Multihop Paths and Key Predistribution in Sensor Networks

1. Multihop Paths and Key Predistribution in Sensor Networks Guy Rozen

2. Contents • Terminoligy (quick review) • Alternate grid types and metrics • k-hop coverage • Calculation • How to optimize • Complete two-hop coverage

3. Terminology • DD(m) – Distinct distribution set of m points • DD(m,r) – DD(m) with maximal Euclidian distance of r • DD*(m)/ DD*(m,r) – DD(m)/ DD(m,r) on a hexagonal grid • DD(m,r) – Denotes use of the Manhattan metric • DD*(m,r) – Denotes use of the Hexagonal metric • Ck(D) – Maximal value of a k-hop coverage for some DDS D • Scheme 1: Let be a distinct difference configuration. Allocate keys to notes as follows: • Label each node with its position in . • For every ‘shift’ generate a key and assign it to the notes labeled by , for .

4. Alternate grid types and metrics • In a regular square grid, where the plane is tiled with unit squares, sensor coordinates are in fact .

5. Alternate grid types and metrics • In a regular square grid, where the plane is tiled with unit squares, sensor coordinates are in fact . • In a hexagonal grid, where the plane is tiled with hexagons, seonsor coordinates can be depicted as

6. Moving between grid types • The linear bijection transitions from a hexagonal grid to a square one. • Alternatively, can be seen as doing

7. Moving between grid types The linear bijection transitions from a hexagonal grid to a square one. Alternatively, can be seen as doing Theorem 1:

8. Moving between grid types The linear bijection transitions from a hexagonal grid to a square one. Alternatively, can be seen as doing Theorem 1: Proof:

9. Moving between grid types • It is important to note that does not preserve distances. • Theorem 2:

10. Alternate metrics • Manhattan/Lee metric: The distance between two points and is . • For example, a sphere of radius 2: • Theorem 3:

11. Alternate metrics • Hexagonal metric: The distance between two points is the amount of hexagons on the shortest path between the points. • For example, a sphere of radius 2: • Theorem 4:

12. k-Hop Coverage Definition:

13. k-Hop Coverage Definition: Theorem 5:

14. k-Hop Coverage • Definition: • Theorem 5: • Proof: When using Scheme 1, we know that a pair of nodes sharing a key are located at , hence the vector is both a difference vector of D and a one hop path when using Scheme 1. Hence, an l-hop path between paths is composed of difference vectors from D.

15. k-Hop Coverage • Theorem 6: • Proof:

16. Maximal k-hop coverage • First, we define a set of integer m-tuples:

17. Maximal k-hop coverage First, we define a set of integer m-tuples: Simply put, the some of elements in each tuple is zero and the sum of positive elements is k. Some examples for m=3:

18. Maximal k-hop coverage First, we define a set of integer m-tuples: Simply put, the some of elements in each tuple is zero and the sum of positive elements is k. Some examples for m=3: Lemma 7:

19. Maximal k-hop coverage Theorem 8:

20. Maximal k-hop coverage Theorem 8: Proof:

21. Maximal k-hop coverage Proof (cont.):

22. Maximal k-hop coverage Proof (cont.): Corollary 9:

23. Maximal k-hop coverage Proof:

24. Maximal k-hop coverage - bounds We would like to show that Theorem 8’s bound is tight. Naïve approach:

25. Maximal k-hop coverage - bounds We would like to show that Theorem 8’s bound is tight. Naïve approach: Lemma 10:

26. Maximal k-hop coverage - bounds We would like to show that Theorem 8’s bound is tight. Naïve approach: Lemma 10: Proof:

27. Bh Sequences Definition 1: Elements may be used more than once.

28. Bh Sequences and DDC Theorem 11:

29. Bh Sequences and DDC Theorem 11: Proof:

30. Bh Sequences and DDC Proof (cont.):

31. Using Bh sequences to build a DDC Construction 1:

32. Using Bh sequences to build a DDC Construction 1: Proof:

33. Maximal k-hop coverage - bounds Theorem 12:

34. Maximal k-hop coverage - bounds Theorem 12: Proof:

35. Maximal k-hop coverage - bounds Proof (cont.):

36. Maximal k-hop coverage - bounds Proof (cont.): Corollary 13:

37. Maximal k-hop coverage - bounds What is the minimal r so that a with maximal k-hop coverage is possible? We denote this r as .

38. Maximal k-hop coverage - bounds What is the minimal r so that a with maximal k-hop coverage is possible? We denote this r as . Theorem 14:

39. Maximal k-hop coverage - bounds What is the minimal r so that a with maximal k-hop coverage is possible? We denote this r as . Theorem 14: Proof: (Upper bound proven in Theorem 12)

40. Maximal k-hop coverage - bounds Proof (cont.):

41. Maximal k-hop coverage - bounds Proof (cont.): For a hexagonal grid we present an equivalent term . Theorem 15: Proof: Theorem 2 & 14.

42. Maximal k-hop coverage - bounds We will give special attention to the case k=1. Theorem 16:

43. Maximal k-hop coverage - bounds We will give special attention to the case k=1. Theorem 16: Proof:

44. Maximal k-hop coverage - bounds We will give special attention to the case k=1. Theorem 16: Proof: Theorem 17: Proof: Analogous hexagonal result from [2].

45. Maximal k-hop coverage - bounds Finally, using results in [2] we can prove: Theorem 19:

46. Minimal k-hop coverage What is the smallest value for a k-hop coverage?

47. Minimal k-hop coverage What is the smallest value for a k-hop coverage? Theorem 20:

48. Minimal k-hop coverage What is the smallest value for a k-hop coverage? Theorem 20: Proof:

49. Minimal k-hop coverage Lemma 21:

50. Minimal k-hop coverage Lemma 21: Proof: