[PPT] - Reliability of Wireless Sensor Networks Random graph under a PowerPoint Presentation

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Reliability of Wireless Sensor Networks under a Heterogeneous Key Predistribution Scheme Rashad Eletreby∗ and Osman Ya˘ gan

Dept. of ECE

Carnegie Mellon University

Supported by NSF CCF #1617934 and by a generous gift from Persistent Systems, Inc.

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Wireless sensor networks (WSNs)

◮ Distributed collection of sensor nodes that are of

low-cost, small-size, and limited capabilities

◮ Facilitate a broad range of applications, e.g., medical,

environmental, industrial, military, etc.

◮ WSNs may be deployed in hostile environments =

⇒ 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 networks1

1Laurent Eschenauer and Virgil D. Gligor “A key-management scheme for

distributed sensor networks” (ACM CCS ’02)

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Key management

key predistribution: practical option for key distribution of large-scale sensor networks

◮ Single mission key =

⇒ an adversary can compromise the whole network by capturing one node

◮ Pair-wise keys =

⇒ huge memory, severely limits network dynamics, requires n

2

keys in total

◮ Location-dependent key predistribution =

⇒ unknown network topology prior to deployment

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Random key predistribution schemes

Pool of Cryptographic Keys

Random key predistribution scheme ◮ Introduced in the seminal

work of Eschenauer-Gligor1

◮ Each node is assigned K

cryptographic keys at random from a key pool of size P

◮ Two nodes can securely

communicate only if they share a key

1Laurent Eschenauer and Virgil D. Gligor. 2002. “A key-management

scheme for distributed sensor networks” (ACM CCS ’02)

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The heterogeneous random key predistribution scheme

◮ Proposed by Ya˘

gan1 as a generalization to the classical Eschenauer-Gligor scheme

◮ Facilitates networks with varying level of resources,

connectivity and security requirements, e.g., regular nodes and cluster heads

◮ Each node is randomly assigned to one of r possible

classes

◮ A class-i node selects Ki keys at random from a large

key pool of size P

◮ Two nodes can securely communicate only if they share

a key

1O. Ya˘

gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Shared-key connectivity: A crucial requirement

◮ Given the randomness involved =

⇒ A pair of nodes may not share a key = ⇒ Is the resulting network connected?

◮ If the network is connected, then there is a secure path

between every pair of nodes

Pool of Cryptographic Keys Pool of Cryptographic Keys

Connected network Disconnected network How should we adjust the number of keys and the size

f the key pool P to ensure shared-key connectivity?

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Shared-key connectivity: A crucial requirement

Pool of Cryptographic Keys Pool of Cryptographic Keys

Connected network Disconnected network How should we adjust the number of keys and the size

f the key pool P to ensure shared-key connectivity?

◮ Ya˘

gan1 proposed scaling conditions on K1, . . . , Kr, P as functions of the network size n such that the network is connected with high probability as n gets large

1O. Ya˘

gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Shared-key connectivity is not sufficient

◮ Shared-key connectivity is crucial, but it assumes that

all wireless links are available and reliable

◮ A wireless link connecting a pair of key-sharing nodes

may fail for various reasons Shared-key connectivity

=

⇒ Actual network connectivity

Pool of Cryptographic Keys

The failure of this link renders the network disconnected

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The notion of reliability

◮ A network is reliable if it preserves its operation despite

the failure of some wireless links ⇐ ⇒ remains connected

◮ A simple model: Assume that each wireless link fails

with probability 1 − α independently Does the network remain connected?

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Objective

◮ How should the number of keys K1, K2, . . . , Kr and the

size of the key pool P be selected to ensure network reliability against random link-failures? How can we scale K1, K2, . . . , Kr, P, α with the network size n such that

lim

n→∞ P [network reliability] = 1

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Approach: Random graph modeling and analysis

Our approach is based on:

◮ Modeling the network by an appropriate random graph ◮ Establishing scaling conditions on the model parameters

such that the network is reliable with high probability

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Random graphs

◮ Sampling viewpoint: A random graph is a graph that

is obtained by randomly sampling from a collection of graphs, e.g., G(n, m)

◮ Construction viewpoint: Start with a vertex set then

connect edges according to a probabilistic rule, e.g., G(n, p)

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The inhomogeneous random key graph K(n;µ µ µ,K K K, P)1

◮ Models the heterogeneous random key predistribution

scheme

◮ Vertex set V = {v1, . . . , vn} where n denotes the graph

size

◮ Given r classes, each vertex vx is classified as class-i

with probability µi > 0 = ⇒ r

i=1 µi = 1 ◮ A class-i vertex vx is given set Σx of Ki objects drawn

uniformly at random (without replacement) from an

bject pool of size P

Two distinct nodes vx and vy are adjacent, denoted by the event Kxy, if they share an object = ⇒ Kxy := [Σx ∩ Σy = ∅]

1O. Ya˘

gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Edge probability of K(n;µ µ µ,K K K, P)

◮ Let tx denote the class of an arbitrary node vx and

pij := P

Kxy
tx = i, ty = j
◮ P−Kj

Ki

/

P

Ki

=

⇒ probability that a random subset of Ki objects is disjoint from a subset of Kj objects

◮ We have

pij = 1 − P − Kj Ki P Ki

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Edge probability of K(n;µ µ µ,K K K, P)

◮ Let λi denote the edge probability of an arbitrary class-i

node in K. We have λi = P

Kxy
tx = i
=

r

j=1

P

Kxy
tx = i, ty = j
P [ty = j]

=

r

j=1

µjpij

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Erd˝

s-R´

enyi graph G(n; α)

◮ A simple model for random link-failures ◮ Vertex set V = {v1, v2, . . . , vn} ◮ Start with a complete graph on V. Then, remove each

edge independently with probability 1 − α Two distinct nodes vx and vy are adjacent, denoted by the event Cxy if the edge connecting them was not deleted P [Cxy] = α

◮ Class-independent

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The composite graph H(n;µ µ µ,K K K, P, α)

◮ We consider a composite graph

H(n;µ µ µ,K K K, P, α) := K(n;µ µ µ,K K K, P) ∩ G(n; α)

◮ Two nodes are adjacent in H(n;µ

µ µ,K K K, P, α) if and only if they are adjacent in both K(n;µ µ µ,K K K, P) and G(n; α)

T

=

K(n;µ µ µ,K K K, P) H(n;µ µ µ,K K K, P, α)

G(n; α)

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The composite graph H(n;µ µ µ,K K K, P, α)

◮ The intersection of K(n;µ

µ µ,K K K, P) with G(n; α) models the case when each link of K(n;µ µ µ,K K K, P) fails independently with probability 1 − α

◮ If H(n;µ

µ µ,K K K, P, α) is connected = ⇒ K(n;µ µ µ,K K K, P) remains connected despite the random failure of links Network Reliability ≡ Connectivity of H(n;µ µ µ,K K K, P, α)

◮ A graph is connected when there is a path between

every pair of vertices

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Edge probability of H(n;µ µ µ,K K K, P, α)

◮ Recall that tx denotes the class of an arbitrary node vx

Two arbitrary nodes vx and vy are adjacent in the intersecting graph H, denoted by the event Exy, if they are adjacent in both K and G = ⇒ Exy := [Kxy ∩ Cxy]

◮ We have

P

Exy
tx = i, ty = j
= P
Cxy ∩ Kxy
tx = i, ty = j
= P [Cxy] P
Kxy
tx = i, ty = j
= αpij

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Edge probability of H(n;µ µ µ,K K K, P, α)

◮ Let Λi denote the edge probability of an arbitrary

class-i node in H. We have Λi = P

Exy
tx = i
=

r

j=1

P

Exy
tx = i, ty = j
P [ty = j]

