Source: KSII Transactions On Internet And Information Systems
Vol. 3, No. 2, April 2009
Authors: Abedelaziz Mohaisen, DaeHun Nyang,
YoungJae Maeng, KyungHee Lee, Dowon Hong
Presenter: Hsing-Lei Wang
Date: 2010/11/26
1
Outline
 Introduction
 Related Works
 The Proposed Scheme
 Grid-Based Key Pre-Distribution in WSN (3D)
 Analysis
 Conclusions
 Comment
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Introduction

A Grid-Based Key Pre-Distribution Scheme (3D)

Goal: Improve the connectivity and resiliency
Z-Plat
X-Plat
Y-Plat
3
Related Works (Blundo et al. scheme, 1993)

Polynomial-Based Key Pre-Distribution Scheme
Setup Server randomly generates a bivariate t  degree polynomial:
f  x, y  
t
i j
a
x
 ij y ,where f  x, y   f  y, x  .
i , j 0
Computes the polynomial share f  i, y  for each node i.
Each node has a unique ID.
Node i computes f  i, j  by evaluating f  i, y  at point j
Node j computes f  j , i  by evaluating f  j, y  at point i
f  i, j   f  j , i   the common key for both nodes.
4
Related Works (Liu et al. scheme, 2003)

Grid-Based Key Pre-Distribution Scheme (1/2)
Setup:
Assume network size  N  m 2 .
Constructs a m  m grid.
Generates 2  m polynomials
 f  x, y  , f  x, y 
c
i
r
j
i , j  0,......, m 1
Assign the sensor nodes and polynomials
to the grid as figure.
Each node has a unique ID i, j or c, r
Each node stores:  ID, fi c  j, y  , f jr  i, y 
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Related Works (Liu et al. scheme, 2003)

Grid-Based Key Pre-Distribution Scheme (2/2)
Polynomial share Discovery:
Suppose node i ci , ri want to establish
a pair-wise key with node j c j , rj .
Node i checks whether:
ci  c j or ri  rj
If equal, they have a common polynomial:
f cci  x, y  or f rir  x, y 
Use the polynomial share to compute
common key.
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (1/11)
Grid Structure:
Assume network size  N  m3 .
Constructs a m  m  m 3D-grid.
Let X , Y , Z be three axes
X  c0 , c1 ,..., cm 1
Y  r0 , r1 ,..., rm 1
Z  h0 , h1 ,..., hm 1
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (2/11)
Sensors Assignment:
Assign the sensors to the grid.
Each node has a unique ID  cx , ry , hz
Node Si has the identifier structure
i  cxi || ryi || hzi
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (3/11)
Z-Plat
x, y, constant
Definition of The Plat:
X-Plat
constant, y, z
Y-Plat
x, constant, z
The plat is the virtual shape confined by
all possible values for two variable axes
and a constant value in the third axis.
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (4/11)
f c 2  x, y 
f c1  x, y 
f c 0  x, y 
Key Material Assignment:
Generates 3  m polynomials
 f  x, y  , f  x, y  , f  x, y 
cx
ry
hz
x , y , z  0,......, m 1
Assign the polynomials to the grid
All nodes in the same plat have the same
polynomial.
c2 , y, z
c1 , y, z
c0 , y, z
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (5/11)
f r 0  x, y  f r1  x, y 
f r 2  x, y 
x, r2 , z
Key Material Assignment:
Generates 3  m polynomials
 f  x, y  , f  x, y  , f  x, y 
cx
ry
hz
x , y , z  0,......, m 1
Assign the polynomials to the grid
All nodes in the same plat have the same
polynomial.
x, r1 , z
x, r0 , z
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (6/11)
x, y, h2
f h 2  x, y 
Key Material Assignment:
Generates 3  m polynomials
 f  x, y  , f  x, y  , f  x, y 
cx
f h1  x, y 
x, y, h1
f h0  x, y 
ry
hz
x , y , z  0,......, m 1
Assign the polynomials to the grid
All nodes in the same plat have the same
polynomial.
x, y, h0
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (7/11)
Polynomial shares:
Each node Si with identifier i  cxi || ryi || hzi was assigned to
three polynomial f cxi  x, y  , f
ryi
 x, y  and f h  x, y 
zi
For each node Si , the server computes the polynomial shares:
g cxi  f cxi  i, y 
g
ryi
 f
ryi
g hzi  f hzi
 i, y 
 i, y 

r
Each node Si will stores: identifier i, g cxi , g yi , g hzi

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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (8/11)
Direct Key Establishment:
Suppose two nodes Si and S j want to communicate.
Si with identifier i  cxi || ryi || hzi
S j with identifier j  cxj || ryj || hzj
Two nodes exchange their identifier.
If cxi  cxj or ryi  ryj or hzi  hzj ,
The two nodes compute common key
by the polynomial share.
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (9/11)
x, y, h2
Direct Key Establishment:
Example :
hzi  hzj  h2
f
h2
 x, y 
Si
Sj
k ij
The two nodes belong to h2 -plat,
Si computes f h2  i, j  by evaluating f h2  i, y  with identifier j
S j computes f h2  j , i  by evaluating f h2  j , y  with identifier i
The common key kij  f h2  i, j   f h2  j , i 
15
The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (10/11)
Indirect Key Establishment:
If the two nodes Si and S j do not belong to the same plat
Let S is a intermediate node with identifier  .
 must satisfied one of the following condition:
a.   cx  i  cx and   ry  j  ry or   hz  j  hz 
b.   ry  i  ry and   cx  j  cx or   hz  j  hz 
c.   hz  i  hz and   cx  j  cx or   ry  j  ry 
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The Proposed Scheme

3D Grid-Based Key Pre-Distribution Scheme (11/11)
Indirect Key Establishment:
2, 2, 2 S j
S
Example :
  cx  i  cx and   hz  j  hz
Si and S j
generate indirect keys with S :
ki  f cx  i,    f cx  , i 
Si 0,1,0
k j  f hz  j ,    f hz  , j 
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Analysis
 Connectivity
18
Analysis
 Resiliency
19
Comparison
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Conclusions
 Original contribution:
 Introduce a grid-based key pre-distribution scheme that
utilizes the notion of plats on grid
 Plat-based polynomial assignment
 The advantages of the proposed scheme
 Guarantees higher connectivity
 More possible intermediate nodes, better resiliency
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Comment
 In the Key Establishment phase, the authors do not
describe how the sensor node find the intermediate
node for indirect key establishment.
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