Editing Cyclic redundancy check

{{Short description|Error-detecting code for detecting data changes}}
{{Use dmy dates|date=October 2023}}

A '''cyclic redundancy check''' ('''CRC''') is an [[Error correcting code|error-detecting code]] commonly used in digital [[Telecommunications network|networks]] and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short ''check value'' attached, based on the remainder of a [[Polynomial long division|polynomial division]] of their contents. On retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption. CRCs can be used for [[Error detection and correction|error correction]] (see [[Mathematics of cyclic redundancy checks#Bitfilters|bitfilters]]).<ref>{{cite web|url=http://www.drdobbs.com/an-algorithm-for-error-correcting-cyclic/184401662|title=An Algorithm for Error Correcting Cyclic Redundance Checks|website=drdobbs.com|access-date=28 June 2017|archive-url=https://web.archive.org/web/20170720165847/http://www.drdobbs.com/an-algorithm-for-error-correcting-cyclic/184401662|archive-date=20 July 2017|url-status=dead}}</ref>

CRCs are so called because the ''check'' (data verification) value is a ''redundancy'' (it expands the message without adding [[Entropy (information theory)|information]]) and the [[algorithm]] is based on [[Cyclic code|''cyclic'' codes]]. CRCs are popular because they are simple to implement in binary [[Computer hardware|hardware]], easy to analyze mathematically, and particularly good at detecting common errors caused by [[Noise (electronics)|noise]] in transmission channels.  Because the check value has a fixed length, the [[Function (mathematics)|function]] that generates it is occasionally used as a [[hash function]].

== Introduction ==
CRCs are based on the theory of [[cyclic code|cyclic]] [[error-correcting code]]s. The use of [[systematic code|systematic]] cyclic codes, which encode messages by adding a fixed-length check value, for the purpose of error detection in communication networks, was first proposed by [[W. Wesley Peterson]] in 1961.<ref name="PetersonBrown1961">{{Cite journal
|last1 = Peterson |first1=W. W. |last2 =Brown |first2=D. T.
|date=January 1961
| title = Cyclic Codes for Error Detection
| journal = Proceedings of the IRE
| doi = 10.1109/JRPROC.1961.287814
| volume = 49
| issue = 1
| pages = 228–235
|s2cid=51666741 }}</ref>
Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of [[burst error]]s: contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many [[communication channel]]s, including magnetic and optical storage devices. Typically an ''n''-bit CRC applied to a data block of arbitrary length will detect any single error burst not longer than ''n'' bits, and the fraction of all longer error bursts that it will detect is approximately {{nowrap|(1 − 2<sup>−''n''</sup>)}}.

Specification of a CRC code requires definition of a so-called [[generator polynomial]]. This polynomial becomes the [[divisor]] in a [[polynomial long division]], which takes the message as the [[division (mathematics)|dividend]] and in which the [[quotient]] is discarded and the [[remainder]] becomes the result.  The important caveat is that the polynomial [[coefficient]]s are calculated according to the arithmetic of a [[finite field]], so the addition operation can always be performed bitwise-parallel (there is no carry between digits).

In practice, all commonly used CRCs employ the finite field of two elements, [[GF(2)]]. The two elements are usually called 0 and 1, comfortably matching computer architecture.

A CRC is called an ''n''-bit CRC when its check value is ''n'' bits long. For a given ''n'', multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree ''n'', which means it has {{nowrap|''n'' + 1}} terms. In other words, the polynomial has a length of {{nowrap|''n'' + 1}}; its encoding requires {{nowrap|''n'' + 1}} bits. Note that most polynomial specifications either drop the [[Most significant bit|MSb]] or [[Least significant bit|LSb]], since they are always 1. The CRC and associated polynomial typically have a name of the form CRC-''n''-XXX as in the [[#table|table]] below.

The simplest error-detection system, the [[parity bit]], is in fact a 1-bit CRC: it uses the generator polynomial&nbsp;{{nowrap|''x'' + 1}} (two terms),<ref name=Ergen-2008>{{cite book|chapter=2.3.3 Error Detection Coding|title=Mobile Broadband|last=Ergen|first=Mustafa|publisher=[[Springer Nature|Springer]]|date=21 January 2008|pages=29–30|doi=10.1007/978-0-387-68192-4_2|isbn=978-0-387-68192-4}}</ref> and has the name CRC-1.

== Application ==
A CRC-enabled device calculates a short, fixed-length binary sequence, known as the ''check value'' or ''CRC'', for each block of data to be sent or stored and appends it to the data, forming a ''codeword''.

