[patch v5 2/8] crc32movetodocumentation.diff
From: Bob Pearson
Date: Wed Aug 10 2011  18:43:57 EST
Moved a nice but 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 slicingbyn
algorithm.
Signedoffby: George Spelvin <linux@xxxxxxxxxxx>
Signedoffby: Bob Pearson <rpearson@xxxxxxxxxxxxxxxxxxxxx>

Documentation/00INDEX  2 Documentation/crc32.txt  183 ++++++++++++++++++++++++++++++++++++++++++++++++
lib/crc32.c  127 
3 files changed, 185 insertions(+), 127 deletions()
Index: infiniband/lib/crc32.c
===================================================================
 infiniband.orig/lib/crc32.c
+++ infiniband/lib/crc32.c
@@ 209,133 +209,6 @@ u32 __pure crc32_be(u32 crc, unsigned ch
EXPORT_SYMBOL(crc32_le);
EXPORT_SYMBOL(crc32_be);
/*
 * 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 multiple 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.
 */

#ifdef UNITTEST
#include <stdlib.h>
Index: infiniband/Documentation/crc32.txt
===================================================================
 /dev/null
+++ infiniband/Documentation/crc32.txt
@@ 0,0 +1,183 @@
+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.
+
+Like all division, the remainder is always smaller than the divisor.
+To produce a 32bit CRC, the divisor is actually a 33bit 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 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, you proceed one digit (bit) at a time.
+Each step of the division, 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 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,
+
+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 littleendian 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 bitreversed) 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 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.
+
+To reduce the number of conditional branches, software commonly uses
+the byteatatime table method, popularized by Dilip V. Sarwate,
+"Computation of Cyclic Redundancy Checks via Table LookUp", Comm. ACM
+v.31 no.8 (August 1998) p. 10081013.
+
+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,
+and the correct multiple of the polynomial to subtract is found using
+a 256entry lookup table indexed by the high 8 bits.
+
+(The table entries are simply the CRC32 of the given onebyte messages.)
+
+When space is more constrained, smaller tables can be used, e.g. two
+4bit shifts followed by a lookup in a 16entry 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 Slicingby8 Algorithm",
+ftp://download.intel.com/technology/comms/perfnet/download/slicingby8.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 48bit intermediate remainder. Rather than doing a single
+lookup in a 65536entry table, the two high bytes are looked up in
+two different 256entry 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 nonzero 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 lookup and thus performs almost
+twice as fast as the basic Sarwate algorithm.
+
+This can be extended to "slicing by 4" using 4 256entry 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 32bit shift
+leaves the loworder bits of the intermediate remainder zero, the
+final CRC is simply the XOR of the 4 table lookups.
+
+But this still enforces sequential execution: a second group of table
+lookups cannot begin until the previous groups 4 table lookups 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 lookups
+in parallel. Each step, the 32bit 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 lookups 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 nonzero 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.
+
Index: infiniband/Documentation/00INDEX
===================================================================
 infiniband.orig/Documentation/00INDEX
+++ infiniband/Documentation/00INDEX
@@ 104,6 +104,8 @@ cpuidle/
 info on CPU_IDLE, CPU idle state management subsystem.
cputopology.txt
 documentation on how CPU topology info is exported via sysfs.
+crc32.txt
+  brief tutorial on CRC computation
cris/
 directory with info about Linux on CRIS architecture.
crypto/

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