A Key Reshuffling Scheme for Wireless Sensor Networks

Ashok Kumar Das

Department of Computer Science and Engineering,Indian Institute of Technology, Kharagpur 721 302, India

akdas@cse.iitkgp.ernet.in

Abstract. In this paper, we propose an improved alternative for the path key es-tablishment phase of the bootstrapping protocol in a sensor network. This schemehas better connectivity than the path key establishment. The communication over-head for our scheme is comparable with that for the path key establishment. More-over, the guarantee of good connectivity allows us to start with bigger key pools,thereby improving the resilience against node capture.

1 Introduction

Recent advances in wireless communications and electronics have enabled the devel-opment of low-cost, low-power, multifunctional sensor nodes that are small in size andcommunicate untethered in short distances. These tiny sensor nodes, which consist ofsensing, data processing, and communicating components, leverage the idea of sensornetworks. Thus, the sensor networks give a significant improvement over the traditionalsensors.

A sensor network consists of many tiny computing sensor nodes that are scatteredin an area for the purpose of sensing some data and transmitting data to nearby basestations for further processing. The transmission between the sensor nodes is done byshort range radio communications. The base station is assumed to be computationallywell-equipped whereas the sensor nodes are resource-starved. Such networks are usedin many applications including tracking of objects in an enemy’s area for military pur-poses, distributed seismic measurements, pollution tracking, monitoring fire and nu-clear power plants, tacking patients, engineering and medical explorations like wildlifemonitoring, etc. Mostly for military purposes, data collected by the sensor nodes needto be encrypted before transmitting to the neighboring nodes and the base stations.

Each sensor node contains a primitive processor featuring very low computing speedand only small amount of programmable memory. Sensor node is battery-powered andis expected to operate for only few days. Moreover, sensor nodes have the ability tocommunicate with each other and the base stations by short range wireless radio tran-simission at low bandwidth and over small communication ranges (typical example isat most 30 meters i.e., 100 feet).

Due to the above limitations of a sensor network it is not feasible to use public-keyroutines [1,2,3]. [4] says that RSA is almost 1000 times slower than the DES [5,6,7].Now-a-days RC5 [8] is much more faster than the DES. Hence, a symmetric cipher isthe only viable option for encryption/decryption of secret data. As a result, one can use

S. Jajodia and C. Mazumdar (Eds.): ICISS 2005, LNCS 3803, pp. 205–216, 2005.c© Springer-Verlag Berlin Heidelberg 2005

206 A.K. Das

RC5 algorithm for secret communications between the sensor nodes. But setting up thesymmetric keys among communicating nodes is a challenging job in a sensor network.

Several sensor nodes are deployed throughout a sensor field very densely. Therefore,careful handling of topology maintenance is required. However, the topology changesdue to the three phases. In predeployment and deployment phase, sensor nodes can bedeployed from a truck or a plane in the sensor field. In post-deployment phase, topologymay change after deployment because of irregularities in the sensor field like obstaclesor due to jamping, noise, available energy of the nodes, malfunctioning, addition of newsensor nodes, mobility of the sensor nodes, etc. In redeployment of additional nodesphase, additional sensor nodes can be redeployed due to the faulty or rogue sensor nodes.

The rest of the paper is organized as follows. We briefly discuss the overviews ofsome existing key pre-distribution schemes in Section 2. In this section, we focus ourdiscussion on the basic random key pre-distribution scheme (the EG scheme) [9] pro-posed by Eschenauer and Gligor and the polynomial-pool based key pre-distributionscheme (the poly-pool scheme) [10] proposed by Liu and Ning. The EG scheme sup-ports a large network, but it is not quite resilient against node capture. On the otherhand, the poly-pool scheme supports reasonable network size and provides better se-curity than the EG scheme. Some other schemes include the q-composite scheme [11],the random pairwise keys scheme [11] proposed by Chan et al. and the matrix-poolbased scheme due to Du et al. [12]. The q-composite scheme is an improved versionof the EG scheme and it is slightly more resilient against node capture than the EGscheme. The random pairwise keys scheme does not support a big network, whereasit is perfectly secure against node capture. The matrix-based scheme has performancecomparable with the poly-pool scheme. In Section 3, we propose a new scheme whichis considered as an alternative for the path key establishment phase. We call our schemeas the key reshuffling scheme. In this section, we first develop our scheme and after thatwe provide a theoretical analysis for our scheme over the EG scheme as well as thepoly-pool scheme. In Section 4, we report our simulation results of the network con-nectivity measurement over the EG scheme as well as the poly-pool scheme. In Section5, we compare our scheme with the path key establishment. Section 6 highlights someadvantages achieved from our scheme. Finally, Section 7 concludes the paper.

