Outline for April 24, 2013

1. Bell-LaPadula: formal model
   1. Set of requests is R
   2. Set of decisions is D
   3. W ⊆ R × D × V × V is motion from one state to another.
   4. System Σ(R, D, W, z₀) ⊆ X × Y × Z such that (x, y, z) ∈ Σ(R, D, W, z₀) iff (x_t, y_t, z_t, z_t−1) ∈ W for each t ∈ T; latter is an action of system
   5. Theorem: Σ(R, D, W, z₀) satisfies the simple security condition for any initial state z₀ that satisfies the simple security condition iff W satisfies the following conditions for each action (r_i, d_i, (b′, m′, f′, h′), (b, m, f, h)):
      1. each (s, o, x) ∈ b′−b satisfies the simple security condition relative to f′ (i.e., x is not read, or x is read and f_s(s) dom f_o(o); and
      2. if (s, o, x) ∈ b does not satisfy the simple security condition relative to f′, then (s, o, x) ≠ b′
   6. Theorem: Σ(R, D, W, z₀) satisfies the *-property relative to S′ ⊆ S for any initial state z₀ that satisfies the *-property relative to S′ iff W satisfies the following conditions for each (r_i, d_i, (b′, m′, f′, h′), (b, m, f, h)):
      1. for each s ∈ S′, any (s, o, x) ∈ b′−b satisfies the *-property with respect to f′; and
      2. for each s ∈ S′, if (s, o, x) ∈ b does not satisfy the *-property with respect to f′, then (s, o, x) ≠ b′
   7. Theorem: Σ(R, D, W, z₀) satisfies the ds-property iff the initial state z₀ satisfies the ds-property and W satisfies the following conditions for each (r_i, d_i, (b′, m′, f′, h′), (b, m, f, h)):
      1. if (s, o, x) ∈ b′−b, then x ∈ m′[s, o]; and
      2. if (s, o, x) ∈ b and x ∈ m′[s, o], then (s, o, x) ≠ b′
   8. Basic Security Theorem: A system Σ(R, D, W, z₀) is secure iff z₀ is a secure state and W satisfies the conditions of the above three theorems for each action.
2. Using the model
   1. Define ssc-preserving, *-property-preserving, ds-property-preserving
   2. Define relation W(ω)
   3. Show conditions under which rules are ssc-preserving, *-property-preserving, ds-property-preserving
   4. Show when adding a state preserves those properties
   5. Example instantiation: get-read for Multics
3. Tranquility
   1. Strong tranquility
   2. Weak tranquility
4. System Z and the controversy