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Full Communication In Transversal Design Based

Key Predistribution Schemes Using Deterministic

Merging Block Strategy

Pinaki SarkarDepartment of Mathematics,

Jadavpur University, Kolkata, India.

Telephone: (+91) 9433531020

Email: pinakisark@gmail.com

AbstractConstraints in resources of the constituent sensors ofany Wireless Sensor Network (WSN) are most challenging aspectswhile designing security models for such networks. Naturallyone prefers less expensive symmetric key cryptography overpublic key techniques during communication among nodes. In

such cryptosystems, both the communicating parties must possessthe same cryptographic key prior to message exchange. Thisis normally achieved by key predistribution (KPD). One suchscheme based on Transversal Design was proposed by Lee andStinson in 2005. However the mentioned scheme is devoid offull communication among the nodes. The aforesaid weaknessgives rise to multi-hop communication involving other node(s)which reduces efficiency of such communication. Deterministi-cally merging nodes results in a smaller network having biggerblocks result in full communication between the blocks. The de-terministic merging block strategy can be thought as preassigningdeterministic paths for any pair of non communicating nodes ofthe original KPD which lags full communication.

Index TermsSecurity, Key predistribution, Transversal De-signs, Deterministic Merging Blocks, Communication Probability.

I. INTRODUCTION

Wireless sensor networks (WSN) are one of the most popu-

lar ad-hoc mobile networks in recent times. Such networks typ-

ically constitute of a Key Distribution Server (KDS) or Base

Station (BS) and identical ordinary sensors (or nodes or sensor

nodes or motes). As such there is no structure to networks

comprising only of these two types of nodes. Such networks

are known as Distributed Sensor Network (DWSN). However

sometimes provisions are made for some special nodes termed

as Cluster Heads (CHs) having certain extra capabilities which

are normally used to provide a sort ofhierarchy to a network

by subdividing it into (small) clusters. Naturally such networks

are called Hierarchical Wireless Sensor Networks (HWSN).Each entity constituting a WSN typically consists of a

(battery) power unit, a processing unit, a storage unit and

a wireless transceiver. Capacities of each such unit in any

ordinary node is quite limited for any WSN. For HWSN, the

capacities and power of the CHs may vary while the KDS of

any WSN is usually quite powerful. As the name suggests,

communication in wireless sensor networks is achieved using

radio frequencies. Resource constrained nodes can communi-

cate with each other only within a limited range having center

as the node and small radius termed as Radio Frequency range

or radius of communication or physical layer of [12]. This

range or radius is generally same for ordinary sensors and may

be varied for CHs. While the KDS has quite a large radius of

communication.In spite of all the weaknesses in the basic building blocks of

WSNs, these networks have several military applications like

monitoring enemy movements, etc. Besides they are utilized

for other scientific purposes like smoke detection, wild fire

detection, seismic activity monitoring etc.

A. Related Works

In all it applications, a WSN once deployed works unat-

tended for long duration of time while its constituent nodes

deals with lot of sensory information. So during exchange

of message among the nodes, though suitable cryptographic

techniques need to be used, one needs to avoid heavy or costly

computations due to constraints in resources in them. Thesefacts restricts the use of existing costly public key security

protocols like [5], [10], [15] for message exchange. Thus we

are forced to revert to symmetric key encryption where both

the communicating parties must possess the same key prior

to message exchange. Using standard online key exchange

technique involving public parameters has to be avoided as

heavy and costly computations are involved. Treating a node

as Trusted Authority (T.A) is highly risky as capture of that

node will make the entire system vulnerable. Hence schemes

like Kerberos [16] can not be implemented for security in

WSN. This leads one to adopt various Key Predistribution

(KPD) techniques.

Eschenauer & Gligor in their work [9] suggested the pio-neering idea ofpredistribution of keys into the sensors. Their

idea can broadly be divided into two steps:

Keys are (randomly) preloaded into the sensors prior to

deployment.

Key establishment: this phase consists of:

Shared key discovery: establishing shared common

key among the nodes and

Path key establishment: establishing path via other

node(s) between a given pair of nodes that do not

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share any common key.

Random preloading of keys meant that the key rings or key

chains were formed randomly. Key establishment was done

using challenge and response technique. Schemes that follow

similar random strategy are called random key predistribution

schemes. Some more examples of such schemes are [4], [6],

[14]. Camptepe & Yener presents an excellent survey of such

schemes in their technical report [2].On the other hand there exists KPD schemes based on

deterministic approach involving Mathematical tools. Most

of such schemes are mainly based on different types of

combinatorial designs like Transversal Designs (TD(k,n)). Any

standard book of Combinatorial Design Theory like [20], [19]

provides detailed study of all such combinatorial designs.

