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Francesco Mecca 2020-04-03 18:12:32 +02:00
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tesi/conv.py
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@ -3,7 +3,7 @@ import re
from sys import argv
allsymbols = json.load(open('./unicode-latex.json'))
mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', '∈', 'ε','₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π']
mysymbols = ['≡', '≠', '≼', '→', '←', '⊀', '⋠', '≺', '∀', '∈', 'ε','₀', '₂', '₁', '₃', 'ₐ', 'ₖ', 'ₘ', 'ₙ', 'ᵢ', 'ⁱ', '⋮', 'ₛ', 'ₜ', '≃', '⇔', '∧', '∅', 'ℕ', 'ⱼ', 'ʲ', '⊥', 'π', 'α', 'β', '∞', 'σ', '≤', '⊈', '∧', '∨', '∃', '⇒' ]
extrasymbols = {'〚': '\llbracket', '〛': '\rrbracket'}
symbols = {s: allsymbols[s] for s in mysymbols}

777
tesi/referenze/blexim.txt Normal file
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@ -0,0 +1,777 @@
==Phrack Inc.==
Volume 0x0b, Issue 0x3c, Phile #0x0a of 0x10
|=--------------------=[ Basic Integer Overflows ]=----------------------=|
|=-----------------------------------------------------------------------=|
|=-------------------=[ by blexim <blexim@hush.com> ]=-------------------=|
1: Introduction
1.1 What is an integer?
1.2 What is an integer overflow?
1.3 Why can they be dangerous?
2: Integer overflows
2.1 Widthness overflows
2.1.1 Exploiting
2.2 Arithmetic overflows
2.2.1 Exploiting
3: Signedness bugs
3.1 What do they look like?
3.1.1 Exploiting
3.2 Signedness bugs caused by integer overflows
4: Real world examples
4.1 Integer overflows
4.2 Signedness bugs
--[ 1.0 Introduction
In this paper I'm going to describe two classes of programming bugs which
can sometimes allow a malicious user to modify the execution path of an
affected process. Both of these classes of bug work by causing variables
to contain unexpected values, and so are not as "direct" as classes which
overwrite memory, e.g. buffer overflows or format strings. All the
examples given in the paper are in C, so a basic familiarity with C is
assumed. A knowledge of how integers are stored in memory is also useful,
but not essential.
----[ 1.1 What is an integer?
An integer, in the context of computing, is a variable capable of
representing a real number with no fractional part. Integers are typically
the same size as a pointer on the system they are compiled on (i.e. on a 32
bit system, such as i386, an integer is 32 bits long, on a 64 bit system,
such as SPARC, an integer is 64 bits long). Some compilers don't use
integers and pointers of the same size however, so for the sake of
simplicity all the examples refer to a 32 bit system with 32 bit integers,
longs and pointers.
Integers, like all variables are just regions of memory. When we talk
about integers, we usually represent them in decimal, as that is the
numbering system humans are most used to. Computers, being digital, cannot
deal with decimal, so internally to the computer integers are stored in
binary. Binary is another system of representing numbers which uses only
two numerals, 1 and 0, as opposed to the ten numerals used in decimal. As
well as binary and decimal, hexadecimal (base sixteen) is often used in
computing as it is very easy to convert between binary and hexadecimal.
Since it is often necessary to store negative numbers, there needs to be a
mechanism to represent negative numbers using only binary. The way this is
accomplished is by using the most significant bit (MSB) of a variable to
determine the sign: if the MSB is set to 1, the variable is interpreted as
negative; if it is set to 0, the variable is positive. This can cause some
confusion, as will be explained in the section on signedness bugs, because
not all variables are signed, meaning they do not all use the MSB to
determine whether they are positive or negative. These variable are known
as unsigned and can only be assigned positive values, whereas variables
which can be either positive or negative are called unsigned.
----[ 1.2 What is an integer overflow?
Since an integer is a fixed size (32 bits for the purposes of this paper),
there is a fixed maximum value it can store. When an attempt is made to
store a value greater than this maximum value it is known as an integer
overflow. The ISO C99 standard says that an integer overflow causes
"undefined behaviour", meaning that compilers conforming to the standard
may do anything they like from completely ignoring the overflow to aborting
the program. Most compilers seem to ignore the overflow, resulting in an
unexpected or erroneous result being stored.
