Secure Communication in Dynamic Wireless Ad hoc Networks

TL;DR

This paper introduces a new secure communication scheme for wireless ad hoc networks with eavesdroppers, analyzing its connectivity properties and proposing methods to improve security and reduce delay.

Contribution

A novel time-varying secure connectivity graph scheme for ad hoc networks, with analysis of connectivity metrics and strategies to enhance security and efficiency.

Findings

Average incoming and outgoing edges are characterized.

Secure connectivity time scales linearly with distance.

Packet splitting and routing improve security and reduce delay.

Abstract

We consider a wireless ad hoc network in the presence of eavesdroppers (EDs), where the nodes are distributed according to independent Poisson point processes (PPPs). The legitimate nodes follow the half-duplex mode of operation employing the slotted ALOHA protocol for transmission. For such a network, a novel communication scheme that induces a time-varying secure connectivity graph (SCG) is proposed, and the connectivity behavior of this induced SCG is studied. In particular, for a legitimate node in the network, we analyze (i) the average number of incoming edges and the average number of outgoing edges; (ii) the time to nearest-neighbor secure connectivity; and (iii) a condition on the EDs' density that allows information percolation, {\ie}, a condition for the existence of a `giant' component. The average time for secure connectivity among the nodes in this giant component is shown to scale linearly with the Euclidean distance. Further, we show that by splitting the packets into two sub-packets and routing each sub-packet along paths that are sufficiently far apart can (a) potentially improve secure connectivity and (b) reduce the overall delay incurred in exchanging packets between any two legitimate nodes in the giant component.

Equations

$$T_c^S=\arg \underset{k}{\min }\left[1(o\in {\Phi }_t(k))\prod _{x\in {\Phi }_r(k)}(1-1(o\stackrel{(1)}{\to }x,\mathbf{S}))\right],$$

$$N_{t,0}:=\sum _{x\in {\Phi }_r}\text{1}\{0\stackrel{(1)}{\to }x\}\text{1}\{\text{no EDs in }\mathbb{B}(0,{\beta }_e\Vert x\Vert )\}.$$

$$\mathbb{E}{N}_{t,0}=\frac{{\lambda }_l(1-p)[1-\exp (-({\lambda }_{l}p{\beta }_l^2+{\lambda }_e{\beta }_e^2)\pi {\eta }^2)]}{({\lambda }_{l}p{\beta }_l^2+{\lambda }_e{\beta }_e^2)}.$$

$$\underset{\eta \to \infty }{\lim }\mathbb{E}{N}_{t,0}$$

$$\underset{{\beta }_l\to 0}{\lim }\mathbb{E}{N}_{t,0}$$

$$N_{r,0}:=\sum _{y\in {\Phi }_t}\text{1}\{y\stackrel{(1)}{\to }0\}\text{1}\{\text{no EDs in }\mathbb{B}(y,{\beta }_e\Vert y\Vert )\}.$$

$$\displaystyle\mathbb{E}\{T_{NNC}^{S}\}=\left\{\begin{array}[]{cc}\frac{\lambda_{l}}{\lambda_{l}-\lambda_{e}\beta_{e}^{2}-\frac{p}{1-p}\lambda_{l}\gamma(\beta_{l})},&p<\frac{\lambda_{l}-\lambda_{e}\beta_{e}^{2}}{\lambda_{l}\gamma(\beta_{l})-\lambda_{l}+\lambda_{e}\beta_{e}^{2}}\\ \infty&\text{otherwise},\end{array}\right.$$

$$\gamma (x)=x^2-\frac{1}{\pi }\left\{x^2{\cos }^{-1}\frac{x}{2}+{\cos }^{-1}\left(1-\frac{x^2}{2}\right)-\frac{x}{2}\sqrt{4-x^2}\right\}$$