For a brief outline, one may refer to [12]. For the sake of

completeness, discussion on combinatorial designs will be

presented in section III.

Camptepe & Yener [1] were first to propose a deterministic

KPD scheme where keys are preloaded and later established

based on generalized quadrangles & symmetric BIBDs. Lee

& Stinson [11], [12], [13] demonstrated that deterministicdesigns have better shared key discovery and path key estab-

lishment than their random counterparts. Following the initial

work of Camptepe & Yener [1], numerous deterministic KPD

schemes based on combinatorial designs have been proposed.

Some such schemes can be found in [7], [11], [12], [13], [17],

[18].

Some existing deterministic KPDs should be viewed differ-

ently for better understanding like the scheme in [17] proposed

recently by Sarkar & Chowdhury . This scheme is designed on

the basis of unique factorization of polynomials over a Finite

Field. It also addresses the problem of selective node attack

(refer [12], [8]) by adapting a novel technique suggested first

by Sarkar et al. [18].

B. Contributions in this paper

Proposed KPD scheme by Lee & Stinson in [12] lacks full

communication among nodes. So when two nodes does not

share a common key wants to communicate, a path has to

be sought. This leads to increased cost of communication and

trusting other (intermediate) nodes, which affects networks

security. Thus one is prompted to think of remedial strategy

to this problem. In this direction, one of the pioneering

contribution is due to Chakrabarti et al. [3] who for the first

time proposed the idea of merging nodes. Their strategy was

to randomly merge z nodes of Lee & Stinsons design [12]

to end up with a network reduced z times with increased,though not full communication among the blocks.

Motivated by Chakrabarti et al.s idea, the current work

presents a deterministic merging technique where exactly two

(2) nodes of Lee & Stinsons KPD scheme in [12] are merged.Corollary 2 of theorem 1 in section V proves that merging two

nodes of Lee & Stinsons scheme [12] with certain property re-

sults in full communication among the newly formed (merged)

blocks. The deterministic merging concept can be thought of

as preassigning communication paths for nodes not sharing

any common key in the original scheme. Thus, though the

size of network is halved, efficiency is bound to improve.

II. TERMINOLOGIES NOTATIONS AN D PRELIMINARIES

Some standard notations, terminologies and preliminaries

that shall be followed throughout the paper are being briefed

here. Unless otherwise stated in a section, these notation and

symbols are reserved for the terms mentioned here.

A. Terminologies and Notations to be used in this paper

The term Uncompromised nodes means nodes that are

not compromised. The word communication is sometimes

abbreviated to com.. Communication model/scheme and

Key predistribution (KPD) model/scheme mean the same &

denoted by KPD. Some commonly used notations are:

N denotes total number of nodes.

q: a prime or prime power which is the index of the finite

field (Fq) over which the Transversal Design proposed by

Lee & Stinson [12] is defined.

k: number of keys in the key ring of each node. This is

inherited from the original KPD of [12].

Other than these, some notations related to combinatorial

designs and resiliency will be defined before their use in

sections II-B and VII respectively.

B. Basics of Combinatorial Design

This section briefly describes some basic notion ofcombina-

torial design necessary for understanding Lee & Stinson [12]

scheme. Elaborate discussions can be found in any standard

book on Combinatorics like Stinsons [19] or [20].

Consider a finite set X. Then a set system or design is a

pair (X, A) where A is a set of subsets of X called blocks.Elements of X are called varieties. A (v,b,r,k) 1 designbased on X satisfies the following conditions:

|X | = v, |A| = b. Each subset in A contains exactly k elements (rank). Each variety in X occurs in r many blocks (degree).

Further a (v,b,r,k) 1 design is called a configurations ifany two of its blocks intersect in at most one point.

Group-divisible design of type gu and block size k is a

triple (X, H,A), where

1) X is a finite set with |X | = gu.2) H is a partition of X into u parts, that is, H =

{H1, H2,H3, . . . , H u} with X = H1 H2 H3 . . . Hu, |Hi| = g 1 i u and Hi Hj = 1 i =j u.

3) A is the collection of blocks of X having the follow-ing properties: |H A| 1 H H, A A,given any pair of varieties x Hi, y Hj withi = j unique A A such that x, y A.

Transversal Designs TD(k,n) are special type of group-

divisible designs with g = n, u = k, and k = k as wellas (nk,n2,n ,k) configuration. Relation between BIBDs,group-divisible designs and transversal designs can be found

in books on Combinatorial Designs like [20], [19] while [12,

section III] briefs the topic nicely.

