How is a CRC32 checksum calculated?

Question

Maybe I'm just not seeing it, but CRC32 seems either needlessly complicated, or insufficiently explained anywhere I could find on the web.

I understand that it is the remainder from a non-carry-based arithmetic division of the message value, divided by the (generator) polynomial, but the actual implementation of it escapes me.

I've read A Painless Guide To CRC Error Detection Algorithms, and I must say it was not painless. It goes over the theory rather well, but the author never gets to a simple "this is it." He does say what the parameters are for the standard CRC32 algorithm, but he neglects to lay out clearly how you get to it.

The part that gets me is when he says "this is it" and then adds on, "oh by the way, it can be reversed or started with different initial conditions," and doesn't give a clear answer of what the final way of calculating a CRC32 checksum given all of the changes he just added.

• Is there a simpler explanation of how CRC32 is calculated?

I attempted to code in C how the table is formed:

for (i = 0; i < 256; i++)
{
    temp = i;

    for (j = 0; j < 8; j++)
    {
        if (temp & 1)
        {
            temp >>= 1;
            temp ^= 0xEDB88320;
        }
        else {temp >>= 1;}
    }
    testcrc[i] = temp;
}

but this seems to generate values inconsistent with values I have found elsewhere on the Internet. I could use the values I found online, but I want to understand how they were created.

Any help in clearing up these incredibly confusing numbers would be very appreciated.

Answer 1

The polynomial for CRC32 is:

x³² + x²⁶ + x²³ + x²² + x¹⁶ + x¹² + x¹¹ + x¹⁰ + x⁸ + x⁷ + x⁵ + x⁴ + x² + x + 1

• Wikipedia
• CRC calculation

Or in hex and binary:

0x 01 04 C1 1D B7
1 0000 0100 1100 0001 0001 1101 1011 0111

The highest term (x³²) is usually not explicitly written, so it can instead be represented in hex just as

0x 04 C1 1D B7

Feel free to count the 1s and 0s, but you'll find they match up with the polynomial, where 1 is bit 0 (or the first bit) and x is bit 1 (or the second bit).

Why this polynomial? Because there needs to be a standard given polynomial and the standard was set by IEEE 802.3. Also it is extremely difficult to find a polynomial that detects different bit errors effectively.

You can think of the CRC-32 as a series of "Binary Arithmetic with No Carries", or basically "XOR and shift operations". This is technically called Polynomial Arithmetic.

• CRC primer, Chapter 5

To better understand it, think of this multiplication:

(x^3 + x^2 + x^0)(x^3 + x^1 + x^0)
= (x^6 + x^4 + x^3
 + x^5 + x^3 + x^2
 + x^3 + x^1 + x^0)
= x^6 + x^5 + x^4 + 3*x^3 + x^2 + x^1 + x^0

If we assume x is base 2 then we get:

x^7 + x^3 + x^2 + x^1 + x^0

• CRC primer Chp.5

Why? Because 3x^3 is 11x^11 (but we need only 1 or 0 pre digit) so we carry over:

=1x^110 + 1x^101 + 1x^100          + 11x^11 + 1x^10 + 1x^1 + x^0
=1x^110 + 1x^101 + 1x^100 + 1x^100 + 1x^11 + 1x^10 + 1x^1 + x^0
=1x^110 + 1x^101 + 1x^101          + 1x^11 + 1x^10 + 1x^1 + x^0
=1x^110 + 1x^110                   + 1x^11 + 1x^10 + 1x^1 + x^0
=1x^111                            + 1x^11 + 1x^10 + 1x^1 + x^0

But mathematicians changed the rules so that it is mod 2. So basically any binary polynomial mod 2 is just addition without carry or XORs. So our original equation looks like:

=( 1x^110 + 1x^101 + 1x^100 + 11x^11 + 1x^10 + 1x^1 + x^0 ) MOD 2
=( 1x^110 + 1x^101 + 1x^100 +  1x^11 + 1x^10 + 1x^1 + x^0 )
= x^6 + x^5 + x^4 + 3*x^3 + x^2 + x^1 + x^0 (or that original number we had)

I know this is a leap of faith but this is beyond my capability as a line-programmer. If you are a hard-core CS-student or engineer I challenge to break this down. Everyone will benefit from this analysis.

So to work out a full example:

   Original message                : 1101011011
   Polynomial of (W)idth 4         :      10011
   Message after appending W zeros : 11010110110000

Now we divide the augmented Message by the Poly using CRC arithmetic. This is the same division as before:

            1100001010 = Quotient (nobody cares about the quotient)
       _______________
10011 ) 11010110110000 = Augmented message (1101011011 + 0000)
=Poly   10011,,.,,....
        -----,,.,,....
         10011,.,,....
         10011,.,,....
         -----,.,,....
          00001.,,....
          00000.,,....
          -----.,,....
           00010,,....
           00000,,....
           -----,,....
            00101,....
            00000,....
            -----,....
             01011....
             00000....
             -----....
              10110...
              10011...
              -----...
               01010..
               00000..
               -----..
                10100.
                10011.
                -----.
                 01110
                 00000
                 -----
                  1110 = Remainder = THE CHECKSUM!!!!

The division yields a quotient, which we throw away, and a remainder, which is the calculated checksum. This ends the calculation. Usually, the checksum is then appended to the message and the result transmitted. In this case the transmission would be: 11010110111110.

• CRC primer, Chapter 7

Only use a 32-bit number as your divisor and use your entire stream as your dividend. Throw out the quotient and keep the remainder. Tack the remainder on the end of your message and you have a CRC32.

Average guy review:

         QUOTIENT
        ----------
DIVISOR ) DIVIDEND
                 = REMAINDER

1. Take the first 32 bits.
2. Shift bits
3. If 32 bits are less than DIVISOR, go to step 2.
4. XOR 32 bits by DIVISOR. Go to step 2.

(Note that the stream has to be dividable by 32 bits or it should be padded. For example, an 8-bit ANSI stream would have to be padded. Also at the end of the stream, the division is halted.)