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

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

Example Cont’ Based on a construction algorithm the blocks are:

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

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)

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:

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)

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