Researches

on

Scheme

of

Pairwise

Key

Establishment

for

DistributedSensor

Networks

Wang

Lei

Fujian

University

Technology

Fuzhou,Funjian,

PR.China

Chen

Zhi-ping

Fujian

University

Technology

Fuzhou,Funjian,

PR.China

Jiang

Xin-hua

Fujian

University

Technology

Fuzhou,Funjian,

PR.China

xhj@csu.edu.cn

ABSTRACT

Security

schemes

of

pairwise

key

establishment,

which

enable

sensors

to

communicate

with

each

other

securely,

play

a

fundamental

role

in

research

on

security

issue

in

wireless

sensor

networks.

A

new

kind

of

cluster

deployed

sensor

networks

distribution

model

is

presented,

and

based

on

which,

an

innovative

Hierarchical

Hypercube

model

H(k,u,m,v,n)

and

the

mapping

relationship

between

cluster

deployed

sensor

networks

and

the

H(k,u,m,v,n)

are

proposed.

By

utilizing

nice

properties

of

H(k,u,m,v,n)

model,

a

new

general

framework

for

pairwise

key

predistribution

and

a

new

pairwise

key

establishment

algorithm

are

designed,

which

combines

the

idea

of

KDC(Key

Distribution

Center)

and

polynomial

pool

schemes.

Furthermore,

the

working

performance

of

the

newly

proposed

pairwise

key

establishment

algorithm

is

seriously

inspected.

Theoretic

analysis

and

experimental

figures

show

that

the

new

algorithm

has

better

performance

and

provides

higher

possibilities

for

sensor

to

establish

pairwise

key,

compared

with

previous

related

works.

Categories

and

Subject

Descriptors

[Computer-Communication-Networks]:

Distributed

Systems-Distributed

applications.

General

Terms:

Security.

INTRODUCTION

Security

communication

is

an

important

requirement

in

many

sensor

network

applications,

so

shared

secret

keys

are

used

between

communicating

nodes

to

encrypt

data.

As

one

of

the

most

fundamental

security

services,

pairwise

key

establishment

enables

the

sensor

nodes

to

communicate

securely

with

each

other

using

cryptographic

techniques.

However,

due

to

the

sensor

nodes'

limited

computational

capabilities,

battery

energy,

and

available

memory,

it

is

not

feasible

for

them

to

use

traditional

pairwise

key

establishment

techniques

such

as

public

key

cryptography

and

key

distribution

center

(KDC).

Several

alternative

approaches

have

been

developed

recently

to

perform

pairwise

key

establishment

on

resource-constrained

sensor

networks

without

involving

the

use

of

traditional

cryptography

Eschenauer

and

Gligor

proposed

a

basic

probabilistic

key

predistribution

scheme

for

pairwise

key

establishment

In

the

scheme,

each

sensor

node

randomly

picks

a

set

of

keys

from

a

key

pool

before

the

deployment

so

that

any

two

of

the

sensor

nodes

have

a

certain

probability

to

share

at

least

one

common

key.

Chan

et

al.

further

extended

this

idea

and

presented

two

key

predistribution

schemes:

a

q-composite

key

pre-distribution

scheme

and

a

random

pairwise

keys

scheme.

The

q-composite

scheme

requires

any

two

sensors

share

at

least

q

pre-distributed

keys.

The

random

scheme

randomly

picks

pair

of

sensors

and

assigns

each

pair

a

unique

random

key

Inspired

by

the

studies

above

and

the

polynomial-based

key

pre-distribution

protocol

Liu

et

al.

further

developed

the

idea

addressed

in

the

previous

works

and

proposed

a

general

framework

of

polynomial

pool-based

key

predistribution

The

basic

idea

can

be

considered

as

the

combination

of

the

polynomial-based

key

pre-distribution

and

the

key

pool

idea

used

in

and

Based

on

such

a

framework,

they

presented

two

pairwise

key

pre-distribution

schemes:

a

random

subset

assignment

scheme

and

a

grid-based

scheme.

A

polynomial

pool

is

used

in

those

schemes,

instead

of

using

a

key

pool

in

the

previous

techniques.

The

random

subset

assignment

scheme

assigns

each

sensor

node

the

secrets

generated

from

a

random

subset

of

polynomials

in

the

polynomial

pool.

The

gridbased

scheme

associates

polynomials

with

the

rows

and

the

columns

of

an

artificial

grid,

assigns

each

sensor

node

to

a

unique

coordinate

in

the

grid,

and

gives

the

node

the

secrets

generated

from

the

corresponding

row

and

column

polynomials.

Based

on

this

grid,

each

sensor

node

can

then

identify

whether

it

can

directly

establish

a

pairwise

key

with

another

node,

and

if

not,

what

intermediate

nodes

it

can

contact

to

indirectly

establish

the

pairwise

key.

A

similar

approach

to

those

schemes

described

by

Liu

et

al

was

independently

developed

by

Du

et

a.

