Question

What is the covering relation of the partial ordering for the poset of security classes defined in Example 25?

Short answer

Covering relations are

\(\begin{array}{*{20}{l}}{(0,\{ {\rm{ spies, moles, double agents }}\} ),}&{(1,\{ {\rm{spies}},{\rm{moles}},{\rm{doubleagents}}\} ),}\\{(0,\{ {\rm{spies}},{\rm{moles}},\} ),}&{(1,\{ {\rm{spies}},{\rm{moles}},\} )}\\{(0,\{ {\rm{spies}}\} ),}&{(1,{\rm{ spies }}\} ),}\\{(2,{\rm{ spies, moles, double agents }}\} ),}&{(3,\{ {\rm{ spies, moles, double agents }}\} ),}\\{(2,\{ {\rm{spies}},{\rm{moles}},\} ),}&{(3,\{ {\rm{ spies, moles, }}\} )}\\{(2,\{ {\rm{spies}}\} ),}&{(3,\{ {\rm{ spies }}\} )}\end{array}\)

Step-by-step solution

Given data

Each piece of information is assigned to a security class, and each security class is represented by a pair (A, C) where \(A\) is an authority level and \(C\) is a category.

The authority levels used are unclassified (0), confidential (1), secret (2), and top secret (3). Categories used in security classes \{spies, moles, double agents\}.

Order of security classes is done by specifying that \((A1,C1) \le (A2,C2)\) if and only if

\(A1 \le A2\) and \(C1 \subseteq C2\). Thus, Information is permitted to flow from security class \((A1,C1)\) into security class \((A2,C2)\) if and only if \((Al,C1) \le (A2,C2)\).

Concept used of covering relations

Let\((S, \le )\)be a poset. We say that an element\(y \in S\)covers an element\(x \in S\)if\(x < y\)and there is no element\(z \in S\)such that\(x < z < y\). The set of pairs\((x,y)\)such that\(y\)covers\(x\)is called the covering relation of\((S, \le )\).

Find the covering relations

As per the question set \(A\) is \(\{ 0,1,2,3\} \)

Set \(C\) is {spies, moles, double agents \(\} .\)

Now we need to form pairs \((x,y)\) such that, if \({x_1},{x_2} \in A\) and \({y_1},{y_2} \in C\), such that information flow is allowed, from \(\left( {{x_1},{y_1}} \right)\) to \(\left( {{x_2},{y_2}} \right)\) when \({x_1} \le {x_2}\) and \({y_1} \subseteq {y_2}\)

Covering relations are then

\(\begin{array}{*{20}{l}}{(0,\{ {\rm{ spies, moles, double agents }}\} ),}&{(1,\{ {\rm{spies}},{\rm{moles}},{\rm{doubleagents}}\} ),}\\{(0,\{ {\rm{spies}},{\rm{moles}},\} ),}&{(1,\{ {\rm{spies}},{\rm{moles}},\} )}\\{(0,\{ {\rm{spies}}\} ),}&{(1,{\rm{ spies }}\} ),}\\{(2,{\rm{ spies, moles, double agents }}\} ),}&{(3,\{ {\rm{ spies, moles, double agents }}\} ),}\\{(2,\{ {\rm{spies}},{\rm{moles}},\} ),}&{(3,\{ {\rm{ spies, moles, }}\} )}\\{(2,\{ {\rm{spies}}\} ),}&{(3,\{ {\rm{ spies }}\} )}\end{array}\)