2 Background : Overview of the Key Predistribution Schemes

Key pre-distribution in sensor networks has received considerable research attentionin recent years [9,11,10,13,14,12]. After a brief overview, we concentrate on the basicrandom key pre-distribution scheme (the EG scheme) [9] and the polynomial pool-based scheme (the poly-pool scheme) [10].

2.1 The Basic Bootstrapping Framework

Key establishment in a sensor network is effected by a three-phase process called boot-strapping. The three stages of bootstrapping are as follows:

– Key predistribution: This step is done before the deployment of the sensor nodesin a particular target field (i.e., a deployment area). A key set-up server chooses a

A Key Reshuffling Scheme for Wireless Sensor Networks 207

pool K of randomly generated keys and assigns to each node u a subset Ku of K.Ku is called the key ring of sensor node u.

– Direct key establishment: Immediately after deployment, each sensor node tries tolocate all other sensor nodes with which it can communicate directly and secretly.Two nodes u and v are called physical neighbors if they are within the communi-cation ranges of one another. u and v are called key neighbors if they share one ormore key(s) in their key rings Ku and Kv. Sensor nodes u and v can secretly and di-rectly communicate with each other if and only if they are both physical neighborsand key neighbors, and they are called as direct neighbors. In direct key establish-ment phase, each sensor node u locates its direct neighbors. For this, u broadcastsits own id and the ids of the keys in its key ring.

– Path key establishment: This is an optional stage and, if executed, adds to theconnectivity of the network. Let two physical neighbors u and v fail to establisha direct link between them in the direct key establishment phase. But there existsa path, say, < u = u0, u1, u2, . . . , uh−1, uh = v > in the network such thateach ui a direct neighbor of ui+1 (i = 0, 1, 2, . . . , h − 1). u generates a randomkey k, encrypts k with the key shared between u = u0 and u1 and transmits thisencrypted key to u1. u1 retrieves k by decrypting the encrypted key using the sharedkey between u and u1 and encrypts k by the key shared between u1 and u2 andtransmits it to u2. This process is continued until the key k reaches the desireddestination v. Therefore, u and v can communicate secretly and directly using kand thereby become direct neighbors. However, the main difficulty in this process isthe discovery of a path between u and v. The communication complexity increasessignificantly with the number h of hops. In practical situations, h is restricted to asmall value, say 2 or 3.

2.2 The Basic Random Key Predistribution Scheme (EG Scheme)

Eschenauer and Gligor in their seminal paper [9] bootstrap the era of research of boot-strapping in sensor networks. This method, henceforth referred to as the EG scheme, isessentially the basic bootstrapping method just described above.

The key set-up server starts with a pool K of randomly generated keys. The number| K |= M is taken to be a small multiple of the network size n. For each sensor nodeu to be deployed, a random subset of m keys is selected and given to u as its key ringKu. Upon deployment, each node discovers its direct neighbors as specified in the basicbootstrapping framework.

