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

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

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OutlineOutlineIntroductionPartially Balanced Incomplete

Block DesignsProposed SchemePerformance EvaluationSecurity AnalysisConclusionComment

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Introduction Introduction (1/3)(1/3)

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

WSN: Wireless Sensor Network c: c1 and c2

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

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(8x7)/2 = 28Connectivity ratio= 27/28 = 0.9642

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

V(c) = = 1/6 = 0.1667

:Sensor node

1,2,4

2,3,5

0,1,3

0,2,6

1,5,6

0,4,5

3,4,5

x

c2

0,3,6

c1

Introduction Introduction (2/3)(2/3)

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)

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

R.H. Bruck, H.J. Ryser, "The nonexistence of certain finite projective planes", Canadian J. Math. vol.1, pp.88–93, 1949

Introduction Introduction (3/3)(3/3)

Goal◦Key agreement

Key Pre-distribution Phase

◦Resilience against node capture attack

◦High connectivity

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Partially Balanced Partially Balanced Incomplete BlockIncomplete Block

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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

0 1 2

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 keys0 1 2

Proposed Scheme Proposed Scheme (1/5)(1/5)

1 2 3

4 5 6

7 8 9

7

r

k

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

1,4,7

2,5,8

3,6,9

1,5,9

2,6,7

3,4,8

1,6,8

2,4,9

3,5,7

Si,j ={(x, xi + j mod r) : 0 ≦ x <

k}i

j

0 1 2

0 1 2

Colbourn, C. J. and Dinitz, J. H. (Eds.). CRC Handbook of Combinatorial Designs. Boca Raton, FL: CRC Press, p. 112, 1996.

Proposed Scheme Proposed Scheme (2/5)(2/5)

Shared-key establishement phase

xi+j≡ xi’+j’ mod rx(i-i’)≡ j’-j mod r If(i≠ i’) and (x≡ (j’-j)(i-i’)-1 mod r)

◦ Then have a common key

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0 1 2

1,4,7

2,5,8

3,6,9

1,5,9

2,6,7

3,4,8

1,6,8

2,4,9

3,5,7

i

j

0 1 2

1,4,7

2,5,8

1,5,9

2,6,7 key identity = H(Key)

H(.): one way hash function

Key identity ignore

0,0

0,1

1,01,1

Proposed Scheme Proposed Scheme (3/4)(3/4)

Path-key establishment phase

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1,4,7

2,5,8,4

1,5,9

0,0

0,1

1,0

2,6,7

1,1

E1[4]

E5[4]

Performance Evaluation (1/3)

Grid-based

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RF radius

Performance Evaluation (2/3)

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v: key-pool size b: number of sensor nodes RF: radio frequencyr: number of nodes in which a given key occurs k: number of keys in a nodeS: sensor node number of nodes connected

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).

Performance Evaluation (3/3)

Connectivity ratio

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k: number of keys in a node

Resilience against node capture attack

Security Analysis (1/2)

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

Security Analysis (2/2)

Resilience against node capture attack

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

Conclusionthey analyze the connectivity of

the network taking the RF radius into account

Transversal Design is useful

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CommentCommentSuitable for small WSNs2ρ (ρ + 1) is not accuracy

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