Full text

I Introduction

Wireless ad hoc networks continue to receive significant attention due to its remarkable features of information exchange between the constituent nodes in the network. As the size of the ad hoc networks grows, the physical-layer issues especially interference prohibit the existence of a direct link between the nodes in the network. A suitable alternative is multi-hop networks, where the intermediate nodes could act as relays to improve the overall throughput. However, it has been observed that, owing to channel impairments, the time required to establish a communication link between any two nodes (path-formation time) could be longer than the number of hops in the network resulting in undesirable delays [Ganti2014]. The delay analysis of wireless ad hoc networks has been of great interest in the community; see, for example, [Tabet2010, Baccelli2011, Khabbaz2013, Ren2014, Ganti2014, Luo2015, Deng2017, Zhong2017, Cui2018, Li2018, Yang2018].

An added dimension to the delay problem arises due to eavesdroppers (EDs) with the malicious intent of trying to disrupt the communication between two nodes. When there is sufficient knowledge of the presence of EDs, the path-formation mechanism to establish connection between the source and destination nodes should also take into account the security aspect. The delay performance of wireless networks in the presence of EDs has been well reported [Haenggi2008, Zhou2011, Zhou2011a, Ao2011, Pinto2012, Capar2012, Lee2012, Zhou2012, Shi2012, Gau2012, Wang2013, Zhao2013, Zhao2014a, Zhao2014, Vaze2014, Zhao2015, Zhong2018]. However, to the best of our knowledge, the characterization of average delay in secure communication using the dynamic graph model is less understood. In this paper, we make progress on this problem by studying the impact of EDs on the average delay incurred for single/multi-hop communication between a pair of nodes by considering the dynamic graph model of wireless ad hoc networks. Specifically, we consider the dynamically varying secrecy graph under a protocol similar to the one proposed in [Ganti2014].

It is standard practice to model a dynamic wireless network as a time-varying graph; see, for example, [Vaze2015]. Given a time varying wireless network, the presence and absence of links in the corresponding dynamic graph can be defined using the channel condition between any two nodes. The long-range connectivity between any two nodes in such a dynamic graph can thus be analyzed using tools from percolation theory [Grimmett1999], which is typically used to study the behavior of a connected cluster of nodes in a random graph. The existence of the so-called ‘giant component’ is critical to get a proper handle on the connectivity of nodes in the random graph. Essentially, the giant component in a dynamic graph facilitates the study of communication between nodes located far apart, especially to characterize the time delay between the nodes involved in communication. In this paper, we concern ourselves with the analysis of time delay between nodes located in the giant component in the presence of EDs.

In our model, both legitimate nodes and EDs are assumed to be distributed according to independent Poisson Point Processes (PPPs) with different densities. The legitimate nodes follow the half-duplex mode of operation employing the slotted ALOHA protocol for transmission. We will study the impact of the transmission probability of a node on the average time required for secure-connectivity. Specifically, we will characterize the average delay of the following two packet forwarding schemes:

1. (1)

Direct-path-based approach: Two nodes can securely communicate with each other within a radius of $\eta>0$ satisfying the interference constraint with no EDs in the vicinity of the transmitter. Using slotted ALOHA this model induces a secure-connectivity graph (SCG) with time-varying links. A packet is forwarded from the source to the destination via the minimum delay path in the SCG. For a legitimate node in the network, we analyze the average number of incoming edges (in-degree) and the average number of outgoing edges (out-degree) in the static SCG, and the average time for the nearest neighbor connectivity of a node in the dynamic SCG in the presence of EDs. Different from [Ganti2014], we present a condition on the legitimate node density for the existence of a ‘giant component’ (defined in Section IV) of securely-connected links. An upper bound on the average delay incurred between any two nodes in the giant component is derived. We show that even in the presence of EDs, the average delay scales linearly with the Euclidean distance. 2. (2)

Two secure path-based approach: Here, there exists two types of paths between two legitimate nodes in the SCG, namely, a direct path or a split path. A packet is divided into two independent sub-packets and sent to the destination along two different paths. Assume that the secure communication is achieved if EDs cannot decode both the packets. If the two paths are “sufficiently far apart”, packets securely reach the destination. Unlike the direct-path approach, it suffices to maintain security only in the vicinity of the communicating pair. Using percolation theory, we show that the secure-connectivity of the graph can be increased, while at the same time the overall delay can be reduced for any pair of nodes in the giant component. Similar to the direct-path approach, the delay scales linearly with the Euclidean distance. This approach is similar in spirit to network coding.

