Sensornet

A Key Predistribution Scheme for Distributed Sensors using Nets

Deepak Kumar Dalai

and Pinaki Sarkar

School of Mathematical Sciences, National Institute of Science Education and Research, Bhubaneswar 752 050, India

Department of Computer Science and Automation, Indian Institute of Science, Bengaluru 560 012, India

Keywords:

Partial Spread, Net, Combinatorial Designs, Key Predistribution Scheme, Distributed Sensor Networks.

Abstract:

Key management is an essential functionality for developing secure cryptosystems; particularly for implemen-

tations to low cost devices of a distributed sensor networks (DSN)–a prototype of Internet of Things (IoT).

Low cost leads to constraints in various resources of constituent devices of a IoT (sensors of a DSN); thereby

restricting implementations of computationally heavy public key cryptosystems. This leads to adaptation of

the novel key predistribution trick in symmetric key platform to efﬁciently tackle the problem of key man-

agement for these resource starved networks. After a few initial proposals based on random graphs, most

key predistribution schemes (KPS) use deterministic (combinatorial) approaches to assure essential design

properties. Combinatorial designs like a (v,b,r,k)−conﬁguration which forms a µ(v,b,r,k)−CID are effective

schemes to design KPS (Lee and Stinson, 2005). A net in a vector space is a set of cosets of certain kind of

subspaces called partial spread. A µ(v,b,r, k)−CID can be formed from a net. In this paper, we propose a key

predistribution scheme for DSN, named as sensornet, using net. Effectiveness of sensornet in term of crucial

metrics in comparison to other prominent schemes has been theoretically established.

1 INTRODUCTION

Distributed (Wireless) Sensor Networks (DSN) are

regarded as revolutionary information gathering sys-

tems owing to their easy deployment and ﬂexible

topology. They are decentralized with numerous

low-cost identical resource starved wireless devices,

called sensors or nodes, that deal with sensory data.

They are considered to be a nice prototype of Internet

of Things (IoT) which is a sophisticated concept that

aims to connect our world. This has boosted the study

of such distributed networks in modern times.

Prominent scientiﬁc applications of IoT are smart

homes, smart cities, smart grids, smart water net-

works, agriculture, health-care, etc. Of particular in-

terest are applications of DSN to networks where se-

curity is a premium; For instance, security may be es-

sential for certain sensitive scientiﬁc and military net-

works that are meant for (i) military surveillance, (ii)

force protection arenas, (iii) self healing mineﬁelds,

and so on. Primary tasks of devices of an IoT in any

such application are to collect information from their

surrounding, process and forward them to other de-

vices. Depending on speciﬁc applications, they may

be further required to (i) track and/or classify an ob-

ject, (ii) determine parametric value(s) of a given lo-

cation, etc. These sensitive task for such critical ap-

plications create the necessity of secure message ex-

change among the low cost IoT devices.

1.1 Type of Cryptosystem for KPS

Constraints in resources of constituent ordinary de-

vices of any IoT (like sensors of DSN) make us opt for

symmetric key cryptosystems (SKC) over their public

key counterparts while designing security protocols

for such networks. SKC require both the sender and

receiver(s) to possess same encryption-decryption key

before message exchange. Standard online key ex-

change techniques that involve public parameters are

generally avoided due to their heavy computations.

One can think of two trivial key distribution tech-

niques. First is to assign a single key for entire net-

work devices. Second is to think of assigning pair-

wise distinct (symmetric) keys for every pair of de-

vices. Former method is completely vulnerable to

single point failure (compromise of even one sensor

reveals this single system key). Whereas, the second

strategy overloads the memory of each sensor; since

N − 1 keys are required to be stored per sensor for

a network of size N . This is particularly impractical

Dalai D. and Sarkar P.

Sensornet - A Key Predistribution Scheme for Distributed Sensors using Nets.

DOI: 10.5220/0006132700490058

In Proceedings of the 6th International Conference on Sensor Networks (SENSORNETS 2017), pages 49-58

ISBN: 421065/17

for large networks (i.e., large value of N ).

Treating a node (or a few) as Trusted Authority

(TA) is risky. This also makes the network prone to

single point failure as capture of this authority (sen-

sor) acting as a TA makes the system vulnerable.

Thereby schemes like Kerberos are avoided while de-

signing secure key management schemes for DSN.

These facts emphasizes the importance of employ-

ing an adequate key management scheme for such

networks. This stalemate situation was wittily over-

come in 2002 by Eschenauer and Gligor by introduc-

ing the concept of key predistribution that involves ap-

plications of SKC to sensor networks. Any key pre-

distribution scheme primarily execute the following:

• Key Distribution: Prior to deployment, keys are

preloaded into sensors to form their keyrings or

key chains from the collection of all network keys,

called key pool. Each system key is marked with

an unique identiﬁer (key id). Certain schemes (Ruj

and Roy, 2008) consider (node id) as an unique

function of all the key ids. These key or, node id

are used during key establishment.

• Key Establishment: The preloaded keys are es-

tablished by a two steps process, described below:

(i) Shared key discovery phase establishes

the shared common key(s) among the participant

nodes. This may be achieved by broadcasting the

key ids of all keys contained in the nodes (or node

id). On receiving each other’s key ids, the sensors

tally (or equate) them to trace their mutual shared

key id(s), hence common shared key(s).

(ii) Path key establishment phase establishes

an optimized path key between a pair of nodes that

do not share a common key. This process involves

intermediate nodes. Refer to common intersection

designs in Section 3.