----[ 1.3 Why can they be dangerous?
Integer overflows cannot be detected after they have happened, so there is
not way for an application to tell if a result it has calculated previously
is in fact correct. This can get dangerous if the calculation has to do
with the size of a buffer or how far into an array to index. Of course
most integer overflows are not exploitable because memory is not being
directly overwritten, but sometimes they can lead to other classes of bugs,
frequently buffer overflows. As well as this, integer overflows can be
difficult to spot, so even well audited code can spring surprises.
--[ 2.0 Integer overflows
So what happens when an integer overflow does happen? ISO C99 has this to
say:
"A computation involving unsigned operands can never overflow,
because a result that cannot be represented by the resulting unsigned
integer type is reduced modulo the number that is one greater than
the largest value that can be represented by the resulting type."
NB: modulo arithmetic involves dividing two numbers and taking the
remainder,
e.g.
10 modulo 5 = 0
11 modulo 5 = 1
so reducing a large value modulo (MAXINT + 1) can be seen as discarding the
portion of the value which cannot fit into an integer and keeping the rest.
In C, the modulo operator is a % sign.
</NB>
This is a bit wordy, so maybe an example will better demonstrate the
typical "undefined behaviour":
We have two unsigned integers, a and b, both of which are 32 bits long. We
assign to a the maximum value a 32 bit integer can hold, and to b we assign
1. We add a and b together and store the result in a third unsigned 32 bit
integer called r:
a = 0xffffffff
b = 0x1
r = a + b
Now, since the result of the addition cannot be represented using 32 bits,
the result, in accordance with the ISO standard, is reduced modulo
0x100000000.
r = (0xffffffff + 0x1) % 0x100000000
r = (0x100000000) % 0x100000000 = 0
Reducing the result using modulo arithmetic basically ensures that only the
lowest 32 bits of the result are used, so integer overflows cause the
result to be truncated to a size that can be represented by the variable.
This is often called a "wrap around", as the result appears to wrap around
to 0.
----[ 2.1 Widthness overflows
So an integer overflow is the result of attempting to store a value in a
variable which is too small to hold it. The simplest example of this can
be demonstrated by simply assigning the contents of large variable to a
smaller one:
/* ex1.c - loss of precision */
#include <stdio.h>
int main(void){
int l;
short s;
char c;
l = 0xdeadbeef;
s = l;
c = l;
printf("l = 0x%x (%d bits)\n", l, sizeof(l) * 8);
printf("s = 0x%x (%d bits)\n", s, sizeof(s) * 8);
printf("c = 0x%x (%d bits)\n", c, sizeof(c) * 8);
return 0;
}
/* EOF */
The output of which looks like this:
nova:signed {48} ./ex1
l = 0xdeadbeef (32 bits)
s = 0xffffbeef (16 bits)
c = 0xffffffef (8 bits)
Since each assignment causes the bounds of the values that can be stored in
each type to be exceeded, the value is truncated so that it can fit in the
variable it is assigned to.
It is worth mentioning integer promotion here. When a calculation
involving operands of different sizes is performed, the smaller operand is
"promoted" to the size of the larger one. The calculation is then
performed with these promoted sizes and, if the result is to be stored in
the smaller variable, the result is truncated to the smaller size again.
For example:
int i;
short s;
s = i;
A calculation is being performed with different sized operands here. What
happens is that the variable s is promoted to an int (32 bits long), then
the contents of i is copied into the new promoted s. After this, the
contents of the promoted variable are "demoted" back to 16 bits in order to
be saved in s. This demotion can cause the result to be truncated if it is
greater than the maximum value s can hold.
------[ 2.1.1 Exploiting
Integer overflows are not like most common bug classes. They do not allow
direct overwriting of memory or direct execution flow control, but are much
more subtle. The root of the problem lies in the fact that there is no way
for a process to check the result of a computation after it has happened,
so there may be a discrepancy between the stored result and the correct
result. Because of this, most integer overflows are not actually
exploitable. Even so, in certain cases it is possible to force a crucial
variable to contain an erroneous value, and this can lead to problems later
in the code.
