Minimizing Detection Probability Routing in Ad Hoc Networks Using Directional Antennas

In a hostile environment, it is important for a transmitter to make its wireless transmission invisible to adversaries because an adversary can detect the transmitter if the received power at its antennas is strong enough. This paper deﬁnes a detection probability model to compute the level of a transmitter being detected by a detection system at arbitrary location around the transmitter. Our study proves that the probability of detecting a directional antenna is much lower than that of detecting an omnidirectional antenna if both the directional and omnidirectional antennas provide the same E ﬀ ective Isotropic Radiated Power (EIRP) in the direction of the receiver. We propose a Minimizing Detection Probability (MinDP) routing algorithm to ﬁnd a secure routing path in ad hoc networks where nodes employ directional antennas to transmit data to decrease the probability of being detected by adversaries. Our study shows that the MinDP routing algorithm can reduce the total detection probability of deliveries from the source to the destination by over 74%.

In a wireless network, nodes communicate with others through shared wireless medium, which makes the communications more susceptible to passive eavesdropping and malicious traffic analysis [1]. An adversary may eavesdrop network in order to discover the location of the transmitter. These adversaries are referred as detection systems. If the power received by a detection system is strong enough, the detection system can distinguish the transmission signals from the electromagnetic noise, and it becomes aware of the existence of a transmitter. If more than two detection systems detect a transmitter in a synchronous manner, they are able to compute the transmitter's position with localization algorithms and go to find the transmitter and catch it. Hence, transmission with low detection probability is very important in an untrustworthy network.
Typically, the assumption for ad hoc networks is that nodes are equipped with omnidirectional antennas, which can transmit and receive signals in all horizontal directions [2,3]. However, a directional antenna can get antenna gain in the main lobe direction, thus transmitters can use the directional antenna to transmit signals farther away than omnidirectional antennas with the same transmit power, or transmit signals to a receiver while using less transmit power [2,4].
The work in [5][6][7][8] mentioned that directional antennas can reduce the detection probability, but no study has been conducted to compare the detection probability of directional and omnidirectional antennas. On the other hand, using directional antennas to achieve secure routing has not been studied yet.
Researchers in the past have done much fundamental research on directional antennas in wireless networks that focused on medium-access control, spatial reuse, efficient power consumption, network capacity, and so forth. The work in [9][10][11][12][13] proposed adaptive Medium-Access Control (MAC) protocols to improve IEEE 802.11. These adaptive MAC protocols attempted to limit the disadvantages of IEEE 802.11 in spatial use. Power is another constrained source in some ad hoc network scenarios because in these cases the power for the antenna comes from batteries, which are energy-constrained. Sometimes, nodes equipped with batteries-powered antennas cannot recharge frequently. This is another reason for using directional antennas. Authors of [14,15] described the advantages of using directional antennas to reduce power consumption in ad hoc networks. As directional antennas can increase spatial use [16], more than one directional antenna can send data at the same time. Directional antennas can also increase network capacity [17,18].
In this paper, we address the work we have done on routing path selection to reduce the transmitter's probability of being detected by adversaries in ad hoc networks. This paper is organized as follows. Section 2 introduces the antenna model. We introduce the detection probability model in Section 3 and our minimizing detection probability routing algorithm in Section 4. In Section 5, we review some related work about anonymous routing and secure routing protocols. Finally, we conclude our work in Section 6.