Rather

than

on

Blundo's

scheme

their

approach

is

based

on

Blom's

scheme

In

some

cases,

it

is

essentially

equivalent

to

the

one

in

All

of

those

schemes

above

improve

the

security

over

the

basic

probabilistic

key

pre-distribution

scheme.

However,

the

pairwise

key

establishment

problem

in

sensor

networks

is

still

not

well

solved.

For

the

basic

probabilistic

and

the

q-composite

key

predistribution

schemes,

as

the

number

of

compromised

nodes

increases,

the

fraction

of

affected

pairwise

keys

increases

quickly.

As

a

result,

a

small

number

of

compromised

nodes

may

affect

a

large

fraction

of

pairwise

keys

Though

the

random

pairwise

keys

scheme

doses

not

suffer

from

the

above

security

problem,

it

incurs

a

high

memory

overhead,

which

increases

linearly

with

the

number

of

nodes

in

the

network

if

the

level

of

security

is

kept

constant

For

the

random

subset

assignment

scheme,

it

suffers

higher

communication

and

computation

overheads.

In

Liu

proposed

a

new

hypercube-based

pairwise

key

predistribution

scheme

which

extends

the

grid-based

scheme

from

a

two

dimensional

grid

to

a

multi-dimensional

hypercube.

The

analysis

shows

that

hypercube-based

scheme

keeps

some

attractive

properties

of

the

grid-based

scheme,

including

the

guarantee

of

establishing

pairwise

keys

and

the

resilience

to

node

compromises.

Also,

when

perfect

security

against

node

compromise

is

required,

the

hypercube-based

scheme

can

support

a

larger

network

by

adding

more

dimensions

instead

of

increasing

the

storage

overhead

on

sensor

nodes.

Though

hypercube-based

scheme

(we

consider

the

grid-based

scheme

is

a

special

case

of

hypercube-based

scheme)

has

many

attractive

properties,

it

requires

any

two

nodes

in

sensor

networks

can

communication

directly

with

each

other.

This

strong

assumption

is

impractical

in

most

of

the

actual

applications

of

the

sensor

networks.

In

this

paper,

we

present

a

kind

of

new

cluster-based

distribution

model

of

sensor

networks,

and

for

which,

we

propose

a

new

pairwise

key

pre-distribution

scheme.

The

main

contributions

of

this

paper

are

as

follows:

Combining

the

deployment

knowledge

of

sensor

networks

and

the

polynomial

pool-based

key

pre-distribution,

we

setup

a

clusterbased

topology

that

is

practical

with

the

real

deployment

of

sensor

networks.

Based

on

the

topology,

we

propose

a

novel

cluster

distribution

based

hierarchical

hypercube

model

to

establish

the

pairwise

key.

The

key

contribution

is

that

our

scheme

dose

not

require

the

assumption

of

all

nodes

can

directly

communicate

with

each

other

as

the

previous

schemes

do,

and

it

still

maintains

high

probability

of

key

establishment,

low

memory

overhead

and

good

security

performance.

We

develop

a

kind

of

new

pairwise

key

establishment

algorithm

with

our

hierarchical

hypercube

model.

The

structure

of

this

paper

is

arranged

as

follows:

In

section

a

new

distribution

model

of

cluster

deployed

sensor

networks

is

presented.

In

section

a

new

Hierarchical

Hypercube

model

is

proposed.

In

section

the

mapping

relationship

between

the

clusters

deployed

sensor

network

and

Hierarchical

Hypercube

model

is

discussed.

In

section

and

section

new

pairwise

key

establishment

algorithm

are

designed

based

on

the

Hierarchical

Hypercube

model

and

detailed

analyses

are

described.

Finally,

section

presents

a

conclusion.

PRELIMINARY

Definition

(Key

Predistribution):

The

procedure,

which

is

used

to

encode

the

corresponding

encryption

and

decryption

algorithms

in

sensor

nodes

before

distribution,

is

called

Key

Predistribution.

Definition

(Pairwise

Key):

For

any

two

nodes

A

and

B,

if

they

have

a

common

key

E,

then

the

key

E

is

called

a

pairwise

key

between

them.

Definition

(Key

Path):

For

any

two

nodes

and

Ak,

when

there

has

not

a

pairwise

key

between

them,

if

there

exists

a

path

and

there

exists

at

least

one

pairwise

key

between

the

nodes

Ai

and

Aj

for

and

then

the

path

consisted

of

is

called

a

Key

Path

between

and

Ak.

Definition

(n-dimensional

Hypercube):

An

n-dimensional

Hypercube

(or

n−cube)

H(v,n)

is

a

topology

with

the

following

properties:

It

is

consisted

of

edges,

Each

node

can

be

coded

as

a

string

with

n

positions

such

as

where

Any

two

nodes

are

called

neighbors,

which

means

that

there

is

an

edge

between

them,

iff

there

is

just

one

position

different

between

their

node

codes.

MODEL

OF

CLUSTERS

DEPLOYED

SENSOR

NETWORKS

In

some

actual

applications

of

sensor

networks,

sensors

can

be

deployed

through

airplanes.

