On providing location privacy for mobile sinks in wireless sensor networks

Abstract

A common practice in sensor networks is to collect sensing data and report them to the sinks or to some pre-defined data rendezvous points via multi-hop communications. Attackers may locate the sink easily by reading the destination field in the packet header or predicting the arrival of the sink at the rendezvous points, which opens up vulnerabilities to location privacy of the sinks. In this paper, we propose a random data collection scheme to protect the location privacy of mobile sinks in wireless sensor networks. Data are forwarded along random paths and stored at the intermediate nodes probabilistically in the network. The sinks will move around randomly to collect data from the local nodes occasionally, which prevents the attackers from predicting their locations and movements. We analyze different kind of attacks threatening the location privacy of the sinks in sensor networks. We also evaluate the delivery rate, data collection delay and protection strength of our scheme by both analysis and simulations. Both analytical and simulation results show that our scheme can protect location privacy of mobile sinks effectively, while providing satisfactory data collection services.

Appendices

Appendix 1: Calculation of probability p _a of receiving new data

We calculate the average probability p _a of a sensor to get a new piece of data in each time slot. A node can get new data either by generating itself or receiving a copy of data generated by its surrounding neighbors (within L hops). We consider that each node has equal probability p _s to generate a new piece of data in one time slot. For each source node s, its surrounding nodes within L hops will have a decreasing probability to receive a copy of that data.

Let p _e , p _s , p _w , and p _n be the probability of forwarding the data to the next hop following the four directions (east, south, west, and north respectively). If the data is generated by node s at coordinates (x,y), then the probability of the neighboring nodes to receive the data is calculated as below. Note that the results in brackets are calculated with p _e , p _s , p _w and p _n all equal to 0.25.

In the first step:

$$ \begin{aligned} p(x+1,y) &= p_e (=0.25)\\ p(x, y-1) &= p_s (=0.25)\\ p(x-1, y) &= p_w (=0.25)\\ p(x, y+1) &= p_n (=0.25) \end{aligned} $$

In the second step:

$$ \begin{aligned} &p(x+1, y+1) = p_e p_n + p_n p_e (=0.125)\\ &p(x-1, y+1) = p_w p_n + p_n p_w (=0.125)\\ &p(x+1, y-1) = p_e p_s + p_s p_e (=0.125)\\ &p(x-1, y-1) = p_w p_s + p_s p_w (=0.125)\\ &p(x,y+2) = p_n p_n (=0.0625)\\ &p(x, y-2) = p_s p_s (=0.0625)\\ &{\ldots} \end{aligned} $$

The calculation is continued until it reaches L steps.

Let p(h) be the probability of receiving a data copy at the surrounding nodes that are h hops away from s, where h ≤ L. If the nodes are uniformly distributed in a network (i.e. grid structure), each node will have the same number of 1-hop, 2-hop, $\ldots,\, L$-hop neighbors, represented as $N(1), N(2), \ldots, N(L)$. Note that we ignore the boundary effect here in calculating the average probability as the nodes at the boundary are the minority in a large-scale network. The total probability of a node to receive a data copy will then become ∑ ^L _h=1 p(h) N(h). Given that the nodes have equal probability p _s to be data source, they will have equal probability to receive a data copy from their neighboring nodes in a long run.

Note that there could be multiple data sources in one time slot. Also, the location of the data sources are randomly generated, which could be different across different time slots. Since the nodes have equal probability of receiving new data, the average probability of a node to receive a new data in one time slot can be calculated by the total number of data copies generated in one time slot divided by the total number of nodes in the network, i.e. $p_{a} = \frac{p_{d} N_{s} N_{c}}{N_{s}}$ as shown in Eq. (1).

Appendix 2: Calculation of sink visiting probability

We show that the sink has equal probability to visit the nodes if it walks with random steps in the grid in a long run. Let us consider that the sink moves in a n by n grid with its location denoted as (i, j) corresponding to the row and column of the grid point. The movement of the sink can be described by a Markov chain, with an initial state π₀, a vector of size n ² to indicate the starting position of the sink, and a transition probability matrix P of size n ² by n ². $P=[p_{(i,j),(i^{\prime},j^{\prime})}]$ ($i,j,i^{\prime},j^{\prime}=0,1,2,\ldots n$) contains the transition probabilities between these n ² grid points. P takes the following form:

$$ P = \left( \begin{array}{l} p(0,0) \\ p(0,1) \\ \vdots \\ p(i,j) \\ \vdots \\ p(n,n)\\ \end{array}\right) $$

Consider a row $P(i,j)=[p_{(i,j),(i^{\prime},j^{\prime})}]\, (i^{\prime},j^{\prime}=0,1,2,\ldots n)$ and given that grid point (i, j) has four neighbors,

$$ p_{(i,j),(i^{\prime},j^{\prime})} = \left\{ \begin{array}{ll} 0.25 &\quad \hbox{if } (i^{\prime},j^{\prime})=(i-1,j),(i+1,j), (i,j-1), (i,j+1)\\ 0 & \quad \hbox{else.}\\ \end{array} \right. $$

If grid point (i, j) is at the corner of the grid, it will have only two neighbors and have a probability of 0.5 moving to each of them. Similarly, if (i, j) is on the side of the grid, it will have a probability of 0.33 to move to each of its three neighbors. The probability $p_{(i,j),(i^{\prime},j^{\prime})}$ in the vector P(i, j) represents the probability that the sink will move from location (i, j) to location $(i^{\prime},j^{\prime}).$ All transition probabilities to non-neighbor nodes will be equal to 0.

We calculate π _m  = π₀ P ^m with randomly chosen initial states π₀. π _m shows the probability that the sink is at a location after moving m random steps. We observed that π _m converges to two alternating states when m tends to infinity. This is mainly due to the moving pattern of the sink which stops every odd number of steps. In our analysis we omit the boundary effects and only focus on the middle nodes (as there are the majority). By taking the average of probability in the two alternative states, we find that there is a equal probability of 1/n ² for the sink to be at a specific location in each time interval.