# Concolic Fuzzing¶

We have previously 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.

While taints are helpful, uninterpreted strings is only one of the attack vectors. Can we say anything more about the properties of variables at any point in the execution? For example, can we say for sure that a function will always receive the buffers with the correct length? Concolic execution offers a solution.

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.
(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.)

With concolic execution, one can collect the constraints that an execution path encounters, and use it to answer questions about the program behavior at any point we prefer along the program execution path. We can further use concolic execution to enhance fuzzing.

In this chapter, we explore in depth how to execute a Python function concolically, and how concolic execution can be used to enhance fuzzing.

**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.

We first set up our infrastructure so that we can make use of previously defined functions.

```
import fuzzingbook_utils
```

## Synopsis¶

To use the code provided in this chapter, write

```
>>> from fuzzingbook.ConcolicFuzzer import <identifier>
```

and then make use of the following features.

This chapter defines two main classes: `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. It can be used as follows.

We first obtain the constraints using `ConcolicTracer`

.

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

These constraints are added to the concolic fuzzer as follows:

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

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

```
>>> scf = SimpleConcolicFuzzer()
>>> for i in range(10):
>>> v = scf.fuzz()
>>> if v is None:
>>> break
>>> print(repr(v))
>>> with ExpectError():
>>> with ConcolicTracer() as _:
>>> _[cgi_decode](v)
>>> scf.add_trace(_, v)
' '
'+\\x00'
'++\\x00'
'+++\\x00'
'+\\x00%\\x00'
'+\\x00\\x00%\\x00'
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
'+\\x00\\x00\\x00\\x00\\x00\\x00\\x00\\x00%B\\x00'
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
'+\\x00\\x00\\x00\\x00\\x00\\x00\\x00\\x00%b\\x00'
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
'+\\x10\\x01\\x08\\x01\\x01\\x02\\x01\\x08%\\x000\\x00'
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
'+ \\x10\\x00@%\\x00E\\x00'
Traceback (most recent call last):
File "<ipython-input-287-2a3454213b54>", line 9, in <module>
_[cgi_decode](v)
File "<ipython-input-33-635d2b1b13c2>", line 3, in __call__
self.result = self.fn(*self.concolic(args))
File "<ipython-input-241-630ee123bed8>", line 42, in cgi_decode
raise ValueError("Invalid encoding")
ValueError: Invalid encoding (expected)
```

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 of the constraints encountered to the grammar, enabling deeper fuzzing. It is used as follows:

```
>>> from InformationFlow import INVENTORY_GRAMMAR, SQLException
>>> cgf = ConcolicGrammarFuzzer(INVENTORY_GRAMMAR)
>>> cgf.prune_tokens(prune_tokens)
>>> for i in range(10):
>>> query = cgf.fuzz()
>>> print(query)
>>> with ConcolicTracer() as _:
>>> with ExpectError():
>>> try:
>>> res = _[db_select](query)
>>> print(repr(res))
>>> except SQLException as e:
>>> print(e)
>>> cgf.update_grammar(_)
>>> print()
insert into Y (M871X,k1NK8,s) values ('$')
Table ('Y') was not found
insert into :5 (Xs4i,B,E5) values ('z')
Table (':5') was not found
select A,S from n
Table ('n') was not found
delete from vehicles where F(_)<.(c!=d-F*v<K(:)*./d)
Invalid WHERE ('F(_)<.(c!=d-F*v<K(:)*./d)')
delete from vehicles where F==A*o(F)-z<s+FA(a,m,B)
Invalid WHERE ('F==A*o(F)-z<s+FA(a,m,B)')
update vehicles set C1=L,X=s,f=I,l=O where S!=m
Column ('C1') was not found
insert into vehicles (f2,L3ucoo6p,JJI) values ('5')
Column ('f2') was not found
insert into vehicles (z,Rd,a) values ('zno',93.64)
Column ('z') was not found
delete from months where (u+K<_!=k*S)==w-O+d!=O==y
Invalid WHERE ('(u+K<_!=k*S)==w-O+d!=O==y')
select eV from vehicles where --8*:!=o>_0(V)-q(T,z)*((Y))
Invalid WHERE ('(--8*:!=o>_0(V)-q(T,z)*((Y)))')
```

## Tracking Constraints¶

In the chapter on information flow, we have seen how dynamic taints can be used to direct fuzzing by indicating which part of input reached interesting places. However, dynamic taint tracking is limited in the information that it can propagate. For example, we might want to explore what happens when certain properties of the input changes.

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

that returns the *factorial value* of its input.

```
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`

.

```
factorial(5)
```

Is this sufficient to explore all the features of the function? How do we know? One way to verify that we have explored all features is to look at the coverage obtained. First we need to extend the `Coverage`

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

```
from Coverage import Coverage
```

```
import inspect
```

```
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.

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

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

```
from ControlFlow import PyCFG, CFGNode, to_graph, gen_cfg
```

```
from graphviz import Source, Graph
```

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

We see that the path `[1, 2, 4, 6, 8, 9, 10, 11, 12]`

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

, `[4, 5]`

and `[6, 7]`

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?

## Concolic Execution¶

One way 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.

```
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.

```
src[1]
```

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

. Since the next line taken is line (4), we know that at this point in the execution path, the predicate was`false`

.

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

We notice that this is one of the predicates where the `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.

As we mentioned in the introduction to the chapter, 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. 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.

## SMT Solvers¶

To solve these constraints, one can use a *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.

```
import z3
```

To ensure that the string constraints we use in this chapter are successfully evaluated, we need to specify the `z3str3`

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

```
assert z3.get_version() >= (4, 8, 6, 0)
z3.set_option('smt.string_solver', 'z3str3')
z3.set_option('timeout', 30 * 1000) # milliseconds
```

Encoding the constraint requires declaring a corresponding symbolic variable to the input `n`

.

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

Remember the constraint `(n < 0)`

from line 2 in `factorial()`

? We can now encode the constraint as follows.

```
zn < 0
```

We previously traced `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.

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

The `z3.solve()`

method can also be used to check if the constraints are satisfiable, and if they are, provide 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:

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

This is *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.

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

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

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

Indeed, both `x = 5`

and `x = 1/2`

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

Similarly, we can ask *Z3* for an input that satisfies the constraint encoded in line 2 of `factorial()`

so that we take the `if`

branch.

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

That is, if one uses `-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.

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

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

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 (4) we encounter a new predicate
`n == 0`

, for which we again took the false branch.

```
src[4]
```

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

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

- If we continue to line (6), we encounter another predicate, for which again, we took the
`false`

branch

```
src[6]
```

The predicates encountered so far are as follows

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

To take the branch at (6), we essentially have to obey the predicates until that point, but invert the last predicate.

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

## A Concolic Tracer¶

Given that there is a symbolic context under which the program is executed (that is the symbolic variables that are used in the program execution) we define a context manager called `ConcolicTracer`

that keeps track of the context.

The `ConcolicTracer`

accepts a single argument which contains the declarations for the symbolic variables seen so far, and the pre-conditions if any.

```
class ConcolicTracer:
def __init__(self, context=None):
self.context = context if context is not None else ({}, [])
self.decls, self.path = self.context
```

We add the `enter`

and `exit`

methods for the context manager.

```
class ConcolicTracer(ConcolicTracer):
def __enter__(self):
return self
def __exit__(self, exc_type, exc_value, tb):
return
```

We use introspection to determine the arguments to the function, which is hooked into the `getitem`

method.

```
class ConcolicTracer(ConcolicTracer):
def __getitem__(self, fn):
self.fn = fn
self.fn_args = {i: None for i in inspect.signature(fn).parameters}
return self
```

Finally, the function itself is invoked using the `call`

method.

```
class ConcolicTracer(ConcolicTracer):
def __call__(self, *args):
self.result = self.fn(*self.concolic(args))
return self.result
```

For now, we define `concolic()`

as a transparent function. It will be modified to produce symbolic variables later.

