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diff --git a/Documentation/crc32.txt b/Documentation/crc32.txt deleted file mode 100644 index 8a6860f33b4e..000000000000 --- a/Documentation/crc32.txt +++ /dev/null @@ -1,189 +0,0 @@ -================================= -brief tutorial on CRC computation -================================= - -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. |