Supposing

that

the

deployment

rounds

of

sensors

are

k,

and

the

communication

radius

of

any

sensors

is

r,

then

the

sensors

deployed

in

the

same

round

can

be

regarded

as

belonging

to

a

same

Cluster.

We

assign

a

unique

cluster

number

l

l

k)

for

each

cluster.

Supposing

that

the

sensors

form

a

connected

graph

in

any

cluster

after

deployment

through

airplanes,

and

then

the

presents

an

actual

model

of

clusters

deployed

sensor

networks.

An

actual

model

of

clusters

deployed

sensor

networks.

From

it

is

easy

to

know

that,

for

a

given

node

A,

there

exist

lots

of

nodes

in

the

same

cluster

of

A,

which

can

be

communicated

directly

with

A,

since

the

nodes

are

deployed

densely

in

a

cluster.

But

there

exist

much

less

nodes

in

a

cluster

neighboring

to

the

cluster

of

A,

which

can

be

communicated

directly

with

A.

since

the

two

clusters

are

not

deployed

at

the

same

time.

HIERARCHICAL

HYPERCUBE

MODEL

Definition

(k-levels

Hierarchical

Hypercube):

Let

there

are

N

nodes

totally,

then

a

k-levels

Hierarchical

Hypercube

named

H(k,u,m,v,n)

can

be

constructed

as

follows:

The

N

nodes

are

divided

into

k

clusters

averagely,

and

the

[N/k]

nodes

in

any

cluster

are

connected

into

an

n-dimensional

Hypercube:

In

the

n-dimensional

Hypercube,

any

node

is

encoded

as

which

are

called

In-Cluster-Hypercube-Node-Codes,

where

n

kN

],[j]

equals

to

an

integer

not

less

than

j.

So

we

can

obtain

k

such

kind

of

different

hypercubes.

The

k

different

hypercubes

obtained

above

are

encoded

as

which

are

called

Out-Cluster-Hypercube-Node-Codes,

where

m

k

And

the

nodes

in

the

k

different

hypercubes

are

connected

into

m-dimensional

hypercubes

according

to

the

following

rules:

The

nodes

with

same

In-Cluster-Hypercube-Node-Codes

and

different

Out-ClusterHypercube-Node-Codes

are

connected

into

an

m-dimensional

hypercube.

(The

graph

constructed

through

above

steps

is

called

a

k-levels

Hierarchical

Hypercube

abbreviated

as

H(k,u,m,v,n).)

Any

node

A

in

H(k,u,m,v,n)

can

be

encoded

as

(i,

j),

where

is

the

In-Cluster-HypercubeNode-Code

of

node

A,

and

is

the

Out-Cluster-Hypercube-Node-Code

of

node

A.

Obviously,

the

H(k,u,m,v,n)

model

has

the

following

good

properties:

Property

The

diameter

of

H(k,u,m,v,n)

model

is

m+n.

Proof:

Since

the

diameter

of

n-dimensional

hypercube

is

n,

and

the

diameter

of

m-dimensional

hypercube

is

m,

so

it

is

easy

to

know

that

the

diameter

of

H(k,u,m,v,n)

model

is

m+n

from

the

definition

Property

The

distance

between

any

two

nodes

and

in

H(k,u,m,v,n)

model

is

d(A,B)=

where

dh

represents

the

Hamming

distance.

Proof:

Since

the

distance

between

any

two

nodes

in

hypercube

equals

to

the

Hamming

distance

between

them,

so

it

is

obvious

that

the

theorem

conclusion

stands

from

definition

MAPPING

CLUSTERS

DEPLOYED

SENSOR

NETWORKS

TO

H(K,U,M,V,N)

Obviously,

from

the

description

in

section

and

we

can

know

that

the

clusters

deployed

sensor

network

can

be

mapped

into

a

klevels-

hierarchical

hypercube

model

as

follows:

At

first,

the

k

clusters

in

the

sensor

network

can

be

mapped

into

k

different

levels

(or

hypercubes)

in

the

k-levels-

hierarchical

hypercube

model.

Then,

the

sensor

nodes

in

each

cluster

can

be

encoded

with

the

In-Cluster-Hypercube-Node-Codes,

and

the

sensor

nodes

in

the

k

different

clusters

with

the

same

In-ClusterHypercube-Node-Codes

can

be

encoded

with

the

Out-ClusterHypercube-Node-Codes

according

to

the

definition

respectively.

Consequently,

the

whole

sensor

network

has

been

mapped

into

a

k-levels-

hierarchical

hypercube

model.

H(K,U,M,V,N)

MODEL-BASED

PAIRWISE

KEY

PREDISTRIBUTION

ALGORITHM

FOR

SENSOR

NETWORKS

In

order

to

overcome

the

drawbacks

of

polynomial-based

and

polynomial

pool-based

key

predistribution

algorithms,

this

paper

proposed

an

innovative

H(k,u,m,v,n)

model-based

key

predistribution

scheme

and

pairwise

key

establishment

algorithm,

which

combines

the

advantages

of

polynomial-based

and

key

pool-based

encryption

schemes,

and

is

based

on

the

KDC

and

polynomials

pool-based

key

predistribution

models.

The

new

H(k,u,m,v,n)

model-based

pairwise

key

establishment

algorithm

includes

three

main

steps:

Generation

of

the

polynomials

pool

and

key

predistribution,

Direct

pairwise

key

discovery,

Path

key

discovery.

Generation

of

Polynomials

Pool

and

Key

Predistribution

Supposing

that,

the

sensor

network

includes

N

nodes,

and

is

deployed

through

k

different

rounds.

Then

we

can

predistribute

keys

for

each

sensor

node

on

the

basis

of

the

H(k,u,m,v,n)

model

as

follows:

Step

Key

setup

server

randomly

generates

a

bivariate

polynomials

pool

such

as

the

following:

F={

f

i

iiil

n

(x,y),

f

j

jjjinii

m

(x,y)

iii

n

i

n,

l

k;

jjj

m

j

m}

with

vn

+[N/vn

different

t-degree

bivariate

polynomials

over

a

finite

field

Fq,

and

then

assigns

a

unique

polynomial

ID

to

each

bivariate

polynomial

in

F.

Step

In

each

round,

key

setup

server

assigns

a

unique

node

ID:

to

each

sensor

node

from

small

to

big,

where

Step

key

setup

server

assigns

a

unique

cluster

ID:

l

to

all

the

sensor

nodes

deployed

in

the

same

round,

where

l

k.

Step

key

setup

server

predistributes

m+n

bivariate

polynomials

f

iiil

n

f

n

iiil

n

(in,y);

f

jjinii

m

f

m

jjinii

m

jm,y)

and

the

corresponding

polynomial

IDs

to

the

sensor

node

deployed

in

the

lth

round

and

with

ID

Direct

Pairwise

Key

Discovery

If

the

node

in

the

sensor

network

wants

to

establish

pairwise

key

with

a

node

B

then

node

A

can

establish

pairwise

key

with

the

node

B

trough

the

following

methods.

Firstly,

node

A

computes

out

the

distance

between

itself

and

node

B:

d=

where

and

If

then

node

A

obtains

the

direct

pairwise

key

between

itself

and

node

B

according

to

the

following

theorem

Theorem

For

any

two

sensor

nodes

and

B

in

the

sensor

network,

supposing

that

the

distance

between

nodes

A

and

B

is

d=

where

and

If

then

there

exists

a

direct

pairwise

key

between

nodes

A

and

B.

Poof:

Since

then

there

is

or

If

From

there

is

nodes

A,

B

belong

to

the

same

cluster.

Supposing

that

nodes

A,

B

belong

to

the

same

cluster

l,

then

from

There

is

only

one

position

different

between

and

Let

it=i"t,

when

t

and

in

i"n

f

n

iiil

n

(in,i"n)=

f

n

iiil

n

(i"n,in).

So,

there

exists

a

direct

pairwise

key

f

n

iiil

n

(in,i"n)

between

nodes

A

and

B.

If

From

There

is

only

one

position

different

between

and

Let

jt=j"t,

when

t

and

jm

j"m.

Since

equals

to

f

m

jjjinii

m

(jm,

j"m)=

f

m

jjji

nii

m

(j"m,jm).

So,

there

exists

a

direct

pairwise

key

f

m

jjjinii

m

(jm,

j"m)

between

nodes

A

and

B.

According

to

theorem

we

present

the

detailed

description

of

the

direct

pairwise

key

discovery

algorithm

as

follows:

Step

Obtain

the

node

IDs

and

cluster

IDs

of

the

source

node

A

and

destination

node

B;

Step

Compute

out

the

distance

between

nodes

A

and

B:

d=

Step

If

then

select

out

a

common

polynomial

share

of

nodes

A

and

B

from

f

iiil

n

f

n

iiil

n

to

establish

direct

pairwise

key;

Step

If

then

select

out

a

common

polynomial

share

of

nodes

A

and

B

from

f

jjinii

m

f

m

jjjinii

m

to

establish

direct

pairwise

key;

Step

Otherwise,

there

exists

no

direct

pairwise

key

between

nodes

A

and

B.

And

then

turn

to

the

following

path

key

discovery

process.

Path

Key

Discovery

If

then

node

A

can

establish

path

key

with

node

B

according

to

the

following

theorem

Theorem

For

any

two

sensor

nodes

and

B

in

the

sensor

network,

supposing

that

the

distance

between

nodes

A

and

B

is

d=

where

and

If

then

there

exists

a

path

key

between

nodes

A

and

B.

Proof:

Let

then

we

can

think

that

it

i't,

when

t

a;

but

it=i't,

when

t>a;

and

jt

j't,

when

t

b;

but

jt=j't,

when

t>b.

Obviously,

nodes

belong

to

the

same

cluster.

So,

according

to

the

supposing

condition

of

The

nodes

in

the

same

cluster

form

a

connected

graph,

there

is

a

route

among

those

nodes.

