And, just when I thought I understood the crc32 stuff, here's an even better
explanation/code/etc. With thanks.
Matt
Original Message
From: linux@horizon.com [mailto:linux@horizon.com]
Sent: Friday, October 12, 2001 10:06 PM
Subject: Here's tableoptimized crc32 code for both ways...
I think just having these in the kernel unconditionally is best.
This version allows various space/time tradeoffs.
For a RAM kernel, the initialization code is smaller than the tables,
and can be made initcode. For a ROM kernel, it would make sense
to precompute the tables and compile them in.
#include <stddef.h> /* For size_t */
typedef unsigned _u32;
/*
* This code is in the public domain; copyright abandoned.
* Liability for nonperformance of this code is limited to the amount
* you paid for it. Since it is distributed for free, your refund will
* be very very small. If it breaks, you get to keep both pieces.
*/
/*
* There are multiple 16bit CRC polynomials in common use, but this is
* *the* standard CRC32 polynomial, first popularized by Ethernet.
* x^32+x^26+x^23+x^22+x^16+x^12+x^11+x^10+x^8+x^7+x^5+x^4+x^2+x^1+x^0
*/
#define CRCPOLY_LE 0xedb88320
#define CRCPOLY_BE 0x04c11db7
/* How many bits at a time to use. Requires a table of 4<<CRC_xx_BITS
bytes. */
#define CRC_LE_BITS 8
#define CRC_BE_BITS 4 /* Half the speed, 960 bytes less space */
/*
* Littleendian CRC computation. Used with serial bit streams sent
* lsbitfirst. Be sure to use cpu_to_le32() to append the computed CRC.
*/
#if CRC_LE_BITS > 8  CRC_LE_BITS < 1  CRC_LE_BITS & CRC_LE_BITS1
# error CRC_LE_BITS must be a power of 2 between 1 and 8
#endif
#if CRC_LE_BITS == 1
/*
* In fact, the tablebased code will work in this case, but it can be
* simplified by inlining the table in ?: form.
*/
void
crc32init_le(void)
{/* noop */;}
_u32
crc32_le(_u32 crc, unsigned char const *p, size_t len)
{
int i;
while (len) {
crc ^= *p++;
for (i = 0; i < i; i++)
crc = (crc >> 1) ^ (crc & 1 ? CRCPOLY_LE : 0);
}
return crc;
}
#else /* Tablebased approach */
_u32 crc32table_le[1<<CRC_LE_BITS];
/*
* crc is the crc of the byte i; other entries are filled in based on the
* fact that crctable[i^j] = crctable[i] ^ crctable[j].
*
* Note that the _init functions never write anything but the final correct
* value to each table entry, so they're safe to call repeatedly, even if
* someone else is currently using the table.
*/
void
crc32init_le(void)
{
unsigned i, j;
_u32 crc = 1;
crc32table_le[0] = 0;
for (i = 1 ; i < 1<<CRC_LE_BITS; i <<= 1) {
crc = (crc >> 1) ^ (crc & 1 ? CRCPOLY_LE : 0);
for (j = 0; j < i; j++)
crc32table_le[i+j] = crc ^ crc32table_le[j];
}
}
_u32
crc32_le(_u32 crc, unsigned char const *p, size_t len)
{
while (len) {
# if CRC_LE_BITS == 8
crc = (crc >> 8) ^ crc32table_le[(crc ^ *p++) & 255];
# elif CRC_LE_BITS == 4
crc ^= *p++;
crc = (crc >> 4) ^ crc32table_le[crc & 15];
crc = (crc >> 4) ^ crc32table_le[crc & 15];
# elif CRC_LE_BITS == 2
crc ^= *p++;
crc = (crc >> 2) ^ crc32table_le[crc & 3];
crc = (crc >> 2) ^ crc32table_le[crc & 3];
crc = (crc >> 2) ^ crc32table_le[crc & 3];
crc = (crc >> 2) ^ crc32table_le[crc & 3];
# endif
}
return crc;
}
#endif
/*
* Bigendian CRC computation. Used with serial bit streams sent
* msbitfirst. Be sure to use cpu_to_be32() to append the computed CRC.
