crc32.txt: standardize document format
Each text file under Documentation follows a different format. Some doesn't even have titles! Change its representation to follow the adopted standard, using ReST markups for it to be parseable by Sphinx: - Add a title for the document; - Mark literal blocks. While here, replace a comma by a dot at the end of a paragraph. Signed-off-by: Mauro Carvalho Chehab <mchehab@s-opensource.com> Signed-off-by: Jonathan Corbet <corbet@lwn.net>
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A brief CRC tutorial.
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=================================
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brief tutorial on CRC computation
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=================================
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A CRC is a long-division remainder. You add the CRC to the message,
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and the whole thing (message+CRC) is a multiple of the given
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@ -8,7 +10,8 @@ remainder computed on the message+CRC is 0. This latter approach
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is used by a lot of hardware implementations, and is why so many
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protocols put the end-of-frame flag after the CRC.
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It's actually the same long division you learned in school, except that
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It's actually the same long division you learned in school, except that:
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- We're working in binary, so the digits are only 0 and 1, and
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- When dividing polynomials, there are no carries. Rather than add and
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subtract, we just xor. Thus, we tend to get a bit sloppy about
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@ -40,11 +43,12 @@ throw the quotient bit away, but subtract the appropriate multiple of
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the polynomial from the remainder and we're back to where we started,
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ready to process the next bit.
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A big-endian CRC written this way would be coded like:
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for (i = 0; i < input_bits; i++) {
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multiple = remainder & 0x80000000 ? CRCPOLY : 0;
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remainder = (remainder << 1 | next_input_bit()) ^ multiple;
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}
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A big-endian CRC written this way would be coded like::
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for (i = 0; i < input_bits; i++) {
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multiple = remainder & 0x80000000 ? CRCPOLY : 0;
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remainder = (remainder << 1 | next_input_bit()) ^ multiple;
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}
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Notice how, to get at bit 32 of the shifted remainder, we look
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at bit 31 of the remainder *before* shifting it.
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@ -54,25 +58,26 @@ the remainder don't actually affect any decision-making until
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32 bits later. Thus, the first 32 cycles of this are pretty boring.
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Also, to add the CRC to a message, we need a 32-bit-long hole for it at
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the end, so we have to add 32 extra cycles shifting in zeros at the
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end of every message,
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end of every message.
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These details lead to a standard trick: rearrange merging in the
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next_input_bit() until the moment it's needed. Then the first 32 cycles
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can be precomputed, and merging in the final 32 zero bits to make room
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for the CRC can be skipped entirely. This changes the code to:
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for the CRC can be skipped entirely. This changes the code to::
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for (i = 0; i < input_bits; i++) {
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remainder ^= next_input_bit() << 31;
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multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
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remainder = (remainder << 1) ^ multiple;
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}
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for (i = 0; i < input_bits; i++) {
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remainder ^= next_input_bit() << 31;
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multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
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remainder = (remainder << 1) ^ multiple;
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}
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With this optimization, the little-endian code is particularly simple:
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for (i = 0; i < input_bits; i++) {
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remainder ^= next_input_bit();
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multiple = (remainder & 1) ? CRCPOLY : 0;
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remainder = (remainder >> 1) ^ multiple;
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}
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With this optimization, the little-endian code is particularly simple::
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for (i = 0; i < input_bits; i++) {
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remainder ^= next_input_bit();
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multiple = (remainder & 1) ? CRCPOLY : 0;
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remainder = (remainder >> 1) ^ multiple;
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}
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The most significant coefficient of the remainder polynomial is stored
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in the least significant bit of the binary "remainder" variable.
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@ -81,23 +86,25 @@ be bit-reversed) and next_input_bit().
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As long as next_input_bit is returning the bits in a sensible order, we don't
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*have* to wait until the last possible moment to merge in additional bits.
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We can do it 8 bits at a time rather than 1 bit at a time:
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for (i = 0; i < input_bytes; i++) {
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remainder ^= next_input_byte() << 24;
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for (j = 0; j < 8; j++) {
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multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
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remainder = (remainder << 1) ^ multiple;
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}
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}
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We can do it 8 bits at a time rather than 1 bit at a time::
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Or in little-endian:
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for (i = 0; i < input_bytes; i++) {
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remainder ^= next_input_byte();
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for (j = 0; j < 8; j++) {
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multiple = (remainder & 1) ? CRCPOLY : 0;
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remainder = (remainder >> 1) ^ multiple;
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for (i = 0; i < input_bytes; i++) {
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remainder ^= next_input_byte() << 24;
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for (j = 0; j < 8; j++) {
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multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
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remainder = (remainder << 1) ^ multiple;
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}
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}
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Or in little-endian::
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for (i = 0; i < input_bytes; i++) {
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remainder ^= next_input_byte();
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for (j = 0; j < 8; j++) {
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multiple = (remainder & 1) ? CRCPOLY : 0;
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remainder = (remainder >> 1) ^ multiple;
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}
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}
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}
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If the input is a multiple of 32 bits, you can even XOR in a 32-bit
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word at a time and increase the inner loop count to 32.
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