In the previous chapter, we have introduced *mutation-based fuzzing*, a technique that generates fuzz inputs by applying small mutations to given inputs. In this chapter, we show how to *guide* these mutations towards specific goals such as coverage. The algorithms in this chapter stem from the popular American Fuzzy Lop (AFL) fuzzer, in particular from its AFLFast and AFLGo flavors. We will explore the greybox fuzzing algorithm behind AFL and how we can exploit it to solve various problems for automated vulnerability detection.

**Prerequisites**

- Reading the introduction on mutation-based fuzzing is recommended.

The algorithms in this chapter stem from the popular American Fuzzy Lop (AFL) fuzzer.

*mutation-based fuzzer*. Meaning, AFL generates new inputs by slightly modifying a seed input (i.e., mutation), or by joining the first half of one input with the second half of another (i.e., splicing).

*greybox fuzzer* (not blackbox nor whitebox). Meaning, AFL leverages coverage-feedback to learn how to reach deeper into the program. It is not entirely blackbox because AFL leverages at least *some* program analysis. It is not entirely whitebox either because AFL does not build on heavyweight program analysis or constraint solving. Instead, AFL uses lightweight program instrumentation to glean some information about the (branch) coverage of a generated input.
If a generated input increases coverage, it is added to the seed corpus for further fuzzing.

We start with discussing the most important parts we need for mutational testing and goal guidance.

We introduce specific classes for mutating a seed.

`Mutator`

class. Given a seed input `inp`

, the mutator returns a slightly modified version of `inp`

. In the chapter on greybox grammar fuzzing, we extend this class to consider the input grammar for smart greybox fuzzing.

In [6]:

```
class Mutator:
"""Mutate strings"""
def __init__(self) -> None:
"""Constructor"""
self.mutators = [
self.delete_random_character,
self.insert_random_character,
self.flip_random_character
]
```

For insertion, we add a random character in a random position.

In [7]:

```
class Mutator(Mutator):
def insert_random_character(self, s: str) -> str:
"""Returns s with a random character inserted"""
pos = random.randint(0, len(s))
random_character = chr(random.randrange(32, 127))
return s[:pos] + random_character + s[pos:]
```

In [8]:

```
class Mutator(Mutator):
def delete_random_character(self, s: str) -> str:
"""Returns s with a random character deleted"""
if s == "":
return self.insert_random_character(s)
pos = random.randint(0, len(s) - 1)
return s[:pos] + s[pos + 1:]
```

In [9]:

```
class Mutator(Mutator):
def flip_random_character(self, s: str) -> str:
"""Returns s with a random bit flipped in a random position"""
if s == "":
return self.insert_random_character(s)
pos = random.randint(0, len(s) - 1)
c = s[pos]
bit = 1 << random.randint(0, 6)
new_c = chr(ord(c) ^ bit)
return s[:pos] + new_c + s[pos + 1:]
```

The main method is `mutate`

which chooses a random mutation operator from the list of operators.

In [10]:

```
class Mutator(Mutator):
def mutate(self, inp: Any) -> Any: # can be str or Seed (see below)
"""Return s with a random mutation applied. Can be overloaded in subclasses."""
mutator = random.choice(self.mutators)
return mutator(inp)
```

In [11]:

```
Mutator().mutate("good")
```

Out[11]:

'cood'

Now we introduce a new concept; the *power schedule*. A power schedule distributes the precious fuzzing time among the seeds in the population. Our objective is to maximize the time spent fuzzing those (most progressive) seeds which lead to higher coverage increase in shorter time.

We call the likelihood with which a seed is chosen from the population as the seed's *energy*. Throughout a fuzzing campaign, we would like to prioritize seeds that are more promising. Simply said, we do not want to waste energy fuzzing non-progressive seeds. We call the procedure that decides a seed's energy as the fuzzer's *power schedule*. For instance, AFL's schedule assigns more energy to seeds that are shorter, that execute faster, and yield coverage increases more often.

First, there is some information that we need to attach to each seed in addition to the seed's data. Hence, we define the following `Seed`

class.

In [13]:

```
class Seed:
"""Represent an input with additional attributes"""
def __init__(self, data: str) -> None:
"""Initialize from seed data"""
self.data = data
# These will be needed for advanced power schedules
self.coverage: Set[Location] = set()
self.distance: Union[int, float] = -1
self.energy = 0.0
def __str__(self) -> str:
"""Returns data as string representation of the seed"""
return self.data
__repr__ = __str__
```

The power schedule that is implemented below assigns each seed the same energy. Once a seed is in the population, it will be fuzzed as often as any other seed in the population.

In Python, we can squeeze long for-loops into much smaller statements.

`lambda x: ...`

returns a function that takes`x`

as input. Lambda allows for quick definitions unnamed functions.`map(f, l)`

returns a list where the function`f`

is applied to each element in list`l`

.`random.choices(l, weights)[0]`

returns element`l[i]`

with probability in`weights[i]`

.

