ. The term in the denominator follows immediately by observing Figure 3 and by imagining R located inside one of the shaded areas. The area over which noise-generating nodes are located is upper bounded by πr n. Hence, P R > a PN > a, a, where N Poissonσn. Here, σn = πr n = conclude π/ν. From Theorem 3., we P R > σn < e/ σn The right hand side goes to 0 as n. Hence, w.h.p., the sum Rk R h k is upper bounded by a sum of σn iid exponential random variables. Noting that the sum of iid exponential random variables is an Erlang random variable and employing Chernoff bounds for the same see Appendix A, we obtain P h k > σn R k R < /e σn Thus, with probability at least I near σn = γ cn α e/ σn /e σn which goes to as n, since σn. We next derive an upper bound for I f ar. We first note that relative to node R, the transmitters in the eight closest cells i.e. the shaded cells in Figure 5 from R are at Euclidean distance at least dcn, where d is a constant and will be determined later. The transmitters in the 6 next closest cells are at Euclidean distance at least 3dcn and so on. In other words, relative to R, the other transmitters are located along the boundaries of concentric squares of increasing size. Therefore, I f ar h k t= R k R t t dcn α where R t denotes the set of noise-generating nodes located along the boundary of the t-th concentric square from R i. Once again, it is easy to see that P R t > a PN > a, a where N Poisson8tσn. From Theorem 3., we conclude I f ar 3σn dcn α t= t t α Notice that the sum converges for α >. A trite calculation shows that this sum can be upper bounded by β = +/α and hence, β I f ar 3 dcn α { } Letting A denote the event I f ar > P ndβ and dcn α using the union bound, we get PA < e/ 8tσn +/e 6tσn t= e/ 8σn /e6σn = + 8σn e/ /e 6σn 0 as n. Substituting from and 3 into and further simplification yields: S cn N α 0 P n d + γ + β d α w.h.p Noting that P n d dcn α and setting d 8βγ /α, we get lim S γ w.h.p

8 3.3.3 Receive SINR in Draining and Delivery Phases Using an analysis very similar to the one in previous section, we show that the receive SINR in the draining and delivery phases exceeds γ. The following analysis applies to both draining and delivery phases. From Section 3.. and Section 3..3, we know that any transmitter-receiver pair is separated by a Euclidean distance at most κcnm. Further, the fading again on each transmitter-receiver link is at least /. κ m Consider a node R i transmitting to another node R. Let S denote the SINR at R. Then, P n d κ m / κcnm α S N 0 + P n di near + P n di f ar Further, the distance from noise-generating nodes to R in each phase is guaranteed to be at least κcnm 8. Proceeding exactly as in Section 3.3., we obtain α I near γ κcnm w.h.p α and β I f ar dcn w.h.p α where β is as defined in Section Letting d = κm and noting that m = n/ /, it can easily be checked that lim S lim κ m α γ + w.h.p β α/ Note that the above choice of d ensures that simultaneous transmissions are d = κm cells apart, which is sufficient to ensure a : mapping between a source node and its eligible entry points and also between an exit point and eligible relay nodes, as discussed in Sections 3.. and 3..3 respectively. Remark: We conclude this discussion by explaining the usage of separate transmission powers for the highway and draining/delivery phases unlike [, ]. In our set up, we require the received SINR to be greater than a fixed decoding threshold γ for all transmissions. If we only cared about the achievable rate on a given link as done in [, ], it can be seen that a single transmit power suffices for all the three phases in our construction too. 3. Time Division Multiplexing Scheme Before finishing the construction for per-node achievable throughput, we establish the following lemma which uniformly bounds the number of nodes per cell. Let N i be the number of nodes in cell ĉ i. LEMMA 3.5. lim PN i m, i = PROOF. The proof follows from the union bound and Chernoff bounds. PN i m, i m PN i > m e e 3 m m as n tends to infinity. 