Depending on whether the above processes are

probabilistic or deterministic, such schemes are clas-

siﬁed into two types: (a) random and (b) determinis-

tic. Sections 2.1 and 2.2 present a brief overview of

individual type of schemes.

1.2 Paper Organization

Observing the signiﬁcant advantages of deterministic

KPS during key management for low cost distributed

networks, we set out to propose one such scheme. Our

proposal uses net partial spreads (or, nets) in a ﬁnite

vector space that have been well studied combinato-

rially and as such, we name the scheme as sensornet.

After a brief literature survey on KPS in Section 2, we

present preliminaries of combinatorial set systems in

KPS, partial spreads and nets in Section 3. Section 4

presents the design of our scheme sensornet that ad-

here to the desirable criteria set out in Section 2.4.

We analyze sensornet in terms of various performance

metrics set out in Section 5 and thereby establish our

scheme’s efﬁciency in comparison to prominent pro-

posals. We brieﬂy summarize our work in Section 6

while stating related future research directions.

2 A BRIEF SURVEY OF KPS

This section presents a state-of-the art survey of

prominent KPS. We split survey into three stage:

(i) ﬁrst generation random KPS, (ii) deterministic

KPS, and (iii) advantages of later type over former.

Thereby, we justify proposal of our new deterministic

KPS adhering to design criteria set out in Section 2.4.

2.1 Random Key Predistribution

Schemes (RKPS)

First generation KPS rely on random graph theory

pioneered by Erd

os and R

enyi (Erd

os and R

enyi,

1960) to preload (symmetric) cryptographic keys into

sensors. Therefore, keyrings are formed randomly.

This leads to probabilistic key sharing and establish-

ment. Later is achieved by either broadcast of key ids

or challenge and response Refer to (Eschenauer and

Gligor, 2002, Section 2.1). Earlier, Blom proposed

the ﬁrst key distribution scheme (Blom, 1985) in pub-

lic key settings meant for resourceful ad hoc net-

works. Blom’s schemes uses pairs of public-private

matrices for key distribution. It cannot be applied to

resource constraint sensor networks due to its heavy

memory requirement to store huge vectors. Several

researchers use variants of Blom’s schemes to pro-

pose both random and deterministic KPS for DSN.

C¸ amptepe and Yener (C¸ amtepe and Yener, 2005) pro-

vides an excellent survey of the random KPS.

2.2 Deterministic Key Predistribution

Schemes (DKPS)

First deterministic KPS were proposed simultaneous

by C¸ amtepe and Yener (C¸ amtepe and Yener, 2004),

Lee and Stinson (Lee and Stinson, 2005) and Wei

and Wu (Wei and Wu, 2004) in 2004. Wei and Wu

(Wei and Wu, 2004) combines subset-based schemes

with existing key distribution schemes such as (Blom,

1985) to obtain multiple key spaces. C¸ amtepe and

Yener (C¸ amtepe and Yener, 2004) exploits combi-

natorial designs like symmetric Balanced Incomplete

Block Designs (BIBD), generalized quadrangles and

SENSORNETS 2017 - 6th International Conference on Sensor Networks

projective planes (see (Lee and Stinson, 2005; Lee

and Stinson, 2008; Paterson and Stinson, 2014)). The

scheme of Lee and Stinson (Lee and Stinson, 2005)

uses quadratic equation solving and can be viewed

as a scalable extension of their later proposal us-

ing Transversal Design (T D(k, p)). This work fur-

ther summarizes the necessary conditions for a com-

binatorial design to yield a deterministic KPS. Cer-

tain KPS exploit special structures like Reed Solomon

code based KPS (Ruj and Roy, 2008) that permit alter-

nate combinatorial description (Bag et al., 2012; Pa-

terson and Stinson, 2014). In the same light, we show

our scheme derived from the net of partial spreads (or

nets) can yield nice combinatorial properties meant

for designing deterministic KPS.

2.3 Advantages of DKPS over RKPS

Deterministic schemes have certain advantages over

their random counterparts. For instance, a desired

property of a randomized scheme may occur only

with a certain probability whereas they can be proven

to hold in a deterministic scheme (refer to (Lee and

Stinson, 2005; Lee and Stinson, 2008; Paterson and

Stinson, 2014)). This led to proposals of numer-

ous deterministic KPS (Lee and Stinson, 2005; Lee

and Stinson, 2008; Bag et al., 2012) etc. using vari-

ous combinatorial tricks. Further the predictable na-

ture of these combinatorial structures has been efﬁ-

ciently exploited to address design weaknesses of cer-

tain prominent KPS. For instance (Dhar and Sarkar,

2011; Bag et al., 2012) primarily address the connec-

tivity aspect of (Ruj and Roy, 2008) by deterministic

and random approaches respectively.

Contrary to these observations, Ruj and Pal (Ruj

and Pal, 2016) state that random graph models are

well suited for ‘scalability’ and ‘resilience’. Thereby,

they justify their proposals of random graph based

preferential attachment models with degree bounds.

They design various network using their model. All

of their designs suffers from highly skewed load dis-

tribution, poor connectivity and resiliency; and hence,

are inappropriate for (distributed) IoT applications.

In fact, sensitive IoT applications require proto-

cols to yield equal distribution of tasks among peers.

Moreover, to reduce hops and hence potential risks

from node capture, it is more important to have con-

nected networks that can not be guaranteed by ran-

dom schemes. So we opt for deterministic protocols

for security applications in IoT networks that assure

predictable (high) connectivity; despite most of them

having restricted scaling operations. This is a major

area of study for most (deterministic) KPS proposals,

including ours (recalled in Section 6).