In [14]:

```
class PowerSchedule:
"""Define how fuzzing time should be distributed across the population."""
def __init__(self) -> None:
"""Constructor"""
self.path_frequency: Dict = {}
def assignEnergy(self, population: Sequence[Seed]) -> None:
"""Assigns each seed the same energy"""
for seed in population:
seed.energy = 1
def normalizedEnergy(self, population: Sequence[Seed]) -> List[float]:
"""Normalize energy"""
energy = list(map(lambda seed: seed.energy, population))
sum_energy = sum(energy) # Add up all values in energy
assert sum_energy != 0
norm_energy = list(map(lambda nrg: nrg / sum_energy, energy))
return norm_energy
def choose(self, population: Sequence[Seed]) -> Seed:
"""Choose weighted by normalized energy."""
self.assignEnergy(population)
norm_energy = self.normalizedEnergy(population)
seed: Seed = random.choices(population, weights=norm_energy)[0]
return seed
```

In [15]:

```
population = [Seed("A"), Seed("B"), Seed("C")]
schedule = PowerSchedule()
hits = {
"A": 0,
"B": 0,
"C": 0
}
```

In [16]:

```
for i in range(10000):
seed = schedule.choose(population)
hits[seed.data] += 1
```

In [17]:

```
hits
```

Out[17]:

{'A': 3387, 'B': 3255, 'C': 3358}

Looks good. Every seed has been chosen about a third of the time.

We'll start with a small sample program of six lines. In order to collect coverage information during execution, we import the `FunctionCoverageRunner`

class from the chapter on mutation-based fuzzing.

The `FunctionCoverageRunner`

constructor takes a Python `function`

to execute. The function `run`

takes an input, passes it on to the Python `function`

, and collects the coverage information for this execution. The function `coverage()`

returns a list of tuples `(function name, line number)`

for each statement that has been covered in the Python `function`

.

`crashme()`

function raises an exception for the input "bad!". Let's see which statements are covered for the input "good".

In [19]:

```
def crashme(s: str) -> None:
if len(s) > 0 and s[0] == 'b':
if len(s) > 1 and s[1] == 'a':
if len(s) > 2 and s[2] == 'd':
if len(s) > 3 and s[3] == '!':
raise Exception()
```

In [20]:

```
crashme_runner = FunctionCoverageRunner(crashme)
crashme_runner.run("good")
list(crashme_runner.coverage())
```

Out[20]:

[('crashme', 2), ('run_function', 132)]

`crashme`

, the input "good" only covers the if-statement in line 2. The branch condition `len(s) > 0 and s[0] == 'b'`

evaluates to False.

Let's integrate both the mutator and power schedule into a fuzzer. We'll start with a blackbox fuzzer -- which does *not* leverage any coverage information.

Our `AdvancedMutationFuzzer`

class is an advanced and *parameterized* version of the `MutationFuzzer`

class from the chapter on mutation-based fuzzing. It also inherits from the Fuzzer class. For now, we only need to know the functions `fuzz()`

which returns a generated input and `runs()`

which executes `fuzz()`

a specified number of times. For our `AdvancedMutationFuzzer`

class, we override the function `fuzz()`

.

`AdvancedMutationFuzzer`

is constructed with a set of initial seeds, a mutator, and a power schedule. Throughout the fuzzing campaign, it maintains a seed corpus called `population`

. The function `fuzz`

returns either an unfuzzed seed from the initial seeds, or the result of fuzzing a seed in the population. The function `create_candidate`

handles the latter. It randomly chooses an input from the population and applies a number of mutations.

In [22]:

```
class AdvancedMutationFuzzer(Fuzzer):
"""Base class for mutation-based fuzzing."""
def __init__(self, seeds: List[str],
mutator: Mutator,
schedule: PowerSchedule) -> None:
"""Constructor.
`seeds` - a list of (input) strings to mutate.
`mutator` - the mutator to apply.
`schedule` - the power schedule to apply.
"""
self.seeds = seeds
self.mutator = mutator
self.schedule = schedule
self.inputs: List[str] = []
self.reset()
def reset(self) -> None:
"""Reset the initial population and seed index"""
self.population = list(map(lambda x: Seed(x), self.seeds))
self.seed_index = 0
def create_candidate(self) -> str:
"""Returns an input generated by fuzzing a seed in the population"""
seed = self.schedule.choose(self.population)
# Stacking: Apply multiple mutations to generate the candidate
candidate = seed.data
trials = min(len(candidate), 1 << random.randint(1, 5))
for i in range(trials):
candidate = self.mutator.mutate(candidate)
return candidate
def fuzz(self) -> str:
"""Returns first each seed once and then generates new inputs"""
if self.seed_index < len(self.seeds):
# Still seeding
self.inp = self.seeds[self.seed_index]
self.seed_index += 1
else:
# Mutating
self.inp = self.create_candidate()
self.inputs.append(self.inp)
return self.inp
```

In [23]:

```
seed_input = "good"
mutation_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule())
print(mutation_fuzzer.fuzz())
print(mutation_fuzzer.fuzz())
print(mutation_fuzzer.fuzz())
```

good gDoodC /

Let's see how many statements the mutation-based blackbox fuzzer covers in a campaign with n=30k inputs.