2.3 The Polynomial Pool-Based Key Predistribution Scheme

Liu and Ning’s polynomial-pool based key distribution scheme (the poly-pool scheme)[10] can be described as follows. Let Fq = GF (q) be a finite field with a q (either aprime or 2m for some positive integer m) just big enough to accomodate a symmetricencryption key. Let f(X, Y ) ∈ Fq[X, Y ] be a t-degree bivariate symmetric polynomiali.e., f(X, Y ) = f(Y, X). A polynomial share of f is a univariate polynomial fα(X) =f(X, α) for some α ∈ Fq . Thus, we have:

fα(β) = f(β, α) = f(α, β) = fβ(α). (1)

208 A.K. Das

If two shares fα and fβ of the same polynomial f are given to two nodes, they cancome up with the common value f(α, β) ∈ Fq as a shared key between them. If (t+1)or more shares of f are known, one can easily reconstruct f(X, Y ) uniquely just byusing the Lagrange’s Interpolation formula. Thus, the disclosure of upto t shares doesnot reveal the polynomial f to an adversary and uncompromised shared keys based onf remains completely secure. If t = 0, then this scheme degenerates to the EG scheme.

The key set-up server selects a random pool K of s symmetric bivariate polynomialsin Fq[X, Y ] each of degree t in X and Y . Some ids α1, α2, . . . , αn ∈ Fq are alsogenerated for the sensor nodes in the network, where n is the network size. For eachsensor node u in the network, s′ polynomials f1, f2, . . . , fs′ are randomly picked upfrom K and the polynomial shares f1(X, α), f2(X, α), . . . , fs′(X, α) are loaded in thekey ring Ku of u, where α is the id of u. Upon deployment, each sensor u transmits theids of the polynomials, the shares of which reside in its key ring. Two physical neighborsu and v having shares of some common polynomial(s) can establish a pairwise keybetween them.

3 Key Reshuffling Scheme

It is seen that due to random selection of keys for the key rings of the sensor nodesfrom a big pool of keys K, the key rings remain some unused keys which are no longeruseful for establishing a pairwise key with the physical neighbors. Hence, it is necessaryto find a proper destination alongwith a path with which the source node can share itsunused key(s) with the destination(s). Our scheme provides two sensor nodes which arenot physical neighbors to establish a key so that they can communicate secretly.

This is accomplished by the following algorithm as follows.

3.1 Algorithm KeyReshuffling

1. for each node u in the sensor network do2. for each unused key k in its key ring do3. Transmit k securely to u’s all direct neighbors.4. Each direct neighbor transmits k to its direct neighbors.5. Continue this process until the key k is found for a destination v in less

than or equal to a pre-determined hop threshold.6. If k is found in v’s key ring, then v transmits an acknowledgement to u by

indicating the matching key k and its id.

Since the network communication overhead of the key reshuffling scheme increasesas the number of hops increases, the number of hops of the key reshuffling scheme maybe restricted to 2 or 3.

3.2 Analysis of the Key Reshuffling Scheme

We analyze our scheme over the EG scheme [9] and the poly-pool scheme [10].

Network Connectivity of the Key Reshuffling Scheme over the EG Scheme: Let Mand m be the key-pool size and the key-ring size respectively. Let the key neighborhood

A Key Reshuffling Scheme for Wireless Sensor Networks 209

graph Gkey = (V, E) on n sensor nodes (| V |= n) in which a link (u, v) ∈ E existsbetween the two nodes u and v if and only if these nodes be key neighbors.

Let p′ denote the probability that a link exists between two randomly selected sensornodes of this graph Gkey . Due to Erdos and Renyi [15], the probability that the Gkey isconnected is given by

Pc = e−e−γ

(2)

where γ is any real constant and p′ = ln nn + γ

n as n → ∞. When we fix Pc with a veryhigh value, say, 0.99999, the expected degree of each node in Gkey is

d′ = p′(n − 1) =(n − 1)

n[ln n − ln(− ln(Pc))]. (3)

Pc is known as the global connectivity.But in practice, the direct neighborhood graph G = Gdirect is not random, because

it depends on the geographical area where the sensor nodes are distributed. However, ifwe assume that the above results for random graphs hold for G also, then we get:

d′ = n′p (4)

where n′ is the expected number of physical neighbors of each node, and p is the prob-ability that two physical neighbors share one or more keys from their key rings. p iscalled the local connectivity.