3.. Time Division Multiplexing Scheme Finally, we achieve a per-node throughput of Ωn / using the following TDM scheme:. We divide time into frames, each frame comprising one or more time slots. We allocate a fraction /3 of total time frames to each of draining, highway and delivery phases.. In the draining phase, we divide each time frame into groups of d time slots, where d = κm. Each cell transmits in one out of d = Θ m time slots. Further, the time slots allocated to a given cell are shared among the nodes in the cell in a round robin fashion. Each time slot is divided into three smaller mini-slots, two of which are used for entry point selection, followed by one for data transmission. Note that the mini-slots used for entry point selection are much smaller than the one for data transmission. From Lemma 3.5, we know that each cell has fewer than m nodes w.h.p. Hence, the per-node throughput during each of draining and delivery phases is Ω 3 m. Since m = n/ /, the per-node throughput is Ω n/ 3 w.h.p. 3. The delivery phase is identical to the draining phase except that each time slot is divided into five minislots. The first three mini-slots are used for selecting a relay to deliver the packet from an exit point to a destination node. The last two mini-slots are used for data transmission. It is not hard to see that the per-node throughput remains the same as that for the draining phase.. In the highway phase, we once again divide each time slot into groups of d time slots, where d is now a constant equal to 8βγ /α. Once again, each time slot is divided into three mini-slots, two for relay selection followed by one for data transmission. As shown in Lemma 3., each highway is accessed by at most n nodes. This yields a pernode throughput of Ω n / in the highway phase. From the above it is clear that the highway phase is the bottleneck phase, and hence, the per-node achievable throughput is Ω n /.

9 INDEPENDENT EAVESDROPPERS We next derive conditions on the intensity of eavesdropper point process, for different constraints on the combined eavesdropper throughput. We start with the case where eavesdroppers are independent and consider collaborating eavesdroppers in Section 5.. Eavesdropping Model Regardless of whether the eavesdroppers collaborate or not, we consider the following model for interception of packets by the eavesdroppers.. We assume an eavesdropper with infinite computational power that can potentially decode messages anywhere in the network. We further allow each eavesdropper to decode multiple messages at the same time.. We assume an eavesdropper cannot use looks from multiple hops to ointly decode a message, but rather has to obtain a received SINR greater than γ for some hop of the message. One mechanism for ensuring this is described in [0].. Combined Eavesdropper Throughput Our approach towards deriving an upper bound on the combined eavesdropper throughput, Λ E n, is to first determine the probability that an individual eavesdropper E can decode a given transmission. The sum of these probabilities over all transmissions in the network in a given time slot yields an upper bound on the throughput of the eavesdropper, denoted as Λ n. The product of Λ n and the total number of eavesdroppers, therefore, yields an upper bound on Λ E n. We note that the upper bound on Λ E n is the same across all time slots. Let S denote the received SINR at an eavesdropper E due to transmission by a node R i located inside cell ĉ i. Then, P n d h i /di α h i /di α S = N 0 + P n d Rk R h k /dk α Rk R h k /dk α where R denotes the set of noise-generating nodes in the current time slot. Regardless of E s location, the transmission scheme described in Section 3.3 ensures that there is at least one transmitting node within a Euclidean distance at most dcn from E at every time slot, where cn denotes the cell size and is equal to /. Recall from Section 3.3 that d denotes the separation in number of cells between simultaneously transmitting nodes and is different for the highway and draining/delivery phases. Letting R d denote the set of noise-generating nodes within a Euclidean distance at most dcn, we obtain h i /di α S Rk R d h k /dk α We now derive a lower bound for the sum Rk R d h k. Since the noise-generating nodes are chosen identically in the highway and draining/delivery phases, the lower bound is applicable for each of the phases. Consider a transmitting node, say R, inside