*/
#if CRC_BE_BITS > 8  CRC_BE_BITS < 1  CRC_BE_BITS & CRC_BE_BITS1
# error CRC_BE_BITS must be a power of 2 between 1 and 8
#endif
#if CRC_BE_BITS == 1
/*
* In fact, the tablebased code will work in this case, but it can be
* simplified by inlining the table in ?: form.
*/
void
crc32init_be(void)
{/*noop*/;}
_u32
crc32_be(_u32 crc, unsigned char const *p, size_t len)
{
int i;
while (len) {
crc ^= *p++ << 24;
for (i = 0; i < i; i++)
crc = (crc << 1) ^ (crc & 0x80000000 ? CRCPOLY_BE :
0);
}
return crc;
}
#else /* Tablebased approach */
_u32 crc32table_be[256];
void
crc32init_be(void)
{
unsigned i, j;
_u32 crc = 0x80000000;
crc32table_le[0] = 0;
for (i = 1<<(CRC_BE_BITS1); i; i >>= 1) {
crc = (crc << 1) ^ (crc & 0x80000000 ? CRCPOLY_BE : 0);
for (j = 0; j < 1<<CRC_BE_BITS; j += 2*i)
crc32table_le[i+j] = crc ^ crc32table_le[j];
}
}
_u32
crc32_be(_u32 crc, unsigned char const *p, size_t len)
{
while (len) {
# if CRC_BE_BITS == 8
crc = (crc << 8) ^ crc32table_be[(crc >> 24) ^ *p++];
# elif CRC_BE_BITS == 4
crc ^= *p++ << 24;
crc = (crc << 4) ^ crc32table_be[crc >> 28];
crc = (crc << 4) ^ crc32table_be[crc >> 28];
# elif CRC_BE_BITS == 2
crc ^= *p++ << 24;
crc = (crc << 2) ^ crc32table_be[crc >> 30];
crc = (crc << 2) ^ crc32table_be[crc >> 30];
crc = (crc << 2) ^ crc32table_be[crc >> 30];
crc = (crc << 2) ^ crc32table_be[crc >> 30];
# endif
}
return crc;
}
#endif
/*
* A brief CRC tutorial.
*
* A CRC is a longdivision 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 endofframe 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.
*
* A 32bit CRC polynomial is actually 33 bits long. But 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 floatingpoint 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 errordetecting properties, this should correspond to the
* order they're actually sent. For example, standard RS232 serial is
* littleendian; 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, the remainder is always smaller than
* the divisor (the CRC polynomial) you're dividing by. 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, it's 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 bigendian 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 decisionmaking 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 32bitlong hole for it at
* the end, so we have to add 32 extra cycles shifting in zeros at the
* end of every message,
*
* So the standard trick is to rearrage 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 littleendian code is simpler:
* for (i = 0; i < input_bits; i++) {
* remainder ^= next_input_bit();
* multiple = (remainder & 1) ? CRCPOLY : 0;
* remainder = (remainder >> 1) ^ multiple;
* }
*
* Note that the other details of endianness have been hidden in CRCPOLY
* (which must be bitreversed) and next_input_bit().
*
* However, as long as next_input_bit is returning the bits in a sensible
* order, we can actually do the merging 8 or more bits at a time rather
* than one 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 littleendian:
* 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 32bit
* 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 byteatatime and adding bitatatime processing
* for any fractional bytes at the end.
*
* The only remaining optimization is to the byteatatime table method.
* 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 40bit (rather than a 33bit) intermediate remainder,
* but again the multiple of the polynomial to subtract depends only on
* the high bits, the high 8 bits in this case.
*
* The multile we need in that case is the low 32 bits of a 40bit
* value whose high 8 bits are given, and which is a multiple of the
* generator polynomial. This is simply the CRC32 of the given
* onebyte message.
*
* Two more details: normally, appending zero bits to a message which
* is already a multiple of a polynomial produces a larger multiple of that
* polynomial. 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 nonzero value.
*
* The same problem applies to zero bits prepended to the message, and
* a similar solution is used. Instead of starting 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.
*/

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