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.
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.
A CRC is pretty simple; you take a polynomial represented as bits and the data, and divide the polynomial into the data (or you represent the data as a polynomial and do the same thing). The remainder, which is between 0 and the polynomial is the CRC. Your code is a bit hard to understand, partly because it's incomplete: temp and testcrc are not declared, so it's unclear what's being indexed, and how much data is running through the algorithm.
The way to understand CRCs is to try to compute a few using a short piece of data (16 bits or so) with a short polynomial -- 4 bits, perhaps. If you practice this way, you'll really understand how you might go about coding it.
If you're doing it frequently, a CRC is quite slow to compute in software. Hardware computation is much more efficient, and requires just a few gates.
In order to reduce crc32 to taking the reminder you need to:
In code this is:
func CRC32 (file []byte) uint32 {
for i , v := range(file) {
file[i] = bits.Reverse8(v)
}
for i := 0; i < 4; i++ {
file[i] ^= 0xFF
}
// Add padding
file = append(file, []byte{0, 0, 0, 0}...)
newReminder := bits.Reverse32(reminderIEEE(file))
return newReminder ^ 0xFFFFFFFF
}
where reminderIEEE is the pure reminder on GF(2)[x]
In addition to the Wikipedia Cyclic redundancy check and Computation of CRC articles, I found a paper entitled Reversing CRC - Theory and Practice* to be a good reference.
There are essentially three approaches for computing a CRC: an algebraic approach, a bit-oriented approach, and a table-driven approach. In Reversing CRC - Theory and Practice*, each of these three algorithms/approaches is explained in theory accompanied in the APPENDIX by an implementation for the CRC32 in the C programming language.
* PDF Link
Reversing CRC – Theory and Practice.
HU Berlin Public Report
SAR-PR-2006-05
May 2006
Authors:
Martin Stigge, Henryk Plötz, Wolf Müller, Jens-Peter Redlich
The polynomial for CRC32 is:
x32 + x26 + x23 + x22 + x16 + x12 + x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1
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.
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
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.
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
(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.)
For IEEE802.3, CRC-32. Think of the entire message as a serial bit stream, append 32 zeros to the end of the message. Next, you MUST reverse the bits of EVERY byte of the message and do a 1's complement the first 32 bits. Now divide by the CRC-32 polynomial, 0x104C11DB7. Finally, you must 1's complement the 32-bit remainder of this division bit-reverse each of the 4 bytes of the remainder. This becomes the 32-bit CRC that is appended to the end of the message.
The reason for this strange procedure is that the first Ethernet implementations would serialize the message one byte at a time and transmit the least significant bit of every byte first. The serial bit stream then went through a serial CRC-32 shift register computation, which was simply complemented and sent out on the wire after the message was completed. The reason for complementing the first 32 bits of the message is so that you don't get an all zero CRC even if the message was all zeros.
I spent a while trying to uncover the answer to this question, and I finally published a tutorial on CRC-32 today: CRC-32 hash tutorial - AutoHotkey Community
In this example from it, I demonstrate how to calculate the CRC-32 hash for the ASCII string 'abc':
calculate the CRC-32 hash for the ASCII string 'abc':
inputs:
dividend: binary for 'abc': 0b011000010110001001100011 = 0x616263
polynomial: 0b100000100110000010001110110110111 = 0x104C11DB7
011000010110001001100011
reverse bits in each byte:
100001100100011011000110
append 32 0 bits:
10000110010001101100011000000000000000000000000000000000
XOR the first 4 bytes with 0xFFFFFFFF:
01111001101110010011100111111111000000000000000000000000
'CRC division':
01111001101110010011100111111111000000000000000000000000
100000100110000010001110110110111
---------------------------------
111000100010010111111010010010110
100000100110000010001110110110111
---------------------------------
110000001000101011101001001000010
100000100110000010001110110110111
---------------------------------
100001011101010011001111111101010
100000100110000010001110110110111
---------------------------------
111101101000100000100101110100000
100000100110000010001110110110111
---------------------------------
111010011101000101010110000101110
100000100110000010001110110110111
---------------------------------
110101110110001110110001100110010
100000100110000010001110110110111
---------------------------------
101010100000011001111110100001010
100000100110000010001110110110111
---------------------------------
101000011001101111000001011110100
100000100110000010001110110110111
---------------------------------
100011111110110100111110100001100
100000100110000010001110110110111
---------------------------------
110110001101101100000101110110000
100000100110000010001110110110111
---------------------------------
101101010111011100010110000001110
100000100110000010001110110110111
---------------------------------
110111000101111001100011011100100
100000100110000010001110110110111
---------------------------------
10111100011111011101101101010011
remainder: 0b10111100011111011101101101010011 = 0xBC7DDB53
XOR the remainder with 0xFFFFFFFF:
0b01000011100000100010010010101100 = 0x438224AC
reverse bits:
0b00110101001001000100000111000010 = 0x352441C2
thus the CRC-32 hash for the ASCII string 'abc' is 0x352441C2
Then there is always Rosetta Code, which shows crc32 implemented in dozens of computer languages. https://rosettacode.org/wiki/CRC-32 and has links to many explanations and implementations.
Source: Stackoverflow.com