When a codeword is received or read, the device either compares its check value with one freshly calculated from the data block, or equivalently, performs a CRC on the whole codeword and compares the resulting check value with an expected ''residue'' constant.

If the CRC values do not match, then the block contains a data error.

The device may take corrective action, such as rereading the block or requesting that it be sent again. Otherwise, the data is assumed to be error-free (though, with some small probability, it may contain undetected errors; this is inherent in the nature of error-checking).<ref name="ritter-1986">{{Cite journal|first=Terry|last=Ritter|title=The Great CRC Mystery|url=http://www.ciphersbyritter.com/ARTS/CRCMYST.HTM|access-date=21 May 2009|journal=[[Dr. Dobb's Journal]]|date=February 1986|volume=11|issue=2|pages=26–34, 76–83|archive-date=16 April 2009|archive-url=https://web.archive.org/web/20090416095111/http://www.ciphersbyritter.com/ARTS/CRCMYST.HTM|url-status=live}}</ref>

== Data integrity ==
CRCs are specifically designed to protect against common types of errors on communication channels, where they can provide quick and reasonable assurance of the [[data integrity|integrity]] of messages delivered. However, they are not suitable for protecting against intentional alteration of data.

Firstly, as there is no authentication, an attacker can edit a message and recompute the CRC without the substitution being detected. When stored alongside the data, CRCs and cryptographic hash functions by themselves do not protect against ''intentional'' modification of data. Any application that requires protection against such attacks must use cryptographic authentication mechanisms, such as [[message authentication code]]s or [[digital signatures]] (which are commonly based on [[cryptographic hash]] functions).

Secondly, unlike cryptographic hash functions, CRC is an easily reversible function, which makes it unsuitable for use in digital signatures.<ref name="stigge-reversecrc">{{Cite web|last1=Stigge|first1=Martin|last2=Plötz|first2=Henryk|last3=Müller|first3=Wolf|last4=Redlich|first4=Jens-Peter|title=Reversing CRC – Theory and Practice|date=May 2006|page=17 |id=SAR-PR-2006-05 |publisher=Humboldt University Berlin|url=http://sar.informatik.hu-berlin.de/research/publications/SAR-PR-2006-05/SAR-PR-2006-05_.pdf|access-date=4 February 2011|quote=The presented methods offer a very easy and efficient way to modify your data so that it will compute to a CRC you want or at least know in advance.|url-status=dead|archive-url=https://web.archive.org/web/20110719042902/http://sar.informatik.hu-berlin.de/research/publications/SAR-PR-2006-05/SAR-PR-2006-05_.pdf|archive-date=19 July 2011}}</ref>

Thirdly, CRC satisfies a relation similar to that of a [[linear function]] (or more accurately, an [[affine function]]):<ref>{{cite web |title=algorithm design – Why is CRC said to be linear? |url=https://crypto.stackexchange.com/a/34013 |website=Cryptography Stack Exchange |access-date=5 May 2019}}</ref>

:<math>\operatorname{CRC}(x \oplus y) = \operatorname{CRC}(x) \oplus \operatorname{CRC}(y) \oplus c</math>

where <math>c</math> depends on the length of <math>x</math> and <math>y</math>. This can be also stated as follows, where <math>x</math>, <math>y</math> and <math>z</math> have the same length

:<math>\operatorname{CRC}(x \oplus y \oplus z) = \operatorname{CRC}(x) \oplus \operatorname{CRC}(y) \oplus \operatorname{CRC}(z);</math>

as a result, even if the CRC is encrypted with a [[stream cipher]] that uses [[Exclusive or|XOR]] as its combining operation (or [[Block cipher modes of operation|mode]] of [[block cipher]] which effectively turns it into a stream cipher, such as OFB or CFB), both the message and the associated CRC can be manipulated without knowledge of the encryption key; this was one of the well-known design flaws of the [[Wired Equivalent Privacy]] (WEP) protocol.<ref name="wep">{{Cite journal | last1=Cam-Winget | first1=Nancy | last2=Housley | first2=Russ | last3=Wagner | first3=David | last4=Walker | first4=Jesse | title=Security Flaws in 802.11 Data Link Protocols | journal=Communications of the ACM | volume=46 | issue=5 | pages=35–39 | date=May 2003 | doi=10.1145/769800.769823 | url=http://www.cs.berkeley.edu/~daw/papers/wireless-cacm.pdf | citeseerx=10.1.1.14.8775 | s2cid=3132937 | access-date=1 November 2017 | archive-date=26 May 2013 | archive-url=https://web.archive.org/web/20130526173938/http://www.cs.berkeley.edu/~daw/papers/wireless-cacm.pdf | url-status=live }}</ref>

== Computation ==
{{unreferenced section|date=July 2016}}
{{Main|Computation of cyclic redundancy checks}}

To compute an ''n''-bit binary CRC, line the bits representing the input in a row, and position the ({{nowrap|''n'' + 1}})-bit pattern representing the CRC's divisor (called a "[[polynomial]]") underneath the left end of the row.