The fuzzer function `runs(crashme_runner, trials=n)`

generates `n`

inputs and executes them on the `crashme`

function via the `crashme_runner`

. As stated earlier, the `crashme_runner`

also collects coverage information.

In [25]:

```
n = 30000
```

In [26]:

```
blackbox_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule())
start = time.time()
blackbox_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n)
end = time.time()
"It took the blackbox mutation-based fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[26]:

'It took the blackbox mutation-based fuzzer 0.35 seconds to generate and execute 30000 inputs.'

`(all_coverage, cumulative_coverage)`

where `all_coverage`

is the set of statements covered by all inputs, and `cumulative_coverage`

is the number of statements covered as the number of executed inputs increases. We are just interested in the latter to plot coverage over time.

In [27]:

```
_, blackbox_coverage = population_coverage(blackbox_fuzzer.inputs, crashme)
bb_max_coverage = max(blackbox_coverage)
"The blackbox mutation-based fuzzer achieved a maximum coverage of %d statements." % bb_max_coverage
```

Out[27]:

'The blackbox mutation-based fuzzer achieved a maximum coverage of 2 statements.'

The following generated inputs increased the coverage for our `crashme`

example.

In [28]:

```
[seed_input] + \
[
blackbox_fuzzer.inputs[idx] for idx in range(len(blackbox_coverage))
if blackbox_coverage[idx] > blackbox_coverage[idx - 1]
]
```

Out[28]:

['good', 'bo']

**Summary***. This is how a blackbox mutation-based fuzzer works. We have integrated the *mutator* to generate inputs by fuzzing a provided set of initial seeds and the *power schedule* to decide which seed to choose next.

In contrast to a blackbox fuzzer, a greybox fuzzer like AFL *does* leverage coverage information. Specifically, a greybox fuzzer adds to the seed population generated inputs which increase code coverage.

The method `run()`

is inherited from the Fuzzer class. It is called to generate and execute exactly one input. We override this function to add an input to the `population`

that increases coverage. The greybox fuzzer attribute `coverages_seen`

maintains the set of statements, that have previously been covered.

In [29]:

```
class GreyboxFuzzer(AdvancedMutationFuzzer):
"""Coverage-guided mutational fuzzing."""
def reset(self):
"""Reset the initial population, seed index, coverage information"""
super().reset()
self.coverages_seen = set()
self.population = [] # population is filled during greybox fuzzing
def run(self, runner: FunctionCoverageRunner) -> Tuple[Any, str]: # type: ignore
"""Run function(inp) while tracking coverage.
If we reach new coverage,
add inp to population and its coverage to population_coverage
"""
result, outcome = super().run(runner)
new_coverage = frozenset(runner.coverage())
if new_coverage not in self.coverages_seen:
# We have new coverage
seed = Seed(self.inp)
seed.coverage = runner.coverage()
self.coverages_seen.add(new_coverage)
self.population.append(seed)
return (result, outcome)
```

Let's take our greybox fuzzer for a spin.

In [30]:

```
seed_input = "good"
greybox_fuzzer = GreyboxFuzzer([seed_input], Mutator(), PowerSchedule())
start = time.time()
greybox_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n)
end = time.time()
"It took the greybox mutation-based fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[30]:

'It took the greybox mutation-based fuzzer 0.35 seconds to generate and execute 30000 inputs.'

Does the greybox fuzzer cover more statements after generating the same number of test inputs?

In [31]:

```
_, greybox_coverage = population_coverage(greybox_fuzzer.inputs, crashme)
gb_max_coverage = max(greybox_coverage)
"Our greybox mutation-based fuzzer covers %d more statements" % (gb_max_coverage - bb_max_coverage)
```

Out[31]:

'Our greybox mutation-based fuzzer covers 2 more statements'

Our seed population for our example now contains the following seeds.

In [32]:

```
greybox_fuzzer.population
```

Out[32]:

[good, bo, baof, bad4u]

In [33]:

```
%matplotlib inline
```

In [35]:

```
line_bb, = plt.plot(blackbox_coverage, label="Blackbox")
line_gb, = plt.plot(greybox_coverage, label="Greybox")
plt.legend(handles=[line_bb, line_gb])
plt.title('Coverage over time')
plt.xlabel('# of inputs')
plt.ylabel('lines covered');
```

***Summary***. We have seen how a greybox fuzzer "discovers" interesting seeds that can lead to more progress. From the input `good`

, our greybox fuzzer has slowly learned how to generate the input `bad!`

which raises the exception. Now, how can we do that even faster?