Observe that the structure of the combinatorial ob-

jects used to design deterministic KPS can not di-

rectly model networks of any speciﬁed size N . Usu-

ally, such structures result in designs having a speciﬁc

pattern in the number of resultant blocks; viz. a prime

power etc. Since N can be any number, a standard

strategy is to consider the least prime power that is

greater than the network size (i.e., p

≥ N ). Then N

subset are randomly selected to form the key rings of

the resultant network nodes. Bose et al. (Bose et al.,

2013) speculate that random removal of blocks may

have a disadvantageous affect on the underlying de-

sign properties and hence become an issue of concern.

Fortunately, this claim of Bose et al. has been suc-

cessfully challenged by Henry et al. (Henry et al.,

2014). Through practical experiments, they establish

that random removal of key rings of a combinatorial

KPS has negligible effect with overwhelming proba-

bility. This work reestablishes the importance of com-

binatorial schemes.

2.4 Desirable Design Criteria

Devices of an IoT (for instance, sensors of a DSN) are

highly prone to damage and/or physical capture. This

is a crucial consideration while designing any KPS.

Primary objectives of any KPS is to ensure that the

resulting network:

1. has less number of keys per node, i.e., sizes of

individual keyrings are less;

2. have large node support, i.e., support large num-

ber of network nodes;

3. has good (ideally full secure) connectivity. Secure

connectivity (or, simply connectivity) is the ratio

of number of (secure) links in eventual network to

all possible links. A pair of nodes are said to be

connected by a (secure) link if there exists at least

one secret key between them;

4. is resilient against adversarial attacks. A pre-

vailing method adopted in most existing works

(C¸ amtepe and Yener, 2004; Lee and Stinson,

2005; Bag et al., 2012; Paterson and Stinson,

2014) is to show that the standard resiliency co-

efﬁcient fail(t) is minimized. Our work will fol-

low suit. The quantiﬁer fail(t) measures the ra-

tio of links broken after compromise of t sensors

to the total number of links in the remaining net-

work. Notationally, fail(t) =

, where b

is the

number of links broken when t nodes are compro-

mised and u

is the total number of links among

uncompromised nodes of remaining network.

Sensornet - A Key Predistribution Scheme for Distributed Sensors using Nets

Ideally a KPS should have small keyrings, and

yet support large number of nodes with appreciable

resiliency, scalability and communication probabil-

ity (or connectivity). However, renowned scientists

proved the impossibility of constructing a perfect KPS

that meets all these criteria (Lee and Stinson, 2005;

Paterson and Stinson, 2014). This motivates several

designs that are robust for speciﬁc purpose.

3 PRELIMINARY

This section introduces the deﬁnitions and notations

that are required to describe our scheme; sensornet.

3.1 Combinatorial Set Systems and KPS

Use of different combinatorial designs to obtain de-

terministic KPS was primarily presented in the pa-

per (Lee and Stinson, 2005). After that there have

been several KPS proposals based on combinatorial

designs. A survey on KPS in WSN is available

in (Chen and Chao, 2011). Recently in a more tech-

nical survey, Paterson and Stinson (Paterson and Stin-

son, 2014) present an uniﬁed treatment of prominent

combinatorial designs in terms of partially balanced

t−design. Basic design theoretic concepts are below:

Let X be a ﬁnite set. The elements of X are called

varieties. Each subset of X is termed as a block. Con-

sider A to be a collection of blocks of X . Then the

pair (X ,A) is said to be a set system or, a design.

(X ,A) is regular (of degree r ) if each point is con-

tained in r blocks. (X ,A) is uniform (of rank k) if all

blocks have the same size, say k.

A design (X ,A) is said to form a (v, b,r,k)−design if

• |X | = v and |A| = b;

• it is regular of degree r and uniform of rank k.

A (v, b,r,k)−design forms a (v,b, r, k)−conﬁguration

if any arbitrary pair of blocks intersect in at most

one point. Moreover, if any pairs of varieties occur

in exactly λ block, then a (v,b,r, k)−design forms a

(v, b,r, k, λ)−BIBD (Balanced Incomplete Block De-

signs). These designs can be used to construct various

KPS(see (Lee and Stinson, 2005)) by mapping:

1. the v varieties of X to the set of keys in the scheme

(:=key pool),

2. b to the number of nodes in the system (:=network

size),

3. k to the number of keys per node (:=size of key

rings), and

4. r to the number of nodes sharing a key (:=degree

of the resultant KPS).

The target is to construct KPS with identical burden

on each sensor. This leads to opting for design with

uniform rank (k) and regular degree (r); so that every

key ring is of equal size (k) and same number of nodes

(r) share each key for the resultant network.

The block graph Γ

of the set design (X ,A) is de-

ﬁned with the vertex set A and edge set E

= {(A,B) :

A,B ∈ A and A∩B 6=

0}. If the set design is regular of

degree r and uniform of rank k, then the block graph

is k(r −1)−regular.

A (v,b,r,k)−conﬁguration (X ,A) is said to form

a µ−common intersection design (µ−CID) in case:

|{A

∈ A : A

∩ A

0 and A

∩ A

0}| ≥ µ

whenever A

∩ A

0, ∀ i 6= j. It is important to con-

struct design that maximize the value of µ.