In this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial {{nowrap|''x''<sup>3</sup> + ''x'' + 1}}. The polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients ({{nowrap|1''x''<sup>3</sup> + 0''x''<sup>2</sup> + 1''x'' + 1}}). In this case, the coefficients are 1, 0, 1 and 1.  The result of the calculation is 3 bits long, which is why it is called a 3-bit CRC. However, you need 4 bits to explicitly state the polynomial.

Start with the message to be encoded:

<pre>
11010011101100
</pre>

This is first padded with zeros corresponding to the bit length ''n'' of the CRC. This is done so that the resulting code word is in [[systematic code|systematic]] form. Here is the first calculation for computing a 3-bit CRC:

<pre>
11010011101100 000 <--- input right padded by 3 bits
1011               <--- divisor (4 bits) = x³ + x + 1
------------------
01100011101100 000 <--- result
</pre>

The algorithm acts on the bits directly above the divisor in each step.  The result for that iteration is the bitwise XOR of the polynomial divisor with the bits above it.  The bits not above the divisor are simply copied directly below for that step.  The divisor is then shifted right to align with the highest remaining 1 bit in the input, and the process is repeated until the divisor reaches the right-hand end of the input row. Here is the entire calculation:

<pre>
11010011101100 000 <--- input right padded by 3 bits
1011               <--- divisor
01100011101100 000 <--- result (note the first four bits are the XOR with the divisor beneath, the rest of the bits are unchanged)
 1011              <--- divisor ...
00111011101100 000
  1011
00010111101100 000
   1011
00000001101100 000 <--- note that the divisor moves over to align with the next 1 in the dividend (since quotient for that step was zero)
       1011             (in other words, it doesn't necessarily move one bit per iteration)
00000000110100 000
        1011
00000000011000 000
         1011
00000000001110 000
          1011
00000000000101 000
           101 1
-----------------
00000000000000 100 <--- remainder (3 bits).  Division algorithm stops here as dividend is equal to zero.
</pre>

Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the only bits in the input row that can be nonzero are the n bits at the right-hand end of the row. These ''n'' bits are the remainder of the division step, and will also be the value of the CRC function (unless the chosen CRC specification calls for some postprocessing).

The validity of a received message can easily be verified by performing the above calculation again, this time with the check value added instead of zeroes. The remainder should equal zero if there are no detectable errors.

<pre>
11010011101100 100 <--- input with check value
1011               <--- divisor
01100011101100 100 <--- result
 1011              <--- divisor ...
00111011101100 100

......

00000000001110 100
          1011
00000000000101 100
           101 1
------------------
00000000000000 000 <--- remainder
</pre>

The following [[Python (programming language)|Python]] code outlines a function which will return the initial CRC remainder for a chosen input and polynomial, with either 1 or 0 as the initial padding. Note that this code works with string inputs rather than raw numbers:
<syntaxhighlight lang="python">
def crc_remainder(input_bitstring, polynomial_bitstring, initial_filler):
    """Calculate the CRC remainder of a string of bits using a chosen polynomial.
    initial_filler should be '1' or '0'.
    """
    polynomial_bitstring = polynomial_bitstring.lstrip("0")
    len_input = len(input_bitstring)
    initial_padding = (len(polynomial_bitstring) - 1) * initial_filler
    input_padded_array = list(input_bitstring + initial_padding)
    while "1" in input_padded_array[:len_input]:
        cur_shift = input_padded_array.index("1")
        for i in range(len(polynomial_bitstring)):
            input_padded_array[cur_shift + i] \
            = str(int(polynomial_bitstring[i] != input_padded_array[cur_shift + i]))
    return "".join(input_padded_array)[len_input:]

def crc_check(input_bitstring, polynomial_bitstring, check_value):
    """Calculate the CRC check of a string of bits using a chosen polynomial."""
    polynomial_bitstring = polynomial_bitstring.lstrip("0")
    len_input = len(input_bitstring)
    initial_padding = check_value
    input_padded_array = list(input_bitstring + initial_padding)
    while "1" in input_padded_array[:len_input]:
        cur_shift = input_padded_array.index("1")
        for i in range(len(polynomial_bitstring)):
            input_padded_array[cur_shift + i] \
            = str(int(polynomial_bitstring[i] != input_padded_array[cur_shift + i]))
    return ("1" not in "".join(input_padded_array)[len_input:])
</syntaxhighlight>

