In the chapter on information flow, we have seen how one can use dynamic taints to produce more intelligent test cases than simply looking for program crashes. We have also seen how one can use the taints to update the grammar, and hence focus more on the dangerous methods.

*Concolic execution* offers a solution here. The idea of *concolic execution* over a function is as follows: We start with a sample input for the function, and execute the function under trace. At each point the execution passes through a conditional, we *save the conditional encountered* in the form of *relations between symbolic variables.* Here, a *symbolic variable* can be thought of as a sort of placeholder for the real variable, sort of like the x in solving for x in Algebra. The symbolic variables can be used to specify relations without actually solving them.

**Prerequisites**

- You should have read the chapter on coverage.
- You should have read the chapter on information flow.
- A familiarity with the basic idea of SMT solvers would be useful.

For example, say we have a function `factorial()`

that returns the *factorial value* of its input.

In [4]:

```
def factorial(n):
if n < 0:
return None
if n == 0:
return 1
if n == 1:
return 1
v = 1
while n != 0:
v = v * n
n = n - 1
return v
```

We exercise the function with a value of `5`

.

In [5]:

```
factorial(5)
```

Out[5]:

120

*coverage obtained*. First we need to extend the `Coverage`

class from the chapter on coverage to provide us with coverage arcs.

In [8]:

```
class ArcCoverage(Coverage):
def traceit(self, frame, event, args):
if event != 'return':
f = inspect.getframeinfo(frame)
self._trace.append((f.function, f.lineno))
return self.traceit
def arcs(self):
t = [i for f, i in self._trace]
return list(zip(t, t[1:]))
```

Next, we use the `Tracer`

to obtain the coverage arcs.

In [9]:

```
with ArcCoverage() as cov:
factorial(5)
```

We can now use the coverage arcs to visualize the coverage obtained.

In [11]:

```
to_graph(gen_cfg(inspect.getsource(factorial)), arcs=cov.arcs())
```

Out[11]:

`[1, 2, 5, 8, 11, 12, 13, 14]`

is covered (green) but sub-paths such as `[2, 3]`

, `[5, 6]`

and `[8, 9]`

are unexplored (red). What we need is the ability to generate inputs such that the `True`

branch is taken at `2`

. How do we do that?

One way to cover additional branches is to look at the execution path being taken, and collect the *conditional constraints* that the path encounters. Then we can try to produce inputs that lead us to taking the non-traversed path.

First, let us step through the function.

In [12]:

```
lines = [i[1] for i in cov._trace if i[0] == 'factorial']
src = {i + 1: s for i, s in enumerate(
inspect.getsource(factorial).split('\n'))}
```

- The line (1) is simply the entry point of the function. We know that the input is
`n`

, which is an integer.

In [13]:

```
src[1]
```

Out[13]:

'def factorial(n):'

- The line (2) is a predicate
`n < 0`

. Since the next line taken is line (5), we know that at this point in the execution path, the predicate was`false`

.

In [14]:

```
src[2], src[3], src[4], src[5]
```

Out[14]:

(' if n < 0:', ' return None', '', ' if n == 0:')

`true`

branch was not taken. How do we generate a value that takes the `true`

branch here? One way is to use symbolic variables to represent the input, encode the constraint, and use an *SMT Solver* to solve the negation of the constraint.

`x`

in solving for `x`

in Algebra. These variables can be used to encode constraints placed on the variables in the program. We identify what constraints the variable is supposed to obey, and finally produce a value that obeys all constraints imposed.

*Satisfiability Modulo Theories* (SMT) solver. An SMT solver is built on top of a *SATISFIABILITY* (SAT) solver. A SAT solver is being used to check whether boolean formulas in first order logic (e.g. `(a | b ) & (~a | ~b)`

) can be satisfied using any assignments for the variables (e.g `a = true, b = false`

). An SMT solver extends these SAT solvers to specific background theories -- for example, *theory of integers*, or *theory of strings*. That is, given a string constraint expressed as a formula with string variables (e.g. `h + t == 'hello,world'`

), an SMT solver that understands *theory of strings* can be used to check if that constraint can be satisfied, and if satisfiable, provide an instantiation of concrete values for the variables used in the formula (e.g `h = 'hello,', t = 'world'`

).

We use the SMT solver Z3 in this chapter.

In [16]:

```
z3_ver = z3.get_version()
```

In [17]:

```
print(z3_ver)
```

(4, 11, 2, 0)

In [18]:

```
assert z3_ver >= (4, 8, 13, 0), \
f"Please install z3-solver 4.8.13.0 or later - you have {z3_ver}"
```

`z3str3`

solver. Further, we set the timeout for Z3 computations to 30 seconds.

In [19]:

```
# z3.set_option('smt.string_solver', 'z3str3')
z3.set_option('timeout', 30 * 1000) # milliseconds
```

`zn`

a placeholder for the Z3 symbolic integer variable `n`

.

In [20]:

```
zn = z3.Int('n')
```

`(n < 0)`

from line 2 in `factorial()`

? We can now encode the constraint as follows.

