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  1 A brief CRC tutorial.
  3 A CRC is a long-division remainder.  You add the CRC to the message,
  4 and the whole thing (message+CRC) is a multiple of the given
  5 CRC polynomial.  To check the CRC, you can either check that the
  6 CRC matches the recomputed value, *or* you can check that the
  7 remainder computed on the message+CRC is 0.  This latter approach
  8 is used by a lot of hardware implementations, and is why so many
  9 protocols put the end-of-frame flag after the CRC.
 11 It's actually the same long division you learned in school, except that
 12 - We're working in binary, so the digits are only 0 and 1, and
 13 - When dividing polynomials, there are no carries.  Rather than add and
 14   subtract, we just xor.  Thus, we tend to get a bit sloppy about
 15   the difference between adding and subtracting.
 17 Like all division, the remainder is always smaller than the divisor.
 18 To produce a 32-bit CRC, the divisor is actually a 33-bit CRC polynomial.
 19 Since it's 33 bits long, bit 32 is always going to be set, so usually the
 20 CRC is written in hex with the most significant bit omitted.  (If you're
 21 familiar with the IEEE 754 floating-point format, it's the same idea.)
 23 Note that a CRC is computed over a string of *bits*, so you have
 24 to decide on the endianness of the bits within each byte.  To get
 25 the best error-detecting properties, this should correspond to the
 26 order they're actually sent.  For example, standard RS-232 serial is
 27 little-endian; the most significant bit (sometimes used for parity)
 28 is sent last.  And when appending a CRC word to a message, you should
 29 do it in the right order, matching the endianness.
 31 Just like with ordinary division, you proceed one digit (bit) at a time.
 32 Each step of the division you take one more digit (bit) of the dividend
 33 and append it to the current remainder.  Then you figure out the
 34 appropriate multiple of the divisor to subtract to being the remainder
 35 back into range.  In binary, this is easy - it has to be either 0 or 1,
 36 and to make the XOR cancel, it's just a copy of bit 32 of the remainder.
 38 When computing a CRC, we don't care about the quotient, so we can
 39 throw the quotient bit away, but subtract the appropriate multiple of
 40 the polynomial from the remainder and we're back to where we started,
 41 ready to process the next bit.
 43 A big-endian CRC written this way would be coded like:
 44 for (i = 0; i < input_bits; i++) {
 45         multiple = remainder & 0x80000000 ? CRCPOLY : 0;
 46         remainder = (remainder << 1 | next_input_bit()) ^ multiple;
 47 }
 49 Notice how, to get at bit 32 of the shifted remainder, we look
 50 at bit 31 of the remainder *before* shifting it.
 52 But also notice how the next_input_bit() bits we're shifting into
 53 the remainder don't actually affect any decision-making until
 54 32 bits later.  Thus, the first 32 cycles of this are pretty boring.
 55 Also, to add the CRC to a message, we need a 32-bit-long hole for it at
 56 the end, so we have to add 32 extra cycles shifting in zeros at the
 57 end of every message,
 59 These details lead to a standard trick: rearrange merging in the
 60 next_input_bit() until the moment it's needed.  Then the first 32 cycles
 61 can be precomputed, and merging in the final 32 zero bits to make room
 62 for the CRC can be skipped entirely.  This changes the code to:
 64 for (i = 0; i < input_bits; i++) {
 65         remainder ^= next_input_bit() << 31;
 66         multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
 67         remainder = (remainder << 1) ^ multiple;
 68 }
 70 With this optimization, the little-endian code is particularly simple:
 71 for (i = 0; i < input_bits; i++) {
 72         remainder ^= next_input_bit();
 73         multiple = (remainder & 1) ? CRCPOLY : 0;
 74         remainder = (remainder >> 1) ^ multiple;
 75 }
 77 The most significant coefficient of the remainder polynomial is stored
 78 in the least significant bit of the binary "remainder" variable.
 79 The other details of endianness have been hidden in CRCPOLY (which must
 80 be bit-reversed) and next_input_bit().
 82 As long as next_input_bit is returning the bits in a sensible order, we don't
 83 *have* to wait until the last possible moment to merge in additional bits.
