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Key Predistribution Using Transversal Design on a Grid of Wireless Sensor Network

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### Key Predistribution Using Transversal Design on a Grid of Wireless Sensor Network

Author: S. Ruj, S. Maitra and B. Roy

Source: Ad Hoc & Sensor Wireless Networks, vol. 5, no. 3-4, pp. 247-264,

2008.

Presenter: Yung-Chih Lu (呂勇志)

Date: 2010/10/08

Outline

- Introduction
- Partially Balanced Incomplete Block Designs
- Proposed Scheme
- Performance Evaluation
- Security Analysis
- Conclusion
- Comment

Introduction (1/3)

- Key Pre-distribution in WSN
- Key pool={0,1,2,3,4,5,6}

2,3,5

(8x7)/2 = 28

Connectivity ratio= 27/28 = 0.9642

E(c)= (7+6+4) / 27 = 0.2593

V(c) =

= 1/6 = 0.1667

x

3,4,5

1,2,4

c1

0,3,6

0,2,6

c2

0,4,5

0,1,3

1,5,6

:Sensor node

WSN: Wireless Sensor Network c: c1 and c2

E(c): Fraction of links broken when c nodes are compromised

V(c): Fraction of nodes disconnected when c nodes are compromised

Introduction (2/3)

R.H. Bruck, H.J. Ryser, "The nonexistence of certain

finite projective planes", Canadian J. Math. vol.1,

pp.88–93, 1949

- Bruck–Chowla–Ryser theorem
- λ-(v,b,r,k)
- q: sum of two square numbers
- q mod 4 = 1 or 2
- If v=b= q2+q+1 , then r=k=q+1
- Example: q=2, λ-(v,b,r,k)=1-(7,7,3,3)
- Key -pool = {0, 1, 2, 3, 4, 5, 6}
- S1=(1,2,4) ．S5=(5,6,1)
- S2=(2,3,5 ) ．S6=(6,0,2)
- S3=(3,4,6 ) ．S7=(0,1,3)
- S4=(4,5,0)

v: key-pool size b: number of sensor nodes

r: number of nodes in which a given key occurs k: number of keys in a node

λ: number of nodes which contain a given pair of keys S: sensor node

Introduction (3/3)

- Goal
- Key agreement
- Key Pre-distribution Phase
- Resilience against node capture attack
- High connectivity

Partially Balanced Incomplete Block

Sushmita Ruj and Bimal Roy, "Key Predistribution

Using Partially Balanced Designs in Wireless

Sensor Networks", ISPA , p.p.431-445, 2007

- Key Pre-distribution
- Block 1: (2, 3, 4, 5, 6, 7) Block 2: (1, 3, 4, 5, 8, 9)
- Block 3: (1, 2, 4, 6, 8, 10) Block 4: (1, 2, 3, 7, 9, 10)
- Block 5: (1, 2, 6, 7, 8, 9) Block 6: (1, 3, 5, 7, 8, 10)
- Block 7: (1, 4, 5, 6, 9, 10) Block 8: (2, 3, 5, 6, 9, 10)
- Block 9: (2, 4, 5, 7, 8, 10) Block 10: (3, 4, 6, 7, 8, 9)

Block: sensor node

Proposed Scheme (1/5)

Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC

Handbook of Combinatorial Designs. Boca

Raton, FL: CRC Press, p. 112, 1996.

- Key Pre-distribution in Transversal Design
- b = r2 ,v = rk , b=v, k=r
- Example: b=32 = 9, v=3×3 = 9.
- Key pool={1,2,3,4,5,6,7,8,9} 。Sensor keys

0 1 2

0 1 2

0 1 2

0 1 2

Si,j ={(x, xi + j mod r) : 0 ≦ x < k}

i

k

r

j

v: key-pool size b: number of sensor nodes

r: number of nodes in which a given key occurs (r is a prime power)

S: sensor node k: number of keys in a node

Proposed Scheme (2/5)

- Shared-key establishement phase
- xi+j≡xi’+j’ mod r
- x(i-i’)≡j’-j mod r
- If(i≠i’) and (x≡(j’-j)(i-i’)-1 mod r)
- Then have a common key

0 1 2

0 1 2

Key identity

1,4,7

0,0

i

ignore

2,5,8

0,1

1,5,9

2,6,7

j

key identity = H(Key)

1,0

1,1

H(.): one way hash function

Performance Evaluation (1/3)

- Grid-based

Performance Evaluation (2/3)

- RF radius

RF radius = ρ

The maximun number of physical neighbors within the RF radius = Bρ =2ρ (ρ + 1)

Number of key-sharing neighbors within the RF radius = Aρ

Connectivity Ratio = Aρ/Bρ

((2ρ+ 1)2 -1)/2 = (4ρ(ρ+ 1))/2=2ρ(ρ+ 1).

v: key-pool size b: number of sensor nodes RF: radio frequency

r: number of nodes in which a given key occurs k: number of keys in a node

S: sensor node number of nodes connected

Security Analysis (1/2)

- Resilience against node capture attack

b: number of sensor nodes b = r2 S: sensor node

r: number of nodes in which a given key occurs k: number of keys in a node

V(c): Fraction of nodes disconnected when c nodes are compromised

Security Analysis (2/2)

- Resilience against node capture attack

b: number of sensor nodes b = r2 S: sensor node

r: number of nodes in which a given key occurs k: number of keys in a node

E(c): Fraction of links broken when c nodes are compromised

Conclusion

- they analyze the connectivity of the network taking the RF radius into account
- Transversal Design is useful

Comment

- Suitable for small WSNs
- 2ρ (ρ + 1) is not accuracy

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