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brief tutorial on CRC computation | |

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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. |