1. Key Predistribution Approach in Wireless Sensor Networks Using LU Matrix
Author: Hangyang Dai and Hongbing Xu
Source: IEEE Sensor Journal, vol. 10, no. 8, pp
, Aug (Impact Factor = 1.581)
Presenter: Yung-Chih Lu
Date: 2010/08/04

2. Outline Introduction Proposed Scheme Performance & Security Evaluation
Conclusion

3. Introduction(1/4) location finding system mobilizer Base station
sensing unit
processing unit
transceiver
Unit
microprocessor
digital/
analog
converter
sensor
storage device
power unit
Power
generation
Wireless Sensor Network
Sensor Architecture

4. Introduction(2/4) LU Matrix
Lower Triangular Matrix　　　　　 Upper Triangular Matrix

5. Introduction(3/4) The type of key agreement protocol Trusted-server:
The trusted server shares a key with every node and transmits session keys to the nodes on quest.
Public-key:
Perform a public-key infrastructure.
Key predistribution:
Keys are distributed to all sensor nodes prior to deployment.

6. Introduction(4/4) Constraints Requirements Limited energy consumption
Low transmission range
Limited Memory overhead
Requirements
High network connectivity
Robust resilience against node capture
Low memory overhead

7. Eschenauer-Gligor Scheme(1/5)
L. Eschenauer and V. Gligor. “A Key-Management Scheme
for Distributed Sensor Networks.” In Proc. 9th ACM Conference
on Computerand Communication Security, pp.41-47, Nov
Key pre-distribution phase
P :key pool size , k = key ring size
Pr[ two key rings share at least a key]
= 1 - Pr[ two nodes do not share any key]
= 1 - (C(P, k) / C(P, k)) × (C(k, 0) × C(P-k, k) / C(P, k)) =
Stirling’s approximation :
Example1:
P=1000 , k=100
Pr =
≒ ×e-83/ ×e-78
≒ 1
Example2:
P=1000 , k=10
≒ ×e-9 / ×e-9
≒ =

8. Eschenauer-Gligor Scheme(2/5)

9. Eschenauer-Gligor Scheme(3/5)
L-Sensor : Low-end sensor
H-Sensor : High-end sensor
Key pre-distribution phase
Key pool
Keys
Key identities
Key ring
(k keys)
:L-Sensor
Key identity = key mod 232
H-Sensor : L-Sensors ID、 L-Sensors key identifiers and Kci
L-Sensor : k keys、 key identifiers and Kci
Kci = EKx(ci) Kx = K1⊕,…, ⊕Kk
ci = H-Sensor ID
H-sensor

10. Eschenauer-Gligor Scheme(4/5)
Shared-key discovery
Step1:Each L-sensor Broadcasts a list of key identities.
Step2:L-sensor runs a challenge-response protocol if L-sensor find the common key.
Key ring
(k keys)
:L-Sensor
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)
H-sensor
Key ring
(k keys)
Key ring
(k keys)
α =
Dki[Eki(α)]
Eki(α)
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)

11. Eschenauer-Gligor Scheme(5/5)
Path-key establishment
Key ring
(k keys)
:L-Sensor
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)
H-sensor
Key ring
(k keys)
Key ring
(k keys)
α =
Dkp[Ekp(α)]
Key ring
(k keys)
Key ring
(k keys)
Key ring
(k keys)
Ekc(kp)
Ekp(α)
Key ring
(k keys)
Key ring
(k keys)
Ekc(kp)

12. Pairwise Key Predistribution Scheme(1/5)
Galois Field
q elements (q is a prime number)
Field is closed under additive and multiplicative operator
The sign is GF(q)
s is a primitive number
Example:
GF(7)
3 mod 7 = 3
3^2 mod 7=2
3^3 mod 7 =6
3^4 mod 7 =4
3^5 mod 7 =5
3^6 mod 7 = 1
The order of 3 is 6
3 is a primitive number
W. Du, J. Deng, Y. S. Han, P. K. Varshney, J. Katz, and A. Khalili.
“A pairwise key predistribution scheme for wireless sensor networks.”
ACM Trans. Inf. Syst. Secur., vol. 8, no. 2, pp. 228–258, 2005.
linear independent
Node k only store the seed sk
k=1,2,…,N