```
class ConcolicTracer(ConcolicTracer):
def concolic(self, args):
return args
```

It can be used as follows

```
with ConcolicTracer() as _:
_[factorial](1)
```

```
_.context
```

The `context`

is empty as we are yet to hook up the necessary infrastructure to `ConcolicTracer`

.

### Concolic Proxy Objects¶

We now define the concolic proxy objects that can be used for concolic tracing. First, we define the `zproxy_create()`

method that given a class name, correctly creates an instance of that class, and the symbolic corresponding variable, and registers the symbolic variable in the context information `context`

.

```
def zproxy_create(cls, sname, z3var, context, zn, v=None):
zv = cls(context, z3var(zn), v)
context[0][zn] = sname
return zv
```

#### A Proxy Class for Booleans¶

First, we define the `zbool`

class which is used to track the predicates encountered. It is a wrapper class that contains both symbolic (`z`

) as well as concrete (`v`

) values. The concrete value is used to determine which path to take, and the symbolic value is used to collect the predicates encountered.

The initialization is done in two parts. The first one is using `zproxy_create()`

to correctly initialize and register the shadow symbolic variable corresponding to the passed argument. This is used exclusively when the symbolic variable needs to be initialized first. In all other cases, the constructor is called with the preexisting symbolic value.

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

Here is how it can be used.

```
with ConcolicTracer() as _:
za, zb = z3.Ints('a b')
val = zbool.create(_.context, 'my_bool_arg', True)
print(val.z, val.v)
_.context
```

##### Negation of Encoded formula¶

The `zbool`

class allows negation of its concrete and symbolic values.

```
class zbool(zbool):
def __not__(self):
return zbool(self.context, z3.Not(self.z), not self.v)
```

Here is how it can be used.

```
with ConcolicTracer() as _:
val = zbool.create(_.context, 'my_bool_arg', True).__not__()
print(val.z, val.v)
_.context
```

##### Registering Predicates on Conditionals¶

The `zbool`

class is being used to track boolean conditions that arise during program execution. It tracks such conditions by registering the corresponding symbolic expressions in the context.

```
class zbool(zbool):
def __bool__(self):
r, pred = (True, self.z) if self.v else (False, z3.Not(self.z))
self.path.append(pred)
return r
```

The `zbool`

class can be used to keep track of boolean values and conditions encountered during the execution. For example, we can encode the conditions encountered by line 6 in `factorial()`

as follows:

First, we define the concrete value (`ca`

), and its shadow symbolic variable (`za`

).

```
ca, za = 5, z3.Int('a')
```

Then, we wrap it in `zbool`

, and use it in a conditional, forcing the conditional to be registered in the context.

```
with ConcolicTracer() as _:
if zbool(_.context, za == z3.IntVal(5), ca == 5):
print('success')
```

We can retrieve the registered conditional as follows.

```
_.path
```

#### A Proxy Class for Integers¶

Next, we define a symbolic wrapper `zint`

for `int`

.
This class keeps track of the `int`

variables used and the predicates encountered in `context`

. Finally, it also keeps the concrete value so that it can be used to determine the path to take. As the `zint`

extends the primitive `int`

class, we have to define a *new* method to open it for extension.

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

As in the case of `zbool`

, the initialization takes place in two parts. The first using `create()`

if a new symbolic argument is being registered, and then the usual initialization.

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

The `int`

value of a `zint`

object is its concrete value.

```
class zint(zint):
def __int__(self):
return self.v
def __pos__(self):
return self.v
```

Using these proxies is as follows.

```
with ConcolicTracer() as _:
val = zint.create(_.context, 'int_arg', 0)
print(val.z, val.v)
_.context
```

The `zint`

class is often used to do arithmetic with, or compare to other `int`

s. These `int`

s can be either a variable or a constant value. We define a helper method `_zv()`

that checks what kind of `int`

a given value is, and produces the correct symbolic equivalent.

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

It can be used as follows

```
with ConcolicTracer() as _:
val = zint.create(_.context, 'int_arg', 0)
print(val._zv(0))
print(val._zv(val))
```

##### Equality between Integers¶

Two integers can be compared for equality using *ne* and *eq*.

```
class zint(zint):
def __ne__(self, other):
z, v = self._zv(other)
return zbool(self.context, self.z != z, self.v != v)
def __eq__(self, other):
z, v = self._zv(other)
return zbool(self.context, self.z == z, self.v == v)
```

We also define *req* using *eq* in case the int being compared is on the left hand side.

```
class zint(zint):
def __req__(self, other):
return self.__eq__(other)
```

It can be used as follows.

```
with ConcolicTracer() as _:
ia = zint.create(_.context, 'int_a', 0)
ib = zint.create(_.context, 'int_b', 0)
v1 = ia == ib
v2 = ia != ib
v3 = 0 != ib
print(v1.z, v2.z, v3.z)
```

##### Comparisons between Integers¶

Integers can also be compared for ordering, and the methods for this are defined below.

```
class zint(zint):
def __lt__(self, other):
z, v = self._zv(other)
return zbool(self.context, self.z < z, self.v < v)
def __gt__(self, other):
z, v = self._zv(other)
return zbool(self.context, self.z > z, self.v > v)
```

We use the comparisons and equality operators to provide the other missing operators.

```
class zint(zint):
def __le__(self, other):
z, v = self._zv(other)
return zbool(self.context, z3.Or(self.z < z, self.z == z),
self.v < v or self.v == v)
def __ge__(self, other):
z, v = self._zv(other)
return zbool(self.context, z3.Or(self.z > z, self.z == z),
self.v > v or self.v == v)
```

These functions can be used as follows.

```
with ConcolicTracer() as _:
ia = zint.create(_.context, 'int_a', 0)
ib = zint.create(_.context, 'int_b', 1)
v1 = ia > ib
v2 = ia < ib
print(v1.z, v2.z)
v3 = ia >= ib
v4 = ia <= ib
print(v3.z, v4.z)
```

##### Binary Operators for Integers¶

We implement relevant arithmetic operators for integers as described in the Python documentation. (The commented out operators are not directly available for `z3.ArithRef`

. They need to be implemented separately if needed. See the exercises for how it can be done.)

```
INT_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__',
]
```

```
def make_int_binary_wrapper(fname, fun, zfun):
def proxy(self, other):
z, v = self._zv(other)
z_ = zfun(self.z, z)
v_ = fun(self.v, v)
if isinstance(v_, float):
# we do not implement float results yet.
assert round(v_) == v_
v_ = round(v_)
return zint(self.context, z_, v_)
return proxy
```

```
INITIALIZER_LIST = []
```

```
def initialize():
for fn in INITIALIZER_LIST:
fn()
```

```
def init_concolic_1():
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))
```

```
INITIALIZER_LIST.append(init_concolic_1)
```

```
init_concolic_1()
```

```
with ConcolicTracer() as _:
ia = zint.create(_.context, 'int_a', 0)
ib = zint.create(_.context, 'int_b', 1)
print((ia + ib).z)
print((ia + 10).z)
print((11 + ib).z)
print((ia - ib).z)
print((ia * ib).z)
print((ia / ib).z)
print((ia ** ib).z)
```

##### Integer Unary Operators¶

We also implement the relevant unary operators as below.

```
INT_UNARY_OPS = [
'__neg__',
'__pos__',
# '__abs__',
# '__invert__',
# '__round__',
# '__ceil__',
# '__floor__',
# '__trunc__',
]
```

```
def make_int_unary_wrapper(fname, fun, zfun):
def proxy(self):
return zint(self.context, zfun(self.z), fun(self.v))
return proxy
```

```
def init_concolic_2():
for fname in INT_UNARY_OPS:
fun = getattr(int, fname)
zfun = getattr(z3.ArithRef, fname)
setattr(zint, fname, make_int_unary_wrapper(fname, fun, zfun))
```

```
INITIALIZER_LIST.append(init_concolic_2)
```

```
init_concolic_2()
```

We can use the unary operators we defined above as follows:

```
with ConcolicTracer() as _:
ia = zint.create(_.context, 'int_a', 0)
print((-ia).z)
print((+ia).z)
```

##### Using an Integer in a Boolean Context¶

An integer may be converted to a boolean context in conditionals or as part of boolean predicates such as `or`

, `and`

and `not`

. In these cases, the `__bool__()`

method gets called. Unfortunately, this method requires a primitive boolean value. Hence, we force the current integer formula to a boolean predicate and register it in the current context.

```
class zint(zint):
def __bool__(self):
# return zbool(self.context, self.z, self.v) <-- not allowed
# force registering boolean condition
if self != 0:
return True
return False
```

It is used as follows

```
with ConcolicTracer() as _:
za = zint.create(_.context, 'int_a', 1)
zb = zint.create(_.context, 'int_b', 0)
if za and zb:
print(1)
```

```
_.context
```

#### Remaining Methods of the ConcolicTracer¶

We now complete some of the methods of the `ConcolicTracer`

.

##### Translating to the SMT Expression Format¶

Given that we are using an SMT Solver z3, it is often useful to retrieve the corresponding SMT expression for a symbolic expression. This can be used as an argument to `z3`

or other SMT solvers.

The format of the SMT expression (SMT-LIB) is as follows:

- Variables declarations in S-EXP format.
E.g. The following declares a symbolic integer variable
`x`

This declares a`(declare-const x Int)`

`bit vector`

`b`

of length`8`

This declares a symbolic real variable`(declare-const b (_ BitVec 8))`

`r`

This declares a symbolic string variable`(declare-const x Real)`

`s`

`(declare-const s String)`

The declared variables can be used in logical formulas that are encoded in *S-EXP* format. For example, here is a logical formula.

```
(assert
(and
(= a b)
(= a c)
(! b c)))
```

Here is another example, using string variables.

```
(or (< 0 (str.indexof (str.substr my_str1 7 19) " where " 0))
(= (str.indexof (str.substr my_str1 7 19) " where " 0) 0))
```

```
class ConcolicTracer(ConcolicTracer):
def smt_expr(self, show_decl=False, simplify=False, path=[]):
r = []
if show_decl:
for decl in self.decls:
v = self.decls[decl]
v = '(_ BitVec 8)' if v == 'BitVec' else v
r.append("(declare-const %s %s)" % (decl, v))
path = path if path else self.path
if path:
path = z3.And(path)
if show_decl:
if simplify:
return '\n'.join([
*r,
"(assert %s)" % z3.simplify(path).sexpr()
])
else:
return '\n'.join(
[*r, "(assert %s)" % path.sexpr()])
else:
return z3.simplify(path).sexpr()
else:
return ''
```

To see how to use `smt_expr()`

, let us consider an example. The `triangle()`

function is used to determine if the given sides to a triangle result in an `equilateral`

triangle, an `isosceles`

triangle, or a `scalene`

triangle. It is implemented as follows.

```
def triangle(a, b, c):
if a == b:
if b == c:
return 'equilateral'
else:
return 'isosceles'
else:
if b == c:
return 'isosceles'
else:
if a == c:
return 'isosceles'
else:
return 'scalene'
```

```
triangle(1, 2, 1)
```

To translate make it run under `ConcolicTracer`

, we first define the arguments. The triangle being defined has sides `1, 1, 1`

. i.e. it is an `equilateral`

triangle.

```
with ConcolicTracer() as _:
za = zint.create(_.context, 'int_a', 1)
zb = zint.create(_.context, 'int_b', 1)
zc = zint.create(_.context, 'int_c', 1)
triangle(za, zb, zc)
print(_.context)
```

We can now call `smt_expr()`

to retrieve the SMT expression as below.

```
print(_.smt_expr(show_decl=True))
```

The collected predicates can also be solved directly using the Python z3 API.

```
z3.solve(_.path)
```

##### Generating Fresh Names¶

While using the proxy classes, we often will have to generate new symbolic variables, with names that have not been used before. For this, we define `fresh_name()`

that always generates unique integers for names.

```
COUNTER = 0
```

```
def fresh_name():
global COUNTER
COUNTER += 1
return COUNTER
```

It can be used as follows

```
fresh_name()
```

```
def reset_counter():
global COUNTER
COUNTER = 0
```

```
class ConcolicTracer(ConcolicTracer):
def __enter__(self):
reset_counter()
return self
def __exit__(self, exc_type, exc_value, tb):
return
```

##### Translating Arguments to Concolic Proxies¶

We had previously defined `concolic()`

as a transparent function. We now provide the full implementation of this function. It inspects a given function's parameters, and infers the parameter types from the concrete arguments passed in. It then uses this information to instantiate the correct proxy classes for each argument.

```
class ConcolicTracer(ConcolicTracer):
def concolic(self, args):
my_args = []
for name, arg in zip(self.fn_args, args):
t = type(arg).__name__
zwrap = globals()['z' + t]
vname = "%s_%s_%s_%s" % (self.fn.__name__, name, t, fresh_name())
my_args.append(zwrap.create(self.context, vname, arg))
self.fn_args[name] = vname
return my_args
```

This is how it gets used:

```
with ConcolicTracer() as _:
_[factorial](5)
```

With the new `concolic()`

method, the arguments to the factorial are correctly associated with symbolic variables, which allows us to retrieve the predicates encountered.

```
_.context
```

As before, we can also print out the SMT expression which can be passed directly to command line SMT solvers.

```
print(_.smt_expr(show_decl=True))
```

We next define methods to evaluate the SMT expression both in Python and from command line.

##### Evaluating the Concolic Expressions¶

We define `zeval()`

to solve the predicates in a context, and return results. It has two modes. The `python`

mode uses `z3`

Python API to solve and return the results. If the `python`

mode is false, it writes the SMT expression to a file, and invokes the command line `z3`

for a solution.

```
class ConcolicTracer(ConcolicTracer):
def zeval(self, python=False, log=False):
r, sol = (zeval_py if python else zeval_smt)(self.path, self, log)
if r == 'sat':
return r, {k: sol.get(self.fn_args[k], None) for k in self.fn_args}
else:
return r, None
```

##### Using the Python API¶

Given a set of predicates that the function encountered, and the tracer under which the function was executed, the `zeval_py()`

function first declares the relevant symbolic variables, and uses the `z3.Solver()`

to provide a set of inputs that would trace the same path through the function.

```
def zeval_py(path, cc, log):
for decl in cc.decls:
if cc.decls[decl] == 'BitVec':
v = "z3.%s('%s', 8)" % (cc.decls[decl], decl)
else:
v = "z3.%s('%s')" % (cc.decls[decl], decl)
exec(v)
s = z3.Solver()
s.add(z3.And(path))
if s.check() == z3.unsat:
return 'No Solutions', {}
elif s.check() == z3.unknown:
return 'Gave up', None
assert s.check() == z3.sat
m = s.model()
return 'sat', {d.name(): m[d] for d in m.decls()}
```

It can be used as follows:

```
with ConcolicTracer() as _:
_[factorial](5)
```

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

That is, given the set of constraints, the assignment `n == 5`

conforms to all constraints.

##### Using the Command Line¶

The `zeval_smt()`

function writes the SMT expression to the file system, and calls the `z3`

SMT solver command line to solve it. The result of SMT expression is again an `sexpr`

. Hence, we first define `parse_sexp()`

to parse and return the correct values.

```
import re
```

```
import subprocess
```

```
SEXPR_TOKEN = r'''(?mx)
\s*(?:
(?P<bra>\()|
(?P<ket>\))|
(?P<token>[^"()\s]+)|
(?P<string>"[^"]*")
)'''
```

```
def parse_sexp(sexp):
stack, res = [], []
for elements in re.finditer(SEXPR_TOKEN, sexp):
kind, value = [(t, v) for t, v in elements.groupdict().items() if v][0]
if kind == 'bra':
stack.append(res)
res = []
elif kind == 'ket':
last, res = res, stack.pop(-1)
res.append(last)
elif kind == 'token':
res.append(value)
elif kind == 'string':
res.append(value[1:-1])
else:
assert False
return res
```

The `parse_sexp()`

function can be used as follows

```
parse_sexp('abcd (hello 123 (world "hello world"))')
```

We now define `zeval_smt()`

which uses the `z3`

command line directly, and uses `parse_sexp()`

to parse and return the solutions to function arguments if any.

```
import tempfile
```

```
def zeval_smt(path, cc, log):
s = cc.smt_expr(True, True, path)
with tempfile.NamedTemporaryFile(mode='w', suffix='.smt') as f:
f.write(s)
f.write("\n(check-sat)")
f.write("\n(get-model)")
f.flush()
if log:
print(s, '(check-sat)', '(get-model)', sep='\n')
output = subprocess.getoutput("z3 " + f.name)
if log:
print(output)
o = parse_sexp(output)
kind = o[0]
if kind == 'unknown':
return 'Gave up', None
elif kind == 'unsat':
return 'No Solutions', {}
assert kind == 'sat'
assert o[1][0] == 'model'
return 'sat', {i[1]: (i[-1], i[-2]) for i in o[1][1:]}
```

We can now use `zeval()`

as follows.

```
with ConcolicTracer() as _:
_[factorial](5)
```

```
_.zeval()
```

Indeed, we get similar results (`n == 5`

) from using the command line as from using the Python API.

#### A Proxy Class for Strings¶

Here, we define the proxy string class `zstr`

. First we define our initialization routines. Since `str`

is a primitive type, we define `new`

to extend it.

```
class zstr(str):
def __new__(cls, context, zn, v):
return str.__new__(cls, v)
```

As before, initialization proceeds with `create()`

and the constructor.

```
class zstr(zstr):
@classmethod
def create(cls, context, zn, v=None):
return zproxy_create(cls, 'String', z3.String, context, zn, v)
def __init__(self, context, z, v=None):
self.context, self.z, self.v = context, z, v
self._len = zint(context, z3.Length(z), len(v))
#self.context[1].append(z3.Length(z) == z3.IntVal(len(v)))
```

We also define `_zv()`

helper to help us with methods that accept another string

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

##### Retrieving Ordinal Value¶

We define `zord`

that given a symbolic one character long string, obtains the `ord()`

for that. It returns two values. The first one is the variable that corresponds to `ord()`

, and second is the predicate that links the variable to the passed in single character string.

```
def zord(context, c):
bn = "bitvec_%d" % fresh_name()
v = z3.BitVec(bn, 8)
context[0][bn] = 'BitVec'
z = (z3.Unit(v) == c)
context[1].append(z)
return v
```

We use it as follows

```
zc = z3.String('arg_%d' % fresh_name())
```

```
with ConcolicTracer() as _:
zi = zord(_.context, zc)
```

The symbolic bitvector is in `zi`

. It is linked to the passed in argument in `context`

```
_.context
```

We can specify what the result of `ord()`

should be, and call `z3.solve()`

to provide us with a solution that will provide the required result.

```
z3.solve(_.path + [zi == 65])
```

##### Translating an Ordinal Value to ASCII¶

Similarly, we can convert the ASCII value back to a single character string using `zchr()`

```
def zchr(context, i):
sn = 'string_%d' % fresh_name()
s = z3.String(sn)
context[0][sn] = 'String'
z = z3.And([s == z3.Unit(i), z3.Length(s) == 1])
context[1].append(z)
return s
```

For using it, we first define a bitvector that is 8 bits long.

```
i = z3.BitVec('bv_%d' % fresh_name(), 8)
```

We can now retrieve the `chr()`

representation as below.

```
with ConcolicTracer() as _:
zc = zchr(_.context, i)
```

```
_.context
```

As before, we can specify what the end result of calling `chr()`

should be to get the original argument.

```
z3.solve(_.path + [zc == z3.StringVal('a')])
```

##### Equality between Strings¶

The equality of `zstr`

is defined similar to that of `zint`

```
class zstr(zstr):
def __eq__(self, other):
z, v = self._zv(other)
return zbool(self.context, self.z == z, self.v == v)
def __req__(self, other):
return self.__eq__(other)
```

The `zstr`

class is used as follows.

```
def tstr1(s):
if s == 'h':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr1]('h')
```

```
_.zeval()
```

It works even if we have more than one character.

```
def tstr1(s):
if s == 'hello world':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr1]('hello world')
```

```
_.context
```

```
_.zeval()
```

##### Concatenation of Strings¶

What if we need to concatenate two strings? We need additional helpers to accomplish that.

```
class zstr(zstr):
def __add__(self, other):
z, v = self._zv(other)
return zstr(self.context, self.z + z, self.v + v)
def __radd__(self, other):
return self.__add__(other)
```

Here is how it can be used. First, we create the wrapped arguments

```
with ConcolicTracer() as _:
v1, v2 = [zstr.create(_.context, 'arg_%d' % fresh_name(), s)
for s in ['hello', 'world']]
if (v1 + ' ' + v2) == 'hello world':
print('hello world')
```

The addition of symbolic variables is preserved in `context`

```
_.context
```

##### Producing Substrings¶

Similarly, accessing substrings also require extra help.

```
class zstr(zstr):
def __getitem__(self, idx):
if isinstance(idx, slice):
start, stop, step = idx.indices(len(self.v))
assert step == 1 # for now
assert stop >= start # for now
rz = z3.SubString(self.z, start, stop - start)
rv = self.v[idx]
elif isinstance(idx, int):
rz = z3.SubString(self.z, idx, 1)
rv = self.v[idx]
else:
assert False # for now
return zstr(self.context, rz, rv)
def __iter__(self):
return zstr_iterator(self.context, self)
```

##### An Iterator Class for Strings¶

We define the iterator as follows.

```
class zstr_iterator():
def __init__(self, context, zstr):
self.context = context
self._zstr = zstr
self._str_idx = 0
self._str_max = zstr._len # intz is not an _int_
def __next__(self):
if self._str_idx == self._str_max: # intz#eq
raise StopIteration
c = self._zstr[self._str_idx]
self._str_idx += 1
return c
def __len__(self):
return self._len
```

Here is how it can be used.

```
def tstr2(s):
if s[0] == 'h' and s[1] == 'e' and s[3] == 'l':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr2]('hello')
```

Again, the context shows predicates encountered.

```
_.context
```

The function `zeval()`

returns a solution for the predicate. Note that the value returned is not exactly the argument that we passed in. This is a consequence of the predicates we have. That is, we have no constraints on what the character value on `s[2]`

should be.

```
_.zeval()
```

##### Translating to Upper and Lower Equivalents¶

A major complication is supporting `upper()`

and `lower()`

methods. We use the previously defined `zchr()`

and `zord()`

functions to accomplish this.

```
class zstr(zstr):
def upper(self):
empty = ''
ne = 'empty_%d' % fresh_name()
result = zstr.create(self.context, ne, empty)
self.context[1].append(z3.StringVal(empty) == result.z)
cdiff = (ord('a') - ord('A'))
for i in self:
oz = zord(self.context, i.z)
uz = zchr(self.context, oz - cdiff)
rz = z3.And([oz >= ord('a'), oz <= ord('z')])
ov = ord(i.v)
uv = chr(ov - cdiff)
rv = ov >= ord('a') and ov <= ord('z')
if zbool(self.context, rz, rv):
i = zstr(self.context, uz, uv)
else:
i = zstr(self.context, i.z, i.v)
result += i
return result
```

The `lower()`

function is similar to `upper()`

except that the character ranges are switched, and the lowercase is above uppercase. Hence, we add the difference to the ordinal to make a character to lowercase.

```
class zstr(zstr):
def lower(self):
empty = ''
ne = 'empty_%d' % fresh_name()
result = zstr.create(self.context, ne, empty)
self.context[1].append(z3.StringVal(empty) == result.z)
cdiff = (ord('a') - ord('A'))
for i in self:
oz = zord(self.context, i.z)
uz = zchr(self.context, oz + cdiff)
rz = z3.And([oz >= ord('A'), oz <= ord('Z')])
ov = ord(i.v)
uv = chr(ov + cdiff)
rv = ov >= ord('A') and ov <= ord('Z')
if zbool(self.context, rz, rv):
i = zstr(self.context, uz, uv)
else:
i = zstr(self.context, i.z, i.v)
result += i
return result
```

Here is how it is used.

```
def tstr3(s):
if s.upper() == 'H':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr3]('h')
```

Again, we use `zeval()`

to solve the collected constraints, and verify that our constraints are correct.

```
_.zeval()
```

Here is a larger example using `upper()`

```
def tstr4(s):
if s.lower() == 'hello world':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr4]('Hello World')
```

```
_.zeval()
```

Again, we obtain the right input value.

##### Checking for String Prefixes¶

We define `startswith()`

.

```
class zstr(zstr):
def startswith(self, other, beg=0, end=None):
assert end is None # for now
assert isinstance(beg, int) # for now
zb = z3.IntVal(beg)
others = other if isinstance(other, tuple) else (other, )
last = False
for o in others:
z, v = self._zv(o)
r = z3.IndexOf(self.z, z, zb)
last = zbool(self.context, r == zb, self.v.startswith(v))
if last:
return last
return last
```

An example.

```
def tstr5(s):
if s.startswith('hello'):
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr5]('hello world')
```

```
_.zeval()
```

```
with ConcolicTracer() as _:
r = _[tstr5]('my world')
```

```
_.zeval()
```

As before, the predicates only ensure that the `startswith()`

returned a true value. Hence, our solution reflects that.

##### Finding Substrings¶

We also define `find()`

```
class zstr(zstr):
def find(self, other, beg=0, end=None):
assert end is None # for now
assert isinstance(beg, int) # for now
zb = z3.IntVal(beg)
z, v = self._zv(other)
zi = z3.IndexOf(self.z, z, zb)
vi = self.v.find(v, beg, end)
return zint(self.context, zi, vi)
```

An example.

```
def tstr6(s):
if s.find('world') != -1:
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr6]('hello world')
```

```
_.zeval()
```

As before, the predicates only ensure that the `find()`

returned a value greater than -1. Hence, our solution reflects that.

##### Remove Space from Ends¶

We next implement `strip()`

.

```
import string
```

```
class zstr(zstr):
def rstrip(self, chars=None):
if chars is None:
chars = string.whitespace
if self._len == 0:
return self
else:
last_idx = self._len - 1
cz = z3.SubString(self.z, last_idx.z, 1)
cv = self.v[-1]
zcheck_space = z3.Or([cz == z3.StringVal(char) for char in chars])
vcheck_space = any(cv == char for char in chars)
if zbool(self.context, zcheck_space, vcheck_space):
return zstr(self.context, z3.SubString(self.z, 0, last_idx.z),
self.v[0:-1]).rstrip(chars)
else:
return self
```

```
def tstr7(s):
if s.rstrip(' ') == 'a b':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr7]('a b ')
print(r)
```

```
_.zeval()
```

```
class zstr(zstr):
def lstrip(self, chars=None):
if chars is None:
chars = string.whitespace
if self._len == 0:
return self
else:
first_idx = 0
cz = z3.SubString(self.z, 0, 1)
cv = self.v[0]
zcheck_space = z3.Or([cz == z3.StringVal(char) for char in chars])
vcheck_space = any(cv == char for char in chars)
if zbool(self.context, zcheck_space, vcheck_space):
return zstr(self.context, z3.SubString(
self.z, 1, self._len.z), self.v[1:]).lstrip(chars)
else:
return self
```

```
def tstr8(s):
if s.lstrip(' ') == 'a b':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr8](' a b')
print(r)
```

```
_.zeval()
```

```
class zstr(zstr):
def strip(self, chars=None):
return self.lstrip(chars).rstrip(chars)
```

Example usage.

```
def tstr9(s):
if s.strip() == 'a b':
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr9](' a b ')
print(r)
```

```
_.zeval()
```

The `strip()`

has generated the right constraints.

##### Splitting Strings¶

We implement string `split()`

as follows.

```
class zstr(zstr):
def split(self, sep=None, maxsplit=-1):
assert sep is not None # default space based split is complicated
assert maxsplit == -1 # for now.
zsep = z3.StringVal(sep)
zl = z3.Length(zsep)
# zi would be the length of prefix
zi = z3.IndexOf(self.z, zsep, z3.IntVal(0))
# Z3Bug: There is a bug in the `z3.IndexOf` method which returns
# `z3.SeqRef` instead of `z3.ArithRef`. So we need to fix it.
zi = z3.ArithRef(zi.ast, zi.ctx)
vi = self.v.find(sep)
if zbool(self.context, zi >= z3.IntVal(0), vi >= 0):
zprefix = z3.SubString(self.z, z3.IntVal(0), zi)
zmid = z3.SubString(self.z, zi, zl)
zsuffix = z3.SubString(self.z, zi + zl,
z3.Length(self.z))
return [zstr(self.context, zprefix, self.v[0:vi])] + zstr(
self.context, zsuffix, self.v[vi + len(sep):]).split(
sep, maxsplit)
else:
return [self]
```

```
def tstr10(s):
if s.split(',') == ['a', 'b', 'c']:
return True
else:
return False
```

```
with ConcolicTracer() as _:
r = _[tstr10]('a,b,c')
print(r)
```

```
_.zeval()
```

##### Trip Wire¶

For easier debugging, we abort any calls to methods in `str`

that are not overridden by `zstr`

.

```
def make_str_abort_wrapper(fun):
def proxy(*args, **kwargs):
raise Exception('%s Not implemented in `zstr`' % fun.__name__)
return proxy
```

```
def init_concolic_3():
strmembers = inspect.getmembers(zstr, callable)
zstrmembers = {m[0] for m in strmembers if len(
m) == 2 and 'zstr' in m[1].__qualname__}
for name, fn in inspect.getmembers(str, callable):
# Omitted 'splitlines' as this is needed for formatting output in
# IPython/Jupyter
if name not in zstrmembers and name not in [
'splitlines',
'__class__',
'__contains__',
'__delattr__',
'__dir__',
'__format__',
'__ge__',
'__getattribute__',
'__getnewargs__',
'__gt__',
'__hash__',
'__le__',
'__len__',
'__lt__',
'__mod__',
'__mul__',
'__ne__',
'__reduce__',
'__reduce_ex__',
'__repr__',
'__rmod__',
'__rmul__',
'__setattr__',
'__sizeof__',
'__str__']:
setattr(zstr, name, make_str_abort_wrapper(fn))
```

```
INITIALIZER_LIST.append(init_concolic_3)
```

```
init_concolic_3()
```

## Examples¶

### Triangle¶

We previously showed how to run `triangle()`

under `ConcolicTracer`

.

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

The predicates are as follows:

```
_.path
```

```
_.zeval()
```

We can modify the predicates if necessary. First, we retrieve the symbolic variables.

```
za, zb, zc = [z3.Int(s) for s in _.context[0].keys()]
```

Then, we pass a modified predicate to `zeval()`

. The key determines which predicate the new predicate will replace.

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

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

The updated predicate returns `isosceles`

as expected.

### Round¶

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

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

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

context.

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

We verify that we were able to capture all the predicates

```
_.context
```

We use `zeval()`

to obtain results.

```
_.zeval()
```

### Absolute Maximum¶

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

as below.

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

It is called by another function `abs_max()`

```
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()`

.

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

As expected, we have the predicates across functions.

```
_.context
```

```
_.zeval()
```

Solving the predicates works as expected.

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

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

```
_.context
```

```
_.zeval()
```

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

in `abs_value()`

).

### Binomial Coefficient¶

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.

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

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

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

```
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`

.

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

Then call `zeval()`

to evaluate.

```
_.zeval()
```

### Database¶

For a larger example using the Concolic String class `zstr`

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

```
from InformationFlow import DB, sample_db, update_inventory
```

We first populate our database.

```
from GrammarMiner import VEHICLES # minor dependency
```

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

```
db.db
```

We are now ready to fuzz our `DB`

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

slightly.

```
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()`

.

```
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.

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

The predicates encountered during the execution are as follows:

```
_.path
```

We can use `zeval()`

as before to solve the constraints.

