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Contents
• Balanced Incomplete Block Design (BIBD)
& Projective Plane• Generalized Quadrangle (GQ)• Mapping and Construction• Analysis
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
• Balanced Incomplete Block Design (BIBD)
& Projective Plane• Generalized Quadrangle (GQ)• Mapping and Construction• Analysis
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
• Balanced Incomplete Block Design (BIBD)
& Projective Plane• Generalized Quadrangle (GQ)• Mapping and Construction• Analysis
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
• Balanced Incomplete Block Design (BIBD)
& Projective Plane• Generalized Quadrangle (GQ)• Mapping and Construction• Analysis
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