In [21]:

```
zn < 0
```

Out[21]:

n < 0

`factorial(5)`

. We saw that with input `5`

, the execution took the `else`

branch on the predicate `n < 0`

. We can express this observation as follows.

In [22]:

```
z3.Not(zn < 0)
```

Out[22]:

¬(n < 0)

`z3.solve()`

method checks if the constraints are satisfiable; if they are, it also provides values for variables such that the constraints are satisfied. For example, we can ask Z3 for an input that will take the `else`

branch as follows:

In [23]:

```
z3.solve(z3.Not(zn < 0))
```

[n = 0]

*a solution* (albeit a trivial one). SMT solvers can be used to solve much harder problems. For example, here is how one can solve a quadratic equation.

In [24]:

```
x = z3.Real('x')
eqn = (2 * x**2 - 11 * x + 5 == 0)
z3.solve(eqn)
```

[x = 5]

Again, this is *one solution*. We can ask z3 to give us another solution as follows.

In [25]:

```
z3.solve(x != 5, eqn)
```

[x = 1/2]

Indeed, both `x = 5`

and `x = 1/2`

are solutions to the quadratic equation $2x^2 -11x + 5 = 0$

*Z3* for an input that satisfies the constraint encoded in line 2 of `factorial()`

so that we take the `if`

branch.

In [26]:

```
z3.solve(zn < 0)
```

[n = -1]

`-1`

as an input to `factorial()`

, it is guaranteed to take the `if`

branch in line 2 during execution.

Let us try using that with our coverage. Here, the `-1`

is the solution from above.

In [27]:

```
with cov as cov:
factorial(-1)
```

In [28]:

```
to_graph(gen_cfg(inspect.getsource(factorial)), arcs=cov.arcs())
```

Out[28]:

Ok, so we have managed to cover a little more of the graph. Let us continue with our original input of `factorial(5)`

:

- In line (5) we encounter a new predicate
`n == 0`

, for which we again took the false branch.

In [29]:

```
src[5]
```

Out[29]:

' if n == 0:'

The predicates required, to follow the path until this point are as follows.

In [30]:

```
predicates = [z3.Not(zn < 0), z3.Not(zn == 0)]
```

- If we continue to line (8), we encounter another predicate, for which again, we took the
`false`

branch

In [31]:

```
src[8]
```

Out[31]:

' if n == 1:'

The predicates encountered so far are as follows

In [32]:

```
predicates = [z3.Not(zn < 0), z3.Not(zn == 0), z3.Not(zn == 1)]
```

In [33]:

```
last = len(predicates) - 1
z3.solve(predicates[0:-1] + [z3.Not(predicates[-1])])
```

[n = 1]

What we are doing here is tracing the execution corresponding to a particular input `factorial(5)`

, using concrete values, and along with it, keeping *symbolic shadow variables* that enable us to capture the constraints. As we mentioned in the introduction, this particular method of execution where one tracks concrete execution using symbolic variables is called *Concolic Execution*.

How do we automate this process? One method is to use a similar infrastructure as that of the chapter on information flow, and use the Python inheritance to create *symbolic proxy objects* that can track the concrete execution.

Let us now define a class to *collect* symbolic variables and path conditions during an execution. The idea is to have a `ConcolicTracer`

class that is invoked in a `with`

block. To execute a function while tracing its path conditions, we need to *transform* its arguments, which we do by invoking functions through a `[]`

item access.

This is a typical usage of a `ConcolicTracer`

:

```
with ConcolicTracer as _:
_.[function](args, ...)
```

After execution, we can access the symbolic variables in the `decls`

attribute:

```
_.decls
```

whereas the `path`

attribute lists the precondition paths encountered:

```
_.path
```

The `context`

attribute contains a pair of declarations and paths:

```
_.context
```

If you read this for the first time, skip the implementation and head right to the examples.

We previously showed how to run `triangle()`

under `ConcolicTracer`

.

In [203]:

```
with ConcolicTracer() as _:
print(_[triangle](1, 2, 3))
```

scalene

The symbolic variables are as follows:

In [204]:

```
_.decls
```

Out[204]:

{'triangle_a_int_1': 'Int', 'triangle_b_int_2': 'Int', 'triangle_c_int_3': 'Int'}

The predicates are as follows:

In [205]:

```
_.path
```

Out[205]:

[Not(triangle_a_int_1 == triangle_b_int_2), Not(triangle_b_int_2 == triangle_c_int_3), Not(triangle_a_int_1 == triangle_c_int_3)]

`zeval()`

, we solve these path conditions and obtain a solution. We find that Z3 gives us three distinct integer values:

In [206]:

```
_.zeval()
```

Out[206]:

('sat', {'a': ('0', 'Int'), 'b': (['-', '2'], 'Int'), 'c': (['-', '1'], 'Int')})

`triangle()`

works with negative length values, too, even if real triangles only have positive lengths.)

If we invoke `triangle()`

with these very values, we take the *exact same path* as the original input:

In [207]:

```
triangle(0, -2, -1)
```

Out[207]:

'scalene'

*negate* individual conditions – and thus take different paths.
First, we retrieve the symbolic variables.

In [208]:

```
za, zb, zc = [z3.Int(s) for s in _.decls.keys()]
```

In [209]:

```
za, zb, zc
```

Out[209]:

(triangle_a_int_1, triangle_b_int_2, triangle_c_int_3)

`zeval()`

. The key (here: `1`

) determines which predicate the new predicate will replace.

In [210]:

```
_.zeval({1: zb == zc})
```

Out[210]:

('sat', {'a': ('1', 'Int'), 'b': ('0', 'Int'), 'c': ('0', 'Int')})

In [211]:

```
triangle(1, 0, 1)
```

Out[211]:

'isosceles'

`isosceles`

as expected. By negating further conditions, we can systematically explore all branches in `triangle()`

.

`ConcolicTracer`

on our example program `cgi_decode()`

from the chapter on coverage. Note that we need to rewrite its code slightly, as the hash lookups in `hex_values`

can not be used for transferring constraints yet.

In [212]:

```
def cgi_decode(s):
"""Decode the CGI-encoded string `s`:
* replace "+" by " "
* replace "%xx" by the character with hex number xx.
Return the decoded string. Raise `ValueError` for invalid inputs."""
# Mapping of hex digits to their integer values
hex_values = {
'0': 0, '1': 1, '2': 2, '3': 3, '4': 4,
'5': 5, '6': 6, '7': 7, '8': 8, '9': 9,
'a': 10, 'b': 11, 'c': 12, 'd': 13, 'e': 14, 'f': 15,
'A': 10, 'B': 11, 'C': 12, 'D': 13, 'E': 14, 'F': 15,
}
t = ''
i = 0
while i < s.length():
c = s[i]
if c == '+':
t += ' '
elif c == '%':
digit_high, digit_low = s[i + 1], s[i + 2]
i = i + 2
found = 0
v = 0
for key in hex_values:
if key == digit_high:
found = found + 1
v = hex_values[key] * 16
break
for key in hex_values:
if key == digit_low:
found = found + 1
v = v + hex_values[key]
break
if found == 2:
if v >= 128:
# z3.StringVal(urllib.parse.unquote('%80')) <-- bug in z3
raise ValueError("Invalid encoding")
t = t + chr(v)
else:
raise ValueError("Invalid encoding")
else:
t = t + c
i = i + 1
return t
```

In [213]:

```
with ConcolicTracer() as _:
_[cgi_decode]('')
```

In [214]:

```
_.context
```

Out[214]:

({'cgi_decode_s_str_1': 'String'}, [Not(0 < Length(cgi_decode_s_str_1))])

In [215]:

```
with ConcolicTracer() as _:
_[cgi_decode]('a%20d')
```

`decls`

attribute. This is a mapping of symbolic variables to types.

In [216]:

```
_.decls
```

Out[216]:

{'cgi_decode_s_str_1': 'String'}

The extracted path conditions can be found in the `path`

attribute:

In [217]:

```
_.path
```

Out[217]:

[0 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 0, 1) == "+"), Not(str.substr(cgi_decode_s_str_1, 0, 1) == "%"), 1 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 1, 1) == "+"), str.substr(cgi_decode_s_str_1, 1, 1) == "%", Not(str.substr(cgi_decode_s_str_1, 2, 1) == "0"), Not(str.substr(cgi_decode_s_str_1, 2, 1) == "1"), str.substr(cgi_decode_s_str_1, 2, 1) == "2", str.substr(cgi_decode_s_str_1, 3, 1) == "0", 4 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 4, 1) == "+"), Not(str.substr(cgi_decode_s_str_1, 4, 1) == "%"), Not(5 < Length(cgi_decode_s_str_1))]

`context`

attribute holds a pair of `decls`

and `path`

attributes; this is useful for passing it into the `ConcolicTracer`

constructor.

In [218]:

```
assert _.context == (_.decls, _.path)
```

In [219]:

```
_.zeval()
```

Out[219]:

('sat', {'s': ('A%20B', 'String')})

*Negating* some of these constraints will yield different paths taken, and thus greater code coverage. This is what our concolic fuzzers (see later) do. Let us go and negate the first constraint, namely that the first character should *not* be a `+`

character:

In [220]:

```
_.path[0]
```

Out[220]:

0 < Length(cgi_decode_s_str_1)

To compute the negated string, we have to construct it via z3 primitives:

In [221]:

```
zs = z3.String('cgi_decode_s_str_1')
```

In [222]:

```
z3.SubString(zs, 0, 1) == z3.StringVal('a')
```

Out[222]:

str.substr(cgi_decode_s_str_1, 0, 1) = "a"

`zeval()`

with the path condition to be changed obtains a new input that satisfies the negated predicate:

In [223]:

```
(result, new_vars) = _.zeval({1: z3.SubString(zs, 0, 1) == z3.StringVal('+')})
```

In [224]:

```
new_vars
```

Out[224]:

{'s': ('+%20A', 'String')}

In [225]:

```
(new_s, new_s_type) = new_vars['s']
```

In [226]:

```
new_s
```

Out[226]:

'+%20A'

We can validate that `new_s`

indeed takes the new path by re-running the tracer with `new_s`

as input:

In [227]:

```
with ConcolicTracer() as _:
_[cgi_decode](new_s)
```

In [228]:

```
_.path
```

Out[228]:

[0 < Length(cgi_decode_s_str_1), str.substr(cgi_decode_s_str_1, 0, 1) == "+", 1 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 1, 1) == "+"), str.substr(cgi_decode_s_str_1, 1, 1) == "%", Not(str.substr(cgi_decode_s_str_1, 2, 1) == "0"), Not(str.substr(cgi_decode_s_str_1, 2, 1) == "1"), str.substr(cgi_decode_s_str_1, 2, 1) == "2", str.substr(cgi_decode_s_str_1, 3, 1) == "0", 4 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 4, 1) == "+"), Not(str.substr(cgi_decode_s_str_1, 4, 1) == "%"), Not(5 < Length(cgi_decode_s_str_1))]

By negating further conditions, we can explore more and more code.

Here is a function that gives you the nearest ten's multiplier

In [229]:

```
def round10(r):
while r % 10 != 0:
r += 1
return r
```

As before, we execute the function under the `ConcolicTracer`

context.

In [230]:

```
with ConcolicTracer() as _:
r = _[round10](1)
```

We verify that we were able to capture all the predicates:

In [231]:

```
_.context
```

Out[231]:

({'round10_r_int_1': 'Int'}, [0 != round10_r_int_1%10, 0 != (round10_r_int_1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1 + 1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)%10, 0 != (round10_r_int_1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)%10, Not(0 != (round10_r_int_1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1)%10)])

We use `zeval()`

to obtain more inputs that take the same path.

In [232]:

```
_.zeval()
```

Out[232]:

('sat', {'r': (['-', '9'], 'Int')})

Do our concolic proxies work across functions? Say we have a function `max_value()`

as below.

In [233]:

```
def abs_value(a):
if a > 0:
return a
else:
return -a
```

It is called by another function `abs_max()`

In [234]:

```
def abs_max(a, b):
a1 = abs_value(a)
b1 = abs_value(b)
if a1 > b1:
c = a1
else:
c = b1
return c
```

Using the `Concolic()`

context on `abs_max()`

.

In [235]:

```
with ConcolicTracer() as _:
_[abs_max](2, 1)
```

As expected, we have the predicates across functions.

In [236]:

```
_.context
```

Out[236]:

({'abs_max_a_int_1': 'Int', 'abs_max_b_int_2': 'Int'}, [0 < abs_max_a_int_1, 0 < abs_max_b_int_2, abs_max_a_int_1 > abs_max_b_int_2])

In [237]:

```
_.zeval()
```

Out[237]:

('sat', {'a': ('2', 'Int'), 'b': ('1', 'Int')})

Solving the predicates works as expected.

Using negative numbers as arguments so that a different branch is taken in `abs_value()`

In [238]:

```
with ConcolicTracer() as _:
_[abs_max](-2, -1)
```

In [239]:

```
_.context
```

Out[239]:

({'abs_max_a_int_1': 'Int', 'abs_max_b_int_2': 'Int'}, [Not(0 < abs_max_a_int_1), Not(0 < abs_max_b_int_2), -abs_max_a_int_1 > -abs_max_b_int_2])

In [240]:

```
_.zeval()
```

Out[240]:

('sat', {'a': (['-', '1'], 'Int'), 'b': ('0', 'Int')})

The solution reflects our predicates. (We used `a > 0`

in `abs_value()`

).

For a larger example that uses different kinds of variables, say we want to compute the binomial coefficient by the following formulas

$$ ^nP_k=\frac{n!}{(n-k)!} $$$$ \binom nk=\,^nC_k=\frac{^nP_k}{k!} $$we define the functions as follows.

In [241]:

```
def factorial(n): # type: ignore
v = 1
while n != 0:
v *= n
n -= 1
return v
```

In [242]:

```
def permutation(n, k):
return factorial(n) / factorial(n - k)
```

In [243]:

```
def combination(n, k):
return permutation(n, k) / factorial(k)
```

In [244]:

```
def binomial(n, k):
if n < 0 or k < 0 or n < k:
raise Exception('Invalid values')
return combination(n, k)
```

As before, we run the function under `ConcolicTracer`

.

In [245]:

```
with ConcolicTracer() as _:
v = _[binomial](4, 2)
```

Then call `zeval()`

to evaluate.

In [246]:

```
_.zeval()
```

Out[246]:

('sat', {'n': ('4', 'Int'), 'k': ('2', 'Int')})

`zstr`

, we use the DB class from the chapter on information flow.

In [247]:

```
if __name__ == '__main__':
if z3.get_version() > (4, 8, 7, 0):
print("""Note: The following example may not work with your Z3 version;
see https://github.com/Z3Prover/z3/issues/5763 for details.
Consider `pip install z3-solver==4.8.7.0` as a workaround.""")
```

We first populate our database.