```
_.zeval()
```

## Fuzzing with Constraints¶

In this section, we show how to use the infrastructure we built for concolic execution for guiding fuzzing.

### SimpleConcolicFuzzer¶

The `SimpleConcolicFuzzer`

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.

First, we import the `Fuzzer`

interface, and an example program `hang_if_no_space()`

```
from Fuzzer import Fuzzer, hang_if_no_space
```

```
from ExpectError import ExpectTimeout, ExpectError
```

```
import random
```

To make fuzzer work, we need a way to represent decisions made during trace. We keep this in a binary tree where each node represents a decision made, and each leaf represents a complete path. A node in the binary tree is represented by the `TraceNode`

class.

When a new node is added, it represents a decision taken by the parent on some predicate. This predicate is supplied as `smt_val`

, which is `True`

for this child to be reached. Since the predicate is actually present in the parent node, we also carry a member `smt`

which will be updated by the first child to be added.

```
class TraceNode:
def __init__(self, smt_val, parent, info):
# This is the smt that lead to this node
self._smt_val = z3.simplify(smt_val) if smt_val is not None else None
# This is the predicate that this node might perform at a future point
self.smt = None
self.info = info
self.parent = parent
self.children = {}
self.path = None
self.tree = None
self._pattern = None
self.log = True
def no(self): return self.children.get(self.tree.no_bit)
def yes(self): return self.children.get(self.tree.yes_bit)
def get_children(self): return (self.no(), self.yes())
def __str__(self):
return 'TraceNode[%s]' % ','.join(self.children.keys())
```

We add a `PlausibleChild`

class to track the leaf nodes.

```
class PlausibleChild:
def __init__(self, parent, cond, tree):
self.parent = parent
self.cond = cond
self.tree = tree
self._smt_val = None
def __repr__(self):
return 'PlausibleChild[%s]' % (self.parent.pattern() + ':' + self.cond)
```

When the leaf nodes are used to generate new paths, we expect its sibling `TraceNode`

to have been already explored. Hence, we make use of the sibling's values for context `cc`

, and the `smt_val`

from the parent.

```
class PlausibleChild(PlausibleChild):
def smt_val(self):
if self._smt_val is not None:
return self._smt_val
# if the parent has other children, then that child would have updatd the parent's smt
# Hence, we can use that child's smt_value's opposite as our value.
assert self.parent.smt is not None
if self.cond == self.tree.no_bit:
self._smt_val = z3.Not(self.parent.smt)
else:
self._smt_val = self.parent.smt
return self._smt_val
def cc(self):
if self.parent.info.get('cc') is not None:
return self.parent.info['cc']
# if there is a plausible child node, it means that there can
# be at most one child.
sibilings = list(self.parent.children.values())
assert len(sibilings) == 1
# We expect at the other child to have cc
return sibilings[0].info['cc']
```

The `PlausibleChild`

instance is used to generate new paths to explore using `path_expression()`

.

```
class PlausibleChild(PlausibleChild):
def path_expression(self):
path_to_root = self.parent.get_path_to_root()
assert path_to_root[0]._smt_val is None
return [i._smt_val for i in path_to_root[1:]] + [self.smt_val()]
```

The `TraceTree`

class helps us keep track of the binary tree. In the beginning, the root is a sentinel `TraceNode`

instance, and simply have two plausible children as leaves. As soon as the first trace is added, one of the plausible children will become a true child.

```
class TraceTree:
def __init__(self):
self.root = TraceNode(smt_val=None, parent=None, info={'num': 0})
self.root.tree = self
self.leaves = {}
self.no_bit, self.yes_bit = '0', '1'
pprefix = ':'
for bit in [self.no_bit, self.yes_bit]:
self.leaves[pprefix + bit] = PlausibleChild(self.root, bit, self)
self.completed_paths = {}
```

The `add_trace()`

method of the `TraceTree`

provides a way for new traces to be added. It is kept separate from the initialization as we might want to add more than one trace from the same function.

```
class TraceTree(TraceTree):
def add_trace(self, tracer, string):
last = self.root
i = 0
for i, elt in enumerate(tracer.path):
last = last.add_child(elt=elt, i=i + 1, cc=tracer, string=string)
last.add_child(elt=z3.BoolVal(True), i=i + 1, cc=tracer, string=string)
```

To make `add_trace()`

work, we need a little more infrastructure, that we define below.

The `bit()`

method translates a predicate to a bit that corresponds to the decision taken at each predicate. If the `if`

branch is taken, the result is `1`

, while `else`

branch is indicated by `0`

. The pattern indicates the bit-pattern of decisions required to reach the leaf from the root.

```
class TraceNode(TraceNode):
def bit(self):
if self._smt_val is None:
return None
return self.tree.no_bit if self._smt_val.decl(
).name() == 'not' else self.tree.yes_bit
def pattern(self):
if self._pattern is not None:
return self._pattern
path = self.get_path_to_root()
assert path[0]._smt_val is None
assert path[0].parent is None
self._pattern = ''.join([p.bit() for p in path[1:]])
return self._pattern
```

Each node knows how to add a new child, and get the path to root, which is cached.

When we add a child to the root node, it means that there was a decision in the current node, and the child is the result of the decision. Hence, to get the decision being made, we simplify the `smt`

expression, and check if it starts with `not`

. If it does not start with a `not`

, we interpret that as the current decision in the node. If it starts with `not`

, then we interpret that `not(smt)`

was the expression being evaluated in the current node.

We know the first decision made only after going through the program at least once. As soon as the program is traversed, we update the parent with the decision that resulted in the current child.

```
class TraceNode(TraceNode):
def add_child(self, elt, i, cc, string):
if elt == z3.BoolVal(True):
# No more exploration here. Simply unregister the leaves of *this*
# node and possibly register them in completed nodes, and exit
for bit in [self.tree.no_bit, self.tree.yes_bit]:
child_leaf = self.pattern() + ':' + bit
if child_leaf in self.tree.leaves:
del self.tree.leaves[child_leaf]
self.tree.completed_paths[self.pattern()] = self
return None
child_node = TraceNode(smt_val=elt,
parent=self,
info={'num': i, 'cc': cc, 'string': string})
child_node.tree = self.tree
# bit represents the path that child took from this node.
bit = child_node.bit()
# first we update our smt decision
if bit == self.tree.yes_bit: # yes, which means the smt can be used as is
if self.smt is not None:
assert self.smt == child_node._smt_val
else:
self.smt = child_node._smt_val
# no, which means we have to negate it to get the decision.
elif bit == self.tree.no_bit:
smt_ = z3.simplify(z3.Not(child_node._smt_val))
if self.smt is not None:
assert smt_ == self.smt
else:
self.smt = smt_
else:
assert False
if bit in self.children:
# if self.log:
#print(elt, child_node.bit(), i, string)
#print(i,'overwriting', bit,'=>',self.children[bit],'with',child_node)
child_node = self.children[bit]
#self.children[bit] = child_node
#child_node.children = old.children
else:
self.children[bit] = child_node
# At this point, we have to unregister any leaves that correspond to this child from tree,
# and add the plausible children of this child as leaves to be explored. Note that
# if it is the end (z3.True), we do not have any more children.
child_leaf = self.pattern() + ':' + bit
if child_leaf in self.tree.leaves:
del self.tree.leaves[child_leaf]
pprefix = child_node.pattern() + ':'
# Plausible children.
for bit in [self.tree.no_bit, self.tree.yes_bit]:
self.tree.leaves[pprefix +
bit] = PlausibleChild(child_node, bit, self.tree)
return child_node
```

The path to root from any node is computed once and cached.

```
class TraceNode(TraceNode):
def get_path_to_root(self):
if self.path is not None:
return self.path
parent_path = []
if self.parent is not None:
parent_path = self.parent.get_path_to_root()
self.path = parent_path + [self]
return self.path
```

The `SimpleConcolicFuzzer`

is defined with the `Fuzzer`

interface.

```
class SimpleConcolicFuzzer(Fuzzer):
def __init__(self):
self.ct = TraceTree()
self.max_tries = 1000
self.last = None
self.last_idx = None
```

The `add_trace()`

method we defined earlier is used as follows. First, we use a random string to generate the concolic trace.

```
with ExpectTimeout(2):
with ConcolicTracer() as _:
_[hang_if_no_space]('ab d')
```

Next, we initialize and add this trace to the fuzzer.

```
_.path
```

```
scf = SimpleConcolicFuzzer()
scf.ct.add_trace(_, 'ab d')
```

The path we added above can be obtained from the `TraceTree`

as below.

