contents balanced incomplete block design (bibd) & projective plane generalized quadrangle (gq)...

Contents

• Balanced Incomplete Block Design (BIBD)

& Projective Plane• Generalized Quadrangle (GQ)• Mapping and Construction• Analysis

Balanced Incomplete Block Design(BIBD)

• There are v distinct object• There are b blocks• Each block contains exactly k distinct objects• Each object occurs in exactly r different blocks• Every pair of distinct object occurs together in

exactly blocks • Can be expressed as or •

, ,v k , , , ,v b r k

vk br

Symmetric BIBD (or Symmetric Design)

• A BIBD is called Symmetric BIBD (or Symmetric Design) when b=v and therefore r=k

• Symmetric BIBD has 4 properties:– Every block contains k=r objects– Every object occurs in r=k blocks– Every pair of object occurs in blocks– Every pair of blocks intersects on objects

Example

• or

• • There are b=7 blocks and each one contains

k=3 objects• Every objects occurs in r=3 blocks• Every pair of distinct objects occurs in

Blocks• Every pair of blocks intersects in objects

, , 7,3,1v k , , , , 7,7,3,3,1v k r b

{1,2,3,4,5,6,7} | | 7S S v

1

1

Example Cont’

• Based on a construction algorithm the blocks are:

{1,2,3},{1,4,5},{1,6,7},

{2,4,6},{2,5,7},{3,4,7},{3,5,6}

Projective Plane

• Consist of finite set P of points and a set of subsets of P, called lines

• A Projective Plane of order q (q>1) has 4 properties– Every line contains exactly q+1 points– Every point occurs on exactly q+1 lines– There are exactly points – There are exactly lines

2 1q q 2 1q q

Projective Plane cont’

• Theorem: If we consider lines as blocks and points as objects, then Projective Plane of order q is a Symmetric BIBD with parameters:

• Theorem: For every prime power q>1 there exist a Symmetric BIBD (Projective Plane of order q)

2 1, 1,1q q q

2 1, 1,1q q q

Complementary Design

• Theorem: If is a symmetric BIBD, then is also a symmetric BIBD

• Example: Consider

Complementary Design of this design is:

with the following blocks:

, ,D v k

, , 2D v v k v k

, , 7,3,1v k

7,4,2D

{4,5,6,7},{2,3,6,7},{2,3,4,5},

{1,3,5,7},{1,3,4,6},{1,2,5,6},{1,2,4,7}

Contents



Projective Space PG(d,q)

• Dimension d

• Order q

• Constructed from the vector space of dimension d+1 over the field finite F– Objects are subspaces of the vector space– Two objects are incident if one contains the

other

Projective Space PG(d,q)

• Subspace dimensions– Point if dimension 1– Line if dimension 2– Hyperplane if dimension d

• Order of a projective space is one less than the number of points incident in a line

Partial Linear Space

• Arrangement of objects into subsets called lines

• Properties– Every line is incident with at least two points– Any two points are incident with at most one

line

Incidence Structure

• includes– Set of points– Set of lines– Symmetric incidence relation

Point-Line Incidence Relation

• (p,L) is in I if and only if they are incident in the space

Point-Line Incidence Relation

• Axioms– Two distinct points are incident with at most

one line.– Two distinct lines are incident with at most

one point

Generalized Quadrangle• GQ(s,t) is a subset of a special Partial Linear

Space subset called Partial Geometry• Incidence structure S = (P,B,I)

– P set of points– B set of lines– I symmetric point-line incidence relation

satisfying:• A The above Axioms• B Each point is incident with t+1 lines (t>=1)• C Each line is incident with s+1 points (s>=1)

Generalized Quadrangle

I point-line incidence relation satisfying

D

GQ(s,t)

• v = (s+1)(st+1) points

• b = (t+1)(st+1) lines

• Each line includes (s+1) points and each point appears in (t+1) lines

3 known GQ’s

• GQ(q,q) from PG(4,q)

• GQ(q,q²) from PG(5,q)

• GQ(q²,q³) from PG(4,q²)

• In GQ(q,q)– b = v = (q+1)(q²+1)

Example

• GQ(2,2) for q = 2

• v = b = (2+1)(2*2+1) = 15

• Each block contains 2+1 objects

• Each object is contained in 2+1 blocks

Example cont.

Contents



Reminder – A Distributed Sensor Network (DSN)

• There are N sensor nodes

• Each sensor has a key-chain of k keys

• Keys are selected from a set P of key-pool

• 2 sensor nodes need to have q keys in common in their key-chain to secure their communication

Mapping from Symmetric Design to Key Distribution

Construction

• There are several ways to construct Symmetric BIBD of the form

• We will use complete sets of Mutually Orthogonal and Latin Squares (MOLS)

to construct Symmetric BIBD (which can be converted to a projective plane of order q)

2 1, 1,1q q q

Construction

Mapping from GQ to Key Distribution

• There are t+1 lines passing through a point• Each line has s+1 points• Therefore, each line shares a point with exactly

t(s+1) other lines• Moreover, if 2 lines A,B do not share a point

there are s+1 distinct lines which share a point with both.

Mapping from GQ to Key Distribution Cont’

• In terms of Key Distribution that means:– A block shares a key with t(s+1) other blocks– If 2 blocks do not share a key, there are s+1

other blocks sharing a key with both

Parameters

Construction

Contents



Analysis SD

• In a Symmetric Design any pair of blocks share exactly one object

• Key share probability between 2 nodes

• Average Key-Path Length

Analysis SD

• Resilience contradicts with high probability of key sharing

• Resilience is compromised

• Adversary best case – captures q+1 nodes

• Adversary worst case – captures q²+1 nodes

Analysis SD

• The probability that a link is compromised when an attacker captures key-chains

Analysis GQ

• In a GQ(s,t) there are b = (t+1)(st+1) lines and a line intersects with t(s+1) other lines– Each block shares exactly one object with

t(s+1) other blocks– How many blocks does a block share n

objects with?

Analysis GQ

• Probability two blocks share an object

• Adversary worst case– Captures st² + st +1 nodes

• Adversary best case– Captures t+1 nodes

Prominent properties

• SD highest number of object share

• GQ(q,q²) highest number of blocks for fixed block size

• GQ(q²,q³) smallest block size for fixed number of blocks and has highest resilience

Analysis

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