In [250]:

```
db = sample_db()
for V in VEHICLES:
update_inventory(db, V)
```

In [251]:

```
db.db
```

Out[251]:

{'inventory': ({'year': int, 'kind': str, 'company': str, 'model': str}, [{'year': 1997, 'kind': 'van', 'company': 'Ford', 'model': 'E350'}, {'year': 2000, 'kind': 'car', 'company': 'Mercury', 'model': 'Cougar'}, {'year': 1999, 'kind': 'car', 'company': 'Chevy', 'model': 'Venture'}])}

`DB`

class. Hash functions are difficult to handle directly (because they rely on internal C functions). Hence we modify `table()`

slightly.

In [252]:

```
class ConcolicDB(DB):
def table(self, t_name):
for k, v in self.db:
if t_name == k:
return v
raise SQLException('Table (%s) was not found' % repr(t_name))
def column(self, decl, c_name):
for k in decl:
if c_name == k:
return decl[k]
raise SQLException('Column (%s) was not found' % repr(c_name))
```

To make it easy, we define a single function `db_select()`

that directly invokes `db.sql()`

.

In [253]:

```
def db_select(s):
my_db = ConcolicDB()
my_db.db = [(k, v) for (k, v) in db.db.items()]
r = my_db.sql(s)
return r
```

We now want to run SQL statements under our `ConcolicTracer`

, and collect predicates obtained.

In [254]:

```
with ConcolicTracer() as _:
_[db_select]('select kind from inventory')
```

The predicates encountered during the execution are as follows:

In [255]:

```
_.path
```

Out[255]:

[0 == IndexOf(db_select_s_str_1, "select ", 0), 0 == IndexOf(db_select_s_str_1, "select ", 0), Not(0 > IndexOf(str.substr(db_select_s_str_1, 7, 19), " from ", 0)), Not(Or(0 < IndexOf(str.substr(db_select_s_str_1, 7, 19), " where ", 0), 0 == IndexOf(str.substr(db_select_s_str_1, 7, 19), " where ", 0))), str.substr(str.substr(db_select_s_str_1, 7, 19), 10, 9) == "inventory"]

We can use `zeval()`

as before to solve the constraints.

In [256]:

```
_.zeval()
```

Out[256]:

('Gave up', None)

The `SimpleConcolicFuzzer`

class starts with a sample input generated by some other fuzzer. It then runs the function being tested under `ConcolicTracer`

, and collects the path predicates. It then negates random predicates within the path and solves it with Z3 to produce a new output that is guaranteed to take a different path than the original.

`ConcolicTracer`

, above, please first look at the examples before digging into the implementation.

`SimpleConcolicFuzzer`

, let us apply it on our example program `cgi_decode()`

from the `Coverage`

chapter. Note that we cannot use it directly as the hash lookups in `hex_values`

can not be used for transferring constraints yet.

In [293]:

```
with ConcolicTracer() as _:
_[cgi_decode]('a+c')
```

In [294]:

```
_.path
```

Out[294]:

[0 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 0, 1) == "+"), Not(str.substr(cgi_decode_s_str_1, 0, 1) == "%"), 1 < Length(cgi_decode_s_str_1), str.substr(cgi_decode_s_str_1, 1, 1) == "+", 2 < Length(cgi_decode_s_str_1), Not(str.substr(cgi_decode_s_str_1, 2, 1) == "+"), Not(str.substr(cgi_decode_s_str_1, 2, 1) == "%"), Not(3 < Length(cgi_decode_s_str_1))]

In [295]:

```
scf = SimpleConcolicFuzzer()
scf.add_trace(_, 'a+c')
```

*trace tree* shows the path conditions encountered so far. Any blue edge towards a "?" implies that there is a path not yet taken.

In [296]:

```
display_trace_tree(scf.ct.root)
```

Out[296]:

So, we fuzz to get a new path that is not empty.

In [297]:

```
v = scf.fuzz()
print(v)
```

A+

We can now obtain the new trace as before.

In [298]:

```
with ExpectError():
with ConcolicTracer() as _:
_[cgi_decode](v)
```

The new trace is added to our fuzzer using `add_trace()`

In [299]:

```
scf.add_trace(_, v)
```

The updated binary tree is as follows. Note the difference between the child nodes of `Root`

node.

In [300]:

```
display_trace_tree(scf.ct.root)
```

Out[300]:

A complete fuzzer run is as follows:

In [301]:

```
scf = SimpleConcolicFuzzer()
for i in range(10):
v = scf.fuzz()
print(repr(v))
if v is None:
continue
with ConcolicTracer() as _:
with ExpectError(print_traceback=False):
# z3.StringVal(urllib.parse.unquote('%80')) <-- bug in z3
_[cgi_decode](v)
scf.add_trace(_, v)
```

' ' '' '+' '%' '+A' '++' 'AB' '++A' 'A%' '+AB'

IndexError: string index out of range (expected) IndexError: string index out of range (expected)

In [302]:

```
display_trace_tree(scf.ct.root)
```

Out[302]:

**Note.** Our concolic tracer is limited in that it does not track changes in the string length. This leads it to treat every string with same prefix as the same string.

`SimpleConcolicFuzzer`

is reasonably efficient at exploring paths near the path followed by a given sample input. However, it is not very intelligent when it comes to choosing which paths to follow. We look at another fuzzer that lifts the predicates obtained to the grammar and achieves better fuzzing.