```
[i._smt_val for i in scf.ct.root.get_children()[0].get_children()[
0].get_children()[1].get_path_to_root()]
```

Below are the registered leaves that we can explore at this moment.

```
for key in scf.ct.leaves:
print(key, '\t', scf.ct.leaves[key])
```

Next, we need a way to visualize the constructed tree.

```
from GrammarFuzzer import display_tree
```

```
TREE_NODES = {}
```

```
def my_extract_node(tnode, id):
key, node, parent = tnode
if node is None:
# return '? (%s:%s)' % (parent.pattern(), key) , [], ''
return '?', [], ''
if node.smt is None:
return '* %s' % node.info.get('string', ''), [], ''
no, yes = node.get_children()
num = str(node.info.get('num'))
children = [('0', no, node), ('1', yes, node)]
TREE_NODES[id] = 0
return "(%s) %s" % (num, str(node.smt)), children, ''
```

```
def my_edge_attr(dot, start_node, stop_node):
# the edges are always drawn '0:NO' first.
if TREE_NODES[start_node] == 0:
color, label = 'red', '0'
TREE_NODES[start_node] = 1
else:
color, label = 'blue', '1'
TREE_NODES[start_node] = 2
dot.edge(repr(start_node), repr(stop_node), color=color, label=label)
```

```
def display_trace_tree(root):
TREE_NODES.clear()
return display_tree(
('', root, None), extract_node=my_extract_node, edge_attr=my_edge_attr)
```

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

For example, the pattern `00:0`

corresponds to the following predicates.

```
scf.ct.leaves['00:0']
```

```
scf.ct.leaves['00:0'].path_expression()
```

Similarly the pattern `:1`

corresponds to the following predicates.

```
scf.ct.leaves[':1']
```

```
scf.ct.leaves[':1'].path_expression()
```

We can now generate the next input to be generated by looking for the a leaf that is incompletely explored. The idea is to collect all leaf nodes, and choose one at random.

```
class SimpleConcolicFuzzer(SimpleConcolicFuzzer):
def add_trace(self, trace, s):
self.ct.add_trace(trace, s)
def next_choice(self):
#lst = sorted(list(self.ct.leaves.keys()), key=len)
c = random.choice(list(self.ct.leaves.keys()))
#c = lst[0]
return self.ct.leaves[c]
```

We use the `next_choice()`

as follows.

```
scf = SimpleConcolicFuzzer()
scf.add_trace(_, 'ab d')
node = scf.next_choice()
```

```
node
```

```
node.path_expression()
```

We get the next choice for exploration, and expand the path expression, and return it together with a context using `get_newpath()`

```
class SimpleConcolicFuzzer(SimpleConcolicFuzzer):
def get_newpath(self):
node = self.next_choice()
path = node.path_expression()
return path, node.cc()
```

```
scf = SimpleConcolicFuzzer()
scf.add_trace(_, 'abcd')
path, cc = scf.get_newpath()
path
```

#### Fuzz¶

The `fuzz()`

method simply generates new lists of predicates, and solves them to produce new inputs.

```
class SimpleConcolicFuzzer(SimpleConcolicFuzzer):
def fuzz(self):
if self.ct.root.children == {}:
# a random value to generate comparisons. This would be
# the initial value around which we explore with concolic
# fuzzing.
return ' '
for i in range(self.max_tries):
path, last = self.get_newpath()
s, v = zeval_smt(path, last, log=False)
if s != 'sat':
#raise Exception("Unexpected UNSAT")
continue
val = list(v.values())[0]
elt, typ = val
if len(elt) == 2 and elt[0] == '-': # negative numbers are [-, x]
elt = '-%s' % elt[1]
# make sure that we do not retry the tried paths
# The tracer we add here is incomplete. This gets updated when
# the add_trace is called from the concolic fuzzer context.
# self.add_trace(ConcolicTracer((last.decls, path)), elt)
if typ == 'Int':
return int(elt)
elif typ == 'String':
return elt
return elt
return None
```

We now fuzz.

```
scf = SimpleConcolicFuzzer()
scf.fuzz()
```

Here is an example program `cgi_decode()`

. Note that we will not be able to use the `cgi_decode()`

from the `Coverage`

chapter directly as the hash lookups in `hex_values`

can not be used for transferring constraints yet.

```
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 < len(s):
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
```

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

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

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

So, we fuzz to get a new path.

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

We can now obtain the new trace as before.

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

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

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

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

node.

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

A complete fuzzer run is as follows

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

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

**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.

The `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.

### ConcolicGrammarFuzzer¶

The concolic framework can be used directly in fuzzing. First, we extend our `GrammarFuzzer`

with a helper method `tree_to_string()`

such that we can retrieve the derivation tree of the fuzz output. We also define `prune_tree()`

and `coalesce()`

to reduce the depth of sub trees. These methods accept a list of tokens types such that a node belonging to the token type gets converted from a tree to a leaf node by calling `tree_to_string()`

.

```
from InformationFlow import INVENTORY_GRAMMAR, SQLException
```

```
from GrammarFuzzer import GrammarFuzzer
```

```
class ConcolicGrammarFuzzer(GrammarFuzzer):
def tree_to_string(self, tree):
symbol, children, *_ = tree
e = ''
if children:
return e.join([self.tree_to_string(c) for c in children])
else:
return e if symbol in self.grammar else symbol
def prune_tree(self, tree, tokens):
name, children = tree
children = self.coalesce(children)
if name in tokens:
return (name, [(self.tree_to_string(tree), [])])
else:
return (name, [self.prune_tree(c, tokens) for c in children])
def coalesce(self, children):
last = ''
new_lst = []
for cn, cc in children:
if cn not in self.grammar:
last += cn
else:
if last:
new_lst.append((last, []))
last = ''
new_lst.append((cn, cc))
if last:
new_lst.append((last, []))
return new_lst
```

We can now use the fuzzer to produce inputs for our DB.

```
tgf = ConcolicGrammarFuzzer(INVENTORY_GRAMMAR)
while True:
qtree = tgf.fuzz_tree()
query = str(tgf.tree_to_string(qtree))
if query.startswith('select'):
break
```

```
from ExpectError import ExpectError
```

```
with ExpectError():
print(repr(query))
with ConcolicTracer() as _:
res = _[db_select](str(query))
print(repr(res))
```

Our fuzzer returns with an exception. It is unable to find the specified table. Let us examine the predicates it encountered.

```
for i, p in enumerate(_.path):
print(i, p)
```

Note that we can obtain constraints that are not present in the grammar from using the `ConcolicTracer`

. In particular, see how we are able to obtain the condition that the table needs to be `inventory`

(Predicate 11) for the fuzzing to succeed.

How do we lift these to the grammar? and in particular how do we do it automatically? One option we have is to simply switch the last predicate obtained. In our case, the last predicate is (11). Can we simply invert the predicate and solve it again?

```
new_path = _.path[0:-1] + [z3.Not(_.path[-1])]
```

```
new_ = ConcolicTracer((_.decls, new_path))
new_.fn = _.fn
new_.fn_args = _.fn_args
```

```
new_.zeval()
```

Indeed, this will not work as the string lengths being compared to are different.

```
print(_.path[-1])
z3.solve(z3.Not(_.path[-1]))
```

A better idea is to investigate what *string* comparisons are being made, and associate that with the corresponding nodes in the grammar. Let us examine our derivation tree (pruned to avoid recursive structures, and to focus on important parts).

```
from GrammarFuzzer import display_tree
```

```
prune_tokens = [
'<value>', '<table>', '<column>', '<literals>', '<exprs>', '<bexpr>'
]
dt = tgf.prune_tree(qtree, prune_tokens)
display_tree(dt)
```

Can we identify which part of the input was supplied by which part of the grammar? We define `span()`

that can recover this information from the derivation tree. For a given node, let us assume that the start point is known. Then, for processing the children, we proceed as follows: We choose one child at a time from left to right, and compute the length of the child. The length of the children before the current child in addition to our starting point gives the starting point of the current child. The end point for each node is simply the end point of its last children (or the length of its node if it is a leaf).

