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:

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.

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

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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