Two trivial examples • Every node is given the same secret “master key” • Low Memory costs • Compromise of a single node would render the network completely insecure and unreliable • For every pair of nodes and there is a secret key given only to these 2 nodes • Expensive memory costs • Excellent resiliency (security)

Ways to establish pairwise secret keys • Using public key protocols • Expensive computational costs • Increased storage requirements • Establishing a trusted server that can communicate with all the nodes in the network (like Kerberos) • Expensive costs for message relay • Employing key predistribution schemes (also called KPSs)

Related Prior Work • Several schemes were proposed for KPS • The schemes we will be discussing closely rely on previous work • We will mention 7 other schemes

The Basic Scheme Developed by Eschenauer and Gligor 3 Parameters: • n number of nodes • k size of key ring • v size of key space Nodes communicate if they have a shared key • Encryption is done using the shared key

The Basic Scheme n can grow greatly even for medium values of v and k

Randomized Keys are chosen by random Key ring assignment is done by random Deterministic Keys are still chosen by random! Key ring assignment is deterministic Basic scheme: Deterministic vsRandomized Key Rings

Basic scheme: Deterministic vsRandomized Key Rings Deterministic No overhead Combinatorial properties are guaranteed. Shared-key discovery and path key establishments can be done in O(1). Randomized Significant overhead in generating good pasudo-random numbers Combinatorial properties are not guaranteed (such as connectivity) Shared-key discovery and path key establishments – O(???)

q Composite Scheme Generalization of the Basic Scheme Two nodes communicate directly if they have at least q common keys • Encryption key is created using all common keys If q=1 then similar to Basic Scheme, yet different

Camtepe and Yener’s Scheme First scheme to use combinatorial designs called Set Systems Blocks and points

2005 Lee and Stinson’s Scheme Authors of the article Set Systems Linear polynomials over a finite field

Chakrabarti, Maitra, and Roy’s Scheme Start with a certain Set System Form key rings by merging blocks Larger key rings Some performance metrics are improved

Multiple Space Schemes Combine basic KPS (set systems) with older KPS such as Blom[1985] Inner and outer schemes

Multiple Space Schemes Blom [1985]

Hash Chain Schemes Another avenue of research using KPS Good resilience Bad complexity

A Set System • A set systemis a pair(X,A) • A is a finite set of subsets of X called blocks • The degree of a point is the number of blocks containing x • )X,A) is regular if (of degree r) if all points have the same degree r • The rank of (X,A) is the size of the largest block. • If all blocks have the same size, say k, then (X,A) is said to be uniform (of rank k)

Example X={1,2,3,4,5,6,7,8,9} A={123,456,789,147,258,369 159,267,348,168,249,357}

The 3 phases There are 3 basic operation that should be implemented: Key predistribution Shared-key discovery Path-key establishment

Key Predistribution Phase Choose n and k input parameters Center creates a uniform and regular set system with rank k and n blocks Center determines q Assignment algorithm What happens if A is just a set of n random k sized blocks?

Shared-Key Discovery phase • The phase in which 2 nodes determine the common points in the 2 blocks assigned to them • Suggestion: node i would broadcast the k points in to each of its neighbors • Suppose that 2 nodes discover that and have exactly t common points : if t>=q then they can establish a secret key

The secret key h is a public key derivation function (such as SHA-1) We are using all the common keys to derive the pairwise key in order to achieve maximum resiliency!

Path-Key Establishment phase • What happens if 2 nodes in wireless communication rage fail to find sufficient number of common keys in the shared-key discovery phase? • They look for multiple secure links (or hops) to reach each other

Metrics for the Evaluation of KPSs • Network Size (denoted by n) • Key Storage (denoted by k) • Global connectivity • Local connectivity • Resiliency • Complexity of Shared-Key Discovery and Path-Key Establishment

Network Size • The number of nodes in a DSN, which we denote by the parameter n. • The number of nodes is usually between 1,000 and 10,000 nodes (or even higher) • Notice that in some schemes cannot be chosen independently!

Key Storage • The number of keys per node, which we denote by the parameter k • When we use a combinatorial set system as a key ring space, the number of keys per node is equal to the rank of the set system , which is denoted by k

Global Connectivity • The communication capabilities of the network • It is depended on the physical level and the network level • The Physical Level is represented by the physical graph • The Network Level is represented by the block graph • Determined by the structure of the key ring space

They Key-Sharing Graph • It is the intersection between the physical graph and the block graph • We hope that the key sharing graph is connected • We say that the DSN is globally connected if the key sharing graph is a connected graph

Local Connectivity • Refers to the situation where nodes that are physically close to each other can establish a short secure communication path between them • Pr1 – The probability that 2 random nodes share at least q common keys • Pr2 – The probability that 2 nodes in wireless communication range do not share q common keys but there exist a third node that shares q common keys with each of the first 2 nodes

Resiliency • When an adversary captures a number of sensor nodes at random we assume that all the keys of information stored in the nodes are revealed to the adversary. • We want node captures to affect as small a part of the entire network as possible • The resiliency of the network is estimated by fail(s), which is the probability that a link between 2 fixed noncompromised nodes is affected after s other nodes are compromised

Complexity of Shared-Key Discovery and Path-Key Establishment • Shared-Key discovery is often done by having the 2 nodes exchange the list of identifiers of the keys they hold • If the 2 lists are presorted in increasing order of key identifiers then this can be done in time O(k) • By choosing carefully structured key ring space we can obtain an algebraic description of the key rings • In that case we can reduce the computational complexity of shared-key discovery to O(1)!!!

Configurations We’ll have q=1 for the rest of the discussion (v,n,r,k)-designs Necessary condition for existing configuration nk = vr

LEMMA 1 Any vertex (i.e., block) A j in the block graph GA of a (v, n, r, k)-design, (X,A), has degree at most k(r − 1). Further, all vertices in GA have degrees equal to k(r−1) if and only if |Ai ∩ A j| ≤ 1 for all

Configurations A (v, n, r, k)-design is called a (v, n, r, k)-configuration if any two distinct blocks intersect in zero or one point.

LEMMA 2 Suppose we use a (v, n, r, k)-design for a key ring space with intersection threshold q = 1. Then Pr1 ≤ k(r − 1)/(n − 1). Further, Pr1 = k(r− 1)/(n− 1) if and only if the (v, n, r, k)-design is a configuration.

LEMMA 3 A (v, n, r, k)-configuration exists only if nk = vr and v − 1 ≥ r(k − 1).

Complete Block Graphs The block graph of a configuration is a complete graph if and only if k(r − 1) = n − 1 n << k²

μ-Common Intersection Designs Two-hop paths Increase choices for best-match common neighbor

μ-Common Intersection Designs Suppose that (X,A) is a (v, n, r, k)-configuration. We say that (X,A) is a μ-common intersection design (or (v, n, r, k;μ)-CID) provided that whenever Ai ∩ A j = ∅.

Contents

Contents

Presentation Transcript

Contents

Contents

Contents

Contents

Contents

Contents

Contents

CONTENTS

Contents

Contents