 84 We can do it 8 bits at a time rather than 1 bit at a time:
 85 for (i = 0; i < input_bytes; i++) {
 86         remainder ^= next_input_byte() << 24;
 87         for (j = 0; j < 8; j++) {
 88                 multiple = (remainder & 0x80000000) ? CRCPOLY : 0;
 89                 remainder = (remainder << 1) ^ multiple;
 90         }
 91 }
 93 Or in little-endian:
 94 for (i = 0; i < input_bytes; i++) {
 95         remainder ^= next_input_byte();
 96         for (j = 0; j < 8; j++) {
 97                 multiple = (remainder & 1) ? CRCPOLY : 0;
 98                 remainder = (remainder >> 1) ^ multiple;
 99         }
100 }
102 If the input is a multiple of 32 bits, you can even XOR in a 32-bit
103 word at a time and increase the inner loop count to 32.
105 You can also mix and match the two loop styles, for example doing the
106 bulk of a message byte-at-a-time and adding bit-at-a-time processing
107 for any fractional bytes at the end.
109 To reduce the number of conditional branches, software commonly uses
110 the byte-at-a-time table method, popularized by Dilip V. Sarwate,
111 "Computation of Cyclic Redundancy Checks via Table Look-Up", Comm. ACM
112 v.31 no.8 (August 1998) p. 1008-1013.
114 Here, rather than just shifting one bit of the remainder to decide
115 in the correct multiple to subtract, we can shift a byte at a time.
116 This produces a 40-bit (rather than a 33-bit) intermediate remainder,
117 and the correct multiple of the polynomial to subtract is found using
118 a 256-entry lookup table indexed by the high 8 bits.
120 (The table entries are simply the CRC-32 of the given one-byte messages.)
122 When space is more constrained, smaller tables can be used, e.g. two
123 4-bit shifts followed by a lookup in a 16-entry table.
125 It is not practical to process much more than 8 bits at a time using this
126 technique, because tables larger than 256 entries use too much memory and,
127 more importantly, too much of the L1 cache.
129 To get higher software performance, a "slicing" technique can be used.
130 See "High Octane CRC Generation with the Intel Slicing-by-8 Algorithm",
133 This does not change the number of table lookups, but does increase
134 the parallelism.  With the classic Sarwate algorithm, each table lookup
135 must be completed before the index of the next can be computed.
137 A "slicing by 2" technique would shift the remainder 16 bits at a time,
138 producing a 48-bit intermediate remainder.  Rather than doing a single
139 lookup in a 65536-entry table, the two high bytes are looked up in
140 two different 256-entry tables.  Each contains the remainder required
141 to cancel out the corresponding byte.  The tables are different because the
142 polynomials to cancel are different.  One has non-zero coefficients from
143 x^32 to x^39, while the other goes from x^40 to x^47.
145 Since modern processors can handle many parallel memory operations, this
146 takes barely longer than a single table look-up and thus performs almost
147 twice as fast as the basic Sarwate algorithm.
149 This can be extended to "slicing by 4" using 4 256-entry tables.
150 Each step, 32 bits of data is fetched, XORed with the CRC, and the result
151 broken into bytes and looked up in the tables.  Because the 32-bit shift
152 leaves the low-order bits of the intermediate remainder zero, the
153 final CRC is simply the XOR of the 4 table look-ups.
155 But this still enforces sequential execution: a second group of table
156 look-ups cannot begin until the previous groups 4 table look-ups have all
157 been completed.  Thus, the processor's load/store unit is sometimes idle.
159 To make maximum use of the processor, "slicing by 8" performs 8 look-ups
160 in parallel.  Each step, the 32-bit CRC is shifted 64 bits and XORed
161 with 64 bits of input data.  What is important to note is that 4 of
162 those 8 bytes are simply copies of the input data; they do not depend
163 on the previous CRC at all.  Thus, those 4 table look-ups may commence
164 immediately, without waiting for the previous loop iteration.
166 By always having 4 loads in flight, a modern superscalar processor can
167 be kept busy and make full use of its L1 cache.
169 Two more details about CRC implementation in the real world:
171 Normally, appending zero bits to a message which is already a multiple
172 of a polynomial produces a larger multiple of that polynomial.  Thus,
173 a basic CRC will not detect appended zero bits (or bytes).  To enable
174 a CRC to detect this condition, it's common to invert the CRC before
175 appending it.  This makes the remainder of the message+crc come out not
176 as zero, but some fixed non-zero value.  (The CRC of the inversion
177 pattern, 0xffffffff.)
179 The same problem applies to zero bits prepended to the message, and a
180 similar solution is used.  Instead of starting the CRC computation with
181 a remainder of 0, an initial remainder of all ones is used.  As long as
182 you start the same way on decoding, it doesn't make a difference.

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