3.2 Partial Spread and Net

Let IF

be the ﬁnite ﬁeld on p elements where p is a

prime. Denote V

= IF

to be the vector space over the

ﬁeld IF

with zero vector 0. Since the ﬁnite ﬁeld IF

is a vector space over IF

which is isomorphic to IF

(see (Lidl and Niederreiter, 1997)), we interchange

the notation as per its suitability. The isomorphism

mapping can be considered as any mapping from a

basis set of IF

(e.g., {1,α,· ·· ,α

n−1

} where α is a

primitive root in IF

) to a basis set of IF

. We consider

n = 2m to be an even integer throughout the paper.

A partial spread Σ of order s in V

is a set

of pairwise supplementary m−dimensional subspaces

,·· · ,E

of V

i.e., E

∩ E

= {0} for all 1 ≤ i <

j ≤ s. A partial spread Σ is a spread if ∪

i=1

= V

It is well known that a spread of V

exists since m

divides n (Lu, 2008); in which case |Σ| = p

+ 1.

Therefore, from a given spread Σ each of the





choices of s members of Σ provides a partial spread

of V

. Note that a partial spread might not be a sub-

set of a spread (Eisfeld and Storme, 2000). A de-

tailed combinatorial study of spread can be found in

the book (Johnson, 2010; Johnson et al., 2007).

Let E be a subspace of the vector space V

. A

coset of E in V

is of the form α+E = {α+v : v ∈V

}

for an α ∈ V

. The set of cosets make a disjoint par-

tition of V

. The element α is called a coset rep-

resentative of the coset α + E. Since E is an ad-

ditive group, any element from the α + E can be a

coset representative the coset. Given a partial spread

Σ = {E

,·· · ,E

} in V

, let E

be a set of coset rep-

resentatives of subspace E

for 1 ≤ i ≤ s. Then the

set A = {α + E

: α ∈ E

and 1 ≤ i ≤ s} i.e., set of all

cosets of subspaces E

,1 ≤ i ≤ s is called a net in V

See the book (Johnson et al., 2007) for the combina-

torial study of net.

SENSORNETS 2017 - 6th International Conference on Sensor Networks

4 SENSORNET

Sensornet is a proposal for a KPS for distributed

(wireless) sensors. The design of sensornet results

from the forthcoming set design and Theorem 1.

Given a partial spread Σ = {E

,·· · ,E

} in V

let E

be a supplementary subspace of E

in V

for

1 ≤ i ≤ s (i.e., their direct sum E

⊕ E

= V

and

∩ E

= {0}). It can be checked that E

is a set of

coset representatives of E

for 1 ≤ i ≤ s. Note that the

subspaces E

’s in a partial spread are pairwise supple-

mentary. So, any E

, j 6= i can be chosen as E

. Con-

sider the set system (X , A) such that X = V

and and

the set of blocks A = {α + E

: α ∈ E

and 1 ≤ i ≤ s}

which is a net in V

Theorem 1. Given any partial spread Σ, the set de-

sign (X , A) is a µ(p

,sp

,s, p

)−CID where µ =

(s − 1)p

Proof. Here v = |X | = p

. Consider two blocks α +

and β + E

. Now we have the following cases.

1. If i = j, then

(a) α + E

= β + E

if α = β or,

(b) (α + E

) ∩ (β + E

) =

0 if α 6= β.

2. If i 6= j, then we shall show that |(α + E

) ∩ (β +

)| = 1. Since E

and E

are supplementary to

each other, the element α−β ∈V

can be uniquely

expressed as -u+ v where u ∈ E

and v ∈ E

. That

is, α−β = −u + v which implies, α +u = β + v is

the unique element in (α + E

) ∩ (β + E

Therefore, the number of blocks i.e., the number of

cosets is b = sp

and each block contains k = p

el-

ements. Given a subspace E

,i ∈ {1,2,· ·· , s}, each

element u ∈ V

belongs to exactly one coset of E

So, each u ∈ V

belongs to exactly s many blocks

in A. The set design (X ,A) is regular with r = s.

Here, every two distinct blocks intersect each other

by at most one element which implies that (X ,A) is a

,sp

,s, p

)−conﬁguration.

We see that two blocks α+E

and β+E

do not in-

tersect iff i = j and α 6= β i.e., both are distinct cosets

of same subspace E

. For the case of non intersecting

blocks α + E

and β + E

, α 6= β, both blocks inter-

sect all other blocks of the form γ + E

where j 6= i.

Since there are µ = (s−1)p

such blocks γ +E

in A,

(X ,A) is a (s − 1)p

,sp

,s, p

)−CID.

Here, the set of blocks (i.e., A) of the scheme

(X ,A) forms a net in a vector space. We denote the

scheme as sensornet.

It can be checked that the block graph of (X ,A)

is a strongly regular graph with parameters (n =

,r = (s − 1)p

,λ = (s − 2)p

,µ = (s − 1)p

Moreover, the block graph is a complete s−partite

graph. In the study of ﬁnite geometry, the varieties

together with the blocks (i.e., cosets) form the points

and lines of an afﬁne plane. Since two non-parallel

lines (i.e., α + E

and β + E

for i 6= j) intersects at

one point, this set of cosets is called net.

4.1 Example of NETS

There are numerous constructions of spreads and par-

tial spreads that can be found in literature, see (John-

son et al., 2007). Now we present a few spreads S

in IF

, where p is a prime. For a given s, any Σ ⊆ S

such that |Σ| = s forms a partial spread of order s. By

Theorem 1, this partial spread yields a KPS.

Spread I: This is a classical example of a spread

from the additive group of the ﬁnite ﬁeld F

. Since

n = 2m, IF

is a subspace of IF

with respect to a

basis. Let {α

: 1 ≤ i ≤ p

+ 1} be a set of coset rep-

resentative of the cosets of the subgroup IF

∗

of the

multiplicative group IF

∗

. Then the set S

= {S

,1 ≤ i ≤ p

+ 1} is a spread in IF

Spread II: This example of spread is represented in

bivariate form (Bu, 1980). For each α ∈ IF

, deﬁne a

subspace U

of IF

×IF

by U

= {(αu,u)|u ∈ IF

}

and U

∞

= {(u,0)|u ∈ IF

}. The set S

= {U

: α ∈

} ∪U

∞

constitute a spread in IF

× IF

' IF

Spread III: This example of spread is generated from

pre-quasiﬁeld, which is deﬁned as following. A sys-

tem Q = (V, +,◦), with |V | ﬁnite, is a pre-quasiﬁeld

if the following axioms hold:

(1) (V, +) is an abelian group, with identity 0.

(2) (V

∗

,◦) is a quasigroup where V

∗

= V \ {0}. That

is, for any a ∈ V

∗

, the left multiplication operator

a ◦ x and the right multiplication operator x ◦ a are

both bijective from V

∗

to V

∗

(3) ∀x,y,z ∈ V : (x + y) ◦ z = x ◦ z + y ◦ z.

(4) x ◦ 0 = 0, ∀x ∈ V .

Now assuming (IF

,+,◦) is a pre-quasiﬁeld, set

= {(x,a ◦ x) : x ∈ IF

} for any a ∈ IF

and E

∞

{(0,x) : x ∈ IF

}. Then it can be checked that S

III

: a ∈ IF

∪{∞}} is a spread in IF

×IF

(John-

son et al., 2007). Many pre-quasiﬁelds are available in

literature. Refer to (Wu, 2013) for three types of pre-

quasiﬁelds on set IF

and (C¸ es¸melio

glu et al., 2015)

for a pre-quasiﬁeld on set IF

Example 1. Here, we present a simple KPS from the

spread of type S

. Take V

= IF

[x]/(x

+ 1).

Consider the subspace IF

= {0,1,2} and {1, x,x +

1,x + 2} a set of coset representatives of IF

∗

. Then S

= {{0,1,2},{0, x,2x},{0,x + 1,2x +

2},{0,x + 2, 2x + 1}} is a spread in IF

. Consider

Sensornet - A Key Predistribution Scheme for Distributed Sensors using Nets

a partial spread Σ = {E

= {0,1,2}, E

= {0,x, 2x}}

with E

= E

and E

= E

. Therefore, by Theorem 1,

the set X = IF

and the net A = {{0,1, 2},{x, x +

1,x +2},{2x,2x +1,2x +2},{0, x,2x},{1,x +1,2x +

1},{2,x + 2,2x + 2}} forms a KPS (X ,A). The block

graph of (X , A) is the K

3,3

5 ANALYSIS OF SENSORNET

In this section we present the values of some impor-

tant metrics involved in our proposed scheme; sensor-

net.

5.1 Time and Space Complexities for

Key Establishment (T

)

For the key establishment between two nodes, the

nodes need to discover a common key stored between

them. For this purpose the nodes need to broadcast

some data, which is required to trace the common

key between two nodes. Since the sensor nodes have

low memory and computation power, data and time

requirement for key establishment are two very im-

portant factors to design a KPS. In this subsection

we discuss the process of key establishment between

two nodes and associated time and data requirement

of the process. In case of path key establishment (re-

fer to Section 1.1, both the concerned nodes have to

ﬁnd a common neighbor node with whom they dis-

cover their share key and establish connection via this

neighbor. Let denote T

and M

be the time and mem-

ory complexity function for the key establishment in

vector space V

The blocks of sensornet forms a net, i.e., they are

afﬁne spaces. Hence, the nodes can be identiﬁed by

their basis vectors and the key establishment is done

using the node id. Let β

,β

,·· · ,β

be a basis set of

the subspace E

for 1 ≤ i ≤ s. Then, the node α + E

can be identiﬁed by the node id (α,β

,β

,·· · ,β

When the nodes α + E

and β + E

need key estab-

lishment between them the following process can be

followed.

Step 1: The nodes α + E

and β + E

compare the

last m vectors (i.e., the basis vectors) in their node

id. If they are same then follow Step 3 otherwise

follow Step 2.

Step 2: In this case, we have E

6= E

i.e., they

share a common key. Let the common key is

α + u = β + v where u ∈ E

and v ∈ E

. Now we

need to ﬁnd u and v in terms of the basis vectors of

and E

respectively. Here, α − β = v − u ∈ V

Since E

and E

are supplementary subspaces in

, α − β can be uniquely expressed as a linear

combination of the basis vectors of E

and E

. Let

α−β = b

+·· ·+ b

m+1

+·· ·+ b

where b

∈ IF

. That is, α − b

− · ·· − b

β + b

m+1

+ · ·· + b

. Hence, the common

key is α − b

− ··· − b

∈ E

which is β +

m+1

+ · ·· + b

∈ E

. The time complexity

in this step is the time complexity to express α −β

in terms of the basis vectors in a basis i.e., O(n

Step 3: In this case, the E

= E

i.e., they do not

share any common key. In this case, they have e

to establish connection through another node with

whom they share a key. That is, they have to ﬁnd a

node γ + E

where k 6= i. The probability of ﬁnd-

ing such a node using a random pick up is

s−1

which is very high. Since both α + E

and β + E

share a key with γ+E

, each one do the same pro-

cess described in Step 2 with γ + E

to discover

their common key. After that α + E

and β + E

can establish connection through γ + E

. Hence,

the time complexity in this case is too O(n

Therefore, each node has to spend M

= (m + 1) ∗ n ∗

log

p = O(n

) bits of data for broadcasting of their

identiﬁcation and the time complexity to discover the

common key(s) is T

= O(n

). Note that in sensor-

net scheme, nodes have to broadcast only node id i.e.,

O(n

) bits instead of all (i.e., O(rp

) many) key ids

as broadcast by many other proposed schemes.

