Integer arithmetic from a computer point of view

A thing that is often overlooked is the way arithmetic operations work in a computer and specifically in the processing unit: not having a clear idea of how the operations are performed and their limitations can cause very important bugs to happen and also help in case you want to reverse unknown code.

In this I will explore how operation on integers (floating point will be treated in a specific, future, post).

Integer encoding

We are interested of the case of a register containing a number: since the arithmetic inside a CPU is done on registers via the ALU and since the registers are size limited, all the arithmetic operations are intended modulo the size of the registers: so if \(N\) is the number of bits of the registers we can only represent (directly) unsigned values between \(0\) and \(2^N - 1\).

Note: there is a tricky part about the representation of numbers, i.e.

The formula is the following

$$ b = \sum_{i = 0}^{N - 1} b_i 2^i $$

Remember the following properties for binary numbers: completness

$$ \sum_{n = 0}^{N} 2^n = 2^{N + 1} - 1 $$

and shifting - multiplication relation

$$ \eqalign{ 2\cdot2^N &= 2^N + 2^N \cr &= 2^N + \sum_{i = 0}^{N - 1} 2^i + 1 \cr &= 2^N + 2^{N - 1} + 2^{N - 2} + \dots + 2^1 + 2^0 + 1 \cr &= \sum_{i = 0}^N 2^i + 1 \cr &= 2^{N + 1} - 1 + 1 \cr &= 2^{N + 1} \cr } $$

that means that left shifting a binary number is equivalent to multiplying the same number for a power of two (I know, I know, this is obvious).

For signed numbers there is not a unique way to represent them: the quick and dirty way would be to use the most significant bit as sign bit but this has the drawback to have two zeros.

I think that this encoding is not used by anyone in the real world (but I could be wrong), there are more efficient ways.

One's complement

It consists in flipping all the bits of a number, in this way if you define the negative of a given number as the one's complement of it you have the nice property that this two numbers summed are equal to zero. However you have to add one to a subtraction to obtain the correct result

$$ \hbox{one}(x) = 2^N - x $$

$$ -x = \hbox{one}(x) + 1 $$

The problem is that you have two zeros: all bits equal to zero and all equal to one.

Two's complement

It's an extension of the one's complement: to obtain the negative representation of a number you have to take the one's complement and add one: in this way you have an asymmetry between the minimum and maximum number that can be represented, i.e. you can represent values between \(-2^{N - 1}\) and \(2^{N - 1} - 1\). For example with 7 bits you have the interval \((-64, 63)\).

$$ w = - a_{N - 1}\, 2^{N - 1} + \sum_{i = 0}^{N - 2} a_i\,2^i $$

Normally in the code is this the way the negative numbers are represented.

Mind blowing realization is that the two's complement of the lowest integer is itself:

$$ \hbox{two}\left(\tt 0x10000000\right) = {\tt 0x01111111} + 1 = {\tt 0x10000000} $$

Remember that a value into a register is not signed or unsigned by itself, it depends on how is used in the code.

In two's complement representation INT_MIN is less than -INT_MAX. Negation may overflow for signed arithmetic

Operations

We have a representation of numbers in binary that can handle unsigned and signed, but we need to do math with them and we have to manage the limits of having a fixed bit width but at the end of the day is not impossible in practice.

Addition

The simplest operation is the addition: we simply use the rules from the normal math but since we have a fixed number of bits we cannot sum two number and have the result in a register with the same number of bits, indeed if with \(N\) bits we can have as a maximum unsigned value \(2^N-1\) if we sum itself we obtain

$$ \eqalign{ \hbox{max}_u(N) + \hbox{max}_u(N) &= 2^N - 1 + \left(2^N - 1\right) \cr &= 2\cdot2^N - 2 \cr &= 2^{N+1} - 2 \cr &= \hbox{max}_u(N + 1) - 1 \cr } $$

i.e. is possible, with one bit more, to represent more numbers that is possible to obtain (think for example in a register with width 4-bit, you have 7 has a maximum unsigned number, so the maximum result of a sum for this case is 14 that is less that 15, the maximum unsigned possible with 5 bits).