Antenna Model
Antennas are either omnidirectional mode or directional mode [2,3]. Omnidirectional antennas cover 360 degrees and send data in all directions. All nodes in the radiation region can receive the communication signals [2,3]. Omnidirectional antennas spread the electromagnetic energy over a large region, while only small portion is received by the desired receivers, so the omnidirectional transmissions waste a large portion of the transmit power and the network capacity.
Directional transmission can overcome this disadvantage. A directional antenna can form a directional beam pointing at the receiver by concentrating its transmit power into that direction. By pointing the main lobe at the receiver, a directional antenna can get more antenna gain in the direction of the receiver. Directional antennas strongly reduce signal interference in unnecessary directions.
In our antenna model, we assume that an antenna can work in two modes: omnidirectional mode and directional mode. It can send and receive data in both these two modes [2]. If nodes have nothing to transmit, their antennas work in omnidirectional mode to detect signals. A receiver and a transmitter can communicate over a larger distance when both antennas are in directional mode than just one of them is in directional mode while another is in omnidirectional mode.
Effective Isotropic Radiated Power (EIRP) is the gain of a transmitting antenna multiplied by the net power accepted by the antenna from the connected transmitter in a given direction [19]. As the gain and received power are measured in dB, EIRP can be calculated as where P t is the transmit power in dBW, and G t is the antenna gain in dBi (dB = 10 log 10 (x)). Antenna gain refers to an antenna's ability to direct its radiated power in a desired direction, or to receive energy preferentially from a desired direction [4]. It is defined as the ratio of the radiation intensity of an antenna in a given direction to the intensity of the same antenna as it radiates in all directions (isotropically) and has no losses [20]. Antenna gain is expressed in dBi.
For an omnidirectional antenna, because the ratio of the radiation intensity is 1, the antenna gain is 10 log 10 (1) = 0. As a directional antenna concentrates the transmit power into the main lobe direction, the radiation intensity in the main lobe direction is larger than that in other directions and its G t in that direction is much larger than zero. Therefore, the directional antenna can provide the same EIRP in the main lobe direction as that an omnidirectional antenna provides while using much less transmit power than that the omnidirection antennas uses.
No directional antenna is able to radiate all of its energy in one preferred direction. Some is inevitably radiated in other directions. These smaller peaks in Figure 1(b) are referred to as side lobes, commonly specified in dBi down from the main lobe. Figure 2 shows a case directional antenna gain in main lobe, side lobes, and back lobe.
As different antennas have different antenna structures and physical characteristic, their antenna gain functions are different. We use an approximate gain function to fit the directional antenna gain function. This approximate gain function is showed in Figure 3.

Link Budget Equation.
If the power received by a detection system is strong enough, the detection system can distinguish the transmission signals from electromagnetic noise. The ratio of the total received signal power to the total noise which includes thermal and system noise plus total interference is denoted as SNIR [21]. Hence, the detection event occurs if and only if the SNIR is larger than a threshold λ at a detection system.
The equation to compute the total received signal level at the receiver antenna is the following [22]: where P t (dBW) is the transmitter's power level, G t (dBi) is the transmitter's antenna gain in the direction towards the receiver, G r (dBi) is the receiver's antenna gain in the direction of the transmitter, C t is the transmitter's cable attenuation, C r is the receiver's cable attenuation, and Pl is adaptive transmission path loss, which we will discuss carefully later. C t and C r are assumed to be zero here. The total noise level at the receiving unit is where k is Boltzmann constant equal to −228.6 dB(Watts/ (Hertz * Degree Kelvin)). T r is noise temperature at the receiver's antenna and T e is environment noise temperature at the receiver's antenna [22]. The receiving bandwidth is of course matched to communication signal's bandwidth BW. The final term I is the total interference power level. The impact of interference is assumed to be zero in our study. Free-space path loss (FSPL) is the loss in signal strength of an electromagnetic wave that would result from a line of sight path through free space, with no obstacles nearby to cause reflection or diffraction [23]. This loss is calculated using the following formula: pl d, f , n = c + 20 log 10 (d) + 20 log 10 f , where d is the distance from the transmitter to the receiver, the radio frequency is f , and c is a constant that depends of the units of measure for d and f . With the units of measure for d and f listed in Table 1, c = −27.55. Past line of sight, communications is still possible, but there is additional attenuation due to shadowing. Additionally it is well know that the average receive power level, measured in dBW, around a circle at a constant distance from the transmitter and beyond the line of sight is a lognormally distributed random variable. Let Pl(d, f , n) be the path loss when the distance from the receiver to the transmitter is larger than the line of sight distance. We modify the FSPL formula and propose an adaptive path loss formula: Pl d, f , n = −27.55 + n10 log 10 (d) + 20 log 10 f , (5) where n is determined by the terrain type.
In our analysis, the coefficient n is a random variable that depends of the type of terrain, that is, how rugged the terrain is to radio frequency waves. Typical terrain types include open rural, rural trees and rolling hills, suburban, and urban. For each of the terrain types there is an average distance to the edge of the unobstructed line of sight given. Beyond this limit, the value of n is drawn uniformly random between the values listed in Table 2 with the possibility that there are locations that have direct line of sight beyond this average.

Detection Probability
Model. Now we study the issue of the probability that a detection system detects a transmitter. Let the direction of the directional antenna's peak radiation intensity lie on the positive x axis and the star node be a detection system in Figure 4. The distance from the transmitter to the detection system is d and the angle between the direction of the detection system and the direction of the positive x axis is θ. We will use d and θ in the following sections of this paper with the same meanings defined here. We assume that the detection system's antenna works in omnidirectional mode.
4 EURASIP Journal on Wireless Communications and Networking  where G t (θ) is the transmitter's antenna gain function as Figure 3 shows, and G r = 0. As P t , 20 log 10 f , dB(T r + T e ), and dB(BW) are constants, let K = P t + 256.15 − 20 log 10 f − dB(T r + T e ) + dB(BW). (8) Substitute (8) into the definition of the SNIR, the probability of the detection event occuring is Now we discuss the value of n, for each of the terrain types listed in Table 1, there is an average distance to the edge of the unobstructed line of sight given, which we defined as d 0 . When the distance d is smaller than d 0 , we set n equal to 2. If the distance to the transmitter is greater than d 0 , the value of n is a random variable between the values listed in Table 1: where K is given by (8).