Because of the subtlety of these bugs, there is a huge number of situations
in which they can be exploited, so I will not attempt to cover all
exploitable conditions. Instead, I will provide examples of some
situations which are exploitable, in the hope of inspiring the reader in
their own research :)
Example 1:
/* width1.c - exploiting a trivial widthness bug */
#include <stdio.h>
#include <string.h>
int main(int argc, char *argv[]){
unsigned short s;
int i;
char buf[80];
if(argc < 3){
return -1;
}
i = atoi(argv[1]);
s = i;
if(s >= 80){ /* [w1] */
printf("Oh no you don't!\n");
return -1;
}
printf("s = %d\n", s);
memcpy(buf, argv[2], i);
buf[i] = '\0';
printf("%s\n", buf);
return 0;
}
While a construct like this would probably never show up in real life code,
it serves well as an example. Take a look at the following inputs:
nova:signed {100} ./width1 5 hello
s = 5
hello
nova:signed {101} ./width1 80 hello
Oh no you don't!
nova:signed {102} ./width1 65536 hello
s = 0
Segmentation fault (core dumped)
The length argument is taken from the command line and held in the integer
i. When this value is transferred into the short integer s, it is
truncated if the value is too great to fit into s (i.e. if the value is
greater than 65535). Because of this, it is possible to bypass the bounds
check at [w1] and overflow the buffer. After this, standard stack smashing
techniques can be used to exploit the process.
----[ 2.2 Arithmetic overflows
As shown in section 2.0, if an attempt is made to store a value in an
integer which is greater than the maximum value the integer can hold, the
value will be truncated. If the stored value is the result of an
arithmetic operation, any part of the program which later uses the result
will run incorrectly as the result of the arithmetic being incorrect.
Consider this example demonstrating the wrap around shown earlier:
/* ex2.c - an integer overflow */
#include <stdio.h>
int main(void){
unsigned int num = 0xffffffff;
printf("num is %d bits long\n", sizeof(num) * 8);
printf("num = 0x%x\n", num);
printf("num + 1 = 0x%x\n", num + 1);
return 0;
}
/* EOF */
The output of this program looks like this:
nova:signed {4} ./ex2
num is 32 bits long
num = 0xffffffff
num + 1 = 0x0
Note:
The astute reader will have noticed that 0xffffffff is decimal -1, so it
appears that we're just doing
1 + (-1) = 0
Whilst this is one way at looking at what's going on, it may cause some
confusion since the variable num is unsigned and therefore all arithmetic
done on it will be unsigned. As it happens, a lot of signed arithmetic
depends on integer overflows, as the following demonstrates (assume both
operands are 32 bit variables):
-700 + 800 = 100
0xfffffd44 + 0x320 = 0x100000064
Since the result of the addition exceeds the range of the variable, the
lowest 32 bits are used as the result. These low 32 bits are 0x64, which
is equal to decimal 100.
</note>
Since an integer is signed by default, an integer overflow can cause a
change in signedness which can often have interesting effects on subsequent
code. Consider the following example:
/* ex3.c - change of signedness */
#include <stdio.h>
int main(void){
int l;
l = 0x7fffffff;
printf("l = %d (0x%x)\n", l, l);
printf("l + 1 = %d (0x%x)\n", l + 1 , l + 1);
return 0;
}
/* EOF */
The output of which is:
nova:signed {38} ./ex3
l = 2147483647 (0x7fffffff)
l + 1 = -2147483648 (0x80000000)
Here the integer is initialised with the highest positive value a signed
long integer can hold. When it is incremented, the most significant bit
(indicating signedness) is set and the integer is interpreted as being
negative.