```
from GrammarFuzzer import START_SYMBOL
```

```
def span(node, g, node_start=0):
hm = {}
k, cs = node
end_i = node_start
new_cs = []
for c in cs:
chm, (ck, child_start, child_end, gcs) = span(c, g, end_i)
new_cs.append((ck, child_start, child_end, gcs))
end_i = child_end
hm.update(chm)
node_end = end_i if cs else node_start + len(k)
if k in g and k != START_SYMBOL:
hm[k] = (node_start, node_end - node_start)
return hm, (k, node_start, node_end, new_cs)
```

We use it as follows:

```
span_hm, _n = span(dt, INVENTORY_GRAMMAR)
```

```
span_hm
```

We can check if we got the right values as follows.

```
print("query:", query)
for k in span_hm:
start, l = span_hm[k]
print(k, query[start:start + l])
```

Next, we need to obtain all the comparisons made in each predicate. For that, we define two helper functions. The first is `unwrap_substrings()`

that translates multiple calls to `z3.SubString`

and returns the start, and length of the given z3 string expression.

```
def unwrap_substrings(s):
assert s.decl().name() == 'str.substr'
cs, frm, l = s.children()
fl = frm.as_long()
ll = l.as_long()
if cs.decl().name() == 'str.substr':
newfrm, _l = unwrap_substrings(cs)
return (fl + newfrm, ll)
else:
return (fl, ll)
```

We define `traverse_z3()`

that traverses a given z3 string expression, and collects all direct string comparisons to a substring of the original argument.

```
def traverse_z3(p, hm):
def z3_as_string(v):
return v.as_string()
n = p.decl().name()
if n == 'not':
return traverse_z3(p.children()[0], hm)
elif n == '=':
i, j = p.children()
if isinstance(i, (int, z3.IntNumRef)):
return traverse_z3(j, hm)
elif isinstance(j, (int, z3.IntNumRef)):
return traverse_z3(i, hm)
else:
if i.is_string() and j.is_string():
if i.is_string_value():
cs, frm, l = j.children()
if (isinstance(frm, z3.IntNumRef)
and isinstance(l, z3.IntNumRef)):
hm[z3_as_string(i)] = unwrap_substrings(j)
elif j.is_string_value():
cs, frm, l = i.children()
if (isinstance(frm, z3.IntNumRef)
and isinstance(l, z3.IntNumRef)):
hm[z3_as_string(j)] = unwrap_substrings(i)
else:
assert False # for now
elif n == '<' or n == '>':
i, j = p.children()
if isinstance(i, (int, z3.IntNumRef)):
return traverse_z3(j, hm)
elif isinstance(j, (int, z3.IntNumRef)):
return traverse_z3(i, hm)
else:
assert False
return p
```

```
comparisons = {}
for p in _.path:
traverse_z3(p, comparisons)
comparisons
```

All that we need now is to declare string variables that match the substrings in `comparisons`

, and solve for them for each item in the path. For that, we define `find_alternatives()`

.

```
def find_alternatives(spans, cmp):
alts = {}
for key in spans:
start, l = spans[key]
rset = set(range(start, start + l))
for ckey in cmp:
cstart, cl = cmp[ckey]
cset = set(range(cstart, cstart + cl))
# if rset.issubset(cset): <- ignoring subsets for now.
if rset == cset:
if key not in alts:
alts[key] = set()
alts[key].add(ckey)
return alts
```

We use it as follows.

```
alternatives = find_alternatives(span_hm, comparisons)
alternatives
```

So, we have our alternatives for each key in the grammar. We can now update our grammar as follows.

```
INVENTORY_GRAMMAR_NEW = dict(INVENTORY_GRAMMAR)
```

```
for k in alternatives:
INVENTORY_GRAMMAR_NEW[k] = INVENTORY_GRAMMAR_NEW[k] + list(alternatives[k])
```

We made a choice here. We could have completely overwritten the definition of `<table>`

. Instead, we added our new alternatives to the existing definition. This way, our fuzzer will also attempt other values for `<table>`

once in a while.

```
INVENTORY_GRAMMAR_NEW['<table>']
```

Let us try fuzzing with our new grammar.

```
cgf = ConcolicGrammarFuzzer(INVENTORY_GRAMMAR_NEW)
```

```
for i in range(10):
qtree = cgf.fuzz_tree()
query = cgf.tree_to_string(qtree)
print(query)
with ExpectError():
try:
with ConcolicTracer() as _:
res = _[db_select](query)
print(repr(res))
except SQLException as e:
print(e)
print()
```

That is, we were able to reach the dangerous method `my_eval()`

.
In effect, what we have done is to lift parts of predicates to the grammar. The new grammar can generate inputs that reach deeper into the program than before. Note that we have only handled the equality predicate. One can also lift the '<' and '>' comparison operators to the grammar if required.

Compare the output of our fuzzer to the original `GrammarFuzzer`

below.

```
gf = GrammarFuzzer(INVENTORY_GRAMMAR)
for i in range(10):
query = gf.fuzz()
print(query)
with ExpectError():
try:
res = db_select(query)
print(repr(res))
except SQLException as e:
print(e)
print()
```

As can be seen, the original grammar fuzzer is unable to proceed beyond the table verification.

#### All together¶

We implement these methods in `ConcolicGrammarFuzzer`

. The method `update_grammar()`

allows `ConcolicGrammarFuzzer`

to collect feedback from concolic fuzzing, and update the grammar used for fuzzing accordingly.

```
class ConcolicGrammarFuzzer(ConcolicGrammarFuzzer):
def prune_tokens(self, tokens):
self.prune_tokens = tokens
def update_grammar(self, trace):
self.comparisons = {}
for p in trace.path:
traverse_z3(p, comparisons)
alternatives = find_alternatives(self.span_range, comparisons)
if self.log:
print('Alternatives:', alternatives, 'Span:', self.span_range)
new_grammar = dict(self.grammar)
for k in alternatives:
new_grammar[k] = list(set(new_grammar[k] + list(alternatives[k])))
self.grammar = new_grammar
```

The `fuzz()`

method simply generates the derivation tree, computes the span range, and returns the string generated from the derivation tree.

```
class ConcolicGrammarFuzzer(ConcolicGrammarFuzzer):
def fuzz(self):
qtree = self.fuzz_tree()
self.pruned_tree = self.prune_tree(qtree, self.prune_tokens)
query = self.tree_to_string(qtree)
self.span_range, _n = span(self.pruned_tree, self.grammar)
return query
```

To ensure that our approach works, let us update our tables slightly.

```
inventory = db.db.pop('inventory', None)
```

```
db.db['vehicles'] = inventory
db.db['months'] = ({
'month': int,
'name': str
}, [{
'month': i + 1,
'name': m
} for i, m in enumerate([
'jan', 'feb', 'mar', 'apr', 'may', 'jun', 'jul', 'aug', 'sep', 'oct',
'nov', 'dec'
])])
db.db
```

The `ConcolicGrammarFuzzer`

is used as follows.

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

As can be seen, the fuzzer starts with no knowledge of the tables `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.

## Limitations¶

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.

## Lessons Learned¶

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.

## Next Steps¶

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.

## Background¶

The technique of concolic execution was originally used to inform and expand the scope of *symbolic execution* [King *et al*, 1976.], 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. [Cadar *et al*, 2005.]. The concolic execution technique for Python programs used in this chapter was pioneered by PeerCheck [A. Bruni *et al*, 2011.], and Python Error Finder [Damián Barsotti *et al*, 2018.].

## Exercises¶

### Exercise 1: Implment a Concolic Float Proxy Class¶

While implementing the `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.

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

We then implement the initialization methods.

```
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`

.

```
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.

```
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

```
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.

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

```
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.

```
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`

```
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__',
]
```

```
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.

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

```
_.context
```

Finally, we fix the `zint`

binary wrapper to correctly create `zfloat`

when needed.

```
def make_int_binary_wrapper(fname, fun, zfun):
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
```

```
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.

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

```
_.zeval()
```

### Exercise 2: Bit Manipulation¶

Similar to floats, implementing the bit manipulation functions such as `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.

```
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.

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

```
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))
```

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

```
init_concolic_4()
```

Invert is the only unary bit manipulation method.