For the derivation of p, we have the total number of ways to select m keys of a key

ring from the pool K of size M is

(Mm

). Now, for a fixed key ring Ku of node u, the

total number of ways to select Kv of a node v such that Kv does not share a key with

Ku is

(M − m

m

). Thus, we have, p = 1− prob[two nodes do not share a key] and

hence,

p = 1 −

(M − m

m

)(

Mm

) = 1 −m−1∏i=0

M − m − i

M − i. (5)

Let d denote the average number of neighbor nodes that each node can contact.Consider any one of these d neighbors. Let h be the number of hops which be less thanor equal to a maximum value, known as hop threshold (i.e., a predetermined limit). Theprobability that an intermediate node shares a pairwise key with both the source u andthe destination v is p2. As long as one of the d nodes can act as an intermediate node,the source u and the destination v can establish a common key. So the probability thattwo nodes u and v establish a pairwise key is

Ps = 1 − (1 − p)(1 − p2)d. (6)

This equation is true for h = 1.

210 A.K. Das

Let us consider h = 2. Consider y as any one of the d neighbors of v. The probabilitythat y does not share a pairwise key with u and v via d possible intermediate nodes is(1 − p3)d. By considering all d possible neighbors of v, we obtain the probability thattwo nodes establish a pairwise key is

Ps = 1 − (1 − p)(1 − p2)d(1 − p3)d2. (7)

Thus, applying the principle of mathematical induction, we finally obtain the prob-ability that the two sensors share a key in h hops (≤ hop threshold) is given by thefollowing formula:

Pkeyreshuffling = 1 − (1 − p)(1 − p2)d . . . (1 − ph+1)dh

. (8)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

P_k

eyre

shuf

fling

number of hops(h)

p=0.22,d=20p=0.22,d=40p=0.22,d=60p=0.22,d=80

p=0.22,d=100

Fig. 1. Key Reshuffling over the EG scheme. Assume M = 40000, m = 100 so that p = 0.22.

We have considered M = 40000, m = 100 so that p = 0.22 for our analysis.From Figure 1, it is very clear that when the average number of neighbors increasesthen the connectivity also increases. Moreover, we have succeeded in obtaining 100%connectivity in h ≥ 2 hops. Thus, the network remains highly connected after the keyreshuffling scheme in h = 2 hops.

Network Connectivity of the Key Reshuffling Scheme over the Polynomial Pool-Based Scheme: Like the EG scheme [9], the poly-pool scheme can be analyzed underthe framework of random graphs. Let s be the polynomial pool size of the symmetricbivariate polynomials in Fq[X, Y ] where q is a large prime such that it is suitable to fitin sensor’s memory and each polynomial is of degree t in X and Y. Let s′ be the numberof polynomial shares per each node. The local connectivity p is computed as

p = 1 −s′−1∏i=0

s − s′ − i

s − i. (9)

A Key Reshuffling Scheme for Wireless Sensor Networks 211

0

0.2

0.4

0.6

0.8

1

1.2

1.4

0 0.5 1 1.5 2 2.5 3

P_k

eyre

shuf

fling

number of hops(h)

p=0.049,d=20p=0.049,d=40p=0.049,d=60p=0.049,d=80

p=0.049,d=100

Fig. 2. Key Reshuffling over the poly-pool scheme. Assume s = 500, s′ = 5 so that p = 0.049.

Given constraints on the network and the nodes, the desired size s of the polynomialpool can be determined from the above formula.

Now, the probability of two sensor nodes establishing a pairwise key in h hops(≤ hop threshold) for the key reshuffling scheme can be determined similar to thederivation for the EG scheme and hence, it is given by

Pkeyreshuffling = 1 − (1 − p)(1 − p2)d . . . (1 − ph+1)dh

. (10)

For analysis, we have taken s = 500 and s′ = 5. Then, p = 0.049, that is, thenetwork is likely to remain disconnected initially with high probability. From Figure2, it is clear that after executing our scheme we have the high network connectivityfigures when the average number of neighbors increases and h ≥ 2. In other words, thenetwork becomes connected with high probability.

4 Simulation

In this section, we discuss the simulation results of the network connectivity achievedby our scheme over the EG scheme as well as the poly-pool scheme.

We have simulated our scheme for several random deployment models two of whichare shown in Figures 3 and 4. Each dot in the figures represents the deployment locationof a sensor node. Our simulation suggests that the results of our scheme are not muchsensitive with respect to different deployment models.