The concolic framework can be used directly in grammar-based fuzzing. We implement a class `ConcolicGrammarFuzzer`

wihich does this.

The `ConcolicGrammarFuzzer`

is used as follows.

In [336]:

```
cgf = ConcolicGrammarFuzzer(INVENTORY_GRAMMAR)
cgf.prune_tokens(prune_tokens)
for i in range(10):
query = cgf.fuzz()
print(query)
with ConcolicTracer() as _:
with ExpectError(print_traceback=False):
try:
res = _[db_select](query)
print(repr(res))
except SQLException as e:
print(e)
cgf.update_grammar(_)
print()
```

`vehicles`

, `months`

and `years`

, but identifies it from the concolic execution, and lifts it to the grammar. This allows us to improve the effectiveness of fuzzing.

As with dynamic taint analysis, implicit control flow can obscure the predicates encountered during concolic execution. However, this limitation could be overcome to some extent by wrapping any constants in the source with their respective proxy objects. Similarly, calls to internal C functions can cause the symbolic information to be discarded, and only partial information may be obtained.

`SimpleConcolicFuzzer`

and `ConcolicGrammarFuzzer`

. The `SimpleConcolicFuzzer`

first uses a sample input to collect predicates encountered. The fuzzer then negates random predicates to generate new input constraints. These, when solved, produce inputs that explore paths that are close to the original path.

At the heart of both fuzzers lies the concept of a *concolic tracer*, capturing symbolic variables and path conditions as a program gets executed.

`ConcolicTracer`

is used in a `with`

block; the syntax `tracer[function]`

executes `function`

within the `tracer`

while capturing conditions. Here is an example for the `cgi_decode()`

function:

In [337]:

```
with ConcolicTracer() as _:
_[cgi_decode]('a%20d')
```

`decls`

attribute. This is a mapping of symbolic variables to types.

In [338]:

```
_.decls
```

Out[338]:

{'cgi_decode_s_str_1': 'String'}

The extracted path conditions can be found in the `path`

attribute:

In [339]:

```
_.path
```

Out[339]:

`context`

attribute holds a pair of `decls`

and `path`

attributes; this is useful for passing it into the `ConcolicTracer`

constructor.

In [340]:

```
assert _.context == (_.decls, _.path)
```

In [341]:

```
_.zeval()
```

Out[341]:

('sat', {'s': ('A%20B', 'String')})

`zeval()`

function also allows passing *alternate* or *negated* constraints. See the chapter for examples.

In [342]:

```
# ignore
from ClassDiagram import display_class_hierarchy
display_class_hierarchy(ConcolicTracer)
```

Out[342]:

The constraints obtained from `ConcolicTracer`

are added to the concolic fuzzer as follows:

In [343]:

```
scf = SimpleConcolicFuzzer()
scf.add_trace(_, 'a%20d')
```

The concolic fuzzer then uses the constraints added to guide its fuzzing as follows:

In [344]:

```
scf = SimpleConcolicFuzzer()
for i in range(20):
v = scf.fuzz()
if v is None:
break
print(repr(v))
with ExpectError(print_traceback=False):
with ConcolicTracer() as _:
_[cgi_decode](v)
scf.add_trace(_, v)
```

' ' '+' '%' '+A' 'AB' '++' '++A' '+++' 'A' '+A' '+++A'

IndexError: string index out of range (expected)

'+AB' '++' '%' '++AB' '++A+' '+A' '++' '+' '+%'

IndexError: string index out of range (expected) IndexError: string index out of range (expected)

We see how the additional inputs generated explore additional paths.

In [345]:

```
# ignore
display_class_hierarchy(SimpleConcolicFuzzer)
```

Out[345]:

The `SimpleConcolicFuzzer`

simply explores all paths near the original path traversed by the sample input. It uses a simple mechanism to explore the paths that are near the paths that it knows about, and other than code paths, knows nothing about the input.

The `ConcolicGrammarFuzzer`

on the other hand, knows about the input grammar, and can collect feedback from the subject under fuzzing. It can lift some constraints encountered to the grammar, enabling deeper fuzzing. It is used as follows:

In [347]:

```
cgf = ConcolicGrammarFuzzer(INVENTORY_GRAMMAR)
cgf.prune_tokens(prune_tokens)
for i in range(10):
query = cgf.fuzz()
print(query)
with ConcolicTracer() as _:
with ExpectError(print_traceback=False):
try:
res = _[db_select](query)
print(repr(res))
except SQLException as e:
print(e)
cgf.update_grammar(_)
print()
```

select 245 from :2 where r(_)-N+e>n Table (':2') was not found delete from months where Q/x/j/q(p)/H*h-B==cz Invalid WHERE ('Q/x/j/q(p)/H*h-B==cz') insert into vehicles (:b) values (22.72) Column (':b') was not found select i*q!=(4) from vehicles where L*S/l/u/b+b==W delete from vehicles where W/V!=A(f)+t<O*E/S-. Invalid WHERE ('W/V!=A(f)+t<O*E/S-.') update vehicles set kind=d,kind=r,model=w,kind=h,kind=. where g+h!=i!=(L-R) delete from o.8586 where ((5860.5))==az9W Table ('o.8586') was not found delete from months where ((e*c(g)>L+S))==((:+l<f*h)) Invalid WHERE ('((e*c(g)>L+S))==((:+l<f*h))') select h*y/G+i(X)<Z from vehicles where OV!=(b!=w) select (X),S,E from months where s<h(V,v)/p/M/O