Subtraction

Thanks to the two's complement representation, it is possible to do subtraction in the same way we can do normally in arithmetics, i.e. \(a - b = a + \left(-b\right)\) that can be translated as \( a - b = a + \hbox{two}(b) = a + \hbox{one}(b) + 1\).

The only thing not obvious is the carry flag: if you take for example the case where we subtract zero with itself, we obtain a carry although we shouldn't have it in a normal calculation.

Multiplication

Multiplication is straightforward as well, I mean, there are operations from a processor that implement that;

$$ \eqalign{ \hbox{max}_u(N)\cdot\hbox{max}_u(N) &= \left(2^N - 1\right)\cdot\left(2^N - 1\right) \cr &= 2^{2N} - 2\cdot2^N + 1 \cr &= 2^{2N} - 2^{N + 1} + 1 \cr &= 2^{2N} - 1 + 1 - 2^{N + 1} + 1 \cr &= \hbox{max}_u(2N) - 2^{N + 1} + 2\cr &= \hbox{max}_u(2N) - \left(2^{N + 1} - 2\right)\cr &= \hbox{max}_u(2N) - 2\cdot\left(2^{N} - 1\right)\cr &= \hbox{max}_u(2N) - 2\cdot\hbox{max}_u(N)\cr } $$

so we can contain a result for sure if we use two registers for the destination (like the x86 does with the mul operation).

If we take two 4-bit number and we multiply them together we obtain as maximum result \(15\cdot15 = 225\)

Division

This operation is straightforward as well, the only consideration to add is that here we are handling fixed precision (integer) numbers and in the general case dividing two integers can result in not integer numbers.

To clarify the point we need some terminology

$$ {\hbox{dividend}\over\hbox{divisor}} \rightarrow \hbox{dividend} = \hbox{quotient}\times\hbox{divisor} + \hbox{remainder} $$

Since we cannot divide with numbers less than one (generally dividing by zero raise an exception) this means that the quotient cannot be greater than the dividend, so a register can contain all the possible results; moreover, usually another register is used to store the remainder of the operation (that can be seen as the result of the \(\mod \hbox{divisor}\) operation).

Note that to use division, the following condition MUST apply (source)

(b != 0) && (!((a == INT32_MIN) && (b == -1)))

in particular, on x86 this leads to the raising of a SIGFPE signal.

Sign extension

In certain cases could be necessary to do operations between numbers having a different number of bits; if these numbers are unsigned it's not big deal, but if instead we having signed ones we have to sign extend i.e. to complete the bits of the extended number with all 1s.

Let me make an example: if we have a 8-bit register with the decimal value \(-16\), its representation with two's complement will be 0xef; now, if we want to put this value into a 16-bit register and represent the same number, we have to set as most significant byte 0xff, i.e. 0xffef: this because of a nice property of binary numbers, namely

in our case we have the Mth bit used for the sign and suppose we have other \(s\) bits

$$ \eqalign{ -2^M + \sum_{i=s}^{M-1} 2^i + \sum_{i=0}^{s - 1} 2^i &= - 1 \cr -2^M + \sum_{i=s}^{M - 1} 2^i &= \sum_{i=0}^{s - 1} 2^i - 1 \cr -2^M + \sum_{i=s}^{M - 1} 2^i &= 2^{s} - 1 } $$

$$ \eqalign{ \left(-2^N + 2^{N - 1} + 2^{N - 2} + \dots + 2^s\right) - \left(-2^s\right) &= -2^N + 2^{N - 1} + 2^{N - 2} + \dots + 2^s + 2^s \cr &= -2^N + 2^{N - 1} + 2^{N - 2} + \dots + 2^s + \left[\left(\sum_{i = 0}^{s - 1}2^i\right) + 1\right] \cr &= -2^N + 2^{N - 1} + 2^{N - 2} + \dots + 2^s + 2^{s - 1} + 2^{s - 2} + \dots + 2 + 2^0 + 1 \cr &= -2^N + \sum_{i = 0}^{N - 1} 2^i + 1 \cr &= -2^N + 2^N - 1 + 1 \cr &= 0 \cr } $$