a cell ĉ within a Euclidean distance of dcn from E. From Section 3.3., we know that the noise-generating nodes are nodes inside cell ĉ located within a distance of rn = /ν from R, where ν = 3+5α/ πγ. Hence, the number of noise-generating nodes depends on R s location. In the worst-case, R could be located at one of the corners of ĉ i, yielding the smallest number of noise-generating nodes in expectation. Clearly, P R d > a > PN > a a, where N Poissonπr n/. Similar to the analysis in Section 3.3., we use Chernoff bounds to simplify the sum Rk R d h k, which yields h k πr n = 6 R k R d with probability at least ν ν ν3 ν 3 ν where ν = / 7+5α/ γ, ν = ν e/ and ν 3 = ν / e. It is easy to see that this probability goes to as n. We next derive PS > γ, the probability that E decodes a transmission by a node R i located inside cell ĉ i. We will see that the bound we derive for PS > γ is independent of d, and is therefore, applicable across all phases of transmission. We consider the following two cases separately: C R i is located at a Euclidean distance at most dcn from E. We call R i s transmission a near transmission and the denote the receive SINR at E to be S near and the corresponding outage probability as PS near > γ. C R i is located at a Euclidean distance greater than dcn from E. We begin with the analysis of case C. S near h i /di α Rk R d h k /dk α Conditioning on d i and letting A = {d i rn} and B = > γ} respectively, we get {S near PB = PB APA+PB A c PA c PA+PB A c Since we know that d i dcn, PA is upper bounded as PA πr n c n = 3+5α/ γ

10 Thus, PB 6ν 6ν + PB A c PA c + PB A c We next derive S near conditioned on A c. In this case, it is easy to see that d k < d i, for any R R d. Thus, conditioned on A c, we observe that S near h i α Rk R d h k h i α ν where the second inequality follows from. Hence, ν PB A c where ν = / 7+7α/. Thus, P S near > γ 6ν ν + Noting that 6ν <, we can upper bound the right hand side of the above as P S near > γ 6ν ν 6ν + 6ν 6ν + ν Noting that 6ν > ν, we conclude that P S near > γ 5 ν 5 We now proceed to analyze case C. We observe that excluding the 8 nearest transmissions from E, the next 6 transmitters are located at Euclidean distance at least dcn, the next 3 transmitters at Euclidean distance at least dcn and so on. Thus, relative to E the transmitters are placed along boundaries of concentric squares of increasing size, such that the transmitters on the boundary of the t-th t > square have distance at least t dcn from E. Hence, the receive SINR at E due to a transmission from a node R i located on the boundary of the t-th concentric square t > is upper bounded as Thus, S t S t h i /t dcn α Rk R d h k / dcn α h i α t α Rk R d h k h i α t α ν 6 where the second inequality follows from. Hence, P S t > γ ν t α ν t Hence, the aggregate rate at which E intercepts packets for each of the phases is upper bounded w.h.p as Λ n P S near > γ 8t + + t= ν t ν + 6 Z ν + t= 5 8 ν + ν ν 8t + dt ν t ν + 6 ν + 8 ν ν The total number of eavesdroppers, En, is a random variable and depends on λ e n. Assuming that lim λ en, we note that by weak law of large numbers lim P εnλ e n En +εnλ e n =, ε > 0. Hence, the combined eavesdropper throughput Λ E n is upper bounded w.h.p as Λ E n 8+εnλ en 8 ν, ε > 0 7 ν.3 Scaling Laws for Allowable Number of Eavesdroppers We next derive scaling laws characterizing the intesity of the eavesdropper process, λ e n. Let Λ S n = n Λ i n denote the overall network capacity. From Section 3., we obtain Λ S n = nω n / = Ω n/ / i= We derive λ e n under each of the following constraints on Λ E n:. lim PΛ E n = 0 =. lim PΛ E n µ =, for some constant µ, and 3. lim PΛ E n/λ S n = 0 = We therefore obtain the following scaling laws for λ e n:. We note that when λ e n = o ν /n, it follows from 7 that lim PΛ E n = 0 =. The allowable number of eavesdroppers, En = o ν. +ε8 ν +. For lim PΛ E n µ =, λ e n µk ν /n, ν where K = and ε can be chosen arbitrarily close to but strictly greater than 0. Therefore, En µk ν. 3. For lim PΛ E n/λ S n = 0 =,

