Outline for January 29, 2007

1. Greetings and Felicitations!
2. Bell-LaPadula Model: full model
   1. Show categories, refefine clearance and classification
   2. Lattice: poset with ≤ relation reflexive, antisymmetric, transitive; greatest lower bound, least upper bound
   3. Apply lattice
      1. Set of classes SC is a partially ordered set under relation dom with glb (greatest lower bound), lub (least upper bound) operators
      2. Note: dom is reflexive, transitive, antisymmetric
      3. Example: (A, C) dom (A′, C′) iff A ≤ A′ and C ⊆ C′; lub((A, C), (A′, C′)) = (max(A, A′), C ∪ C′), glb((A, C), (A′, C′)) = (min(A, A′), C ∩ C′)
   4. Simple security condition (no reads up), *-property (no writes down), discretionary security property
   5. Basic Security Theorem: if it is secure and transformations follow these rules, it will remain secure
   6. Maximum, current security level
3. BLP: formally
   1. Elements of system: s_i subjects, o_i objects
   2. State space V = B×M×F×H where:
      B set of current accesses (i.e., access modes each subject has currently to each object);
      M access permission matrix;
      F consists of 3 functions: f_s is security level associated with each subject, f_o security level associated with each object, and f_c current security level for each subject;
      H hierarchy of system objects, functions h: O→P(O) with two properties:
      1. If o_i o_j, then h(o_i) ∩ h(o_j) = ∅
      2. There is no set { o₁, ..., o_k } ⊆ O such that for each i, o_i+1 ∈ h(o_i) and o_k+1 = o₁.
   3. Set of requests is R
   4. Set of decisions is D
   5. W ⊆ R×D×V×V is motion from one state to another.
   6. System Σ(R, D, W, z₀) ⊆ X×Y×Z such that (x, y, z) ∈ Σ(R, D, W, z₀) iff (x_t , y_t , zt, z_t-1) ∈ W for each i ∈ T; latter is an action of system
   7. Theorem: Σ(R, D, W, z₀) satisfies the simple security property for any initial state z₀ that satisfies the simple security property 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))
      2. if (s, o, x) ∈ b does not satisfy the simple security condition relative to f′, then (s, o, x) ∉ b′
   8. 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′
      2. for each s ∈ S′, if (s, o, x) ∈ b does not satisfy the *-property with respect to f′, then (s, o, x) ∉ b′