Addition is not the only arithmetic operation which can cause an integer to
overflow. Almost any operation which changes the value of a variable can
cause an overflow, as demonstrated in the following example:
/* ex4.c - various arithmetic overflows */
#include <stdio.h>
int main(void){
int l, x;
l = 0x40000000;
printf("l = %d (0x%x)\n", l, l);
x = l + 0xc0000000;
printf("l + 0xc0000000 = %d (0x%x)\n", x, x);
x = l * 0x4;
printf("l * 0x4 = %d (0x%x)\n", x, x);
x = l - 0xffffffff;
printf("l - 0xffffffff = %d (0x%x)\n", x, x);
return 0;
}
/* EOF */
Output:
nova:signed {55} ./ex4
l = 1073741824 (0x40000000)
l + 0xc0000000 = 0 (0x0)
l * 0x4 = 0 (0x0)
l - 0xffffffff = 1073741825 (0x40000001)
The addition is causing an overflow in exactly the same way as the first
example, and so is the multiplication, although it may seem different. In
both cases the result of the arithmetic is too great to fit in an integer,
so it is reduced as described above. The subtraction is slightly
different, as it is causing an underflow rather than an overflow: an
attempt is made to store a value lower than the minimum value the integer
can hold, causing a wrap around. In this way we are able to force an
addition to subtract, a multiplication to divide or a subtraction to add.
------[ 2.2.1 Exploiting
One of the most common ways arithmetic overflows can be exploited is when a
calculation is made about how large a buffer must be allocated. Often a
program must allocate space for an array of objects, so it uses the
malloc(3) or calloc(3) routines to reserve the space and calculates how
much space is needed by multiplying the number of elements by the size of
an object. As has been previously shown, if we are able to control either
of these operands (number of elements or object size) we may be able to
mis-size the buffer, as the following code fragment shows:
int myfunction(int *array, int len){
int *myarray, i;
myarray = malloc(len * sizeof(int)); /* [1] */
if(myarray == NULL){
return -1;
}
for(i = 0; i < len; i++){ /* [2] */
myarray[i] = array[i];
}
return myarray;
}
This seemingly innocent function could bring about the downfall of a system
due to its lack of checking of the len parameter. The multiplication at
[1] can be made to overflow by supplying a high enough value for len, so we
can force the buffer to be any length we choose. By choosing a suitable
value for len, we can cause the loop at [2] to write past the end of the
myarray buffer, resulting in a heap overflow. This could be leveraged into
executing arbitrary code on certain implementations by overwriting malloc
control structures, but that is beyond the scope of this article.
Another example:
int catvars(char *buf1, char *buf2, unsigned int len1,
unsigned int len2){
char mybuf[256];
if((len1 + len2) > 256){ /* [3] */
return -1;
}
memcpy(mybuf, buf1, len1); /* [4] */
memcpy(mybuf + len1, buf2, len2);
do_some_stuff(mybuf);
return 0;
}
In this example, the check at [3] can be bypassed by using suitable values
for len1 and len2 that will cause the addition to overflow and wrap around
to a low number. For example, the following values:
len1 = 0x104
len2 = 0xfffffffc
when added together would result in a wrap around with a result of 0x100
(decimal 256). This would pass the check at [3], then the memcpy(3)'s at
[4] would copy data well past the end of the buffer.
--[ 3 Signedness Bugs
Signedness bugs occur when an unsigned variable is interpreted as signed,
or when a signed variable is interpreted as unsigned. This type of
behaviour can happen because internally to the computer, there is no
distinction between the way signed and unsigned variables are stored.
Recently, several signedness bugs showed up in the FreeBSD and OpenBSD
kernels, so there are many examples readily available.
----[ 3.1 What do they look like?
Signedness bugs can take a variety of forms, but some of the things to look
out for are:
* signed integers being used in comparisons
* signed integers being used in arithmetic
* unsigned integers being compared to signed integers
Here is classic example of a signedness bug:
int copy_something(char *buf, int len){
char kbuf[800];
if(len > sizeof(kbuf)){ /* [1] */
return -1;
}
return memcpy(kbuf, buf, len); /* [2] */
}
The problem here is that memcpy takes an unsigned int as the len parameter,
but the bounds check performed before the memcpy is done using signed
integers. By passing a negative value for len, it is possible to pass the
check at [1], but then in the call to memcpy at [2], len will be interpeted
as a huge unsigned value, causing memory to be overwritten well past the
end of the buffer kbuf.
Another problem that can stem from signed/unsigned confusion occurs when
arithmetic is performed. Consider the following example:
int table[800];
int insert_in_table(int val, int pos){
if(pos > sizeof(table) / sizeof(int)){
return -1;
}
table[pos] = val;
return 0;
}
Since the line
table[pos] = val;
is equivalent to
*(table + (pos * sizeof(int))) = val;
we can see that the problem here is that the code does not expect a
negative operand for the addition: it expects (table + pos) to be greater
than table, so providing a negative value for pos causes a situation which
the program does not expect and can therefore not deal with.