In

addition,

in

those

nodes,

the

distance

between

any

two

neighboring

nodes

is

so

from

theorem

it

is

easy

to

know

that

there

exists

direct

pairwise

key

between

any

two

neighboring

nodes

among

those

nodes.

For

nodes

jm),…,

since

they

have

the

same

Out-Cluster-Hypercube-Node-Codes

with

the

node

so

nodes

jm),…,

jm)

and

node

B

belong

to

a

same

logical

hypercube.

Obviously,

from

the

supposing

condition

of

The

whole

sensor

network

forms

a

connected

graph,

there

is

a

route

among

those

nodes.

In

addition,

in

those

nodes,

the

distance

between

any

two

neighboring

nodes

is

so

from

theorem

it

is

easy

to

know

that

there

exists

direct

pairwise

key

between

any

two

neighboring

nodes

among

those

nodes.

So,

it

is

obvious

that

there

exists

a

path

key

between

nodes

A

and

B.

According

to

theorem

we

present

the

detailed

description

of

the

path

key

discovery

algorithm

as

follows:

Step

Compute

out

the

intermediate

nodes

and

jm)

from

the

source

node

and

the

destination

node

B

Step

In

those

nodes

series

and

jm),…,

jm),

B

the

neighboring

nodes

select

their

common

polynomial

share

to

establish

direct

pairwise

key.

From

theorem

it

is

easy

to

know

that

any

source

node

A

can

compute

out

a

key

path

P

to

the

destination

node

B

according

to

the

above

algorithm,

when

there

are

no

compromised

nodes

in

the

sensor

network.

Once

the

key

path

P

is

computed

out,

then

node

A

can

send

messages

to

B

along

the

path

P

to

establish

indirect

pairwise

key

with

node

B.

presents

a

example

of

key

path

establishment.

Key

path

establishment

example.

For

example:

In

the

above

node

can

establish

pairwise

key

with

node

through

the

following

key

path:

where

node

F

shall

route

through

nodes

G,

H,

I,

J

to

establish

direct

pairwise

key

with

node

B.

According

to

the

properties

of

H(k,u,m,v,n)

model,

we

can

prove

that

the

following

theorem

by

combing

the

proof

of

theorem

Theorem

Supposing

that

there

exist

no

compromised

nodes

in

the

sensor

network,

and

the

distance

between

node

A

and

B,

then

there

exists

a

shortest

key

path

with

k

distance

between

node

A

and

B

logically.

That

is

to

say,

node

A

can

establish

indirect

pairwise

key

with

node

B

through

intermediate

nodes.

Proof:

Supposing

that

the

distance

between

node

and

B

is

where

Since

d=t,

according

to

the

construction

properties

of

H(k,u,m,v,n),

it

is

easy

to

know

that

there

exist

intermediate

nodes

in

the

logical

space

H(k,u,m,v,n),

which

satisfy

that

the

distance

between

any

two

neighboring

nodes

in

the

nodes

series

A,

B

equals

to

So

according

to

the

theorem

we

can

know

that

nodes

A,

B

form

a

correct

key

path

between

node

A

and

B.

If

any

two

neighboring

nodes

in

the

nodes

series

A,

B

can

communicate

directly,

then

node

A

can

establish

indirect

pairwise

key

with

node

B

through

those

intermediate

nodes.

Dynamic

Path

Key

Discovery

The

path

key

discovery

algorithm

proposed

in

the

above

section

can

establish

a

key

path

correctly,

only

when

there

exist

no

compromised

nodes

in

the

whole

sensor

network,

since

the

key

path

is

computed

out

beforehand.

And

the

proposed

algorithm

cannot

find

an

alternative

key

path

when

there

exist

some

compromised

nodes

or

some

intermediate

nodes

not

in

the

communication

radius,

even

that

there

exists

other

alternative

key

paths

in

the

sensor

network.

From

the

following

example

we

can

know

that

there

are

many

parallel

paths

in

the

H(k,u,m,v,n)

model

for

any

two

given

source

and

destination

nodes,

since

the

H(k,u,m,v,n)

model

is

high

Alternative

key

path

establishment

example.

For

example:

Considering

the

key

path

establishment

example

given

in

the

above

section

based

on

supposing

that

node

has

compromised,

then

from

we

can

know

that

there

exists

another

alternative

key

path

as

which

can

be

used

to

establish

the

indirect

pairwise

key

between

node

A

and

B,

where

node

E

shall

route

through

nodes

D

and

K

to

establish

direct

pairwise

key

with

node

M,

and

node

M

shall

route

through

nodes

N,

O,

G,

H,

I,

J

to

establish

direct

pairwise

key

with

node

B.

Since

the

sensors

are

source

limited,

so

they

are

easy

to

die

or

out

of

the

communication

radius,

therefore

the

algorithm

proposed

in

the

above

section

cannot

guarantee

to

establish

correct

key

path

efficiently.