Some architectures have direct instructions to do that, like the movxs in x86, other instead use multiple operations to do the same

Flags

It's all fine and good but as already said, we have a limited number of bits to represent numbers, so it's possible that some operations couldn't be done correctly: for example, if you want to sum, in a 8bits-register 0xff to any other number, you can't fit the result in the register, you should have one bit more; for this reason in a CPU you have also some flags (i.e. one-bit values) usually contained in an unique register to indicate some particular properties of the last arithmetic operation. Take in mind that not all architectures have it.

Each system has its own nomenclature and specific flags, but I think the minimal set is composed of the following

Carry flag (CF)

Used in unsigned numbers to indicate that the result doesn't fit in the register; for an addition is pretty clear what that means, for a subtraction is a little tricky since this flag can be used for this operation as borrow flag (see wikipedia).

There are two schools of thoughts: some architectures (like x86) use the borrow bit, others (like ARM) use the carry and the relation \( (a - b) = a + \hbox{not}(b) + 1\).

Overflow flag (OF)

Used for signed numbers to indicate that the resulting sign bit is not coherent with the correct result; for example with 4-bit (binary) numbers we can have the following four cases:

$$ \eqalign{ 0100 + 0100 &= 1000 \quad\hbox{overflow} \cr 1000 + 1000 &= 0000 \quad\hbox{overflow} \cr 0100 + 0001 &= 0101 \quad\hbox{no overflow} \cr 1100 + 1100 &= 1000 \quad\hbox{no overflow} \cr } $$

It's important to stress that the OF doesn't indicate an overflow into the sign bit, but that the sign bit is wrong: in the last example above you can see that although there is a carry bit into the sign bit, at the end of the calculation the sign is right.

If you want a more in deep explanation, this post about the OF of the 6502's ALU is amazing.

Zero flag (ZF)

The last operation resulted in a result equal to zero, like subtracting two registers containing the same value or doing the logical and operation between two registers having both zero as value.

Sign flag (SF)

This indicates that the result of the last operation is negative

Flow control

At the end of the day the flags are used primarly to do the so called flow control that in high level languages is implemented via if, while, for, etc...

Each architecture implements this with some particular couple of family of instructions: one family to set the flag, like cmp and test, and another to jump to a particular location depending on the particular values the flags have, like jmp, jne, jnz and so on in x86 or b, bne, ble etc... in ARM.

Take in mind that in an instruction like cmp arg1, arg2 is arg2 that is subtracted from arg1.

For an unsigned comparison is sufficient to look at the CF to understand if a number is greater than another. As convention the terms above and below are used in the related jump.

For a signed number it's trickier: the greater condition is achieved if the sign bit is not set and no overflow happened (i.e. the sign bit is consistent) or if the sign bit is set (i.e. the number is negative) and the overflow happened (making the sign bit wrong). As convention the terms greater and *less are used in the related jump.

For the equal condition is sufficient to check the ZF.

Arithmetics in the C language

Our discussion is about how processors consume numbers but obvioulsy you usually write code in some high-level language like C and this presents with the problem of how the variables we declare in the code are "translated" into assembly language by the compiler and how the different operations between variables interact with each other (taking into consideration also that generally you have variables of different size and "signess").

If you want a really deep dive into this kind of stuff, you need to read "The art of software security assessment", in particular chapter 6.

First of all you have a finite number of type: char, integer and floating point. We have to add the sign/unsigned type for the first twos.

Each type has its own bit-width and generally are "classified" following the scheme below

(obviously the ILP stands for "integer", "long", "pointer" and LL stands for "long long"). Note how in all the systems the char is supposed to be 8-bit wide.

C Language's constructs

It's important to be aware of the terminology: source An rvalue is the value of an expression, such as \(2\), or \((x + 3)\) , or \((x + y) * (a - b)\) . rvalues are not storage space

An lvalue is an expression that describes the location of an object used in the program. The location of the object is the object's lvalue, and the object's rvalue is the value stored at the location described by the lvalue.