Model Analysis.
Assume both the directional and omnidirectional antennas provide the same EIRP in the direction of the receiver. Assume that the omnidirectional antenna's transmit power is 3 watt and the directional antenna's gain function is as Figure 3 shows, so the directional antenna's transmit power is 0.03 watt. We assume that the operational area Ω is a finite area 100 kilometers × 100 kilometers and the terrain is rural-open. We place the transmitter at the center of the operational area. Figure 5 shows the detection probability map of an omnidirectional antenna in the operational area. In this figure, different colors mean different probability values. As omnidirectional antennas radiate signals in all directions equally, the contour lines are almost circles in Figure 5. The detection probability becomes lower and lower with the increase of the distance d. Figure 6 shows the detection probability map of a directional antenna. Only locations in the main lobe direction of the directional antenna have high probabilities to detect the transmitter, the detection probabilities at other directions are very low.
Let A 1 , . . . , A n be a partition of the operational area Ω. Assume that there is only one detection system that is in one of {A i }. According to the total probability theorem, the probability of detecting the transmitter is (11) where Pr(A i ) is the probability of the detection system being in region A i . We assume that the probability of the detection system being in A i are even, Pr(A 1 ) = Pr(A 2 ) = · · · = Pr(A n ). Then the probability of detecting the transmitter is Here we assume that each A i is 1 km × 1 km, which is a small region for directional transmissions. Normally, if two locations are very near, the detection probabilities at these two locations should be almost equal, so we can assume Pr(Detection | A i ) to be the detection probability at the center of A i . Using equation (10), we can calculate the probability of detecting a transmitter at the center of A i . The dp of Figure 5 is 0.36 and dp of Figure 6 is 0.012. This indicates that directional antennas can reduce the detection probability by over 96.7%. Comparing these two figures, we can find that the area where the detection probability being zero in Figure 6 is much larger than that in Figure 5 and the colorful area where the detection probabilities being larger than 0.1 in Figure 6 is much less than that area in Figure 5. This can explain why a directional antenna has the lower detection probability than an omnidirectional antenna if they provide the same EIRP in the direction of receiver.

Minimizing Detection Probability
Routing Algorithm 4.1. Definition. We model adversaries as passive. Adversaries in this model are assumed to be able to receive any transmit- ter's signals but are not able to modify these signals. If a set of adversaries detect a transmitter in a synchronous manner, they may be able to compute the transmitter's position with localization algorithms. It is dangerous to reveal the position information to adversaries, because adversaries may find the transmitter and catch it according to its position.
As directional antennas can transmit signals towards a specific direction, we can employ several directional antennas as relays to bypass a detection system. In Figure 7, node a, b, and c are three network nodes and the black node is a detection system. Assume that node a wants to send data to node c. If node a transmits data to node c directly using directional antenna, as the detection system happens to lie in main lobe direction of node a, it can detect node a with 100% probability. Or, node a can send data to node c via node b as Figure 7(b) shows. As the detection system is not in the main lobe direction of these two directional antennas, the probability of detecting the transmissions at the detection system is very low as Figure 6 indicates.
Assume detection systems and network nodes are scattered within the operational area. To make the relay transmission from the source to the destination more secure, the strategy of our routing algorithm is to Minimize Detection Probability (MinDP) by selecting a routing path with the lowest detection probability rather than the shortest distance or the least power consumption. In Figure (8), the relay transmission path (a → b → c → d → e) is more secure than the path (a → b → c → e). If network nodes know the locations of detection systems, they can use equation (10) to calculate the detection probability. If network nodes do not know the locations of detection systems, they can use equation (12) to calculate the detection probability.
The goal of our routing protocol is to find a secure routing path which has the lowest detection probability throughout the whole delivery process from the source to the destination. Assume that a packet would be delivered from the source to the destination through N hops. If any of these N hops deliveries is detected by a detection system, the detection event occurs. Let TDP be the total detection probability from the source to the destination where P i is the probability of the i hop delivery being detected by all detection systems. Some assumptions for this routing algorithm are as follows.
(1) Assume that there are k network nodes and all of them employ directional antennas to transmit data. (2) The transmit power of a transmitter varies based on the distance from the transmitter to the receiver and the transmit rate.
The formal definition of MinDP routing algorithm is shown in Algorithm 1.