------[ 3.1.1 Exploiting
This class of bug can be problematic to exploit, due to the fact that
signed integers, when interpreted as unsigned, tend to be huge. For
example, -1 when represented in hexadecimal is 0xffffffff. When
interpreted as unsiged, this becomes the greatest value it is possible to
represent in an integer (4,294,967,295), so if this value is passed to
mempcpy as the len parameter (for example), memcpy will attempt to copy 4GB
of data to the destination buffer. Obviously this is likely to cause a
segfault or, if not, to trash a large amount of the stack or heap.
Sometimes it is possible to get around this problem by passing a very low
value for the source address and hope, but this is not always possible.
----[ 3.2 Signedness bugs caused by integer overflows
Sometimes, it is possible to overflow an integer so that it wraps around to
a negative number. Since the application is unlikely to expect such a
value, it may be possible to trigger a signedness bug as described above.
An example of this type of bug could look like this:
int get_two_vars(int sock, char *out, int len){
char buf1[512], buf2[512];
unsigned int size1, size2;
int size;
if(recv(sock, buf1, sizeof(buf1), 0) < 0){
return -1;
}
if(recv(sock, buf2, sizeof(buf2), 0) < 0){
return -1;
}
/* packet begins with length information */
memcpy(&size1, buf1, sizeof(int));
memcpy(&size2, buf2, sizeof(int));
size = size1 + size2; /* [1] */
if(size > len){ /* [2] */
return -1;
}
memcpy(out, buf1, size1);
memcpy(out + size1, buf2, size2);
return size;
}
This example shows what can sometimes happen in network daemons, especially
when length information is passed as part of the packet (in other words, it
is supplied by an untrusted user). The addition at [1], used to check that
the data does not exceed the bounds of the output buffer, can be abused by
setting size1 and size2 to values that will cause the size variable to wrap
around to a negative value. Example values could be:
size1 = 0x7fffffff
size2 = 0x7fffffff
(0x7fffffff + 0x7fffffff = 0xfffffffe (-2)).
When this happens, the bounds check at [2] passes, and a lot more of the
out buffer can be written to than was intended (in fact, arbitrary memory
can be written to, as the (out + size1) dest parameter in the second memcpy
call allows us to get to any location in memory).
These bugs can be exploited in exactly the same way as regular signedness
bugs and have the same problems associated with them - i.e. negative values
translate to huge positive values, which can easily cause segfaults.
--[ 4 Real world examples
There are many real world applications containing integer overflows and
signedness bugs, particularly network daemons and, frequently, in operating
system kernels.
----[ 4.1 Integer overflows
This (non-exploitable) example was taken from a security module for linux.
This code runs in the kernel context:
int rsbac_acl_sys_group(enum rsbac_acl_group_syscall_type_t call,
union rsbac_acl_group_syscall_arg_t arg)
{
...
switch(call)
{
case ACLGS_get_group_members:
if( (arg.get_group_members.maxnum <= 0) /* [A] */
|| !arg.get_group_members.group
)
{
...
rsbac_uid_t * user_array;
rsbac_time_t * ttl_array;
user_array = vmalloc(sizeof(*user_array) *
arg.get_group_members.maxnum); /* [B] */
if(!user_array)
return -RSBAC_ENOMEM;
ttl_array = vmalloc(sizeof(*ttl_array) *
arg.get_group_members.maxnum); /* [C] */
if(!ttl_array)
{
vfree(user_array);
return -RSBAC_ENOMEM;
}
err =
rsbac_acl_get_group_members(arg.get_group_members.group,
user_array,
ttl_array,
arg.get_group_members.max
num);
...
}
In this example, the bounds checking at [A] is not sufficient to prevent
the integer overflows at [B] and [C]. By passing a high enough (i.e.
greater than 0xffffffff / 4) value for arg.get_group_members.maxnum, we
can cause the multiplications at [B] and [C] to overflow and force the
buffers ttl_array and user_array to be smaller than the application
expects. Since rsbac_acl_get_group_members copies user controlled data
to these buffers, it is possible to write past the end of the user_array
and ttl_array buffers. In this case, the application used vmalloc() to
allocate the buffers, so an attempt to write past the end of the buffers
will simply raise an error, so it cannot be exploited. Even so, it
provides an example of what these bugs can look like in real code.
