# Introduction

As I stated in my previous post, my undergraduate thesis is on static program analysis of Android applications to prove malicious behavior. In this article, I try to explain what static program analysis is and isn’t and the rational behind using using it for this research. Oh, and there is code to look at, too!

# What is Static Program Analysis

Static Analysis attempts to predict the future behavior of a program. My research professor has a great introduction to static analysis article that shies away from a lot of the technical aspects of it.

## Types

There are two main types of Static Program Analysis: sound and unsound. The difference between them being that unsound is trying to determine behavior based on a partial modeling of program behavior while sound analysis models all behaviors of a program. While unsound analysis can say something about all lines of code, it will make assumptions that certain portions of the program are either unreachable or unimportant for the analysis, while sound analysis does not make these assumptions and tries to prove and model all behavior.

In other words, unsound analysis uses an incomplete model while sound uses a complete model. Furthermore, sound analysis must begin with a soundness proof of all layers of abstraction. I’ll get into more details about this in later articles.

This research is sound static program analysis.

## Abstract Interpretation and Soundness

We have a deep and unforgiving failure at the core of the assumption that we can prove behavior of a program: the halting problem (undecidability). This ultimately states that it is impossible to prove that a program terminates for all program-input pairs. So, if can’t know if a program terminates how can we determine all paths a program can take? Wouldn’t that cause an infinitely deep tree of paths?

Fortunately, this is only undecidable because in an infinite domain, we can get around this by presenting a finite number of approximations (abstractions), which guarantees that the abstract interpretation of the program will terminate.

Let’s take the following code

This is a loop that will never terminate. Graphed like a tree, it would look a lot like an infinitely long linked list. Graphed like a DFA, if would be a single state looped back on itself.

However, we can abstract this out into a state that matches a type of infinite loop.

# Program Verification

## A complete picture

In static analysis, the more information you have about something, the more likely you’ll find a contradiction. The less information you have about something, the more generalized it is. This relationship comes from Order Theory.

Rewriting this statement in terms of code, let’s give a bunch of information in a control sequence. Since Ruby hasn’t had much love on my blog so far, let’s give it some.

Here we have given enough information to form a contradiction. $% 1)\land (x < 1) %]]>$, so the call to malware.steal_passwords can never be reached.

## Lattices and Partial Order

We can describe relations of information and state by using lattices, which comes from Order Theory. An article, again by my research professor, gives a good overview of the aspects of Order Theory that pertain to Computer Science.

The basic idea of a lattice in Static Analysis, is that you have a top and a bottom. At the top, you know so very little information about a thing as to make it generic - like Object in ruby or python are close to the top. Whereas x from the code above is closer to the bottom of the lattice, having so much information as to cause a contradiction. From this basic idea, we can define the symbol for a lattice as $(L,\sqsubseteq)$ with a supremum $\top$ and a infimum $\bot$ (max and min). Where the order of elements is determined partially (asymmetrically).

If you think of a lattice as an upper and lower bound for your program ($L$), then all elements in your program must be partially ordered such that $x \sqsubseteq \top$ and $\bot \sqsubseteq x$ $\forall x\in L$

As an example:

We can say that $\text{ralph}\sqsubseteq\text{creature}$ and that creature is closer to $\top$ and ralph is closer to $\bot$.

This example shows that it is possible to have a provable definition of the order of inheritance that allows us now to make mathematical assertions and proofs on the inheritance of that program. We know that all MythicalCreature objects can never steal_souls so we can ignore that possibility.

It is one use of a lattice in static program analysis and in a larger context Order Theory. However, there are many other ways of using Order Theory to prove things about a program including monotonism, continuity, and fixed points to name a few.

# Assertions

Our first goal is to have a sound mathematical definition of each element of a program so that we can assert behavior about that program. Since it is completely possible to execute a program in theory that could never happen in practice, it is important to know for certain that it really could never happen.

In math, we can prove that $\sqrt{2}$ is irrational by contradiction, but we have to make assumptions like mutually-prime, natural number numerator/denominator pairs are rational and we must define a strict set of rules for rationality, mathematical operations and what numbers really are. (If you haven’t taken the chance to take a mathematical analysis class, I encourage you to).

Fortunately for Mathematicians, we can and have proven these things, so we can say with certainty that $\sqrt{2}$ is irrational. And fortunately for us, since we have Order Theory, we can prove things about a finite set of elements that make up a program so that we can make assertions about it, like this program will run perfectly 999 times and then start blowing up memory usage. Or as I mentioned in my post on dalvik, we know that the garbage collector will never factor into the concerns of the application, because it runs in a separate environment.

## Security

Security is really a subset of assertions, where the assertions are things like: this program cannot access a remote server, there is no way for a user to leave the bounds of the application from within the application, there is a reachable path of execution that will crash the device

# Caveats

One major issue with static program analysis, is two fold: it needs access to the source and that source should not be obfuscated.

To be fair, the research that we are doing is on Dalvik byte-code, so we could disassemble any application. It is also for the military, so it well within the bounds of operation to know that we will have access to any code that could potentially be added to a device.

The reason why code obfuscation is an issue, is that there are limitations in how you can abstractly interpret and ultimately analyse a program. Take my professor’s example of proving that the result of $4\times -3$ is negative without computing it is $-12$ and turn it into code:

We would convert this to $\alpha(4) \otimes \alpha(-3)$ which would reduce to $\{-\}$, a negative.

What if we made one of the numbers into a variable?

That is easy enough, we could have easy just substituted in or did an environment lookup. (For details on that see substitution and implementing substitution).

What if we make it even harder, by making each number a lambda that creates a number?

Ok, well, that is easy enough, we could just beta reduce. That would be perfect except for the small fact that this quickly becomes undecidable as the obfuscation gets worse. (For implementation details of reduction, see my post on reduction)

So, code obfuscation has the ability to make it impractical to do static program analysis.

# Conclusion

The goals of Static Program Analysis lend themselves well to the field of security. This is particularly true for controlled environments like government. While it is not a be-all-end-all answer for security, it should allow us to make strong assertions about the behavior of Andriod applications.