A preliminary version of this paper can be found in [Bharath2019a]. Notation: $\mathbb{E}\{\cdot\}$ denotes the expectation operator, $\mathbf{1}\{\cdot\}$ is the indicator function. The Euclidian norm is denoted by $\|\cdot\|_{2}$. For any $x,y\in\mathbb{R}^{2}$, $\mathbb{B}(x,y)$ denotes a two-dimensional Euclidian ball of radius $\|y\|_{2}$ centered at $x$. We use $X\stackrel{{\scriptstyle(d)}}{{=}}Y$ to denote that the distributions of random variables $X$ and $Y$ are same.

Section II comprises the system model and our protocol for secure communication. Properties of the connectivity graph resulting from the protocol model are established in Section III. The existence of a giant component and the delay analysis for secure communication for the two schemes is presented in Section IV. Concluding remarks are provided in Section LABEL:sec:sims.

II System Model

The legitimate nodes (denoted by $\Phi_{l}\subseteq\mathbb{R}^{2}$) and the EDs (denoted by $\Phi_{e}\subseteq\mathbb{R}^{2}$) are distributed as independent PPPs with densities $\lambda_{l}\geq 0$ and $\lambda_{e}\geq 0$, respectively, in the two-dimensional Euclidean space $\mathbb{R}^{2}$. Time is slotted and $\Phi_{l}$’s employ the slotted ALOHA protocol for transmission, i.e., they transmit with probability $p$. Due to the half-duplex mode of operation, the set of transmitting and receiving nodes vary across time. In slot $k\in\mathbb{N}$, the set of transmitting and receiving nodes are denoted by $\Phi_{t}(k)\subseteq\Phi_{l}$ and $\Phi_{r}(k)\subseteq\Phi_{l}\setminus\Phi_{t}(k)$, respectively. The following protocol decides whether two legitimate nodes can communicate with each other securely: In slot $k$, a node $x\in\Phi_{t}(k)$ can securely connect to a node $y\in\Phi_{r}(k)$ if it satisfies the following constraints:

1. (C1)

for $\eta>0$, $\|x-y\|_{2}<\eta$, which ensures that the transmitters and receivers are close enough to communicate 2. (C2)

for $\beta_{l}>0$, the disk $\mathbb{B}(y,\beta_{l}\|x-y\|_{2})$ does not contain any transmitting nodes, which ensures that a receiver does not experience interference above a tolerable level. 3. (C3)

for $\beta_{e}>0$, the disk $\mathbb{B}(x,\beta_{e}\|x-y\|_{2})$ does not contain EDs. $\beta_{e}>1$ signifies that an ED can decode the message even if it is at a farther away from the corresponding legitimate receiver.

This connectivity model together with the slotted ALOHA induces a time-varying SCG on the network. The SCG from slot $m$ to slot $n$ denoted by $G(m,n)$ is characterized by $(\Phi_{l},\bigcup_{k=m}^{n}E_{k})$, where $E_{k}:=\{(x,y)\subseteq(\Phi_{t}(k),\Phi_{r}(k)):\mathbf{1}(x\stackrel{{\scriptstyle(1)}}{{\rightarrow}}y)\times\mathbf{1}(\text{no EDs in }\mathbb{B}(x,\beta_{e}\|x-y\|))=1\}$. Here $\mathbf{1}\{x\stackrel{{\scriptstyle(1)}}{{\rightarrow}}y\}$ denotes a single hop connectivity between the two legitimate nodes $x$ and $y$ in the SCG for a given $k$ without the security constraints. The function $\mathbf{1}(\text{no EDs in }\mathbb{B}(x,\beta_{e}\|x-y\|))=1$ if there are no EDs in a circle of radius $\beta_{e}\|x-y\|$ centered at $x$, and [math] otherwise. A causal secure path exists between $x$ and $y$ if there exists $N>0$ time slots $k_{1}<k_{2}<\ldots<k_{N}$ and $N$ nodes $z_{1},z_{2},\ldots,z_{N}\subseteq\Phi_{l}$ s.t. $(x,z_{1})\in E_{k_{1}}$, $(z_{1},z_{2})\in E_{k_{2}},\ldots,$ $(x_{N},y)\in E_{k_{N}}$. Two nodes are securely connected iff there is a causal path between them [Ganti2014]. For SCG at a given $k$, we drop the index and let $\Phi_{t}$ and $\Phi_{r}$ denote the set of transmitting and receiving nodes, respectively.

III Properties of Secure-Connectivity Graph

Let us consider a legitimate node at the origin $o\equiv(0,0)$ (this is typical for PPPs [Chiu2013]). For a dynamic SCG, the minimum time $T_{c}^{S}$ required for this node to securely communicate with another legitimate node in the network using a single hop is expressed as follows:

[TABLE]

where $\mathbf{1}(o\stackrel{{\scriptstyle(1)}}{{\rightarrow}}x,\mathbf{S})$ denotes one hop secure connection between the node at $o\equiv(0,0)$ and the one at $x\in\mathbb{R}^{2}$.

Lemma 1

The average time for one-hop connectivity is infinity, i.e., $\mathbb{E}^{!o}T_{C}^{S}=\infty$.

Proof: Without the security constraint, it is known that the average time for one-hop connectivity $\mathbb{E}^{!o}[T_{C}^{S}]=\infty$, where the expectation is with respect to the palm measure [Ganti2014]. Thus, with security constraint, we have $\mathbb{E}^{!o}T_{C}^{S}=\infty$. $\blacksquare$

Lemma 1 guides us in considering the average time for multi-hop connectivity. Since a multi-hop involves intermediate nodes at different time slots, it is of great interest to study the node degree distribution of the SCG at a given $k$. We next characterize the average in-degree and the out-degree of any node (for SCG at a given $k$), and the time to nearest-neighbor secure-connectivity (for dynamic SCG). Later, we will invoke these to characterize the average delay incurred in securely transmitting a packet between two legitimate nodes.

III-A Average Node Degree of the SCG for a given $k$

Consider a transmitter at the origin, and let $N_{t,0}$ denote its secure out-degree, i.e., the number of nodes that it can securely connect to is given by

[TABLE]

The average out-degree is characterized by the following theorem.

Theorem 1

For the network of legitimate nodes and EDs distributed as a PPP with densities $\lambda_{l}$ and $\lambda_{e}$, respectively, and with legitimate nodes employing the slotted ALOHA protocol, the average secure out degree is given by

[TABLE]

Proof: See Appendix LABEL:app:outdegreeProof. $\blacksquare$

When $\lambda_{e}=0$, we obtain the expression for the average out-degree without EDs derived in [Ganti2014]. The average out-degree is a decreasing function of $\lambda_{e}$. The interference-limited regime and the noise-limited regime, which are obtained letting $\eta\rightarrow\infty$ and $\beta_{l}\rightarrow 0$, respectively, yield [Ganti2014]

[TABLE]

In the interference-limited regime, as $\lambda_{l}\rightarrow\infty$, the number of adjacent nodes and the fraction of the interfering nodes go to $\infty$. This counter-effect results in a constant average out-degree. From (4), as $\lambda_{l}\rightarrow\infty$, the average out-degree $\rightarrow\frac{1-p}{p\beta_{l}^{2}}$. However, in the noise-limited regime, as $\lambda_{l}\rightarrow\infty$, the number of adjacent nodes $\rightarrow\infty$, and hence the average out degree also $\rightarrow\infty$ [see (5)]. In both cases, as $\lambda_{e}\rightarrow\infty$, the average out-degree $\rightarrow 0$. Next, we present the average secure in-degree of a receiving node at the origin. Let $N_{r,0}$ denote the in-degree, i.e., the number of nodes from which the node at the origin can receive data securely:

[TABLE]

The average in-degree of a typical node at the origin is given by the following theorem.