5.2 Key-node Ratio (σ)

The key-node ratio is deﬁned as σ =

. This ratio

provides idea about the storage requirement of the

scheme at each node with respect to the total num-

ber of nodes. With this metric we can compare the

storage requirement of the schemes from different de-

signs. It is desirable for this ratio σ to be as small as

possible as lesser amount of memory required for key

storage at each node. In sensornet (X ,A), key-node

ratio σ =

|Σ|

. If the size of partial spread

is larger, then the storage requirement to store keys in

sensornet is lesser.

5.3 Resiliency(fail(t))

The schemes need to be well equipped to perform

against adversarial attacks. To this end, the standard

resiliency metric fail(t) need to be minimized. This

is prevalent method adopted by most existing works

(Bag et al., 2012; C¸ amtepe and Yener, 2004; Lee and

Stinson, 2004; Lee and Stinson, 2005; Paterson and

Stinson, 2014). The quantiﬁer fail(t) measures the

SENSORNETS 2017 - 6th International Conference on Sensor Networks

probability that a random link between two sensor

nodes is broken due to the compromise of t other ran-

dom nodes. Notationally, fail(t) =

, where b

the number of links broken when t nodes are compro-

mised and u

is the total number of links among un-

compromised nodes of remaining network. Theorem

2 is due to Lee and Stinson (Lee and Stinson, 2005,

Section VIII) provides the formula to compute fail(t)

for any (v,b,r,k,1)−conﬁguration.

Theorem 2. For any (v,b, r,k,1)−conﬁguration, the

value of the metric fail(t) on random compromise of

t nodes is given by:

fail(t) = 1 −



b − r

b − 2



. (1)

Corollaries 1 is an immediate outcomes of substi-

tuting in Equation 1, the values of b and r, that sen-

sornet achieves.

Corollary 1. The value of the resilience fail(t) for

the set design (X ,A) of the scheme sensornet, which

is a (p

,sp

,s, p

)−conﬁguration is

fail(t) = 1 −



− s

− 2



In particular,

fail(1) =

s − 2

− 2

−

− 2

(sp

− 2)

≈ p

−m

The metric fail(1) = O(p

−m

) i.e., if a node N is

compromised, then the probability that a link (which

is not incident with N) fails is O(p

−m

). Here, the size

of the partial spread (i.e., s) has no signiﬁcant effect

on fail(1). As an example, if n = 10 and p = 2 (i.e.,

there are 2

≈ 1000 many nodes) then the value of

fail(1) ≈ 0.03.

5.4 Connectivity(p

)

We say two blocks in a set system are connected by

e−links (or, are at a distance e) if the shortest path

between them in the block graph includes e edges.

Hence, we deﬁne the metric connectivity (or, connec-

tion probability) p

of the network to be the probabil-

ity that two nodes (placed in physical neighborhood)

are connected by e−links for a positive integer e.

Observe that the value of e for a µ−CID with µ > 1

is either 1 (if they share a key) or 2 (if they do not

share a key). The formulae for p

and p

are pro-

vided in (Lee and Stinson, 2005, Section VI), which

are being formally restated in the following theorem.

Let η denote the number of nodes in the intersection

of the physical neighborhood of two given nodes.

Theorem 3. The value of the metric connectivities of

a µ(v, b,r, k)−CID are given by p

k(r−1)

b−1

and p

≈

(1 − p

) ×



1 −



b−µ−2

b−2



The following corollary is an immediate outcome

for our scheme by substituting the values of b, r,k and

λ in Theorem 3.

Corollary 2. The value of the metric connectivities

for the set system (X ,A), which is a (s − 1)p

−

,sp

,s, p

)−CID are

≈ 1 −

and p

≈

− 1

η+1

Proof. Now putting the value of b = sp

,r = s,k =

and µ = (s − 1)p

in p

and p

, we have

(s − 1)

− 1

− p

− 1

= 1 −

− 1

≈ 1 −

and p

≈



1 −



− (s − 1)p

− 2





1 −



− 2



≈



1 −



− 1

η+1

The metric p

≈ 1 −

i.e., the connectivity in-

creases if the size of spread increases. Here, the size

of base ﬁeld (i.e., the value of p) has no signiﬁcant

effect on p

. As an example, if n = 10, p = 2 (i.e.,

there are 2

≈ 1000 many nodes) and s = 2

then the

value of p

= 1 − 2

−5

5.5 Comparative Study

This subsection presents a comparative study of sen-

sornet with prominent existing works with respect to

connectivity, resilience and network scaling. Perfor-

mance of sensornet with respect to other metric like

storage, etc. has been discussed in previous section.

