Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Reliability of Wireless Sensor Networks Random graph under a Heterogeneous Key Predistribution models Composite random Scheme graphs Main results Ongoing and Rashad Eletreby ∗ and Osman Ya˘ future work gan Dept. of ECE Carnegie Mellon University Supported by NSF CCF #1617934 and by a generous gift from Persistent Systems, Inc. 1 / 26
Wireless sensor networks (WSNs) Rashad Eletreby ∗ and Osman Ya˘ gan ◮ Distributed collection of sensor nodes that are of Motivation low-cost, small-size, and limited capabilities Random graph models ◮ Facilitate a broad range of applications, e.g., medical, Composite random graphs environmental, industrial, military, etc. Main results Ongoing and ◮ WSNs may be deployed in hostile environments = ⇒ future work eavesdropping and node-capture attacks are possible Cryptographic protection is needed ◮ Asymmetric (Public-key) cryptosystems = ⇒ excessive energy consumption and computation overhead ◮ Symmetric cryptosystems = ⇒ faster, energy-efficient, feasible choice for securing wireless sensor networks 1 1 Laurent Eschenauer and Virgil D. Gligor “A key-management scheme for distributed sensor networks” (ACM CCS ’02) 2 / 26
Key management Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models key predistribution : practical option for key distribution of Composite random large-scale sensor networks graphs Main results ◮ Single mission key = ⇒ an adversary can compromise Ongoing and the whole network by capturing one node future work ◮ Pair-wise keys = ⇒ huge memory, severely limits � n � network dynamics, requires keys in total 2 ◮ Location-dependent key predistribution = ⇒ unknown network topology prior to deployment 3 / 26
Random key predistribution schemes Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models ◮ Introduced in the seminal Composite random Pool of work of Eschenauer-Gligor 1 graphs Cryptographic Keys Main results ◮ Each node is assigned K Ongoing and cryptographic keys at future work random from a key pool of size P ◮ Two nodes can securely Random key predistribution communicate only if they scheme share a key 1 Laurent Eschenauer and Virgil D. Gligor. 2002. “A key-management scheme for distributed sensor networks” (ACM CCS ’02) 4 / 26
The heterogeneous random key predistribution Rashad Eletreby ∗ and Osman Ya˘ gan scheme Motivation gan 1 as a generalization to the classical Random graph ◮ Proposed by Ya˘ models Eschenauer-Gligor scheme Composite random graphs ◮ Facilitates networks with varying level of resources, Main results connectivity and security requirements, e.g., regular Ongoing and future work nodes and cluster heads ◮ Each node is randomly assigned to one of r possible classes ◮ A class- i node selects K i keys at random from a large key pool of size P ◮ Two nodes can securely communicate only if they share a key 1 O. Ya˘ gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016 5 / 26
Shared-key connectivity: A crucial requirement Rashad Eletreby ∗ and Osman Ya˘ gan ◮ Given the randomness involved = ⇒ A pair of nodes Motivation ⇒ Is the resulting network may not share a key = Random graph models connected? Composite random ◮ If the network is connected, then there is a secure path graphs between every pair of nodes Main results Ongoing and future work Pool of Pool of Cryptographic Cryptographic Keys Keys Connected network Disconnected network How should we adjust the number of keys and the size of the key pool P to ensure shared-key connectivity? 6 / 26
Shared-key connectivity: A crucial requirement Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Pool of Pool of Cryptographic Cryptographic Random graph Keys Keys models Composite random graphs Main results Ongoing and future work Connected network Disconnected network How should we adjust the number of keys and the size of the key pool P to ensure shared-key connectivity? gan 1 proposed scaling conditions on K 1 , . . . , K r , P ◮ Ya˘ as functions of the network size n such that the network is connected with high probability as n gets large 1 O. Ya˘ gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016 7 / 26