Evaluation.
Assume the experimental area is 100 km × 100 km and detection systems and network nodes are scattered within the operational area randomly. We compare the total detection probability of MinDP routing algorithm using directional antennas with that of shortest path rouging using omnidirectional antennas. We randomly select two nodes as the source and the destination of each routing. Figure 9 shows the TDP function of hops. In this figure, the TDP of Shortest path routing using omni-direction antennas increases rapidly, while the TDP of MinDP routing algorithm increases adagio. In a scenario where the number of detection systems is given, the TDP of Shortest path routing is much higher than that of MinDP routing algorithm. It is reasonable that the more detection systems are within the experiment area, the higher total detection probability is. We can know from this figure that the transmission from the source to the destination using omni-directional antennas will be detected by detection systems definitely when the number of detection systems is larger than 3 and the number of hops is larger than 2. The average TDP of Shortest path routing is 0.953 and the average TDP of MinDP routing algorithm is 0.244. Hence, the MinDP routing algorithm using directional antennas can reduce the total detection probability by over 74%.

Related Work
Many protocols have been proposed to provide anonymity in Internet, such as Crowds [24], Onion [25]. For ad hoc networks, although a number of papers about secure routing have been proposed, such as SEAD [26], ARAN [27], AODV-S [28], only a few papers are about anonymous routing issue and few of them talk about directional antennas and locations. Zhu et al. proposed a secure routing protocol ASR for MANET [29] to realize anonymous data transmission. ASR makes sure that adversaries are not able to know the source and the destination from data packets. ASR considers the anonymity of addresses of the source and the destination in a packet but not the physical location of the source. In ASR, their solution make use of the shared secrets between any two consecutive nodes. The goal of ASR is to hide the source and destination information from data packets but not to protect the transmission from being detected by hostile detection systems.
ANODR is an secure protocol for mobile Ad hoc networks to provide route anonymity and location privacy [30]. For route anonymity, ANODR prevents strong adversaries from tracing a packet flow back to its source or destination; for location privacy, ANODR ensures that adversaries cannot discover the real identities of local transmitters. However, the location privacy ANODR provides is the identity of sender, not the physical location privacy.
Zhang et al. proposed an anonymous on-demand routing protocol, MASK, for MANET [31]. In MASK, nodes authenticate their neighboring nodes without revealing their identities to establish pairwise secret keys. By utilizing the secret keys, MASK achieves routing and forwarding task without disclosing the identities of participating nodes.
Most secure routing protocols and anonymous routing protocols employ authentication and secret key approaches Let PATH note the selected path and AvailablePath save all possible routing paths Min = 1 for i = 1 to k for j = 1 to k if i != j Calculate dp(node i → node j ) end if end for end for / * Generate all available routing paths and save routing paths to AvailablePath. A path is nodes sequence like path 1 → path 2 → · · · → path * x / GeneratePath(AvailablePath) while AvailablePath != Empt y path = GetPath(AvailablePath) / * Calculate the total detection probability (TDP) of path * / TDP = 1 − (1 − dp(path 1 → path 2 )) · · · (1 − dp(path {x−1} → path x )) if TDP < Min then Min = TDP PATH = path end if DeletePath(AvailablePath,path) / * delete path from AvailablePath * / end while PATH is the selected routing path Algorithm 1 to ensure the security. In a real wireless network, there is no clear transmission range, hostile detection systems can detect the transmitter's signals even if it is very far away from the transmitter. In this scenario, the detection system does not need to pass the authentication, they just detect signals. Hence, authentication cannot thwart hostile detection.

Conclusions
In an untrustworthy network, it is very important for the transmitter to avoid being detected by adversaries. In this paper, we propose a detection probability model to calculate the probability of detecting a transmitter at any location around the transmitter. Since signals from omnidirectional antennas are radiated in all directions, hostile nodes at any location can receive these electromagnetic waves, they have probabilities to tell signals from noises. A directional antenna could form a directional beam pointing to the receiver, and only nodes in the main lobe beam region can receive signals well. If a directional antenna employs less transmit power than an omnidirectional antenna but provides the same EIRP to the receiver, the directional antenna can reduce the detection probability by over 96.7%. Therefore, we prefer to employ directional antennas to relay data from the source to the destination. Minimizing Detection Probability (MinDP) routing algorithm we proposed can select a routing path that has the lowest total detection probability. The simulation results show that the MinDP routing algorithm can reduce the TDP by over 74% so as to provide high security and concealment for transmitters.