5.5.1 Connectivity and Resiliency Tradeoff

The schemes with high connectivity (i.e., p

) and re-

siliency (i.e., fail(1)) as small as possible are pre-

ferred. As unfortunately, both the metrics behave in

opposite way, it is a fundamental problem of trading

off connectivity against resiliency. In (Dong et al.,

2011; Paterson and Stinson, 2014), the ratio ρ =

fail(1)

is considered for the comparison of several

combinatorial designs. Therefore, the larger value

of ρ conﬁrms of higher connectivity and lower re-

siliency. It is desirable that the ratio ρ be as large as

possible.

Sensornet - A Key Predistribution Scheme for Distributed Sensors using Nets

Table 1: Comparison of asymptotic behavior of different schemes.

Scheme No. of nodes p

fail(1) ρ =

fail(1)

(X ,A) (sensornet) N = sp

1 −

−m

= N

−1

(1 −

T D(2,k,q), k = cq N = q

= N

−1

1/2

(Paterson and Stinson, 2014)

T D(3,k,q), k = cq,c < 1 N = q

c(2−c)

2(1−c)

(2−c)

−1

c(2−c)

4(1−c)

1/3

(Paterson and Stinson, 2014) = c −

T D(3,k,q),k = q N = q

1/2 5N

−2

(Paterson and Stinson, 2014)

T D(4,k,q),k = cq N = q

c(c

−3c+6)

3(c

−2c+2)

−3c+6

−1

c(c

−3c+6)

18(c

−2c+2)

(Paterson and Stinson, 2014)

Symmetric BIBD N = q

+ q + 1 1 N

−1

(C¸ amtepe and Yener, 2004)

RS code based (Ruj and Roy, 2008) N = q

q−1

q+1

−1

MB designs for T D(2kq) or N = q

/2 1 (2N )

−1

(2N )

RS code (Bag et al., 2012)

There have been several proposals for determin-

istic key predistribution schemes for wireless sen-

sor networks based on various types of combinatorial

structures such as designs and codes. The paper (Pa-

terson and Stinson, 2014) proposes a general frame-

work by unifying those structures into a new design,

termed as “partially balanced t-designs(PBtD)”. Al-

though, our scheme sensornet falls into 2 − (v, k, λ

b,λ

= r)−PBtD as a conﬁguration, the generaliza-

tion does not consider µ−CIDs. Hence, being a

µ−CID, sensornet does not classify as PBtD by their

description (Paterson and Stinson, 2014). There are

few comparison tables of different schemes are pro-

vided in (Paterson and Stinson, 2014). In the fol-

lowing, we take data of TD(t,k, Q) with intersection

threshold η = 1 from the paper (Paterson and Stin-

son, 2014) along with other designs to compare with

the scheme sensornet.

Let consider the number of nodes in all the com-

pared scheme is N . Now we shall compare the

asymptotic behavior of metrics p

,fail(1) and the ra-

tio ρ. The comparison is displayed in Table 1.

From this comparison table it is clear that the

asymptotic behavior of the ratio ρ of sensornet (X , A)

is similar or better than all other schemes except the

scheme T D(3, k,q),k = q and Merging Block (MB)

design of (Bag et al., 2012). Former scheme needs

computation of some number theoretic problems dur-

ing key agreement; while the later has signiﬁcantly

less (merging) block support (halved). Moreover, in

our scheme (X ,A), the shared key discovery is done

with time complexity O((log

N )

) and the amount

of data need to be broadcast is O((log

N )

). This is

an added advantage over most KPS that require key

id comparisons during key discovery.

5.5.2 Scalability Comparison

Sensornet (X , A) can support large networks. This is

because the choice n and respectively m and/or s are

unbounded in theory. This may help in scaling net-

works designed by our schemes (preﬁx large values).

Scalability, otherwise is a major challenge in

most deterministic KPS. For instance the schemes

(C¸ amtepe and Yener, 2004; Lee and Stinson, 2004;

Lee and Stinson, 2005; Lee and Stinson, 2008; Ruj

and Roy, 2008) have restricted scaling. This owes

to the fact that key establishment for these network

require general solutions of polynomials. Therefore

the complexity of the key establishment process in-

creases with increment in degree of these polynomi-

als. Random schemes can scalable arbitrarily (Ruj

and Pal, 2016); at the expense of desirable parameters

like connectivity, resilience, storage (key-node ratio),

etc. Therefore we opt deterministic schemes while de-

signing KPS (Paterson and Stinson, 2014). Also refer

to Section 2.3.

6 CONCLUSION

Realizing the need of deterministic KPS with desir-

able properties (set out in Section 2.4) to address the

problem of key management in low cost networks,

we propose one such scheme. Since the scheme is

constructed using nets in a vector space, we named

as sensornet. Key establishment of sensornet is a

SENSORNETS 2017 - 6th International Conference on Sensor Networks

great advantage over many other schemes. Although

sensornet suffers from lack of full connectivity, it is

very close to full connectivity for large size of par-

tial spread. Moreover, the generic computations in

Section 5.4 establish that connectivity of sensornet is

good (either direct or 1−hop path connectivity), it is

preferable to have full connectivity or at least a de-

terministic path in case of 1−hop connectivity. The

sophisticated MB designs of (Bag et al., 2012; Dhar

and Sarkar, 2011) establishes a deterministic 1−hop

connectivity for the Reed Solomon code based KPS

(Ruj and Roy, 2008). These heavily design dependent

works can certainly open the doors for future research

by considering similar constructions over sensornet in

place of other combinatorial design based schemes.

REFERENCES

Bag, S., Dhar, A., and Sarkar, P. (2012). 100% connectivity

for location aware code based kpd in clustered wsn:

Merging blocks. In Information Security Conference,

ISC 2012, number 7483 in Lecture Notes in Computer

Science, pages 136–150. Springer-Verlag.