In

this

section,

we

will

propose

a

dynamic

path

key

discovery

algorithm

as

follows,

which

can

improve

the

probability

of

key

path

effectively:

Algorithm

I:

Dynamic

key

path

establishment

algorithm

based

on

H(k,u,m,v,n)

model

for

cluster

deployed

sensor

networks.

Input:

Sub-sensor

network

H(k,u,m,v,n),

which

has

some

compromised

/fault

sensors

and

fault

links,

And

two

reachable

nodes

and

in

H(k,u,m,v,n),

where

a"t

b"t,

a"t=b"t,

t

>s.

Output:

A

correct

key

path

from

node

A

to

B

in

H(k,u,m,v,n).

Step

Obtain

the

code

strings

of

node

A

and

B:

A

B

where

aj,

bj

a"j,

b"j

Step

If

then

node

A

can

find

a

route

to

B

according

to

the

routing

algorithms

of

hypercube

Step

Otherwise,

node

A

can

find

a

route

to

according

to

the

Algorithm

I

or

Algorithm

II.

Then

let

and

each

node

It

in

the

above

nodes

series

find

a

route

to

its

neighboring

node

on

the

basis

of

the

location

information

(Detailed

routing

algorithms

based

on

location

information

can

see

the

Step

Algorithm

exits.

If

such

kind

of

a

correct

key

path

exists,

then

through

which

node

A

can

establish

an

indirect

pairwise

key

with

node

B.

Otherwise,

node

A

fails

to

establish

an

indirect

pairwise

key

with

node

B.

And

node

A

will

tries

again

to

establish

an

indirect

pairwise

key

with

node

B

some

time

later.

ALGORITHM

ANALYSES

Practical

Analyses

According

to

the

former

description

and

analyses,

it

is

easy

to

know

that

the

above

newly

proposed

algorithm

has

the

following

properties:

Property

When

there

exist

no

fault

and

compromised

nodes,

by

using

new

pairwise

key

predistribution

scheme

based

on

H(k,u,m,v,n)

model,

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

can

be

estimated

as

where

N

is

the

total

number

of

nodes

in

the

sensor

network,

and

N=um

vn

Proof:

Since

the

predistributed

pairwise

keys

for

any

node

FA

f

iiil

n

f

n

iiil

n

(in,y);

f

jjinii

m

,y),…,

f

m

jjinii

m

jm,y)

in

the

newly

proposed

algorithm.

Obviously,

in

the

logical

hypercube

formed

by

the

nodes

in

the

same

cluster

of

node

A,

there

are

nodes,

which

have

direct

pairwise

key

with

node

A.

And

in

the

logical

hypercube

formed

by

the

nodes

in

different

clusters

from

that

of

node

A,

there

are

nodes,

which

have

direct

pairwise

key

with

node

A.

Therefore,

there

are

totally

nodes,

which

have

direct

pairwise

key

with

node

A.

So,

the

probability

of

pairwise

key

establishment

between

any

two

nodes

can

be

estimated

as

since

the

whole

sensor

network

has

N

sensor

nodes

in

all.

presents

the

comparision

between

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

and

the

dimension

n,

when

the

sensor

network

has

different

total

nodes,

and

use

the

new

pairwise

key

predistribution

scheme

based

on

model.

Number

of

Dimension

ProbabilitytoEstablishDirectKey

N

Comparision

between

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

and

the

dimension

n,

when

the

sensor

network

has

different

total

nodes,

and

use

the

new

pairwise

key

predistribution

scheme

based

on

model.

From

it

is

easy

to

know

that

by

using

new

pairwise

key

predistribution

scheme

based

on

H(k,u,m,v,n)

model,

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

decreases

with

the

increasing

of

the

scale

of

the

sensor

networks,

and

in

addition,

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

decreases

with

the

increasing

of

the

dimension

n,

when

the

scale

of

the

sensor

network

is

fixed.

Theorem

Supposing

that

the

total

sensors

is

N

in

the

sensor

network,

then

when

u

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes,

when

using

the

key

distribution

scheme

based

on

the

hypercube

model

H(v,p),

is

smaller

than

that

when

using

the

key

distribution

scheme

based

on

the

H(k,u,m,v,n)

model.

Proof:

Since

u

v,

then

we

can

let

u=vt

where

t

Since

the

total

number

of

nodes

in

H(v,p)

is

vp

=N,

the

total

number

of

nodes

in

H(k,u,m,v,n)

is

um

vn

=N.

Let

p=x+n,

then

there

is

um

*vn

vx

vn

um

=vx

x=tm.

From

the

property

it

is

easy

to

know

that

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

can

be

estimated

as

According

to

the

description

in

it

is

well

know

that

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

can

be

estimated

as

P"=

Next,

we

will

prove

that

m(vt

Construct

a

function

as

f(t)=

vt

where

t

When

it

is

obvious

that

there

is

f(t)=

vt

and

f"(t)=t

So,

there

is

f(t)

vt

m(vt

Therefore,

the

conclusion

of

the

theorem

stands.

As

for

the

conclusion

of

theorem

we

give

an

example

to

illustrate.

