# January 28, 2019 Outline

Reading: Chapters from revised text §5.2–5.3

1. Elements of system: siS subjects, oiO 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: fs is security level associated with each subject, fo security level associated with each object, and fc current security level for each subject;
H hierarchy of system objects, functions h: OP(O) with two properties:
1. If oioj, then h(oi) ∩ h(oi) = ∅
2. There is no set { o1, …, ok } ⊆ O such that for each i, oi+1h(oi) and ok+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 (xt, yt, zt, zt−1) ∈ W for each tT; latter is an action of system
7. 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, d, (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 fs(s) dom fo(o)); and
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, d, (b′, m′, f′, h′), (b, m, f, h)):
1. for each sS′, any (s, o, x) ∈ b′ − b satisfies the *-property with respect to f′; and
2. for each sS′, if (s, o, x) ∈ b does not satisfy the *-property with respect to f′, then (s, o, x) ∉ b
9. 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, d, (b′, m′, f′, h′), (b, m, f, h)):
1. if (s, o, x) ∈ b′ − b, then xm′[s, o]; and
2. if (s, o, x) ∈ b and xm′[s, o], then (s, o, x) ∉ b
10. 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

 Matt Bishop Office: 2209 Watershed Science Phone: +1 (530) 752-8060 Email: mabishop@ucdavis.edu