TypeError: 'NotImplementedType' object is not callable (expected) TypeError: 'NotImplementedType' object is not callable (expected) TypeError: 'NotImplementedType' object is not callable (expected) TypeError: 'NotImplementedType' object is not callable (expected)

In [348]:

```
# ignore
display_class_hierarchy(ConcolicGrammarFuzzer)
```

Out[348]:

Concolic execution can often provide more information than taint analysis with respect to the program behavior. However, this comes at a much larger runtime cost. Hence, unlike taint analysis, real-time analysis is often not possible.

Similar to taint analysis, concolic execution also suffers from limitations such as indirect control flow and internal function calls.

Predicates from concolic execution can be used in conjunction with fuzzing to provide an even more robust indication of incorrect behavior than taints, and can be used to create grammars that are better at producing valid inputs.

A costlier but stronger alternative to concolic fuzzing is symbolic fuzzing. Similarly, search based fuzzing can often provide a cheaper exploration strategy than relying on SMT solvers to provide inputs slightly different from the current path.

The technique of concolic execution was originally used to inform and expand the scope of *symbolic execution* \cite{king1976symbolic}, a static analysis technique for program analysis. Laron et al. cite{Larson2003} was the first to use the concolic execution technique.

The idea of using proxy objects for collecting constraints was pioneered by Cadar et al. \cite{cadar2005execution}. The concolic execution technique for Python programs used in this chapter was pioneered by PeerCheck \cite{PeerCheck}, and Python Error Finder \cite{Barsotti2018}.

`zint`

binary operators, we asserted that the results were `int`

. However, that need not be the case. For example, division can result in `float`

. Hence, we need proxy objects for `float`

. Can you implement a similar proxy object for `float`

and fix the `zint`

binary operator definition?

**Solution.** The solution is as follows.

As in the case of `zint`

, we first open up `zfloat`

for extension.

In [349]:

```
class zfloat(float):
def __new__(cls, context, zn, v, *args, **kw):
return float.__new__(cls, v, *args, **kw)
```

We then implement the initialization methods.

In [350]:

```
class zfloat(zfloat):
@classmethod
def create(cls, context, zn, v=None):
return zproxy_create(cls, 'Real', z3.Real, context, zn, v)
def __init__(self, context, z, v=None):
self.z, self.v = z, v
self.context = context
```

The helper for when one of the arguments in a binary operation is not `float`

.

In [351]:

```
class zfloat(zfloat):
def _zv(self, o):
return (o.z, o.v) if isinstance(o, zfloat) else (z3.RealVal(o), o)
```

Coerce `float`

into bool value for use in conditionals.

In [352]:

```
class zfloat(zfloat):
def __bool__(self):
# force registering boolean condition
if self != 0.0:
return True
return False
```

Define the common proxy method for comparison methods

In [353]:

```
def make_float_bool_wrapper(fname, fun, zfun):
def proxy(self, other):
z, v = self._zv(other)
z_ = zfun(self.z, z)
v_ = fun(self.v, v)
return zbool(self.context, z_, v_)
return proxy
```

We apply the comparison methods on the defined `zfloat`

class.

In [354]:

```
FLOAT_BOOL_OPS = [
'__eq__',
# '__req__',
'__ne__',
# '__rne__',
'__gt__',
'__lt__',
'__le__',
'__ge__',
]
```

In [355]:

```
for fname in FLOAT_BOOL_OPS:
fun = getattr(float, fname)
zfun = getattr(z3.ArithRef, fname)
setattr(zfloat, fname, make_float_bool_wrapper(fname, fun, zfun))
```

Similarly, we define the common proxy method for binary operators.

In [356]:

```
def make_float_binary_wrapper(fname, fun, zfun):
def proxy(self, other):
z, v = self._zv(other)
z_ = zfun(self.z, z)
v_ = fun(self.v, v)
return zfloat(self.context, z_, v_)
return proxy
```

And apply them on `zfloat`

In [357]:

```
FLOAT_BINARY_OPS = [
'__add__',
'__sub__',
'__mul__',
'__truediv__',
# '__div__',
'__mod__',
# '__divmod__',
'__pow__',
# '__lshift__',
# '__rshift__',
# '__and__',
# '__xor__',
# '__or__',
'__radd__',
'__rsub__',
'__rmul__',
'__rtruediv__',
# '__rdiv__',
'__rmod__',
# '__rdivmod__',
'__rpow__',
# '__rlshift__',
# '__rrshift__',
# '__rand__',
# '__rxor__',
# '__ror__',
]
```

In [358]:

```
for fname in FLOAT_BINARY_OPS:
fun = getattr(float, fname)
zfun = getattr(z3.ArithRef, fname)
setattr(zfloat, fname, make_float_binary_wrapper(fname, fun, zfun))
```

These are used as follows.

