mirror of
https://git.kernel.org/pub/scm/linux/kernel/git/stable/linux.git
synced 2024-08-27 03:10:12 +00:00
![Bob Pearson](/assets/img/avatar_default.png)
Move a long comment from lib/crc32.c to Documentation/crc32.txt where it will more likely get read. Edited the resulting document to add an explanation of the slicing-by-n algorithm. [djwong@us.ibm.com: minor changelog tweaks] [akpm@linux-foundation.org: fix typo, per George] Signed-off-by: George Spelvin <linux@horizon.com> Signed-off-by: Bob Pearson <rpearson@systemfabricworks.com> Signed-off-by: Darrick J. Wong <djwong@us.ibm.com> Signed-off-by: Andrew Morton <akpm@linux-foundation.org> Signed-off-by: Linus Torvalds <torvalds@linux-foundation.org>
182 lines
8.5 KiB
Text
182 lines
8.5 KiB
Text
A brief CRC tutorial.
|
|
|
|
A CRC is a long-division remainder. You add the CRC to the message,
|
|
and the whole thing (message+CRC) is a multiple of the given
|
|
CRC polynomial. To check the CRC, you can either check that the
|
|
CRC matches the recomputed value, *or* you can check that the
|
|
remainder computed on the message+CRC is 0. This latter approach
|
|
is used by a lot of hardware implementations, and is why so many
|
|
protocols put the end-of-frame flag after the CRC.
|
|
|
|
It's actually the same long division you learned in school, except that
|
|
- We're working in binary, so the digits are only 0 and 1, and
|
|
- When dividing polynomials, there are no carries. Rather than add and
|
|
subtract, we just xor. Thus, we tend to get a bit sloppy about
|
|
the difference between adding and subtracting.
|
|
|
|
Like all division, the remainder is always smaller than the divisor.
|
|
To produce a 32-bit CRC, the divisor is actually a 33-bit CRC polynomial.
|
|
Since it's 33 bits long, bit 32 is always going to be set, so usually the
|
|
CRC is written in hex with the most significant bit omitted. (If you're
|
|
familiar with the IEEE 754 floating-point format, it's the same idea.)
|
|
|
|
Note that a CRC is computed over a string of *bits*, so you have
|
|
to decide on the endianness of the bits within each byte. To get
|
|
the best error-detecting properties, this should correspond to the
|
|
order they're actually sent. For example, standard RS-232 serial is
|
|
little-endian; the most significant bit (sometimes used for parity)
|
|
is sent last. And when appending a CRC word to a message, you should
|
|
do it in the right order, matching the endianness.
|
|
|
|
Just like with ordinary division, you proceed one digit (bit) at a time.
|
|
Each step of the division you take one more digit (bit) of the dividend
|
|
and append it to the current remainder. Then you figure out the
|
|
appropriate multiple of the divisor to subtract to being the remainder
|
|
back into range. In binary, this is easy - it has to be either 0 or 1,
|
|
and to make the XOR cancel, it's just a copy of bit 32 of the remainder.
|
|
|
|
When computing a CRC, we don't care about the quotient, so we can
|
|
throw the quotient bit away, but subtract the appropriate multiple of
|
|
the polynomial from the remainder and we're back to where we started,
|
|
ready to process the next bit.
|
|
|
|
A big-endian CRC written this way would be coded like:
|
|
for (i = 0; i < input_bits; i++) {
|
|
multiple = remainder & 0x80000000 ? CRCPOLY : 0;
|
|
remainder = (remainder << 1 | next_input_bit()) ^ multiple;
|
|
}
|
|
|
|
Notice how, to get at bit 32 of the shifted remainder, we look
|
|
at bit 31 of the remainder *before* shifting it.
|
|
|
|
But also notice how the next_input_bit() bits we're shifting into
|
|
the remainder don't actually affect any decision-making until
|
|
32 bits later. Thus, the first 32 cycles of this are pretty boring.
|
|
Also, to add the CRC to a message, we need a 32-bit-long hole for it at
|
|
the end, so we have to add 32 extra cycles shifting in zeros at the
|
|
end of every message,
|
|
|
|
These details lead to a standard trick: rearrange merging in the
|
|
next_input_bit() until the moment it's needed. Then the first 32 cycles
|
|
can be precomputed, and merging in the final 32 zero bits to make room
|
|
for the CRC can be skipped entirely. This changes the code to:
|
|
|
|
for (i = 0; i < input_bits; i++) {
|
|
remainder ^= next_input_bit() << 31;
|
|
multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
|
|
remainder = (remainder << 1) ^ multiple;
|
|
}
|
|
|
|
With this optimization, the little-endian code is particularly simple:
|
|
for (i = 0; i < input_bits; i++) {
|
|
remainder ^= next_input_bit();
|
|
multiple = (remainder & 1) ? CRCPOLY : 0;
|
|
remainder = (remainder >> 1) ^ multiple;
|
|
}
|
|
|
|
The most significant coefficient of the remainder polynomial is stored
|
|
in the least significant bit of the binary "remainder" variable.
|
|
The other details of endianness have been hidden in CRCPOLY (which must
|
|
be bit-reversed) and next_input_bit().
|
|
|
|
As long as next_input_bit is returning the bits in a sensible order, we don't
|
|
*have* to wait until the last possible moment to merge in additional bits.
