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Computer SecurityFoundational Results

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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?

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ReminderIn 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.

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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.

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Security – Leaking rightsLet 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 s0 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.

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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

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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 questionDoes there exist an algorithm for determining whether a given protection system (with initial state s0) is safe with respect to a generic right r ?

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Theorem 1

There exists an algorithm that will determine whether a given mono-operational protection system with initial protection state s0 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.

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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.

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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.

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The Take-Grant protection modelCan 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).

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Transitions: rewriting rules

• Take rule• Grant rule• Create rule• Remove rule

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Take rule

tz y x z y

x

( a set of rules) x takes ( to y) from z

t

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Grant rule

gz y x z y

x

z grants ( to y) to x

g

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Create rule

x y

x

x creates ( to new vertex) y

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Remove rule

y x y

x

x removes ( to) y

-

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Theorem 3.11

Let G0 be a protection graph containing exactly one subject vertex and no edge and let R be a set of rights. Then

G0 * 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.

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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?

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Closing the Gap

1. The Schematic Protection Model (SPM)

2. The Extended Schematic Protection Model (ESPM)

3. Typed Access Matrix Models (TAMS)

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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.

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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.

ExampleAnne 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.

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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.

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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 →TFor details see textbook.