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Common Intersection Design (CID): maximal CID

Suppose that X, A is a v,b,r,kconfiguration. X, A is saidto be a common intersection design (CID) if:

|{A A : Ai A = and Aj A = }|

wheneverAiAj = . For the sake of consistency, one defines = whenever Ai Aj = , i, j.

For any given set of parametric values ofv,b,r,k

, suchthat a configuration can be obtained with them, one would

like to construct a configuration with maximum possible .

This maximal value of will be denoted . Theorem 14. of

[12, section IV] basically establishes such TD(k, n) designsare k(k 1) CID

III. KEY PRED ISTRIBUTION SCHEME BASED ON

TRANSVERSAL DESIGN OF [12]

Suppose that p is prime and 2 k p. Then one canconstruct a TD(k, p) in the following manner:

Define X = {0, 1, 2, . . . , k 1} Zp For 0 x k 1, define Hx = x Zp

Now define H = {H

x : 0 x k 1}. NodeNi,j are assigned with the key identifiers {(x, (ix+j)mod p) : 0 x k1} for every ordered pair (i, j) Zp Zp.

Finally let N = {Ni,j : (i, j) Zp Zp}. Then (X, H,N) forms a TD(k, p) as has been proved

in Theorem 6. of [12, section IIIA].

Without loss of generality, this KPD scheme can be

extended to q = pr, where pprime & rany positiveinteger by simply replacing p by q in the above steps.

The following Tables I and II have been constructed for

a sample network having q = 4 = 22 nodes by taking thevalues of k = q and k = q 1 respectively. The tables

gives alternative groupdivisible form of presentation of theirequivalent transversal forms for the same parametric values

based on Lee & Stinson [12] scheme. N0 to N15 denotes

the nodes with ids ranging from 0 to 15 whose polynomialsare represented in the column immediately below it. Key ids

contained in the nodes are presented in the columns below

each node. V C denoted the distinct Variety ClassesH1, H2, H3, where Hd = {(i, d) : 0 i 3} for d = 1, 2, 3.One notes that the scheme under consideration is a q(q 1)CID or (q 1)(q 2)CID according to k = q or k = q 1(see CID discussion in section II-B). Thus for nodes not

sharing any key, there are enough nodes which can play the

role of the intermediate node in multi-hop (2-hop) process.

This encourages one to search for a deterministic design with

exactly two merged nodes per big (new) block yielding full

communication among the blocks.

IV. WEAKNESS OF THE KPD IN [12]

Apart from other possible weaknesses, the Transversal De-

sign based KPD presented in [12] lacks full communication

among nodes as has been exhibited in section III. As for

example, Ni, Ni+1,Ni+2 and Ni+3,i = 0, 2, 3, . . . q 1can not communicate directly as they do not ever share any

common key in the original KPD of [12]. This results in

multi-hop communications among the nodes which increases

cost of communication and makes the system more prone to

adversarial attacks. Thus the efficiency and security of message

exchange of the network can be grossly affected.

V. DETERMINISTIC MERGING BLOCK STRATEGY

Lack of direct communication for any arbitrarily chosen

pair of nodes can be tackled by merging certain number of

nodes yielding a network with fewer blocks, each possessing

larger key rings. With this increased number of keys per block,

one may expect improved communication between any given

pair of blocks. Chakrabarti et al. [3] suggested the novel

idea of random merging z nodes of a TD(k,n) based KPD

proposed by Lee & Stinson [12] having similar weakness.

Though communication probability of the resultant model was

much improved, full communication was not assured. Other

than this, the authors of [3] have not thoroughly explained

many aspects of the design, like the heuristic in [3, section 4].

Moreover merging blocks concept is best thought as ensuring

some path between nodes which are not connected. This can

not ever be guaranteed by a random strategy.

These observations prompted the current author to think of

deterministic merging of nodes of the scheme in [12] for the

case k = q and k = q 1. Deterministic merging of nodesensures all the bigger blocks of merged design have full

communication amidst themselves. Thus a definite path can be

ensured between non communicating nodes of original KPD

in [12].

To explain the merging strategy, it is easier to understand

the exact key sharing picture for sample network of size

N = q2 for q = 4, 5 with k = q and q 1. Table I

and II corresponds to a network having q2 = 42 = 16nodes. From the tables, it is evident can be partitioned into

q = 4 classes each containing q = 4 nodes on the basisof their key sharing. The former Table I corresponds to the

case when k = q = 4 and the later Table IIis for thecase k = q 1 = 3. The separation of these classes areindicated by double column partitioning lines after each set of

4 nodes: N0, N1,N2, N3; N4, N5,N6, N7; N8, N9,N10, N11;& N12,N13, N14, N15. Every class has the property that the

coefficient ofy in their respective polynomials is same. Thus

they equate each others polynomials i+ ly with 0 i 3 forsome fixed l = 0, 1, 2 or 3 resulting in no common solution,hence no common key. This motivates one to visualize the key

sharing of the 16 nodes, N0 to N15, like a square-grid aspresented in Figure 1. Any pair of nodes, other than the ones

lying in the same row shares exactly 1 key for the case k = qas equating any 2 nodes polynomial in here implies solvingfor (j j)y = (i i) over F4 with 0 i = i, j = j 3which is feasible. As for the case k = q1, each nodes sharesa common key with all other expect for its own row and the

nodes its preceding column. For example, N0 does not share

common keys with N1, N2, N3 and N7,N11, N15 (N3 counted

once).