11 λ e n = o En = o n ν n ν /. /. Therefore,. Eavesdroppers With Multiple Antennas We next allow the eavesdropper E to have more than one receive antenna. In particular, we let each eavesdropper have access to a fixed number of, say q, antennas. A = {d i rn} and B = {SE near Similar to Section., we define the events > γ}. Then, PB PA+PB A c ν + PB Ac Similar to 5 in Section., we obtain by conditioning on A c that S near q l= h il α ν where h il denotes the fading gain from node R i to the l- th antenna of E. Noting that sum of q iid exponential random variables is a q-stage Erlang random variable and applying Chernoff bounds for the same see Appendix A, we obtain PB A c e ν γ α q ν γ α q e q ν 5 q where ν 5 = ν γ/ α+ and the second inequality follows from the fact [] that Hence, PS near E x x x, x > 0 > γ ν + eq ν 5 e q + ν 5 Once again, the analysis in Section. can be trivially extended to show that, for t > Therefore, P S t S t > γ q l= h il α t α ν 5t α ν ν 5t and Λ n 8eq + ν 5 ν 5 Therefore the combined eavesdropper throughput Λ E n is upper bounded w.h.p as: Λ E n 8+ελ ene q + ν 5 ν, ε > 0 5 From the above expression, it is adequately clear that apart from a change in constants, the scaling laws derived in Section.3 trivially extend to the case when nodes have q antennas..5 Lower Bound on Eavesdropper Throughput So far, we focused on deriving conditions for λ e n that guarantee the secrecy of the system as per the various secrecy metrics considered. We next derive conditions on λ e n that is sufficient to break the secrecy of the system. In particular, we obtain sufficient conditions for λ e n such that:. lim PΛ E n µ =, for some constant µ, and. lim PΛ E n/λ S n = µ =, for some constant µ. Once again, we will see that Λ E n is independent of d and hence the phase of operation. We begin by noting that regardless of E s position, at every time slot there exists at least one transmitting node, say R i, such that d i dcn. where S = P n d h i /d α i N 0 + P n di near + P n di f ar 8 h k I near = d α R k R d k h k I f ar = R k R dk α R d and R denote the set of noise-generating nodes at Euclidean distance at most dcn and the rest of the network respectively. Let A denote the event {d R,E > rn, R T }, where T denotes the set of transmitting nodes at a Euclidean distance at most dcn from E. Then, PS > γ PS > γ APA 9 Noting that T 8 and applying the union bound, we get PA 8 πr n cn = 5α/ > We next derive PS > γ A by obtaining a lower bound on S under the condition that the event A holds.

12 When A holds, it is easy to see that d k > d i /, R R d. Hence, I near α+3 d α i R k R d h k Similar to the analysis in Section., we can show that the sum ν6 h k 3πr n = R k R d with probability at least ν7 ν8 where ν 6 = / 5α/ γ, ν 7 = ν 6 /e/ and ν 8 = ν 6 e//. Therefore, I near α+3 ν6 di α We next derive an upper bound on I f ar. Once again, we observe that relative to E, the transmitting nodes are placed along the boundaries of concentric squares of inreasing size. Let I t f ar denote the interference due to nodes located along the t-th square. Letting R t represent the set of these nodes, we get I t f ar R k R t h k /d α k The transmission scheme described in Section 3.3 ensures that there are at most 8t transmitting nodes along the t-th concentric square, each of which is at a distance at least t dcn from E. Applying Chernoff bounds, we obtain I t f ar 3tπr n t ν6 t dcn α = t dcn α with probability at least tν7 tν8 Hence, I f ar t= ν6 tdcn α = t + t= ν6 ζ dcn α for some constant ζ. The equality follows from the fact that the sum t t + /t α converges for α >. The upper bound on I f ar holds with probability at least tν7 tν8 + Noting that each of the summands above is a geometric series and upon further simplification, it is easy to see that the above probability goes to as n. Substituting from and into 8 and further simplification yields: S N 0 d α i P n d + α+3 h i h i N 0 α + α+ ζ+8 ν6 + d α i dcn α ν6 ν6 ζ We thus see that the lower bound of S is independent of d. Therefore, PS > γ A e γn 0 α ν9 3 where ν 9 = ζ + 8/ 3α/. Substituting from 3 and 0 into 9, we get PS > γ 7e γn 0 α ν9 8 We thus obtain the following conditions for λ e n:. When λ e n = O ν 9/n, it follows that lim PΛ En = µ =, for some constant µ. In terms of number of eavesdroppers, En = O ν 9 is sufficient to achieve a constant throughput. /, it follows that. When λ e n = O n ν 9 lim Pλ En/λ S n = µ =, for some constant µ. / n Thus, En = O is sufficient to ν 9 achieve an eavesdropper throughput that scales with the overall network capacity. 