Another example of a recent real world integer overflow vulnerability
was the problem in the XDR RPC library (discovered by ISS X-Force). In this
case, user supplied data was used in the calculation of the size of a
dynamically allocated buffer which was filled with user supplied data. The
vulnerable code was this:
bool_t
xdr_array (xdrs, addrp, sizep, maxsize, elsize, elproc)
XDR *xdrs;
caddr_t *addrp; /* array pointer */
u_int *sizep; /* number of elements */
u_int maxsize; /* max numberof elements */
u_int elsize; /* size in bytes of each element */
xdrproc_t elproc; /* xdr routine to handle each element */
{
u_int i;
caddr_t target = *addrp;
u_int c; /* the actual element count */
bool_t stat = TRUE;
u_int nodesize;
...
c = *sizep;
if ((c > maxsize) && (xdrs->x_op != XDR_FREE))
{
return FALSE;
}
nodesize = c * elsize; /* [1] */
...
*addrp = target = mem_alloc (nodesize); /* [2] */
...
for (i = 0; (i < c) && stat; i++)
{
stat = (*elproc) (xdrs, target, LASTUNSIGNED); /* [3] */
target += elsize;
}
As you can see, by supplying large values for elsize and c (sizep), it
was possible to cause the multiplication at [1] to overflow and cause
nodesize to be much smaller than the application expected. Since
nodesize was then used to allocate a buffer at [2], the buffer could be
mis-sized leading to a heap overflow at [3]. For more information on this
hole, see the CERT advisory listed in the appendix.
----[ 4.2 Signedness bugs
Recently, several signedness bugs were brought to light in the freebsd
kernel. These allowed large portions of kernel memory to be read by
passing
negative length paramters to various syscalls. The getpeername(2) function
had such a problem and looked like this:
static int
getpeername1(p, uap, compat)
struct proc *p;
register struct getpeername_args /* {
int fdes;
caddr_t asa;
int *alen;
} */ *uap;
int compat;
{
struct file *fp;
register struct socket *so;
struct sockaddr *sa;
int len, error;
...
error = copyin((caddr_t)uap->alen, (caddr_t)&len, sizeof (len));
if (error) {
fdrop(fp, p);
return (error);
}
...
len = MIN(len, sa->sa_len); /* [1] */
error = copyout(sa, (caddr_t)uap->asa, (u_int)len);
if (error)
goto bad;
gotnothing:
error = copyout((caddr_t)&len, (caddr_t)uap->alen, sizeof (len));
bad:
if (sa)
FREE(sa, M_SONAME);
fdrop(fp, p);
return (error);
}
This is a classic example of a signedness bug - the check at [1] did not
take into account the fact that len could be negative, in which case the
MIN macro would always return len. When this negative len parameter was
passed to copyout, it was interpretted as a huge positive integer which
caused copyout to copy up to 4GB of kernel memory to user space.
--[ Conclusion
Integer overflows can be extremely dangerous, partly because it is
impossible to detect them after they have happened. If an integer overflow
takes place, the application cannot know that the calculation it has
performed is incorrect, and it will continue under the assumption that it
is. Even though they can be difficult to exploit, and frequently cannot be
exploited at all, they can cause unepected behaviour, which is never a good
thing in a secure system.
--[ Appendix
CERT advisory on the XDR bug:
http://www.cert.org/advisories/CA-2002-25.html
FreeBSD advisory: http://online.securityfocus.com/advisories/4407
|=[ EOF ]=---------------------------------------------------------------=|

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@ -636,6 +636,131 @@ pattern is not exhaustive or some patterns are shadowed by precedent ones.
Symbolic execution is a widely used techniques in the field of
computer security.
It allows to analyze different execution paths of a program
simultanously while tracking which inputs trigger the execution of
different parts of the program.
Inputs are modelled symbolically rather than taking "concrete" values.
A symbolic execution engine keeps track of expressions and variables
in terms of these symbolic symbols and attaches logical constraints to every
branch that is being followed.