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Fig. 4. Deterministic Merging strategy: general case with N = q = pr

nodes of scheme in [12].

is a direct generalization of the case q = 22 = as explainedabove through Figure 1 For qodd, taking combination of

two row would have left out one row, so top three row are

combined separately. As such Figure 4 explains the merging

for this case.

A. Assured full communication: theoretical results

Equating the polynomials of the 4 nodes constituting anytwo merged blocks, one readily sees that:

Theorem 1. The proposed deterministic Merging Block Strat-

egy where two nodes of the Transversal Design based KPD

scheme in [12] are clubbed to form the merged blocks results

in full communication among the merged blocks.

Proof: Consider any two arbitrary blocks A and B. It is

evident from the construction that at least node from blockA

will never lie in the horizontal line as well as the vertical line

of either of the two nodes the other blockB (refer figures 1,3 2 and 4 for q = 4, 2r, 5 and for general case respectively).This implies that these two nodes will have a common key and

hence the blocks A and B can communicate through this key.

As the two blocks were arbitrarily chosen, one is assured of

full communication in the new network consisting of blocks

constructed by merging two nodes in the manner explained

above (again refer figures 1, 3 2 and 4 for q = 4, 2r, 5 and forgeneral case respectively).

Theorem 2. The resulting Merged Block Design has a min-

imum of one (1) to a maximum of four (4) common keysbetween any two given pair of (merged) blocks.

Proof: Any two nodes can share at most one key in

original Transversal Design based KPD in [12]. So there are

at most 4 keys common between two blocks. This situationoccurs only if both nodes of the 1st block shares two (2)distinct keys with each node of the 2nd block.

Remark 1. Some of the important features of the merging

block design are as follows:

Thus the resultant merged block design (as suggested

here) has full communication among the blocks through

at least one common keys between any two given pair of

(merged) blocks.

Full communication can not be assured when nodes are

merged randomly to form larger blocks. Probably this

is the main reason why authors of [3] could not justify

several issues in their random merging model.

The current authors feel that it is mandatory to have

inter nodal communication otherwise the entire concept

to achieving full communication by this technique is

lost. The essence of the merging concept is that any

communication is always received by either of the two

constituent nodes of a block. Now if required it can pass

it down to the other node comprising that its block and

make it connected. As such while proposing the merged

block design, this idea was given importance.

It should be highlighted that merging does not mean that

the nodes physically combine to become one. Just that

they are to be treated as one unit.

Therefore the total number links in the merged scheme

is same as that of the original Transversal Design based

KPD of Lee & Stinson in [12]. This fact will be recalledlater while discussing E(S) and V(s) / resiliency.

VI. KEY ESTABLISHMENT PROTOCOL

Nodes are deployed with following preloaded data which

may be utilized for key establishment according to the protocol

that follows:

their key rings containing the cryptographic keys.

key ids corresponding to each key in their.

their respective node identifiers (i, j).

Establishing keys for nodes automatically establishes com-

mon keys between the merged blocks. The algorithm for key

establishment is presented in algorithm 1.

Each node send their node ids immediately after1

deployment.;

On receiving each others ids (i, j), the nodes form2the polynomial ix +j corresponding to the node ids. ;These polynomials are equated for common key ids3

in much the same way as was to be done by nodes in

Lee and Stinsons model [12].;

Since nodes are equipped with the knowledge of the4

merging technique, establishing shared keys between

nodes automatically establishes shared keys between

any pair of blocks.;

Theorem 1 and and its corollary 2 of section V-A5ensures all merged blocks have shared keys. ;

The shared keys thus established can be used for6

message exchange.;

Algorithm 1: Algorithm for Key Establishment.

VII. NETWORK PARAMETERS

Theoretical analysis of some important aspects of the merg-

ing scheme will be present here. Main topics of interests

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Amrita Saha of IIT, Bombay, Mr. Sumit Kumar Pandey of ISI,

Kolkata and Sabyasachi Datta of CU for their active support

in preparation of this paper.

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