Tight Bounds for Connectivity of Random K-out Graphs

Random K-out graphs are used in several applications including modeling by sensor networks secured by the random pairwise key predistribution scheme, and payment channel networks. The random K-out graph with $n$ nodes is constructed as follows. Each node draws an edge towards $K$ distinct nodes selected uniformly at random. The orientation of the edges is then ignored, yielding an undirected graph. An interesting property of random K-out graphs is that they are connected almost surely in the limit of large $n$ for any $K \geq2$. This means that they attain the property of being connected very easily, i.e., with far fewer edges ($O(n)$) as compared to classical random graph models including Erd\H{o}s-R\'enyi graphs ($O(n \log n)$). This work aims to reveal to what extent the asymptotic behavior of random K-out graphs being connected easily extends to cases where the number $n$ of nodes is small. We establish upper and lower bounds on the probability of connectivity when $n$ is finite. Our lower bounds improve significantly upon the existing results, and indicate that random K-out graphs can attain a given probability of connectivity at much smaller network sizes than previously known. We also show that the established upper and lower bounds match order-wise; i.e., further improvement on the order of $n$ in the lower bound is not possible. In particular, we prove that the probability of connectivity is $1-\Theta({1}/{n^{K^2-1}})$ for all $K \geq 2$. Through numerical simulations, we show that our bounds closely mirror the empirically observed probability of connectivity

k-Connectivity in Random Key Graphs with Unreliable Links

Random key graphs form a class of random intersection graphs and are naturally induced by the random key predistribution scheme of Eschenauer and Gligor for securing wireless sensor network (WSN) communications. Random key graphs have received much interest recently, owing in part to their wide applicability in various domains including recommender systems, social networks, secure sensor networks, clustering and classification analysis, and cryptanalysis to name a few. In this paper, we study connectivity properties of random key graphs in the presence of unreliable links. Unreliability of the edges are captured by independent Bernoulli random variables, rendering edges of the graph to be on or off independently from each other. The resulting model is an intersection of a random key graph and an Erdos-Renyi graph, and is expected to be useful in capturing various real-world networks; e.g., with secure WSN applications in mind, link unreliability can be attributed to harsh environmental conditions severely impairing transmissions. We present conditions on how to scale this model's parameters so that i) the minimum node degree in the graph is at least k, and ii) the graph is k-connected, both with high probability as the number of nodes becomes large. The results are given in the form of zeroone laws with critical thresholds identified and shown to coincide for both graph properties. These findings improve the previous results by Rybarczyk on the k-connectivity of random key graphs (with reliable links), as well as the zero-one laws by Yagan on the 1-connectivity of random key graphs with unreliable links.Comment: Published in IEEE Transactions on Information Theor

On the Strength of Connectivity of Inhomogeneous Random K-out Graphs

Random graphs are an important tool for modelling and analyzing the underlying properties of complex real-world networks. In this paper, we study a class of random graphs known as the inhomogeneous random K-out graphs which were recently introduced to analyze heterogeneous sensor networks secured by the pairwise scheme. In this model, first, each of the $n$ nodes is classified as type-1 (respectively, type-2) with probability $0<\mu<1$ (respectively, $1-\mu)$ independently from each other. Next, each type-1 (respectively, type-2) node draws 1 arc towards a node (respectively, $K_n$ arcs towards $K_n$ distinct nodes) selected uniformly at random, and then the orientation of the arcs is ignored. From the literature on homogeneous K-out graphs wherein all nodes select $K_n$ neighbors (i.e., $\mu=0$), it is known that when $K_n \geq2$, the graph is $K_n$-connected asymptotically almost surely (a.a.s.) as $n$ gets large. In the inhomogeneous case (i.e., $\mu>0$), it was recently established that achieving even 1-connectivity a.a.s. requires $K_n=\omega(1)$. Here, we provide a comprehensive set of results to complement these existing results. First, we establish a sharp zero-one law for $k$-connectivity, showing that for the network to be $k$-connected a.a.s., we need to set $K_n = \frac{1}{1-\mu}(\log n +(k-2)\log\log n + \omega(1))$ for all $k=2, 3, \ldots$. Despite such large scaling of $K_n$ being required for $k$-connectivity, we show that the trivial condition of $K_n \geq 2$ for all $n$ is sufficient to ensure that inhomogeneous K-out graph has a connected component of size $n-O(1)$ whp

Model for Secure Data Transmission in Deep Space Networks

The main thrust of space communications to-date has been to provide secure communications between ground mission control and a single spacecraft. Little work has been reported on developing a secure mode of communications in a deep space satellite network. The main objective is to develop an algorithm that can increase the connectivity and security in the communication path of the network.Computer Science Departmen

Efficient Authentication, Node Clone Detection, and Secure Data Aggregation for Sensor Networks

