Multihop Paths and Key Predistribution in Sensor Networks Guy Rozen

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

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 C k (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.

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

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

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

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

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:

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

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

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:

k-Hop Coverage Definition:

k-Hop Coverage Definition: Theorem 5:

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.

k-Hop Coverage Theorem 6: Proof:

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

Theorem 8: Maximal k-hop coverage

Theorem 8: Proof: Maximal k-hop coverage

Proof (cont.): Maximal k-hop coverage

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

Proof: Maximal k-hop coverage

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

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

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

Definition 1: Elements may be used more than once. B h Sequences

Theorem 11: B h Sequences and DDC

Theorem 11: Proof: B h Sequences and DDC

Proof (cont.): B h Sequences and DDC

Construction 1: Using B h sequences to build a DDC

Construction 1: Proof: Using B h sequences to build a DDC

Theorem 12: Maximal k-hop coverage - bounds

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

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

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

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

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

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

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

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

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

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

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

Lemma 21: Minimal k-hop coverage

Lemma 21: Proof: Minimal k-hop coverage

Lemma 21: Proof: Lemma 21 can be used to prove Theorem 21: Minimal k-hop coverage

For a prime, we will show a construction of a with complete 2-hop coverage. That ensures a two-hop path between a point x and any other grid point within a rectangle centered at x. Complete 2-hop coverage HeightWidth

For a prime, we will show a construction of a with complete 2-hop coverage. That ensures a two-hop path between a point x and any other grid point within a rectangle centered at x. Definition 2 (Welch Periodic Array): Equivalent points: Complete 2-hop coverage HeightWidth

Example of an array

Lemma 23: Complete 2-hop coverage

Lemma 23: Proof: Complete 2-hop coverage

From Lemma 23 we conclude: Complete 2-hop coverage

We now define a by using dots from. Complete 2-hop coverage

We now define a by using dots from. Construction 2: ◦ Complete 2-hop coverage

We now define a by using dots from. Construction 2: ◦ Complete 2-hop coverage

We now define a by using dots from. Construction 2: ◦ Complete 2-hop coverage

Meet 

Contained in a square. Has a border region of width 2 which contains exactly 5 points. Has a central region which is a rectangle. The central region contains dots. One column is empty. and there are no other equivalent points.  - Vital statistics

Example of  B’ A’ B A’’A

Lemma 24: Complete 2-hop coverage

Lemma 24: Proof: Complete 2-hop coverage

This is why

Proof (cont.): Complete 2-hop coverage

Motivational boost: Complete 2-hop coverage

Motivational boost: Lemma 25: Complete 2-hop coverage

Motivational boost: Lemma 25: Proof: Complete 2-hop coverage

 illustrated S 11 33 44 22

11 33 44 22

Proof (cont.): Complete 2-hop coverage

Proof (cont.): Complete 2-hop coverage

D 1 to D 1 (or any D x to D x ) 11 33 44 22 

D 1 to D 3 (or D 2 to D 4 ) 11 33 44 22

11 33 44 22

D 1 to D 4 11 33 44 22

D 3 to D 2 11 33 44 22

Lemma 26: Complete 2-hop coverage

Lemma 26: Why do we need this? Complete 2-hop coverage

Lemma 26 motivation: Complete 2-hop coverage

Lemma 26 motivation: Complete 2-hop coverage

Proof of (a)

Lemma 26 motivation: Complete 2-hop coverage

Proof of (b)

Lemma 26 motivation: Complete 2-hop coverage

Proof of (c)

Lemma 26 motivation: Complete 2-hop coverage

Proof of (d) – case one

Proof of (d) – case two No dots

Lemma 26 motivation: Complete 2-hop coverage

Lemma 26 motivation: Complete 2-hop coverage

Lemma 26 motivation: We will now face insurmountable suspense… Complete 2-hop coverage

Proof (of Lemma 26): Complete 2-hop coverage

Proof (of Lemma 26, cont.): Complete 2-hop coverage

Theorem 27: Complete 2-hop coverage

Theorem 27: Proof: Complete 2-hop coverage

We have shown maximal k-hop coverage as We used a construction of to produce a with maximal k-hop coverage and of the order of We have found a bound for (verifying the order above). Could we find tighter bounds? What is the exact value for small k and m? The questions above also hold for the hexagonal grid and the alternate metrics. We have constructed a with complete 2-hop coverage from the center of a rectangular region. The rectangle’s region is of order. Can we find a construction for significantly larger rectangles? For circles? Can we find constructions for k-hop coverage where k>2? Conclusion and open problems