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-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.