Sensor networks are innovative wireless networks consisting of a large number of low-cost, resource-constrained sensor nodes that collect, process, and transmit data in a distributed and collaborative way. There are numerous applications for wireless sensor networks, and security is vital for many of them. However, sensor nodes suffer from many constraints, including low computation capability, small memory, limited energy resources, susceptibility to physical capture, and the lack of infrastructure, all of which impose formidable security challenges and call for innovative approaches. In this thesis, we present our research results on three important aspects of securing sensor networks: lightweight entity authentication, distributed node clone detection, and secure data aggregation. As the technical core of our lightweight authentication proposals, a special type of circulant matrix named circulant-P2 matrix is introduced. We prove the linear independence of matrix vectors, present efficient algorithms on matrix operations, and explore other important properties. By combining circulant-P2 matrix with the learning parity with noise problem, we develop two one-way authentication protocols: the innovative LCMQ protocol, which is provably secure against all probabilistic polynomial-time attacks and provides remarkable performance on almost all metrics except one mild requirement for the verifier's computational capacity, and the HB$^C$ protocol, which utilizes the conventional HB-like authentication structure to preserve the bit-operation only computation requirement for both participants and consumes less key storage than previous HB-like protocols without sacrificing other performance. Moreover, two enhancement mechanisms are provided to protect the HB-like protocols from known attacks and to improve performance. For both protocols, practical parameters for different security levels are recommended. In addition, we build a framework to extend enhanced HB-like protocols to mutual authentication in a communication-efficient fashion. Node clone attack, that is, the attempt by adversaries to add one or more nodes to the network by cloning captured nodes, imposes a severe threat to wireless sensor networks. To cope with it, we propose two distributed detection protocols with difference tradeoffs on network conditions and performance. The first one is based on distributed hash table, by which a fully decentralized, key-based caching and checking system is constructed to deterministically catch cloned nodes in general sensor networks. The protocol performance of efficient storage consumption and high security level is theoretically deducted through a probability model, and the resulting equations, with necessary adjustments for real application, are supported by the simulations. The other is the randomly directed exploration protocol, which presents notable communication performance and minimal storage consumption by an elegant probabilistic directed forwarding technique along with random initial direction and border determination. The extensive experimental results uphold the protocol design and show its efficiency on communication overhead and satisfactory detection probability. Data aggregation is an inherent requirement for many sensor network applications, but designing secure mechanisms for data aggregation is very challenging because the aggregation nature that requires intermediate nodes to process and change messages, and the security objective to prevent malicious manipulation, conflict with each other to a great extent. To fulfill different challenges of secure data aggregation, we present two types of approaches. The first is to provide cryptographic integrity mechanisms for general data aggregation. Based on recent developments of homomorphic primitives, we propose three integrity schemes: a concrete homomorphic MAC construction, homomorphic hash plus aggregate MAC, and homomorphic hash with identity-based aggregate signature, which provide different tradeoffs on security assumption, communication payload, and computation cost. The other is a substantial data aggregation scheme that is suitable for a specific and popular class of aggregation applications, embedded with built-in security techniques that effectively defeat outside and inside attacks. Its foundation is a new data structure---secure Bloom filter, which combines HMAC with Bloom filter. The secure Bloom filter is naturally compatible with aggregation and has reliable security properties. We systematically analyze the scheme's performance and run extensive simulations on different network scenarios for evaluation. The simulation results demonstrate that the scheme presents good performance on security, communication cost, and balance

Security in Distributed, Grid, Mobile, and Pervasive Computing

This book addresses the increasing demand to guarantee privacy, integrity, and availability of resources in networks and distributed systems. It first reviews security issues and challenges in content distribution networks, describes key agreement protocols based on the Diffie-Hellman key exchange and key management protocols for complex distributed systems like the Internet, and discusses securing design patterns for distributed systems. The next section focuses on security in mobile computing and wireless networks. After a section on grid computing security, the book presents an overview of security solutions for pervasive healthcare systems and surveys wireless sensor network security

Smart Wireless Sensor Networks

The recent development of communication and sensor technology results in the growth of a new attractive and challenging area - wireless sensor networks (WSNs). A wireless sensor network which consists of a large number of sensor nodes is deployed in environmental fields to serve various applications. Facilitated with the ability of wireless communication and intelligent computation, these nodes become smart sensors which do not only perceive ambient physical parameters but also be able to process information, cooperate with each other and self-organize into the network. These new features assist the sensor nodes as well as the network to operate more efficiently in terms of both data acquisition and energy consumption. Special purposes of the applications require design and operation of WSNs different from conventional networks such as the internet. The network design must take into account of the objectives of specific applications. The nature of deployed environment must be considered. The limited of sensor nodes� resources such as memory, computational ability, communication bandwidth and energy source are the challenges in network design. A smart wireless sensor network must be able to deal with these constraints as well as to guarantee the connectivity, coverage, reliability and security of network's operation for a maximized lifetime. This book discusses various aspects of designing such smart wireless sensor networks. Main topics includes: design methodologies, network protocols and algorithms, quality of service management, coverage optimization, time synchronization and security techniques for sensor networks