***Try it***. How much coverage would be achieved over time using a blackbox *generation-based* fuzzer? Try plotting the coverage for all three fuzzers. You can define the blackbox generation-based fuzzer as follows.

```
from Fuzzer import RandomFuzzer
blackbox_gen_fuzzer = RandomFuzzer(min_length=4, max_length=4, char_start=32, char_range=96)
```

You can execute your own code by opening this chapter as Jupyter notebook.

***Read***. This is the high-level view how AFL works, one of the most successful vulnerability detection tools. If you are interested in the technical details, have a look at: https://github.com/mirrorer/afl/blob/master/docs/technical_details.txt

Our boosted greybox fuzzer assigns more energy to seeds that promise to achieve more coverage. We change the power schedule such that seeds that exercise "unusual" paths have more energy. With *unusual paths*, we mean paths that are not exercised very often by generated inputs.

In order to identify which path is exercised by an input, we leverage the function `getPathID`

from the section on trace coverage.

The function `getPathID`

returns a unique hash for a coverage set.

In [37]:

```
def getPathID(coverage: Any) -> str:
"""Returns a unique hash for the covered statements"""
pickled = pickle.dumps(sorted(coverage))
return hashlib.md5(pickled).hexdigest()
```

There are several ways to assign energy based on how unusual the exercised path is. In this case, we implement an *exponential power schedule* which computes the energy $e(s)$ for a seed $s$ as follows
$$e(s) = \frac{1}{f(p(s))^a}$$
where

- $p(s)$ returns the ID of the path exercised by $s$,
- $f(p)$ returns the number of times the path $p$ is exercised by generated inputs, and
- $a$ is a given exponent.

In [38]:

```
class AFLFastSchedule(PowerSchedule):
"""Exponential power schedule as implemented in AFL"""
def __init__(self, exponent: float) -> None:
self.exponent = exponent
def assignEnergy(self, population: Sequence[Seed]) -> None:
"""Assign exponential energy inversely proportional to path frequency"""
for seed in population:
seed.energy = 1 / (self.path_frequency[getPathID(seed.coverage)] ** self.exponent)
```

In [39]:

```
class CountingGreyboxFuzzer(GreyboxFuzzer):
"""Count how often individual paths are exercised."""
def reset(self):
"""Reset path frequency"""
super().reset()
self.schedule.path_frequency = {}
def run(self, runner: FunctionCoverageRunner) -> Tuple[Any, str]: # type: ignore
"""Inform scheduler about path frequency"""
result, outcome = super().run(runner)
path_id = getPathID(runner.coverage())
if path_id not in self.schedule.path_frequency:
self.schedule.path_frequency[path_id] = 1
else:
self.schedule.path_frequency[path_id] += 1
return(result, outcome)
```

In [40]:

```
n = 10000
seed_input = "good"
fast_schedule = AFLFastSchedule(5)
fast_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), fast_schedule)
start = time.time()
fast_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n)
end = time.time()
"It took the fuzzer w/ exponential schedule %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[40]:

'It took the fuzzer w/ exponential schedule 0.24 seconds to generate and execute 10000 inputs.'

In [42]:

```
x_axis = np.arange(len(fast_schedule.path_frequency))
y_axis = list(fast_schedule.path_frequency.values())
plt.bar(x_axis, y_axis)
plt.xticks(x_axis)
plt.ylim(0, n)
# plt.yscale("log")
# plt.yticks([10,100,1000,10000])
plt;
```

In [43]:

```
print(" path id 'p' : path frequency 'f(p)'")
fast_schedule.path_frequency
```

path id 'p' : path frequency 'f(p)'

Out[43]:

{'e014b68ad4f3bc2daf207e2498d14cbf': 5612, '0a1008773804033d8a4c0e3aba4b96a0': 2607, 'eae4df5b039511eac56625f47c337d24': 1105, 'b14f545c3b39716a455034d9a0c61b8c': 457, '11529f85aaa30be08110f3076748e420': 219}

How does it compare to our greybox fuzzer with the classical power schedule?

In [44]:

```
seed_input = "good"
orig_schedule = PowerSchedule()
orig_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), orig_schedule)
start = time.time()
orig_fuzzer.runs(FunctionCoverageRunner(crashme), trials=n)
end = time.time()
"It took the fuzzer w/ original schedule %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[44]:

'It took the fuzzer w/ original schedule 0.16 seconds to generate and execute 10000 inputs.'