5 COLLABORATING EAVESDROP- PERS A natural extension of our analysis in Section is to consider the case of collaborating eavesdroppers. In particular, we consider the case of a single eavesdropper E with mn antennas located uniformly at random within the D plane. Further, we allow the eavesdropper to add the signals received by the mn antennas to decode a message. 5. Upper Bound on Combined Eavesdropper Throughput Consider a transmission by a node R i. Let S denote the combined SINR at E and is expressed thus: S = mn l= mn l= P n d h il /d α i N 0 + P n d Rk R h kl /d α k h il /d α i Rk R d h kl /d α k where h il denotes the fading gain from R i to the -th antenna of eavesdropper E and R d denotes the set of

13 noise-generating nodes at a distance at most dcn. Proceeding exactly in Section., we can show that PS near > γ k + P mn = h il α k > γ Applying Markov Inequality to bound the probability on the right hand side, we get PS near > γ k + mn k mn Similarly, we can show that P S t > γ mn k t α k Similar to our analyses in Section, we note that the upper bounds on the probability that E decodes a message is independent of the phase of operation. Hence, the aggregate rate at which E intercepts packets for each of the phases is upper bounded w.h.p as Λ E n 8P S near > γ + t= mn 8mn k + k k 3mn k 8t + P k S t+ > γ 5.. Scaling Laws For Allowable Number of Antennas Based on the analysis in Section 5., we obtain the following scaling laws for mn:. For lim PΛ E n = 0 =, mn = o k µ. For lim PΛ E n µ =, mn 3 3. For lim PΛ E n/λ S n = 0 =,mn = / o n. 5. Lower Bound on Eavesdropper Throughput Similar to Section.5, we again derive sufficient conditions on mn to achieve a desired eavesdropper throughput. Once again, we know that at every time slot there is a transmitting node R i within a distance of dcn from E. Rewriting 8 for the case of an eavesdropper with mn antennas, we get mn S = l= P n d h il /d α i N 0 + P n di near + P n di f ar where I near and I f ar are given by and respectively. Proceeding exactly as described in Section.5, we get S mn l= h il N 0 d α i P n d + α β+8 k5 w.h.p Noting that the numerator is a mn-stage Erlang random variable and applying Chernoff bounds for the same, we see that S mn/ N 0 di α P n d + α β+8 k5 w.h.p mn +α/ N 0 + α+ β+8 k5 w.h.p where the second inequality follows from observing that the first summand in the denominator is smaller than the second. Thus, for the eavesdropper to achieve constant throughput i.e. for lim PΛ E n = =, mn = O k5. 6 DISCUSSION In this section, we discuss applicability of our results to other fading models and the validity of our physical layer model, and finally consider power control as an alternative to noise-generation for achieving secrecy. 6. Other Fading Models Although we assume a quasi-static Rayleigh fading model in our analysis, we note that the analysis and the scaling results apply to other fading models so long as the fading coefficients, h i, can be modeled as iid random variables with an exponentially decaying tail. More formally, we require that P h i > x e qx, x > x for some positive, real parameters q and x. As shown in [7], this assumption is satisfied by Nakagami and Ricean fading models in addition to the Rayleigh model assumed in this paper. 6. Receive Power vs Transmit Power One common source of confusion often arising due to the physical layer models such as the one used in this paper is whether the model allows for the receive power to exceed transmit power. The confusion becomes apparent in a traditional dense network setting [6, 8], in which nodes are placed in a disk of unit area and therefore, allowing internode-distances to be smaller than. Incorporating fading into this setting [8], also allows the receive power to exceed transmit power. The other problem when considering the dense network setting is that as n, the inter-node distances become smaller than half a wavelength, and hence fades between nodes can no longer be assumed independent. Both of these problems are addressed in our work by considering a network in which the number of nodes per