1. Slide #3-1 Chapter 3: Foundational Results Overview Harrison-Ruzzo-Ullman result –Corollaries

2. Slide #3-2 Overview Safety Question HRU Model

3. Slide #3-3 The general question How can we determine whether a system is safe? Is there a generic algorithm that will determine whether a system is safe?

4. Reminder: safe vs secure Slide #3-4 The term safe is used to refer to the abstract model. The term secure is used when referring to implementations

5. Slide #3-5 What Is “Safe”? Adding a generic right r where there was not one is “leaking” If a system S, beginning in initial state s 0, cannot leak right r, it is safe with respect to the right r.

6. Slide #3-6 Safety Question Does there exist an algorithm for determining whether a protection system S with initial state s 0 is safe with respect to a generic right r?

7. Slide #3-7 Mono-Operational Commands Answer: yes Theorem There exists an algorithm that will determine whether given mono-operational protection system with initial state s 0 is safe with respect to a given generic right r.

8. Slide #3-8 Mono-Operational Commands Sketch of proof: Consider the minimal sequence of commands c 1, …, c k needed to leak the right. – Can omit delete, destroy (no commands can test for the absence of rights in an ACM) – Can merge all creates into one Worst case: insert every right into every entry; with s subjects and o objects initially, and n rights. Then (at worst) there are k ≤ n(s+1)(o+1) commands

9. Slide #3-9 General Case Answer: no Theorem It is undecidable whether a given state of a general protection system with is safe for a given right r.

10. Slide #3-10 General Case Sketch of proof: Reduce the halting problem to safety problem

11. Slide #3-11 Turing Machine review Infinite tape in one direction States K, Symbols M (alphabet); distinguished blank b Transition function  (k, m) = (k, m, L) means in state k, symbol m on tape location replaced by symbol m, head moves to left one square, and enters state k Halting state is q f ; TM halts when it enters this state

12. Slide #3-12 Halting problem Determine if an arbitrary TM will enter a halting state q f The Halting problem is known to be undecidable.

13. Slide #3-13 Mapping First we construct a map from the states and symbols of a Turing machine TM to the rights in the access control matrix A. The generic rights are taken to be –the symbols in M and –a set of distinct symbols each representing a state in K. Each cell of TM is a subject. Define a distinguished right own and such that s i owns s i+1

14. Slide #3-14 Mapping A BCD… 1234 head s1s1 s2s2 s3s3 s4s4 s4s4 s3s3 s2s2 s1s1 A B C k D end own Current state is k

15. Slide #3-15 Mapping A BXD… 1234 head s1s1 s2s2 s3s3 s4s4 s4s4 s3s3 s2s2 s1s1 A B X D k 1 end own After  (k, C) = (k 1, X, R) where k is the current state and k 1 the next state

16. Slide #3-16 Command Mapping  (k, C) = (k 1, X, R) at intermediate becomes command c k,C (s 3,s 4 ) if own in A[s 3,s 4 ] and k in A[s 3,s 3 ] and C in A[s 3,s 3 ] then delete k from A[s 3,s 3 ]; delete C from A[s 3,s 3 ]; enter X into A[s 3,s 3 ]; enter k 1 into A[s 4,s 4 ]; end

17. Slide #3-17 Mapping A BXY 1234 head s1s1 s2s2 s3s3 s4s4 s4s4 s3s3 s2s2 s1s1 A B X Y own After  (k 1, D) = (k 2, Y, R) where k 1 is the current state and k 2 the next state s5s5 s5s5 own b k 2 end 5 b

18. Slide #3-18 Command Mapping  (k 1, D) = (k 2, Y, R) at end becomes command crightmost k,C (s 4,s 5 ) if end in A[s 4,s 4 ] and k 1 in A[s 4,s 4 ] and D in A[s 4,s 4 ] then delete end from A[s 4,s 4 ]; create subject s 5 ; enter own into A[s 4,s 5 ]; enter end into A[s 5,s 5 ]; delete k 1 from A[s 4,s 4 ]; delete D from A[s 4,s 4 ]; enter Y into A[s 4,s 4 ]; enter k 2 into A[s 5,s 5 ]; end

19. Slide #3-19 Rest of Proof Protection system exactly simulates a TM –Exactly 1 end right in ACM –1 right in entries corresponds to state –Thus, at most 1 applicable command If TM enters state q f, then right has leaked If safety question decidable, then represent TM as above and determine if q f leaks –Implies halting problem decidable Conclusion: safety question undecidable

20. Slide #3-20 Other Results Set of unsafe systems is recursively enumerable (there is TM that will enumerate all systems) Delete create primitive; then safety question is complete in P-SPACE Delete destroy, delete primitives; then safety question is undecidable –Such systems are called monotonic: the size and complexity only increases. Safety question for mono-conditional, monotonic protection systems is decidable Safety question for mono-conditional protection systems with create, enter, delete (and no destroy) is decidable.

21. Slide #3-21 Key Points Safety problem undecidable Limiting scope of systems can make problem decidable