=

r

j=1

µjαpij = αP

Kxy
tx = i
= αλi

where λi denotes the edge probability of an arbitrary class-i node in K

◮ K1 ≤ . . . ≤ Kr =

⇒ λ1 ≤ . . . ≤ λr = ⇒ Λ1 ≤ . . . ≤ Λr

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

A zero-one law for connectivity of H(n;µ µ µ,K K K, P, α)1

Theorem

Consider a probability distribution µ µ µ = {µ1, . . . , µr} with µi > 0 for i = 1, . . . , r, a scaling K K K, P : N0 = ⇒ Nr+1 , and a scaling α : N0 = ⇒ (0, 1) such that Λ1(n)

edge prob. of class-1 nodes

= αnλ1(n) ∼ clog n n holds for some c > 0. Then, we have lim

n→∞ P[H(n;µ

µ µ,K K Kn, Pn, αn) is connected] =

if c < 1

1 if c > 1 under few extra conditions.

1R. Eletreby and O. Ya˘

gan, “On the network reliability problem of the heterogeneous key predistribution scheme” (IEEE CDC 2016)

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Equivalent scaling

◮ If λ1(n) = o(1), we can express our scaling condition in

simple terms, namely Kmin,nKavg,n Pn ∼ clog n nαn

◮ If all links are reliable, i.e., αn = 1 for n = 1, 2, . . ., our

scaling condition reduces to the scaling condition given by Ya˘ gan1 for the shared-key connectivity Kmin,nKavg,n Pn ∼ clog n n

1O. Ya˘

gan, “Zero-One Laws for Connectivity in Inhomogeneous Random Key Graphs,” in IEEE Transactions on Information Theory, Aug. 2016

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The finite case: Probability of connectivity

◮ Recall that K1,nKavg,n Pn

∼ c log n

nαn

5 10 15 20 25 30 35

K1

0.2 0.4 0.6 0.8 1

Probability of Connectivity α = 0.2 α = 0.4 α = 0.6 α = 0.8

Min. value of K1

for which c > 1 with α = 0.2

Figure: Empirical probability that H(n;µ µ µ,K K K, P, α) is connected with n = 500, µ µ µ = (1/4, 1/4, 1/4, 1/4), K K K = (K1, K1 + 5, K1 + 10, K1 + 15), and P = 104. For each parameter pair (K K K, α), 200 independent samples of the graph H(n;µ µ µ,K K K, P, α) are generated.

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

The finite case: Importance of K1

◮ Recall that K1,nKavg,n Pn

∼ c log n

nαn

0.4 0.5 0.6 0.7 0.8 0.9 1

Link Failure Probability 1 − α

0.2 0.4 0.6 0.8 1

Probability of Connectivity

K1 = 10, K2 = 70 K1 = 20, K2 = 60 K1 = 30, K2 = 50 K1 = 40, K2 = 40

Figure: Empirical probability that H(n;µ µ µ,K K K, P, α) is connected with n = 500, µ µ µ = (1/2, 1/2), and P = 104; we consider four choices of K K K = (K1, K2) each with the same mean. For each parameter pair (K K K, α), 200 independent samples of the graph H(n;µ µ µ,K K K, P, α) are generated.

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Ongoing and future work

◮ A stronger notion of reliability1 =

⇒ Reliability against both link and node failures

◮ If H(n;µ

µ µ,K K K, P, α) is k-connected with high probability, then the network is reliable against i) the random failure

f each link and ii) the failure of any k − 1 nodes

◮ A general model for random link failures2

◮ A link between nodes of class-i and class-j fails with

probability 1 − αij independently = ⇒ α α α = [αij]

◮ H(n;µ

µ µ,K K K, P,α α α) := K(n;µ µ µ,K K K, P) ∩ G(n;α α α)

Inhomogeneous Erd˝

s-R´

enyi Graph

◮ A new approach to modeling complex networks

= ⇒ Using graph union and intersection to generate network models with a rich mixture of structural properties

1IEEE ISIT 2016 and IEEE ISIT 2017 2Allerton 2016 and IEEE ISIT 2017 25 / 26

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Rashad Eletreby∗ and Osman Ya˘ gan Motivation Random graph models Composite random graphs Main results Ongoing and future work

Thank You!

www.andrew.cmu.edu/~reletreb

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