Outline for April 5, 2006

Reading: text, §3.1—3.3.2

1. Greetings and felicitations!
2. What is the safety question?
   1. An unauthorized state is one in which a generic right r could be leaked into an entry in the ACM that did not previously contain r. An initial state is safe for r if it cannot lead to a state in which r could be leaked.
   2. Question: in a given arbitrary protection system, is safety decidable?
   3. Theorem: there is an algorithm that decides whether a given mono-operational system and initial state is safe for a given generic right.
3. General case: It is undecidable whether a given state of a given protection system is safe for a given generic right.
   1. Represent TM as ACM
   2. Reduce halting problem to it
4. Take-Grant
   1. Counterpoint to HRU result
   2. Symmetry of take and grant rights
   3. Islands (maximal subject-only tg-connected subgraphs)
   4. Bridges (as a combination of terminal and initial spans)
5. Sharing
   1. Definition: can•share(r, x, y, G₀) true iff there exists a sequence of protection graphs G₀, ..., G_n such that G₀ ⊢^* G_n using only take, grant, create, remove rules and in G_n, there is an edge from x to y labeled r
   2. Theorem: can•share(r, x, y, G₀) iff there is an edge from x to y labelled r in G₀, or all of the following hold:
      1. there is a vertex y′ with an edge from y′ to y labeled r;
      2. there is a subject y′′ which terminally spans to y′, or y′′ = y′;
      3. there is a subject x′ which initially spans to x, or x′ = x; and
      4. there is a sequence of islands I₁, ..., I_n connected by bridges for which x′ is in I₁ and y′ is in I_n.
6. Model Interpretation
   1. ACM very general, broadly applicable; Take-Grant more specific, can model fewer situations
   2. Theorem: G₀ protection graph with exactly one subject, no edges; R set of rights. Then G₀ ⊢^* G iff G is a finite directed graph containing subjects and objects only, with edges labeled from nonempty subsets of R, and with at least one subject with no incoming edges
   3. Example: shared buffer managed by trusted third party