Theorem 2

For the network of legitimate nodes (employing slotted ALOHA) and EDs distributed as a PPPs with densities $\lambda_{l}$ and $\lambda_{e}$, respectively, the average secure in-degree is given by $\mathbb{E}N_{r,0}=\frac{\lambda_{l}p[1-\exp(-(\lambda_{l}p\beta_{l}^{2}+\lambda_{e}\beta_{e}^{2})\pi\eta^{2})]}{(\lambda_{l}p\beta_{l}^{2}+\lambda_{e}\beta_{e}^{2})}$.

Proof: See Appendix LABEL:app:indegreeProof. $\blacksquare$

The expressions for the average out-degree and the average in-degree differs only by a factor of $\frac{\mathbb{E}N_{r,0}}{\mathbb{E}N_{t,0}}=\frac{p}{1-p}$. When $p=1/2$, the average in-degree is equal to the average out-degree. Thus, the observations made for the average out-degree holds good for the average in-degree as well. In the next subsection, we derive the average time required for a node in the dynamic SCG to securely connect to its nearest neighbor.

III-B Average Time for Secure Nearest Neighbor Connectivity

Due to the slotted ALOHA, a node may require multiple attempts before it can securely connect to its neighbor. Since PPPs are assumed to be stationary, let $T_{NNC}^{S}$ denote the time required for a node at $(0,0)$ to securely connect to its nearest neighbor in the dynamic SCG. An expression for $\mathbb{E}\{T_{NNC}^{s}\}$ is given in the following theorem.

Theorem 3

The mean time for secure nearest neighbor connectivity in the SCG is given by

[TABLE]

where

[TABLE]

if $x<2$, and $x^{2}-1$ if $x\geq 2$.

Proof: See Appendix LABEL:app:Proofnearestneighborseureconnectivity. $\blacksquare$

The average time for secure nearest neighbor connectivity exhibits a phase transition behavior (see Fig. 1). As the relative density $\frac{\lambda_{l}}{\lambda_{e}}$ increases, the average delay decreases, and approaches a value surrounding $1$, which is the minimum delay possible. For higher $p$, the transition happens at higher values of the relative density, because of fewer active receivers. Next, we study the percolation and dissemination of information in the SCG for a given $k$, which will pave the way for characterizing the secure communication delay for legitimate nodes in the giant component of the dynamic SCG.

IV Percolation of Information

We begin with the definition of a giant component, i.e., a connected component with infinitely many nodes.

Definition 1

*We say that a giant component exists if

$\mathcal{G}=\left\{C_{s}\subseteq\Phi_{l}:\left|C_{s}\right|=\infty,\right.$ $\left.\forall(x,y)\in C_{s},~{}\mathbf{1}\{x\rightarrow y,\mathbf{S}\}=1\right\}$ occurs with probability one, where $\mathbf{1}\{x\rightarrow y,\mathbf{S}\}$ indicates the existence of a casual multi-hop secure communication between the two legitimate nodes $x$ and $y$.

Even if two nodes are separated by at most $\eta$ and the interference constraint is satisfied, they may still not be able to securely communicate. This situation may arise, for example, when there are ED nodes located in the vicinity of the transmitting node, leading to the violation of condition (C3) in Sec. II. Thus, for any three consecutive nodes $x$, $y$ and $z$ in the minimum delay path, $\|x-z\|_{2}$ need not be greater than $\eta$. This situation is in contrast to a wireless ad hoc network without EDs, where for the three nodes $x$, $y$ and $z$ in the minimum delay path, $\|x-z\|_{2}>\eta$; otherwise it leads to a contradiction. Thus to facilitate the proof of the existence of a giant component, and the ensuing delay analysis, we have the following definition of $\zeta$-path, $\zeta>0$.*