Type conversions

Here we have the interesting and useful stuffs

value preserving: when the conversion allow to represent all the possible value of the starting type otherwise is called value changing.

The so called integer conversion rank (source) (note that different rank doesn't imply different width of its representation)

usual arithmetic conversions: when an operator needs two integer operands, first check either are floating point, otherwise starts the following procedure

• integer promotions: any type with rank lower than integer is promoted to integer

as described by the C11 6.3.1.1 rule (source here)

if an int can represent all values of the original type (as restricted by the width, for a bit-field), the value is converted to an int; otherwise, it is converted to an unsigned int. These are called the integer promotions.

at this point if the two operands are of the same type then we stop since there is no problem to do the operation, otherwise we need to take into consideration some factors

• same sign, different rank: the narrower type is converted to the wider
• rank(unsigned) >= rank(signed): convert to unsigned type
• rank(unsigned) < rank(signed): there are two cases
  • value preserving: convert both to the signed type
  • value changing: convert both to the corresponding signed type of the unsigned operand

Literal declaration

From the previous section it's obvious that is important to understand precisely the type of a constant to avoid un-expected results; to declare the type of a literal the following suffixes are used

• U or u for unsigned
• L or l for long
• LL or ll for long long

Programmation errors

Out of bounds

Signedness

The signedness can cause two kind of bugs, one pretty logical, like the following where we suppose that

int n = read_some_n();

char buffer[1024];

if (n > 1024) { /* both are integer so no conversion */
    return -1;
}

read(fd, buffer, n);/* here "n" is converted to size_t, i.e. unsigned */

Take in mind that modern operating systems can have some measure to avoid catastrophic event like that, from the man page of read(2):

On  Linux, read() (and similar system calls) will transfer at most
0x7ffff000 (2,147,479,552) bytes, returning the number of bytes actually
transferred.  (This is true on both 32-bit and 64-bit sys‐ tems.)

This avoid that negative numbers casted to size_t can cause harm.

Overflow

Wrap

The following piece of code run indefinetly

    int count = 10;
    while(count-- >= 0U)
        fprintf(stdout, "%c", count);

Conversion

For example left shift << of variables with ranking less than integers are converted to signed integers also in the case they were unsigned so they give unexpected result: for example in this code

uint8_t src = 0x80;
uint64_t dst = src << 24;

dst will contain the value 0xffffffff80000000 since the sign bit is "extended".

Undefined behaviour

Having an asymmetry between the size of the greatest positive and lowest negative in two's complement arithmetics causes some particular behaviour to happen for some functions: take for example the abs() one; this function returns the positive version of (almost) any number, indeed what is the positive value of the lowest negative possible in a given architecture (in two's complement arithmetics)? From the man page we can read that Trying to take the absolute value of the most negative integer is not defined. so the following code is not doing what you would expect when is passed INT_MIN as argument

#define MAX_VALUE 5000

char buffer[MAX_VALUE];

int arg1 = atoi(argv[1]);

if (abs(arg1) < MAX_VALUE) {
    buffer[arg1] = arg2;
}

Links

• A common C integer shifting mistake
• The CARRY flag and OVERFLOW flag in binary arithmetic
• Intel x86 JUMP quick reference
• SEI CERT C Coding Standard
• Condition Codes 1: Condition Flags and Codes from ARM site
• Jumps, flags, and the CMP instruction
• The 6502 overflow flag explained mathematically
• Avoiding Overflow, Underflow, and Loss of Precision "The cardinal rule in numerical analysis is to avoid subtracting nearly equal numbers. The more nearly equal two numbers are, the more precision is lost in the subtraction"
• https://www.cs.utah.edu/~rajeev/cs3810/slides/3810-08.pdf
• https://electronics.stackexchange.com/questions/22410/how-does-division-occur-in-our-computers
• https://gcc.gnu.org/onlinedocs/gcc/Integer-Overflow-Builtins.html
• 64-Bit Programming Models: Why LP64?