Blom, R. (1985). An optimal class of symmetric key gener-

ation systems. In Advances in Cryptology - Eurocrypt

1984, number 209 in Lecture Notes in Computer Sci-

ence, pages 335–338. Springer-Verlag.

Bose, M., Dey, A., and Mukerjee, R. (2013). Key pre-

distribution schemes for distributed sensor networks

via block designs. Design, Codes and Cryptography,

67(1):111–136.

Bu, T. (1980). Partitions of a vector space. Discrete Math-

ematics, 31:79–83.

C¸ amtepe, S. A. and Yener, B. (2004). Combinatorial design

of key distribution mechanisms for wireless sensor

networks. In 9th European Symposium on Research

Computer Security, ESORICS 2004, number 3193 in

Lecture Notes in Computer Science, pages 293–308.

Springer-Verlag.

C¸ amtepe, S. A. and Yener, B. (2005). Key distribution

mechanisms for wireless sensor networks: a sur-

vey. Technical report, Rensselaer Polytechnic In-

stitute. Available at www.cs.rpi.edu/research/pdf/05-

07.pdf.

C¸ es¸melio

glu, A., Meidl, W., and Pott, A. (2015). Bent func-

tions, spreads, and o-polynomials. SIAM Journal of

Discrete Mathematics, 29(2):854–867.

Chen, C. Y. and Chao, H. C. (2011). A survey of key pre-

distribution in wireless sensor networks. Security and

Communication Networks, 7(12).

Dhar, A. and Sarkar, P. (2011). Full communication in a

wireless sensor network by merging blocks of a key

predistribution using reed solomon code. In Proceed-

ings of CCSEA 2011, CS & IT 02, pages 389–400.

Dong, J. W., Pei, D. Y., and Wang, X. L. (2011). A class

of key predistribution schemes based on orthogonal

arrays. Journal of Computer Science and Technology,

23:825–831.

Eisfeld, J. and Storme, L. (2000). (partial) t-spreads and

minimal t-covers in ﬁnite projective spaces. In Lec-

ture notes for the Socrates Intensive Course on Finite

Geometry and its Applications, University of Ghent.

Erd

os, P. and R

enyi, A. (1960). On the evolution of random

graphs. Publication of the Mathematical Institute of

the Hungarian Academy of Sciences.

Eschenauer, L. and Gligor, V. (2002). A key-management

scheme for distributed sensor networks. In Proceed-

ings of 9th ACM conference on computer and commu-

nications security, pages 41–47. ACM press.

Henry, K. J., Paterson, M. B., and Stinson, D. R. (2014).

Practical approaches to varying network size in com-

binatorial key predistribution schemes. In Selected

Areas in Cryptography (SAC) 2013, number 8282 in

Lecture Notes in Computer Science, pages 89–117.

Springer-Verlag.

Johnson, N. L. (2010). Combinatorics of Spreads and Par-

allelisms. CRC Press.

Johnson, N. L., Jha, V., and Biliotti, M. (2007). Handbook

of Finite Translation Planes, volume 289 of Pure and

Applied Mathematics. Chapman & Hall/CRC.

Lee, J. and Stinson, D. R. (2004). Deterministic key pre-

distribution schemes for distributed sensor networks.

In Selected Areas in Cryptography (SAC) 2004, num-

ber 3357 in Lecture Notes in Computer Science, pages

294–307. Springer-Verlag.

Lee, J. and Stinson, D. R. (2005). A combinatorial approach

to key predistribution for distributed sensor networks.

In IEEE Wireless Communications and Networking

Conference, WCNC-2005, pages 1200–1205.

Lee, J. and Stinson, D. R. (2008). On the construc-

tion of practical key predistribution schemes for dis-

tributed sensor networks using combinatorial designs.

ACM Transactions on Information and System Secu-

rity, 11(2):854–867.

Lidl, R. and Niederreiter, H. (1997). Finite ﬁelds. Ency-

clopaedia of mathematics and its applications. Cam-

bridge University Press.

Lu, H. Y. (2008). Partial spreads and hyperbent functions

in odd characteristic. Master’s thesis, Simon Fraser

University.

Paterson, M. B. and Stinson, D. R. (2014). A uniﬁed ap-

proach to combinatorial key predistribution schemes

for sensor networks. Design, Codes and Cryptogra-

phy, 71(3):433–457.

Ruj, S. and Pal, A. (2016). Preferential attachment model

with degree bound and its application to key predistri-

bution in WSN. In IEEE Conference on Advanced In-

formation Networking and Applications, AINA 2016,

pages 677–683.

Ruj, S. and Roy, B. K. (2008). Key predistribution schemes

using codes in wireless sensor networks. In Informa-

tion Security and Cryptology, Inscrypt 2008, number

5487 in Lecture Notes in Computer Science, pages

275–288. Springer-Verlag.

Wei, R. and Wu, J. (2004). Product construction of key

distribution schemes for sensor networks. In Selected

Sensornet - A Key Predistribution Scheme for Distributed Sensors using Nets

Areas in Cryptography (SAC) 2004, number 3357 in

Lecture Notes in Computer Science, pages 280–293.

Springer-Verlag.

Wu, B. (2013). Ps bent functions constructed from ﬁ-

nite pre-quasiﬁeld spreads. Available at http://arxiv.

org/pdf/1308.3355.pdf.

SENSORNETS 2017 - 6th International Conference on Sensor Networks