In [45]:

```
x_axis = np.arange(len(orig_schedule.path_frequency))
y_axis = list(orig_schedule.path_frequency.values())
plt.bar(x_axis, y_axis)
plt.xticks(x_axis)
plt.ylim(0, n)
# plt.yscale("log")
# plt.yticks([10,100,1000,10000])
plt;
```

In [46]:

```
print(" path id 'p' : path frequency 'f(p)'")
orig_schedule.path_frequency
```

path id 'p' : path frequency 'f(p)'

Out[46]:

{'e014b68ad4f3bc2daf207e2498d14cbf': 6581, '0a1008773804033d8a4c0e3aba4b96a0': 2379, 'eae4df5b039511eac56625f47c337d24': 737, 'b14f545c3b39716a455034d9a0c61b8c': 241, '11529f85aaa30be08110f3076748e420': 62}

The exponential power schedule shaves some of the executions of the "high-frequency path" off and adds them to the lower-frequency paths. The path executed least often is either not at all exercised using the traditional power schedule or it is exercised much less often.

Let's have a look at the energy that is assigned to the discovered seeds.

In [47]:

```
orig_energy = orig_schedule.normalizedEnergy(orig_fuzzer.population)
for (seed, norm_energy) in zip(orig_fuzzer.population, orig_energy):
print("'%s', %0.5f, %s" % (getPathID(seed.coverage), # type: ignore
norm_energy, repr(seed.data)))
```

In [48]:

```
fast_energy = fast_schedule.normalizedEnergy(fast_fuzzer.population)
for (seed, norm_energy) in zip(fast_fuzzer.population, fast_energy):
print("'%s', %0.5f, %s" % (getPathID(seed.coverage), # type: ignore
norm_energy, repr(seed.data)))
```

Exactly. Our new exponential power schedule assigns most energy to the seed exercising the lowest-frequency path.

Let's compare them in terms of coverage achieved over time for our simple example.

In [49]:

```
_, orig_coverage = population_coverage(orig_fuzzer.inputs, crashme)
_, fast_coverage = population_coverage(fast_fuzzer.inputs, crashme)
line_orig, = plt.plot(orig_coverage, label="Original Greybox Fuzzer")
line_fast, = plt.plot(fast_coverage, label="Boosted Greybox Fuzzer")
plt.legend(handles=[line_orig, line_fast])
plt.title('Coverage over time')
plt.xlabel('# of inputs')
plt.ylabel('lines covered');
```

As expected, the boosted greybox fuzzer (with the exponential power schedule) achieves coverage much faster.

***Summary***. By fuzzing seeds more often that exercise low-frequency paths, we can explore program paths in a much more efficient manner.

***Try it***. You can try other exponents for the fast power schedule, or change the power schedule entirely. Note that a large exponent can lead to overflows and imprecisions in the floating point arithmetic producing unexpected results. You can execute your own code by opening this chapter as Jupyter notebook.

***Read***. You can find out more about fuzzer boosting in the paper "Coverage-based Greybox Fuzzing as Markov Chain" \cite{boehme2018greybox} and check out the implementation into AFL at [http://github.com/mboehme/aflfast].

Let's compare the three fuzzers on a more realistic example, the Python HTML parser. We run all three fuzzers $n=5k$ times on the HTMLParser, starting with the "empty" seed.

In [51]:

```
# create wrapper function
def my_parser(inp: str) -> None:
parser = HTMLParser() # resets the HTMLParser object for every fuzz input
parser.feed(inp)
```

In [52]:

```
n = 5000
seed_input = " " # empty seed
blackbox_fuzzer = AdvancedMutationFuzzer([seed_input], Mutator(), PowerSchedule())
greybox_fuzzer = GreyboxFuzzer([seed_input], Mutator(), PowerSchedule())
boosted_fuzzer = CountingGreyboxFuzzer([seed_input], Mutator(), AFLFastSchedule(5))
```

In [53]:

```
start = time.time()
blackbox_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n)
greybox_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n)
boosted_fuzzer.runs(FunctionCoverageRunner(my_parser), trials=n)
end = time.time()
"It took all three fuzzers %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[53]:

'It took all three fuzzers 12.60 seconds to generate and execute 5000 inputs.'

How do the fuzzers compare in terms of coverage over time?

In [54]:

```
_, black_coverage = population_coverage(blackbox_fuzzer.inputs, my_parser)
_, grey_coverage = population_coverage(greybox_fuzzer.inputs, my_parser)
_, boost_coverage = population_coverage(boosted_fuzzer.inputs, my_parser)
line_black, = plt.plot(black_coverage, label="Blackbox Fuzzer")
line_grey, = plt.plot(grey_coverage, label="Greybox Fuzzer")
line_boost, = plt.plot(boost_coverage, label="Boosted Greybox Fuzzer")
plt.legend(handles=[line_boost, line_grey, line_black])
plt.title('Coverage over time')
plt.xlabel('# of inputs')
plt.ylabel('lines covered');
```

In [55]:

```
blackbox_fuzzer.inputs[-10:]
```

Out[55]:

[' H', '', '', '`', ' i', '', '(', 'j ', '', '0']

In [56]:

```
greybox_fuzzer.inputs[-10:]
```

Out[56]:

['m*\x08', 'r.5<)h', '</F/aG>iq', '\x11G', '5<n5', 'i&<d$', '/Wi<`<4Gxs', '4$<?B\x16g', '$G<!?Bg', '\x06!|$v']

The greybox fuzzer executes much more complicated inputs, many of which include special characters such as opening and closing brackets and chevrons (i.e., `<, >, [, ]`

). Yet, many important keywords, such as `<html>`

are still missing.

To inform the fuzzer about these important keywords, we will need grammars; in the section on smart greybox fuzzing, we combine them with the techniques above.

***Try it***. You can re-run these experiments to understand the variance of fuzzing experiments. Sometimes, the fuzzer that we claim to be superior does not seem to outperform the inferior fuzzer. In order to do this, you just need to open this chapter as Jupyter notebook.

Sometimes, you just want the fuzzer to reach some dangerous location in the source code. This could be a location where you expect a buffer overflow. Or you want to test a recent change in your code base. How do we direct the fuzzer towards these locations?

In this chapter, we introduce directed greybox fuzzing as an optimization problem.

To provide a meaningful example where you can easily change the code complexity and target location, we generate the maze source code from the maze provided as string. This example is loosely based on an old blog post on symbolic execution by Felipe Andres Manzano (Quick shout-out!).

You simply specify the maze as a string. Like so.

In [57]:

```
maze_string = """
+-+-----+
|X| |
| | --+ |
| | | |
| +-- | |
| |#|
+-----+-+
"""
```

`generate_maze_code()`

. We'll hide the implementation and instead explain what it does. If you are interested in the coding, go here.

In [59]:

```
# ignore
def maze(s: str) -> str:
return "" # Will be overwritten by exec()
```

In [60]:

```
# ignore
def target_tile() -> str:
return ' ' # Will be overwritten by exec()
```

In [61]:

```
maze_code = generate_maze_code(maze_string)
```

In [62]:

```
exec(maze_code)
```

`D`

for down, `U`

for up, `L`

for left, and `R`

for right.

In [63]:

```
print(maze("DDDDRRRRUULLUURRRRDDDD")) # Appending one more 'D', you have reached the target.
```

SOLVED +-+-----+ | | | | | --+ | | | | | | +-- | | | |X| +-----+-+

Each character in `maze_string`

represents a tile. For each tile, a tile-function is generated.

- If the current tile is "benign" (
`), the tile-function corresponding to the next input character (D, U, L, R) is called. Unexpected input characters are ignored. If no more input characters are left, it returns "VALID" and the current maze state.`

- If the current tile is a "trap" (
`+`

,`|`

,`-`

), it returns "INVALID" and the current maze state. - If the current tile is the "target" (
`#`

), it returns "SOLVED" and the current maze state.

***Try it***. You can test other sequences of input characters, or even change the maze entirely. In order to execute your own code, you just need to open this chapter as Jupyter notebook.

To get an idea of the generated code, lets look at the static call graph. A call graph shows the order in which functions can be executed.

In [65]:

```
callgraph(maze_code)
```

Out[65]:

We introduce a `DictMutator`

class which mutates strings by inserting a keyword from a given dictionary:

In [66]:

```
class DictMutator(Mutator):
"""Variant of `Mutator` inserting keywords from a dictionary"""
def __init__(self, dictionary: Sequence[str]) -> None:
"""Constructor.
`dictionary` - a list of strings that can be used as keywords
"""
super().__init__()
self.dictionary = dictionary
self.mutators.append(self.insert_from_dictionary)
def insert_from_dictionary(self, s: str) -> str:
"""Returns `s` with a keyword from the dictionary inserted"""
pos = random.randint(0, len(s))
random_keyword = random.choice(self.dictionary)
return s[:pos] + random_keyword + s[pos:]
```

`DictMutator`

class to append dictionary keywords to the end of the seed and to remove a character from the end of the seed.

In [67]:

```
class MazeMutator(DictMutator):
def __init__(self, dictionary: Sequence[str]) -> None:
super().__init__(dictionary)
self.mutators.append(self.delete_last_character)
self.mutators.append(self.append_from_dictionary)
def append_from_dictionary(self, s: str) -> str:
"""Returns s with a keyword from the dictionary appended"""
random_keyword = random.choice(self.dictionary)
return s + random_keyword
def delete_last_character(self, s: str) -> str:
"""Returns s without the last character"""
if len(s) > 0:
return s[:-1]
return s
```

In [68]:

```
n = 20000
seed_input = " " # empty seed
maze_mutator = MazeMutator(["L", "R", "U", "D"])
maze_schedule = PowerSchedule()
maze_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, maze_schedule)
start = time.time()
maze_fuzzer.runs(FunctionCoverageRunner(maze), trials=n)
end = time.time()
"It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[68]:

'It took the fuzzer 6.95 seconds to generate and execute 20000 inputs.'

