Enhancing Pairwise Key Establishment in Wireless Network Security through Enhanced Polynomial Techniques

1. ITIS 6010/8010 Wireless Network Security Dr. Weichao Wang

2. Pairwise key establishment with guarantee • Problems of basic key pre-distribution and Chan’s improvement • The key establishment is not guaranteed • Tolerance to sensor compromise • Polynomial based key pre-distribution • Random subset assignment approach • Grid based key distribution

3. Polynomial based key distribution • A bivariate t-degree polynomial f(x, y) is generated • It has the property of f(x, y) = f(y, x) • For every sensor i, we can replace x with i and generate a new poly f(i, y) • When sensor i meets sensor j, node i can calculate f(i, j), node j can calculate f(j, i); • The two keys are the same

4. Overhead • Every sensor needs to store a t-degree poly • Evaluation of the polynomial • Robustness • Need at least t+1 nodes to figure out a poly • Problem • Want to further reduce overhead • Improvement • Using a group of polynomials

5. Polynomial pool based key pre-distribution • We generate a pool of bivariate polynomials • When we have only one poly, it returns to the previous method • When all poly are 0-degree, it returns to the basic approach • Each sensor gets a subset of polys • Direct key establishment • Path key establishment

6. Random subset assignment – approach 1 • Every sensor gets a random set of polys • Analysis of key sharing • Directly b/w two sensors • Through one hop neighbors • Similar to the basic approach • Then what is the advantage of using poly to replace a key • ?

7. Grid based key pre-distribution • Guaranteed key establishment • Improved resilience to sensor compromise • “Zero” interaction to figure out the key – except the node identity

8. We have n sensors, n < m * m • Every sensor can be mapped to a unique point in the m*m matrix • Generate 2m polynomial, one for each row and one for each column • For a sensor at position (i, j), the corresponding row and column polys will be given to the node

9. Any two sensors in the same row or column will share a poly – they can derive the key • If the two sensors are not in the same row or column • Locate the node that can establish keys with both nodes

10. Advantages • Storage overhead: every node only stores two polys • A sensor can directly figure out can it establish a key to the other sensor

12. Key pre-distribution based on Blom’s scheme • Improve resilience to sensor compromise • Authentication between sensor pair

13. Blom’s key pre-distribution • Generate a (λ+1) * N matrix G, N is the size of the network, λ is the threshold of tolerance. The matrix is public • Generate a (λ+1) * (λ+1) symmetric matrix D and keep it as secret • A = (D * G)^T, A is a N * (λ+1) matrix • Since D is symmetric, we have A*G = (A*G)^T, so A*G is a symmetric matrix

14. If we let K = A*G, then Kij = Kji • See example of the calculation • Every node i will have ith row of A and ith column of G • When node i and j meet, they exchange the columns of G and calculate Kij and Kji

15. Blom’s scheme guarantees that any two sensors can find a key. But we do not need such dense keys • If we generate multiple Blom’s matrices, each can be viewed as a key space

16. Approach • Generate one matrix G • Generate w matrix D1, D2, ---, Dw, we can calculate A1=(D1 * G)^T, A2=(D2 * G)^T, ---, Aw=(Dw * G)^T. • Every node will select t key spaces and get corresponding information from the matrices. • If two sensors have the same key space, they can generate a key.

17. Analysis of key space sharing • Similar to the basic mechanisms • What is the probability that a key space is compromised? • Need at least (λ+1) sensors holding this key space • When x nodes are broken, the probability that j of them know the key space is:

18. When the key space is not compromised, pairwise keys can be used to authenticate