|
|
We can do it 8 bits at a time rather than 1 bit at a time:
|
|
for (i = 0; i < input_bytes; i++) {
|
|
remainder ^= next_input_byte() << 24;
|
|
for (j = 0; j < 8; j++) {
|
|
multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
|
|
remainder = (remainder << 1) ^ multiple;
|
|
}
|
|
}
|
|
|
|
Or in little-endian:
|
|
for (i = 0; i < input_bytes; i++) {
|
|
remainder ^= next_input_byte();
|
|
for (j = 0; j < 8; j++) {
|
|
multiple = (remainder & 1) ? CRCPOLY : 0;
|
|
remainder = (remainder >> 1) ^ multiple;
|
|
}
|
|
}
|
|
|
|
If the input is a multiple of 32 bits, you can even XOR in a 32-bit
|
|
word at a time and increase the inner loop count to 32.
|
|
|
|
You can also mix and match the two loop styles, for example doing the
|
|
bulk of a message byte-at-a-time and adding bit-at-a-time processing
|
|
for any fractional bytes at the end.
|
|
|
|
To reduce the number of conditional branches, software commonly uses
|
|
the byte-at-a-time table method, popularized by Dilip V. Sarwate,
|
|
"Computation of Cyclic Redundancy Checks via Table Look-Up", Comm. ACM
|
|
v.31 no.8 (August 1998) p. 1008-1013.
|
|
|
|
Here, rather than just shifting one bit of the remainder to decide
|
|
in the correct multiple to subtract, we can shift a byte at a time.
|
|
This produces a 40-bit (rather than a 33-bit) intermediate remainder,
|
|
and the correct multiple of the polynomial to subtract is found using
|
|
a 256-entry lookup table indexed by the high 8 bits.
|
|
|
|
(The table entries are simply the CRC-32 of the given one-byte messages.)
|
|
|
|
When space is more constrained, smaller tables can be used, e.g. two
|
|
4-bit shifts followed by a lookup in a 16-entry table.
|
|
|
|
It is not practical to process much more than 8 bits at a time using this
|
|
technique, because tables larger than 256 entries use too much memory and,
|
|
more importantly, too much of the L1 cache.
|
|
|
|
To get higher software performance, a "slicing" technique can be used.
|
|
See "High Octane CRC Generation with the Intel Slicing-by-8 Algorithm",
|
|
ftp://download.intel.com/technology/comms/perfnet/download/slicing-by-8.pdf
|
|
|
|
This does not change the number of table lookups, but does increase
|
|
the parallelism. With the classic Sarwate algorithm, each table lookup
|
|
must be completed before the index of the next can be computed.
|
|
|
|
A "slicing by 2" technique would shift the remainder 16 bits at a time,
|
|
producing a 48-bit intermediate remainder. Rather than doing a single
|
|
lookup in a 65536-entry table, the two high bytes are looked up in
|
|
two different 256-entry tables. Each contains the remainder required
|
|
to cancel out the corresponding byte. The tables are different because the
|
|
polynomials to cancel are different. One has non-zero coefficients from
|
|
x^32 to x^39, while the other goes from x^40 to x^47.
|
|
|
|
Since modern processors can handle many parallel memory operations, this
|
|
takes barely longer than a single table look-up and thus performs almost
|
|
twice as fast as the basic Sarwate algorithm.
|
|
|
|
This can be extended to "slicing by 4" using 4 256-entry tables.
|
|
Each step, 32 bits of data is fetched, XORed with the CRC, and the result
|
|
broken into bytes and looked up in the tables. Because the 32-bit shift
|
|
leaves the low-order bits of the intermediate remainder zero, the
|
|
final CRC is simply the XOR of the 4 table look-ups.
|
|
|
|
But this still enforces sequential execution: a second group of table
|
|
look-ups cannot begin until the previous groups 4 table look-ups have all
|
|
been completed. Thus, the processor's load/store unit is sometimes idle.
|
|
|
|
To make maximum use of the processor, "slicing by 8" performs 8 look-ups
|
|
in parallel. Each step, the 32-bit CRC is shifted 64 bits and XORed
|
|
with 64 bits of input data. What is important to note is that 4 of
|
|
those 8 bytes are simply copies of the input data; they do not depend
|
|
on the previous CRC at all. Thus, those 4 table look-ups may commence
|
|
immediately, without waiting for the previous loop iteration.
|
|
|
|
By always having 4 loads in flight, a modern superscalar processor can
|
|
be kept busy and make full use of its L1 cache.
|
|
|
|
Two more details about CRC implementation in the real world:
|
|
|
|
Normally, appending zero bits to a message which is already a multiple
|
|
of a polynomial produces a larger multiple of that polynomial. Thus,
|
|
a basic CRC will not detect appended zero bits (or bytes). To enable
|
|
a CRC to detect this condition, it's common to invert the CRC before
|
|
appending it. This makes the remainder of the message+crc come out not
|
|
as zero, but some fixed non-zero value. (The CRC of the inversion
|
|
pattern, 0xffffffff.)
|
|
|
|
The same problem applies to zero bits prepended to the message, and a
|
|
similar solution is used. Instead of starting the CRC computation with
|
|
a remainder of 0, an initial remainder of all ones is used. As long as
|
|
you start the same way on decoding, it doesn't make a difference.
|