1. 2/1/20161 Computer Security Foundational Results

2. 2/1/20162 THE GENERAL QUESTION What does secure mean? How can one determine when a computer system is secure? Which policies are secure? What does secure mean?

3. 2/1/20163 Reminder In our model a computer system is represented by a family of states: the set of all protection states P must be a subset of the set of authorized states Q if the system is to be secure. In the previous section we used a primitive, the ACM, to manage a protection system. Protection was in terms or rights and the ACM was the used to relate subjects to objects (also basic primitives). We also discussed protection state transitions and commands, which correspond to (cause) a sequence of state transitions.

4. 2/1/20164 Security If the system dynamically evolves (S,O and A change) how do we guarantee that the system will not enter a state which is not secure (unauthorized). To start with suppose that the system does not have rights such as copy or own, and that the principle of attenuation of principle does not apply.

5. 2/1/20165 Security – Leaking rights Let R be the set of generic (primitive) rights of the system, r  R and let A be the ACM. Definitions 1.If r  R is added to an element of A not already containing r, then r is said to be leaked. 2.Let s 0 be the initial protection state. a. If a system can never leak the right r  R then the system is safe with respect to r. b. If a system can leak r  R then the system is called unsafe with respect to r.

6. 2/1/20166 Security – safe vs secure We use the term safe to refer to the (abstract) model. Secure will be used when referring to implementations. So secure implementations must be modeled on a safe system. Example: safe vs secure --see textbook

7. 2/1/20167 Foundation theorems The model we shall use is based on protecting: — states, — the ACM and — a set of commands essentially the HRU model (discussed in the previous section). Safety question Does there exist an algorithm for determining whether a given protection system (with initial state s 0 ) is safe with respect to a generic right r ?

8. 2/1/20168 Theorem 1 There exists an algorithm that will determine whether a given mono-operational protection system with initial protection state s 0 is safe with respect to a generic right. Proof: see textbook. A mono-operational command invokes a single primitive operation This whole section is a project topic for anybody who is interested in the foundations aspect of Computer Sercurity.

9. 2/1/20169 Theorem 2 It is undecidable whether a given state of a given protection system is safe wrt a generic right. Proof --reduction to the halting problem. The proof is by contradiction. It is shown that an arbitrary Turing Machine can be reduced to the safety problem with the final state corresponding to the leaking of a right. If the safety problem is decidable then one can determine when the Turing machine halts, showing that that the halting problem is decidable, which is false. For details see textbook.

10. 2/1/201610 Theorem 3 The set of unsafe systems is recursively enumerable. (accepted by a TM). That is, we can generate a list of all unsafe protection systems.

11. 2/1/201611 The Take-Grant protection model Can the safety of a protection system with specific rules be established? Answer: Yes: The Take-Grant protection model. This model represents the system by a directed graph, called the protection graph. Vertices are subjects “●” or objects “○”, or both “  ”. Edges are labeled by a set of rights, that the source has over the destination. R contains two distinguished rights: t (take) and g (grant).

12. 2/1/201612 Transitions: rewriting rules Take rule Grant rule Create rule Remove rule

13. 2/1/201613 Take rule t zyxzy  x    (  a set of rules) x takes (  to y) from z t   

14. 2/1/201614 Grant rule g zyxzy  x    z grants (  to y) to x g   

15. 2/1/201615 Create rule xy  x x creates (  to new vertex) y 

16. 2/1/201616 Remove rule yxy  x    x removes (  to) y  --

17. 2/1/201617 Theorem 3.11 Let G 0 be a protection graph containing exactly one subject vertex and no edge and let R be a set of rights. Then G 0 * G iff G is a finite directed acyclic graph with subjects and objects only, with edges labeled for non- empty subsets of R and at least one subject (a trusted entity) having no incoming edge. Proof in textbook. 

18. 2/1/201618 Closing the Gap We can answer the safety question in specific systems, but not for generic systems What is it about the HRU system that makes the safety question undecidable? What characteristics distinguishes a model for which the safety problem is decidable from one in which it is undecidable?

19. 2/1/201619 Closing the Gap 1.The Schematic Protection Model (SPM) 2.The Extended Schematic Protection Model (ESPM) 3.Typed Access Matrix Models (TAMS)

20. 2/1/201620 The Schematic Protection Model (SPM) This model is based on the notion of a protection type: This is: a label for an entity that determines how control rights affect it. Rights are partitioned into sets of Inert rights (RI) and Control rights (RC) Inert rights do not alter the protection state of a system. For example reading a file does not modify which entities have access to the document: so is an RI. However in the Take-Grant model the take rule does, so is in RC.

21. 2/1/201621 The Extended Schematic Protection Model (ESPM) Implicit in the SPM is the assumption of a single parent. ESPM allows for more parents. This problem arises distributed systems. Example Anne and Bill must cooperate to perform a certain task, but do not trust each other. Such tasks may be achieved by using proxies: each create a proxy, and grants the other’s proxy only those rights that are needed to perform the task.

22. 2/1/201622 Typed Access Matrix models (TAMS) The safety properties of SPM and ESPM are implicitly based on protection types. ESPM and HRU are essentially equivalent, ut the safety properties of ESPM are considerably stronger. The TAM model adds the notion of type to the Access Control Matrix model.

23. 2/1/201623 Typed Access Matrix models (TAMS) The type of an entity is fixed when the entity is created. The protection state of a system is defined as: ( S, O, , A ) where, S = set of subjects, O = set of objects, A = the Access Control Matrix, T the set of types and  : O →T For details see textbook.