Supposing

that

the

total

number

of

nodes

in

the

sensor

network

is

and

H(v,p)=

then

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

based

on

the

H(k,u,m,v,n)

model

is

P=

but

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

based

on

the

H(v,p)

model

is

P"=

Supposing

that

the

total

number

of

nodes

in

the

sensor

network

is

N,

illustrates

the

comparison

between

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

based

on

the

H(k,u,m,v,n)

model

and

the

probability

of

direct

pairwise

key

establishment

between

any

two

nodes

based

on

the

H(v,p)

model,

when

and

x

scaleofthesensornetwork

ProbabilitytoEstablishDirectKey

H(k,u,m,v,n)model-based

H(v,p)model-based

Comparison

between

the

probability

of

direct

pairwise

key

establishment

between

H(v,n)

and

H(k,u,m,v,n)

models.

From

it

is

easy

to

know

that

the

theorem

stands.

Theorem

Supposing

that

the

total

sensors

is

N

in

the

sensor

network,

then

the

pairwise

key

distribution

scheme

based

on

the

hypercube

model

H(v,p),

is

only

a

special

case

of

the

pairwise

key

distribution

scheme

based

on

the

H(k,u,m,v,n)

model.

Proof:

As

for

the

pairwise

key

distribution

scheme

based

on

the

H(k,u,m,v,n)

model,

let

which

means

that

the

total

sensor

network

includes

only

one

cluster.

Then

obviously,

the

H(k,u,m,v,n)

model

will

degrade

into

the

H(v,n)

model.

According

to

the

former

anayses

in

this

paper

and

the

definition

of

the

pairwise

key

distribution

scheme

based

on

the

hypercube

model

H(v,p)

in

it

is

easy

to

know

that

the

conclusion

of

the

theorem

stands.

Security

Analyses

By

using

the

pairwise

key

establishment

algorithm

based

on

the

H(k,u,m,v,n)

model,

the

intruders

can

launch

two

kinds

of

attacks:

The

attackers

may

target

the

pairwise

key

between

two

particular

sensor

node,

in

order

to

compromise

the

pairwise

key

between

them,

or

prevent

them

to

establish

pairwise

key.

The

attackers

may

attack

against

the

whole

sensor

network,

inorder

to

decrease

the

probability

of

the

pairwise

key

establishment,

or

increase

the

cost

of

the

pairwise

key

establishment.

Attacks

against

a

Pair

of

sensor

nodes

Supposing

that

the

intruders

want

to

attack

two

particular

sensor

nodes

u,v,

where

u,v

are

all

not

compromised

nodes,

but

the

intruders

want

to

compromise

the

pairwise

key

between

them.

If

u,v

can

establish

direct

pairwise

key,

then

the

only

way

to

compromise

the

key

is

to

compromise

the

common

bivariate

polynomial

f(x,y)

between

u,v.

Since

the

degree

of

the

bivariate

polynomial

f(x,y)

is

t,

so

the

intruders

need

to

compromise

at

least

sensor

nodes

that

have

a

share

of

the

bivariate

polynomial

f(x,y).

If

u,v

can

establish

indirect

pairwise

key

through

intermediate

nodes,

then

the

intruders

need

to

compromise

at

least

one

intermediate

node,

or

compromise

the

common

bivariate

polynomial

f(x,y)

between

two

neighboring

intermediate

nodes.

But

even

if

the

intruders

succeed

to

do

that,

node

u

and

v

can

still

reestablish

indirect

pairwise

key

through

alternative

intermediate

nodes.

Supposing

that

the

intruders

want

to

attack

two

particular

sensor

nodes

u,v,

where

u,v

are

all

not

compromised

nodes,

but

the

intruders

want

to

prevent

them

to

establish

the

pairwise

key.

Then,

the

intruders

need

to

compromise

all

of

the

m+n

bivariate

polynomials

of

node

u

or

v.

Since

the

degree

of

the

bivariate

polynomial

f(x,y)

is

t,

so

for

bivariate

polynomial,

the

intruders

need

to

compromise

at

least

sensor

nodes

that

have

a

share

of

the

given

bivariate

polynomial.

Therefore,

the

intruders

need

to

compromise

sensor

nodes

altogether

to

prevent

u,v

to

establish

the

pairwise

key.

Attacks

against

the

sensor

network

Supposing

that

the

Attackers

know

the

distribution

of

the

polynomials

over

sensor

nodes,

it

may

systematically

attack

the

network

by

compromising

the

polynomials

in

F

one

by

one

in

order

to

compromise

the

entire

network.

Assume

the

fraction

of

the

compromised

polynomials

is

pc,

then

there

are

up

to

N"=pc

vn

v

N

umv

n

n

mn

pc

××N

(m+n)

Sensor

nodes

that

have

at

least

one

compromised

polynomial

share.

Among

all

of

the

remaining

N-

N"

sensor

nodes,

of

them

includes

a

compromised

polynomial

share.