For the EG scheme, we have taken the standard parameters n = 10000, M = 40000,m = 100. Then, we have the local connectivity as p = 0.23, whereas the actual valueis 0.22. Figure 5 clearly shows that both the simulation as well as actual network con-nectivity of our scheme over the EG scheme are very closed.

212 A.K. Das

Fig. 3. A random deployment model

For the poly-pool scheme, we have considered the parameters as s = 500, s′ = 5 andn = 10000. The actual local connectivity becomes 0.0492, while the simulated value is0.0484. Figure 6 clearly also shows that both the simulation as well as actual networkconnectivity of our scheme over the poly-pool scheme are very closed with respect toboth the random deployment models.

Fig. 4. A random deployment model

5 Comparison

In this section, we compare our scheme with the path key establishment over the EGscheme as well as the poly-pool scheme.

For the poly-pool scheme, we have considered the parameters: s = 500, s′ = 5, andn = 10000. Then, we have simulated with respect to the random deployment models

A Key Reshuffling Scheme for Wireless Sensor Networks 213

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

P_k

eyre

shuf

fling

number of hops(h)

Analysis(M=40000,m=100,d=50)Simulation(M=40000,m=100,d=50)

Fig. 5. Simulation v.s. actual results of network connectivity for the key reshuffling scheme overthe EG scheme. Assume M = 40000, m = 100, n = 10000.

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3

P_k

eyre

shuf

fling

number of hops(h)

Analysis(p=0.0492,d=100)Simulation(p=0.0484,d=100)

Fig. 6. Simulation v.s. actual results of network connectivity for the key reshuffling scheme overthe poly-pool scheme. Assume s = 500, s′ = 5, n = 10000.

shown in Figures 3 and 4. Figure 7 shows that our scheme has better connectivity thanwhat the path key establishment guarantees.

Similarly, for the EG scheme we have the parameters: M = 40000, m = 100, n =10000, and d = 50. Again, Figure 8 clearly tells us that our scheme has better perfor-mance than the path key establishment with respect to both random models shown inFigures 3 and 4.

214 A.K. Das

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

Net

wor

k co

nnec

tivity

pro

babi

lity

number of hops(h)

Key Reshuffling(p=0.0484,d=100)Path Key Establishment(p=0.0481,d=100)

Fig. 7. Comparison of network connectivity between the key reshuffling and the path key estab-lishment over the poly-pool scheme. Assume s = 500, s′ = 5, n = 10000.

0

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 3

Net

wor

k co

nnec

tivity

pro

babi

lity

number of hops(h)

Key Reshuffling(p=0.23,d=50)Path Key Establishment(p=0.20,d=50)

Fig. 8. Comparison of network connectivity between the key reshuffling and the path key estab-lishment over the EG scheme. Assume M = 40000, m = 100, n = 10000.

6 Advantages of the Key Reshuffling Scheme

In this section, we highlight some advantages achieved from our scheme.Some advantages of our scheme are as follows:

1. Network connectivityFrom the above results we conclude that our key reshuffling scheme produces ex-cellent performances over the EG scheme as well as the poly-pool based schemewith respect to network connectivity than the path key establishment.

A Key Reshuffling Scheme for Wireless Sensor Networks 215

2. Network resilience against node captureThe resilience figures against node capture are given in the EG scheme [9] andthe poly-pool scheme [10]. To establish a path between a source and a destinationfor key sharing network communication overhead due to extra transmission of un-used keys is necessary. Thus, this scheme improves security at the cost of networkcommunication overhead.

3. Routing decisionsSince our scheme increases connectivity of the network, so each sensor node canroute encrypted sensing data to a proper destination using a predetermined numberof hops path instead of sending to its neighbors. Hence, this allows us to reduce theoverhead cost of routing in the sensor network.

7 Conclusion

We have shown that our proposed scheme produces better performances with respect tothe network connectivity, the network resilience against nodes capture and the routingdecisions between sensor nodes. This scheme, however, improves the security at thecost of network communication overhead. Thus, our scheme would be a more attractivechoice than the path key establishment.

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