Symbolic execution engines are used to track bugs by modelling the
domain of all possible inputs of a program, detecting infeasible
paths, dead code and proving that two code segments are equivalent.
Let's take as example this signedness bug that was found in the
FreeBSD kernel and allowed, when calling the getpeername function, to
read portions of kernel memory.
#+BEGIN_SRC
int compat;
{
struct file *fp;
register struct socket *so;
struct sockaddr *sa;
int len, error;
...
len = MIN(len, sa->sa_len); /* [1] */
error = copyout(sa, (caddr_t)uap->asa, (u_int)len);
if (error)
goto bad;
...
bad:
if (sa)
FREE(sa, M_SONAME);
fdrop(fp, p);
return (error);
}
#+END_SRC
The tree of the execution when the function is evaluated considering
/int len/ our symbolic variable α, sa->sa_len as symbolic variable β
and π as the set of constraints on a symbolic variable:
#+BEGIN_SRC
[1] compat (...) { π_{α}: -∞ < α < ∞ }
|
[2] min (σ₁, σ₂) { π_{σ}: -∞ < (σ₁,σ₂) < ∞ ; π_{α}: -∞ < α < β ; π_{β}: ...}
|
[3] cast(u_int) (...) { π_{σ}: 0 ≤ (σ) < ∞ ; π_{α}: -∞ < α < β ; π_{β}: ...}
|
... // rest of the execution
#+END_SRC
We can see that at step 3 the set of possible values of the scrutinee
α is bigger than the set of possible values of the input σ to the
/cast/ directive, that is: π_{α} ⊈ π_{σ}. For this reason the /cast/ may fail when α is /len/
negative number, outside the domain π_{σ}. In C this would trigger undefined behaviour (signed
overflow) that made the exploitation possible.
Every step of evaluation can be modelled as the following transition:
#+BEGIN_SRC
(π_{σ}, (πᵢ)ⁱ) → (π'_{σ}, (π'ᵢ)ⁱ)
#+END_SRC
if we express the π constraints as logical formulas we can model the
execution of the program in terms of Hoare Logic.
State of the computation is a Hoare triple {P}C{Q} where P and Q are
respectively the /precondition/ and the /postcondition/ that
constitute the assertions of the program. C is the directive being
executed.
The language of the assertions P is:
#+BEGIN_SRC
P ::= true | false | a < b | P₁ ∧ P₂ | P₁ ∨ P₂ | ~P
#+END_SRC
where a and b are numbers.
In the Hoare rules assertions could also take the form of
#+BEGIN_SRC
P ::= ∀i. P | ∃i. P | P₁ ⇒ P₂
#+END_SRC
where i is a logical variable, but assertions of these kinds increases
the complexity of the symbolic engine.
Execution follows the rules of Hoare logic:
- Empty statement :
\begin{verbatim}
————————————
{P}/skip/{P}
\end{verbatim}
- Assignment statement : The truthness of P[a/x] is equivalent to the
truth of {P} after the assignment.
\begin{verbatim}
————————————
{P[a/x]}x:=a{P}
\end{verbatim}
- Composition : c₁ and c₂ are directives that are executed in order;
{Q} is the /midcondition/.
\begin{verbatim}
{P}c₁{R}, {R}c₂{Q}
——————————————————
{P}c₁;c₂{Q}
\end{verbatim}
- Conditional :
\begin{verbatim}
{P∧b}c₁{Q}, {P∧~b}c₂{Q}
————————————————————————
{P}if b then c₁ else c₂{Q}
\end{verbatim}
- Loop : {P} is the loop invariant. After the loop is finished /P/
holds and ~b caused the loop to end.
\begin{verbatim}
{P∧b}c{P}
————————————————————————
{P}while b do c{P∧~b}
\end{verbatim}
Even if the semantics of symbolic execution engines are well defined,
the user may run into different complications when applying such
analysis to non trivial codebases.
For example, depending on the domain, loop termination is not
guaranteed. Even when termination is guaranteed, looping causes
exponential branching that may lead to path explosion or state
explosion.
Reasoning about all possible executions of a program is not always
feasible and in case of explosion usually symbolic execution engines
implement heuristics to reduce the size of the search space.
** Translation validation
Translators, such as translators and code generators, are huge pieces of