We will need to print statistics for several fuzzers. Why don't we define a function for that?

In [69]:

```
def print_stats(fuzzer: GreyboxFuzzer) -> None:
total = len(fuzzer.population)
solved = 0
invalid = 0
valid = 0
for seed in fuzzer.population:
s = maze(str(seed.data))
if "INVALID" in s:
invalid += 1
elif "VALID" in s:
valid += 1
elif "SOLVED" in s:
solved += 1
if solved == 1:
print("First solution: %s" % repr(seed))
else:
print("??")
print("""Out of %d seeds,
* %4d solved the maze,
* %4d were valid but did not solve the maze, and
* %4d were invalid""" % (total, solved, valid, invalid))
```

How well does our good, old greybox fuzzer do?

In [70]:

```
print_stats(maze_fuzzer)
```

It probably didn't solve the maze a single time. How can we make the fuzzer aware how "far" a seed is from reaching the target? If we know that, we can just assign more energy to that seed.

***Try it***. Print the statistics for the boosted fuzzer using the `AFLFastSchedule`

and the `CountingGreyboxFuzzer`

. It will likely perform much better than the unboosted greybox fuzzer: The lowest-probablity path happens to be also the path which reaches the target. You can execute your own code by opening this chapter as Jupyter notebook.

Using the static call graph for the maze code and the target function, we can compute the distance of each function $f$ to the target $t$ as the length of the shortest path between $f$ and $t$.

Fortunately, the generated maze code includes a function called `target_tile`

which returns the name of the target-function.

In [71]:

```
target = target_tile()
target
```

Out[71]:

'tile_6_7'

`get_callgraph`

returns the call graph for the maze code as networkx graph. Networkx provides some useful functions for graph analysis.

In [74]:

```
cg = get_callgraph(maze_code)
for node in cg.nodes():
if target in node:
target_node = node
break
```

In [75]:

```
target_node
```

Out[75]:

'callgraphX__tile_6_7'

We can now generate the function-level distance. The dictionary `distance`

contains for each function the distance to the target-function. If there is no path to the target, we assign a maximum distance (`0xFFFF`

).

The function `nx.shortest_path_length(CG, node, target_node)`

returns the length of the shortest path from function `node`

to function `target_node`

in the call graph `CG`

.

In [76]:

```
distance = {}
for node in cg.nodes():
if "__" in node:
name = node.split("__")[-1]
else:
name = node
try:
distance[name] = nx.shortest_path_length(cg, node, target_node)
except:
distance[name] = 0xFFFF
```

These are the distance values for all tile-functions on the path to the target function.

In [77]:

```
{k: distance[k] for k in list(distance) if distance[k] < 0xFFFF}
```

Out[77]:

{'callgraphX': 1, 'maze': 23, 'tile_2_1': 22, 'tile_2_3': 8, 'tile_2_4': 7, 'tile_2_5': 6, 'tile_2_6': 5, 'tile_2_7': 4, 'tile_3_1': 21, 'tile_3_3': 9, 'tile_3_7': 3, 'tile_4_1': 20, 'tile_4_3': 10, 'tile_4_4': 11, 'tile_4_5': 12, 'tile_4_7': 2, 'tile_5_1': 19, 'tile_5_5': 13, 'tile_5_7': 1, 'tile_6_1': 18, 'tile_6_2': 17, 'tile_6_3': 16, 'tile_6_4': 15, 'tile_6_5': 14, 'tile_6_7': 0}

***Summary***. Using the static call graph and the target function $t$, we have shown how to compute the function-level distance of each function $f$ to the target $t$.

***Try it***. You can try and execute your own code by opening this chapter as Jupyter notebook.

- How do we compute distance if there are multiple targets? (Hint: Geometric Mean).
- Given the call graph (CG) and the control-flow graph (CFG$_f$) for each function $f$, how do we compute basic-block (BB)-level distance? (Hint: In CFG$_f$, measure the BB-level distance to
*calls*of functions on the path to the target function. Remember that BB-level distance in functions with higher function-level distance is higher, too.)

***Read***. If you are interested in other aspects of search, you can follow up by reading the chapter on Search-based Fuzzing. If you are interested, how to solve the problems above, you can have a look at our paper on "Directed Greybox Fuzzing".

Now that we know how to compute the function-level distance, let's try to implement a power schedule that assigns *more energy to seeds with a lower average distance* to the target function. Notice that the distance values are all *pre-computed*. These values are injected into the program binary, just like the coverage instrumentation. In practice, this makes the computation of the average distance *extremely efficient*.

If you really want to know. Given the function-level distance $d_f(s,t)$ of a function $s$ to a function $t$ in call graph $CG$, our directed power schedule computes the seed distance $d(i,t)$ for a seed $i$ to function $t$ as $d(i,t)=\dfrac{\sum_{s\in CG} d_f(s,t)}{|CG|}$ where $|CG|$ is the number of nodes in the call graph $CG$.