13. Pairwise Key Predistribution Scheme(2/5)
Blom’s scheme
D : a symmetric matrix of size
(λ+1)×(λ+1)
G : a matrix of size (λ+1)×N
1X + 1Y = 0 …(1)
3X + 2Y = 0 …(2)
2X + 4Y = 0 …(3)
(2) – 2．(1)
X = 0 …(4)
substitute (4) into (1)
X = Y = 0
Example: N=2 , λ=2 , GF(7)
D．G =
A=(D．G)T =
A．G =
K12 = K21 = 3
[ ]
１　６　２
６　３　５
２　５　２
[ ]
１　１
３　２
２　４
２　０
４　４
０　６
＝ mod 7 .
[ ]
２　４　０
０　４　６
[ ]
２　０
４　４
０　６ T ＝
[ ]
２　４　０
０　４　６
[ ]
１　１
３　２
２　４
[ ]
０　３
３　４ .
＝ mod 7 ﹛
λ-secure property guarantees that no coalition
of up to λ nodes (not including i and j) have
any information about Kij or Kji.

14. Pairwise Key Predistribution Scheme(3/5)
Key pre-distribution phase
Step1:Generating a G matrix
Step2:Generating ω D matrices
D1,…,Dω
Step3:Caculating Ai = (Di．G)T
i = 1,…, ω
Step4:Selecting τ spaces per node
2≦τ<ω
Example: ω=3 , τ=2 ,
each L-sensor store (λ+1)×τ elements
H-sensor
Step3: A1 A2 A3
L1-Sensor
L２-Sensor
Step4
Step4:
A1(1)
A3(1)
Seed sk
k=1,…,N …
Seed: s1
A2(2)
A3(2)
Seed: s2 …

15. Pairwise Key Predistribution Scheme(4/5)
Key agreement phase
Step1:Each L-sensor Broadcasts a message
message = L-sensor’s id + the indices of the spaces + seed
Step2:Two L-sensors can establish a common secret key if they both hold a common key space.
H-sensor
message
A1(1)
A3(1)
A2(2)
A3(2)
Seed: s2
Seed: s1

16. Pairwise Key Predistribution Scheme(5/5)
Pactual[ two nodes share at least a space]
= 1 – Pactual[two nodes do not share any space]

17. [ ] Proposed Scheme(1/6) Blundo Polynomial-based protocol
Setup server randomly generates a symmetric bivariate t-degree polynomial
Example:
f(x,y) = 4x2y2 + x3y1 + x1y3
It’s a symmetric bivariate
3-degree polynomial
over a finite field Fq
[ ]
０　０　１
０　４　０
１　０　０

18. Proposed Scheme(2/6) Blundo Polynomial-based protocol Step1: computes
1: Lu-Sensor ID 2: Lv-Sensor ID
f(1,y) = 4y2 + y1 +y3
f(2,y) = 16y2 + 8y1 + 2y3
Step2:The Setup server loads the sensor node with coefficients
Step3:Each sensor node broadcasts its own ID
Step4:Receiver use ID to compute a shared secret key
Kuv = f(u,v) = f(v,u) = Kvu
K12 = f(1,2) = 26 = f(2,1) = K21
H-sensor
Lu-Sensor
y1 y2 y3
Lv-Sensor 1 4
y1 y2 y3 8 16 2

19. Proposed Scheme(3/6) Polynomial predistribution phase Polynomial pool
(Bivariate
t-degree
Polynomial +
Unique ID)
Assuming that u11=1, u22=2, u33=3

20. Proposed Scheme(4/6) Polynomial predistribution phase
Randomly distribute one row of L and one column of U to each sensor node
r1 : 1st row , c1 : 1st column
r2 : 2nd row , c2 : 2nd column

21. Proposed Scheme(5/6) Shared key establishment phase SB: node B’s ID
To Match or Not To Match?
MAC: message-authentication code
CLR: is a confirmation

22. Proposed Scheme(6/6) Example: ω=3 , τ=2 , …
Step1:Generating a Polynomial
Pool (ω Polynomials)
P1,…,Pω
Step2:Selecting τ polynomials per node
2≦τ<ω
Example: ω=3 , τ=2 ,
each L-sensor store (t+1)×τ
elements
H-sensor
Step1: P1 P2 P3
L1-Sensor
L２-Sensor
Step2:
P1(r1)
P1(c1)
P3(r1)
P3(c1)
P2(r2)
P2(c2)
P3(r2)
P3(c2) …

23. Performance & Security Evaluation(1/4)

24. Performance & Security Evaluation(2/4)
network connectivity

25. Performance & Security Evaluation(3/4)
resilience against node capture
k=400

26. Performance & Security Evaluation(4/4)
memory overhead
Compare with Blundo scheme

27. Conclusion High network connectivity memory space saving
certain threshold
node to node authentication