� Shared-key connectivity is not sufficient Rashad Eletreby ∗ and Osman Ya˘ gan ◮ Shared-key connectivity is crucial , but it assumes that all wireless links are available and reliable Motivation ◮ A wireless link connecting a pair of key-sharing nodes Random graph models may fail for various reasons Composite random graphs = ⇒ Actual network Shared-key connectivity Main results connectivity Ongoing and future work Pool of Cryptographic Keys The failure of this link renders the network disconnected 8 / 26
The notion of reliability Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models Composite random ◮ A network is reliable if it preserves its operation despite graphs the failure of some wireless links ⇐ ⇒ remains Main results connected Ongoing and future work ◮ A simple model: Assume that each wireless link fails with probability 1 − α independently Does the network remain connected? 9 / 26
Objective Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models ◮ How should the number of keys K 1 , K 2 , . . . , K r and the Composite random size of the key pool P be selected to ensure network graphs reliability against random link-failures? Main results Ongoing and future work How can we scale K 1 , K 2 , . . . , K r , P , α with the network size n such that n →∞ P [ network reliability ] = 1 lim 10 / 26
Approach: Random graph modeling and analysis Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Our approach is based on: Main results Ongoing and ◮ Modeling the network by an appropriate random graph future work ◮ Establishing scaling conditions on the model parameters such that the network is reliable with high probability 11 / 26
Random graphs Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs ◮ Sampling viewpoint : A random graph is a graph that Main results is obtained by randomly sampling from a collection of Ongoing and graphs, e.g., G ( n, m ) future work ◮ Construction viewpoint : Start with a vertex set then connect edges according to a probabilistic rule, e.g., G ( n, p ) 12 / 26
The inhomogeneous random key graph Rashad Eletreby ∗ K, P ) 1 and Osman Ya˘ gan K ( n ; µ µ K µ,K Motivation ◮ Models the heterogeneous random key predistribution Random graph scheme models Composite random ◮ Vertex set V = { v 1 , . . . , v n } where n denotes the graph graphs size Main results Ongoing and ◮ Given r classes, each vertex v x is classified as class- i future work ⇒ � r with probability µ i > 0 = i =1 µ i = 1 ◮ A class- i vertex v x is given set Σ x of K i objects drawn uniformly at random (without replacement) from an object pool of size P Two distinct nodes v x and v y are adjacent, denoted by the event K xy , if they share an object = ⇒ K xy := [Σ x ∩ Σ y � = ∅ ] 1 O. Ya˘ gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016 13 / 26
Edge probability of K ( n ; µ µ K K, P ) µ,K Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models ◮ Let t x denote the class of an arbitrary node v x and Composite random � � � graphs � t x = i, t y = j p ij := P K xy Main results � P Ongoing and ◮ � P − K j � � / = ⇒ probability that a random subset of future work K i K i K i objects is disjoint from a subset of K j objects ◮ We have ��� P � P − K j � p ij = 1 − K i K i 14 / 26
Edge probability of K ( n ; µ µ K, P ) K µ,K Rashad Eletreby ∗ and Osman Ya˘ gan Motivation Random graph models ◮ Let λ i denote the edge probability of an arbitrary class- i Composite random node in K . We have graphs Main results � � � � t x = i λ i = P K xy Ongoing and future work r � � � � � t x = i, t y = j = P K xy P [ t y = j ] j =1 r � = µ j p ij j =1 15 / 26
Erd˝ os-R´ enyi graph G ( n ; α ) Rashad Eletreby ∗ and Osman Ya˘ gan Motivation ◮ A simple model for random link-failures Random graph models ◮ Vertex set V = { v 1 , v 2 , . . . , v n } Composite random graphs ◮ Start with a complete graph on V . Then, remove each Main results edge independently with probability 1 − α Ongoing and future work Two distinct nodes v x and v y are adjacent, denoted by the event C xy if the edge connecting them was not deleted P [ C xy ] = α ◮ Class-independent 16 / 26
The composite graph H ( n ; µ µ K, P, α ) K µ,K Rashad Eletreby ∗ and Osman Ya˘ gan Motivation ◮ We consider a composite graph Random graph models H ( n ; µ µ µ,K K K, P, α ) := K ( n ; µ µ µ,K K K, P ) ∩ G ( n ; α ) Composite random ◮ Two nodes are adjacent in H ( n ; µ graphs µ µ,K K K, P, α ) if and only Main results µ K if they are adjacent in both K ( n ; µ µ,K K, P ) and G ( n ; α ) Ongoing and future work T = µ K K ( n ; µ µ,K K, P ) G ( n ; α ) µ K H ( n ; µ µ,K K, P, α ) 17 / 26
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