In [78]:

```
class DirectedSchedule(PowerSchedule):
"""Assign high energy to seeds close to some target"""
def __init__(self, distance: Dict[str, int], exponent: float) -> None:
self.distance = distance
self.exponent = exponent
def __getFunctions__(self, coverage: Set[Location]) -> Set[str]:
functions = set()
for f, _ in set(coverage):
functions.add(f)
return functions
def assignEnergy(self, population: Sequence[Seed]) -> None:
"""Assigns each seed energy inversely proportional
to the average function-level distance to target."""
for seed in population:
if seed.distance < 0:
num_dist = 0
sum_dist = 0
for f in self.__getFunctions__(seed.coverage):
if f in list(self.distance):
sum_dist += self.distance[f]
num_dist += 1
seed.distance = sum_dist / num_dist
seed.energy = (1 / seed.distance) ** self.exponent
```

Let's see how the directed schedule performs against the good, old greybox fuzzer.

In [79]:

```
directed_schedule = DirectedSchedule(distance, 3)
directed_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, directed_schedule)
start = time.time()
directed_fuzzer.runs(FunctionCoverageRunner(maze), trials=n)
end = time.time()
"It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[79]:

'It took the fuzzer 9.02 seconds to generate and execute 20000 inputs.'

In [80]:

```
print_stats(directed_fuzzer)
```

It probably didn't solve a single maze either, but we have more valid solutions. So, there is definitely progress.

Let's have a look at the distance values for each seed.

In [81]:

```
y = [seed.distance for seed in directed_fuzzer.population] # type: ignore
x = range(len(y))
plt.scatter(x, y)
plt.ylim(0, max(y))
plt.xlabel("Seed ID")
plt.ylabel("Distance");
```

Let's normalize the y-axis and improve the importance of the small distance seeds.

The improved directed schedule normalizes seed distance between the minimal and maximal distance. Again, if you really want to know. Given the seed distance $d(i,t)$ of a seed $i$ to a function $t$, our improved power schedule computes the new seed distance $d'(i,t)$ as $$ d'(i,t)=\begin{cases} 1 & \text{if } d(i,t) = \text{minD} = \text{maxD}\\ \text{maxD} - \text{minD} & \text{if } d(i,t) = \text{minD} \neq \text{maxD}\\ \frac{\text{maxD} - \text{minD}}{d(i,t)-\text{minD}} & \text{otherwise} \end{cases} $$ where $$\text{minD}=\min_{i\in T}[d(i,t)]$$ and $$\text{maxD}=\max_{i\in T}[d(i,t)]$$ where $T$ is the set of seeds (i.e., the population).

In [82]:

```
class AFLGoSchedule(DirectedSchedule):
"""Assign high energy to seeds close to the target"""
def assignEnergy(self, population: Sequence[Seed]):
"""Assigns each seed energy inversely proportional
to the average function-level distance to target."""
min_dist: Union[int, float] = 0xFFFF
max_dist: Union[int, float] = 0
for seed in population:
if seed.distance < 0:
num_dist = 0
sum_dist = 0
for f in self.__getFunctions__(seed.coverage):
if f in list(self.distance):
sum_dist += self.distance[f]
num_dist += 1
seed.distance = sum_dist / num_dist
if seed.distance < min_dist:
min_dist = seed.distance
if seed.distance > max_dist:
max_dist = seed.distance
for seed in population:
if seed.distance == min_dist:
if min_dist == max_dist:
seed.energy = 1
else:
seed.energy = max_dist - min_dist
else:
seed.energy = (max_dist - min_dist) / (seed.distance - min_dist)
```

Let's see how the improved power schedule performs.

In [83]:

```
aflgo_schedule = AFLGoSchedule(distance, 3)
aflgo_fuzzer = GreyboxFuzzer([seed_input], maze_mutator, aflgo_schedule)
start = time.time()
aflgo_fuzzer.runs(FunctionCoverageRunner(maze), trials=n)
end = time.time()
"It took the fuzzer %0.2f seconds to generate and execute %d inputs." % (end - start, n)
```

Out[83]:

'It took the fuzzer 16.65 seconds to generate and execute 20000 inputs.'

In [84]:

```
print_stats(aflgo_fuzzer)
```

In contrast to all previous power schedules, this one generates hundreds of solutions. It has generated many solutions.

Let's filter out all ignored input characters from the first solution. The function `filter(f, seed.data)`

returns a list of elements `e`

in `seed.data`

where the function `f`

applied on `e`

returns True.

In [85]:

```
for seed in aflgo_fuzzer.population:
s = maze(str(seed.data))
if "SOLVED" in s:
filtered = "".join(list(filter(lambda c
```