So,

the

remaining

N-

N"

sensor

nodes

can

establish

direct

pairwise

key

by

using

any

one

of

their

polynomial

shares.

However,

the

indirect

pairwise

keys

in

the

remaining

N-

N"

sensor

nodes

may

be

affected.

And

they

may

need

to

re-establish

a

new

indirect

pairwise

key

between

them

by

select

alternative

intermediate

nodes

that

do

not

belong

to

the

N"

compromised

nodes.

Supposing

that

the

scale

of

the

sensor

network

is

presents

the

comparison

between

pc

and

the

number

of

sensor

nodes

with

at

least

one

compromised

polynomial

share

in

sensor

networks

based

on

different

H(k,u,m,v,n)

distribution

models.

From

it

is

easy

to

know

that,

when

the

scale

of

the

sensor

network

is

fixed,

the

number

of

the

affected

sensor

nodes

in

the

sensor

network

increases

with

the

increasing

of

the

number

of

compromised

nodes.

F

rac

tion

of

C

om

prom

is

ed

B

ivariate

P

oly

nom

ialsSensorNodeswithatleastoneCompromisedPolynomialShare

H

H

H

H

the

comparison

between

pc

and

the

number

of

sensor

nodes

with

at

least

one

compromised

polynomial

share

in

sensor

networks

based

on

different

H(k,u,m,v,n)

distribution

models.

Theorem

Supposing

that

the

total

sensors

is

N

in

the

sensor

network,

and

the

fraction

of

compromised

nodes

is

pc,

then

when

u>v,

the

number

of

affected

nodes

of

the

H(v,p)

model

based

key

predistribution

scheme,

is

bigger

than

that

of

the

H(k,u,m,v,n)

model

based

key

predistribution

scheme.

Proof:

Since

the

number

of

affected

nodes

of

the

H(k,u,m,v,n)

model

based

key

predistribution

scheme

is

pc

××N

(m+n),

and

it

is

proved

in

that

the

number

of

affected

nodes

of

the

H(v,p)

model

based

key

predistribution

scheme

is

pc

××N

p.

Let

p=x+n,

then

there

is

um

vn

vx

vn

um

=vx

Since

u>v

x>m

pc

××N

(m+n)<

pc

××N

(x+n)=

pc

××N

p.

Supposing

that

the

scale

of

the

sensor

network

is

presents

the

comparison

between

pc

and

the

number

of

sensor

nodes

with

at

least

one

compromised

polynomial

share

in

sensor

networks

based

on

and

distribution

models.

From

it

is

easy

to

know

that

the

conclusion

of

theorem

is

correct,

and

the

number

of

the

affected

sensor

nodes

in

the

sensor

network

increases

with

the

increasing

of

the

number

of

compromised

nodes,

when

the

scale

of

the

sensor

network

is

fixed.

Fraction

of

Compromised

Bivariate

Polynomials

the

comparison

between

pc

and

the

number

of

sensor

nodes

with

at

least

one

compromised

polynomial

share

in

sensor

networks

based

on

and

distribution

models.

CONCLUSION

A

new

hierarchical

hypercube

model

named

H(k,u,m,v,n)

is

proposed,

which

can

be

used

for

pairwise

key

predistribution

for

cluster

deployed

sensor

networks.

And

Based

on

the

H(k,u,m,v,n)

model,

an

innovative

pairwise

key

predistribution

scheme

and

algorithm

are

designed

respectively,

by

combing

the

good

properties

of

the

Polynomial

Key

and

Key

Pool

encryption

schemes.

The

new

algorithm

uses

the

good

characteristics

of

node

codes

and

high

fault-tolerance

of

H(k,u,m,v,n)

model

to

route

and

predistribute

pairwise

keys,

in

which

nodes

are

not

needed

to

be

able

to

communicate

with

each

other

directly

such

as

that

the

algorithms

proposed

by

shall

need.

So,

the

traditional

pairwise

key

predistribution

algorithm

based

on

hypercube

model

is

only

a

special

case

of

the

new

algorithm

proposed

in

this

paper.

Theoretical

and

experimental

analyses

show

that

the

newly

proposed

algorithm

is

an

efficient

pairwise

key

establishment

algorithm

that

is

suitable

for

the

cluster

deployed

sensor

networks.

ACKNOWLEDGMENTS

Our

thanks

to

ACM

SIGCHI

for

allowing

us

to

modify

templates

they

had

developed,

and

to

nature

science

fund

of

Fujian

province

of

PR.China

under

grant

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O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O B I O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I I I O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O B O O O O O O O B I O O O O O O O O O O O O O O O O O O O B O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O 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O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O B I O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O B O O O O B I I I O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O B I O O O O O O O O B I O O O O O O O B I O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O B I O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O B I O O O O O O O O O O O O O O B I O O O O O O O O O O O O O B I O B I B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O B I O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O O O O B O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O

cluster-based distribution model
node code
high fault-tolerance

sensor network
key pool
key predistribution
hierarchical hypercube model
security
pairwise key establishment algorithm
polynomial key
encryption
pairwise key