In [359]:

```
with ConcolicTracer() as _:
za = zfloat.create(_.context, 'float_a', 1.0)
zb = zfloat.create(_.context, 'float_b', 0.0)
if za * zb:
print(1)
```

In [360]:

```
_.context
```

Out[360]:

({'float_a': 'Real', 'float_b': 'Real'}, [Not(float_a*float_b != 0)])

Finally, we fix the `zint`

binary wrapper to correctly create `zfloat`

when needed.

In [361]:

```
def make_int_binary_wrapper(fname, fun, zfun): # type: ignore
def proxy(self, other):
z, v = self._zv(other)
z_ = zfun(self.z, z)
v_ = fun(self.v, v)
if isinstance(v_, float):
return zfloat(self.context, z_, v_)
elif isinstance(v_, int):
return zint(self.context, z_, v_)
else:
assert False
return proxy
```

In [362]:

```
for fname in INT_BINARY_OPS:
fun = getattr(int, fname)
zfun = getattr(z3.ArithRef, fname)
setattr(zint, fname, make_int_binary_wrapper(fname, fun, zfun))
```

Checking whether it worked as expected.

In [363]:

```
with ConcolicTracer() as _:
v = _[binomial](4, 2)
```

In [364]:

```
_.zeval()
```

Out[364]:

('sat', {'n': ('4', 'Int'), 'k': ('2', 'Int')})

`xor`

involves converting `int`

to its bit vector equivalents, performing operations on them, and converting it back to the original type. Can you implement the bit manipulation operations for `zint`

?

**Solution.** The solution is as follows.

We first define the proxy method as before.

In [365]:

```
def make_int_bit_wrapper(fname, fun, zfun):
def proxy(self, other):
z, v = self._zv(other)
z_ = z3.BV2Int(
zfun(
z3.Int2BV(
self.z, num_bits=64), z3.Int2BV(
z, num_bits=64)))
v_ = fun(self.v, v)
return zint(self.context, z_, v_)
return proxy
```

It is then applied to the `zint`

class.

In [366]:

```
BIT_OPS = [
'__lshift__',
'__rshift__',
'__and__',
'__xor__',
'__or__',
'__rlshift__',
'__rrshift__',
'__rand__',
'__rxor__',
'__ror__',
]
```

In [367]:

```
def init_concolic_4():
for fname in BIT_OPS:
fun = getattr(int, fname)
zfun = getattr(z3.BitVecRef, fname)
setattr(zint, fname, make_int_bit_wrapper(fname, fun, zfun))
```

In [368]:

```
INITIALIZER_LIST.append(init_concolic_4)
```

In [369]:

```
init_concolic_4()
```

Invert is the only unary bit manipulation method.

In [370]:

```
class zint(zint):
def __invert__(self):
return zint(self.context, z3.BV2Int(
~z3.Int2BV(self.z, num_bits=64)), ~self.v)
```

The `my_fn()`

computes `xor`

and returns `True`

if the `xor`

results in a non-zero value.

In [371]:

```
def my_fn(a, b):
o_ = (a | b)
a_ = (a & b)
if o_ & ~a_:
return True
else:
return False
```

Using that under `ConcolicTracer`

In [372]:

```
with ConcolicTracer() as _:
print(_[my_fn](2, 1))
```

True

We log the computed SMT expression to verify that everything went well.

In [373]:

```
_.zeval(log=True)
```

Predicates in path: 0 0 != BV2Int(int2bv(BV2Int(int2bv(my_fn_a_int_1) | int2bv(my_fn_b_int_2))) & int2bv(BV2Int(~int2bv(BV2Int(int2bv(my_fn_a_int_1) & int2bv(my_fn_b_int_2)))))) (declare-const my_fn_a_int_1 Int) (declare-const my_fn_b_int_2 Int) (assert (let ((a!1 (bvnot (bvor (bvnot ((_ int2bv 64) my_fn_a_int_1)) (bvnot ((_ int2bv 64) my_fn_b_int_2)))))) (let ((a!2 (bvor (bvnot (bvor ((_ int2bv 64) my_fn_a_int_1) ((_ int2bv 64) my_fn_b_int_2))) a!1))) (not (= 0 (bv2int (bvnot a!2))))))) (check-sat) (get-model) z3 -t:6000 /var/folders/n2/xd9445p97rb3xh7m1dfx8_4h0006ts/T/tmpk6hn2b0y.smt sat ( (define-fun my_fn_a_int_1 () Int (- 1)) (define-fun my_fn_b_int_2 () Int (- 9223372036854775809)) )

Out[373]:

('sat', {'a': (['-', '1'], 'Int'), 'b': (['-', '9223372036854775809'], 'Int')})

We can confirm from the formulas generated that the bit manipulation functions worked correctly.

`upper()`

and `lower()`

. Can you define the `capitalize()`

, `title()`

, and `swapcase()`

methods?

**Solution.** Solution not yet available.