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Mohamed G. Gouda				Solution to Homework 1
CS 395T: Network Protocol Security		Fall 2006
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1. The id (u,v,w) of each process p[u,v,w] can be viewed as an element of a 
3-dimensional grid. Process p[u,v,w] is assigned every grid key of one of the
following three forms:

	gk(u,v',w'), gk(u',v,w'), and gk(u',v',w)

Process p[u,v,w] is also assigned every direct key of one of the following
three forms.

	dk(u,v,w)(u,v',w'), dk(u,v,w)(u',v,w'), and dk(u,v,w)(u',v',w)

Therefore, each process is assigned 3*(n^2/3) grid keys plus 3*(n^2/3) direct
keys. Thus, each process is assigned a total of 6*(n^2/3) keys which is more 
than the total of 4*(n^1/2) keys assigned to each process in the case of
using a two-dimensional grid.

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2. Assume that we use only two grids (instead of three): AB-grid and BC-grid.
Now assume that two processes p[u',v,w'] and p[u",v,w"], where u' != u" and
w' != w", need to communicate. Then these two processes conclude that they
share only two (direct) keys:

	dk(u',v)(u",v) from the AB grid, and
	dk(v,w')(v,w") from the BC grid.

Unfortunately, these two keys are also known to a third process p[u',v,w"]
and to a fourth process p[u",v,w']. Thus, the two processes p[u',v,w'] and 
p[u",v,w"] cannot communicate securely using their two shared keys.

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3. Use either of two methods, superimposition or decomposition, to develop
logarithmic keying scheme for the given network.

Using Superimposition:
First, develop a logarithmic keying scheme for the acyclic network that 
consists of the following four a-edges:
	{p[0],p[1]},
	{p[0],p[2]},
	{p[0],p[3]}, and
	{p[1],p[4]}.
Second add the remaining three c-edges to the logarithmic keying scheme:
	{p[1],p[3]},
	{p[2],p[3]}, and
	{p[2],p[4]}.

Using Decomposition:
First decompose the network into two acyclic subnetworks, each of degree 2.
One subnetwork has the following four edges:
	{p[0],p[1]},
	{p[1],p[4]},
	{p[4],p[2]}, and
	{p[2],p[3]}.
The other subnetwork has the following three edges.
	{p[1],p[3]},
	{P[3],P[0]}, and
	{p[0],p[2]}.
Second, develop a logarithmic keying scheme for each (acyclic) subnetwork.
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