<syntaxhighlight lang="pycon">
>>> crc_remainder('11010011101100', '1011', '0')
'100'
>>> crc_check('11010011101100', '1011', '100')
True
</syntaxhighlight>

== Mathematics ==
{{More citations needed section|date=July 2016}}
{{Main|Mathematics of cyclic redundancy checks}}
Mathematical analysis of this division-like process reveals how to select a divisor that guarantees good error-detection properties. In this analysis, the digits of the bit strings are taken as the coefficients of a polynomial in some variable ''x''—coefficients that are elements of the finite field [[GF(2)]] (the integers modulo 2, i.e. either a zero or a one), instead of more familiar numbers. The set of binary polynomials is a mathematical [[ring (mathematics)|ring]].

=== Designing polynomials ===
The selection of the generator polynomial is the most important part of implementing the CRC algorithm. The polynomial must be chosen to maximize the error-detecting capabilities while minimizing overall collision probabilities.

The most important attribute of the polynomial is its length (largest degree(exponent) +1 of any one term in the polynomial), because of its direct influence on the length of the computed check value.

The most commonly used polynomial lengths are 9 bits (CRC-8), 17 bits (CRC-16), 33 bits (CRC-32), and 65 bits (CRC-64).<ref name=Ergen-2008/>

A CRC is called an ''n''-bit CRC when its check value is ''n''-bits. For a given ''n'', multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree ''n'', and hence {{nowrap|''n'' + 1}} terms (the polynomial has a length of {{nowrap|''n'' + 1}}). The remainder has length ''n''. The CRC has a name of the form CRC-''n''-XXX.

The design of the CRC polynomial depends on the maximum total length of the block to be protected (data + CRC bits), the desired error protection features, and the type of resources for implementing the CRC, as well as the desired performance. A common misconception is that the "best" CRC polynomials are derived from either [[irreducible polynomial]]s or irreducible polynomials times the factor&nbsp;{{nowrap|1 + ''x''}}, which adds to the code the ability to detect all errors affecting an odd number of bits.<ref name="williams93" /> In reality, all the factors described above should enter into the selection of the polynomial and may lead to a reducible polynomial. However, choosing a reducible polynomial will result in a certain proportion of missed errors, due to the quotient ring having [[zero divisor]]s.

The advantage of choosing a [[Primitive polynomial (field theory)|primitive polynomial]] as the generator for a CRC code is that the resulting code has maximal total block length in the sense that all 1-bit errors within that block length have different remainders (also called [[Syndrome decoding|syndromes]]) and therefore, since the remainder is a linear function of the block, the code can detect all 2-bit errors within that block length. If <math>r</math> is the degree of the primitive generator polynomial, then the maximal total block length is <math>2 ^ {r} - 1 </math>, and the associated code is able to detect any single-bit or double-bit errors.<ref>{{Cite book | last1=Press | first1=WH | last2=Teukolsky | first2=SA | last3=Vetterling | first3=WT | last4=Flannery | first4=BP | year=2007 | title=Numerical Recipes: The Art of Scientific Computing | edition=3rd | publisher=Cambridge University Press | isbn=978-0-521-88068-8 | chapter=Section 22.4 Cyclic Redundancy and Other Checksums | chapter-url=http://numerical.recipes/book.html | access-date=20 August 2024 | archive-date=13 July 2024 | archive-url=https://web.archive.org/web/20240713014419/http://numerical.recipes/book.html | url-status=live }}</ref> However, if we use the generator polynomial <math>g(x) = p(x)(1 + x)</math>, where <math>p</math> is a primitive polynomial of degree <math>r - 1</math>, then the maximal total block length is <math>2^{r - 1} - 1</math>, and the code is able to detect single, double, triple and any odd number of errors.

A polynomial <math>g(x)</math> that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power. The [[BCH code]]s are a powerful class of such polynomials. They subsume the two examples above. Regardless of the reducibility properties of a generator polynomial of degree&nbsp;''r'', if it includes the "+1" term, the code will be able to detect error patterns that are confined to a window of ''r'' contiguous bits. These patterns are called "error bursts".

== Specification ==
The concept of the CRC as an error-detecting code gets complicated when an implementer or standards committee uses it to design a practical system. Here are some of the complications:
*Sometimes an implementation '''prefixes a fixed bit pattern''' to the bitstream to be checked. This is useful when clocking errors might insert 0-bits in front of a message, an alteration that would otherwise leave the check value unchanged.
*Usually, but not always, an implementation '''appends ''n'' 0-bits''' (''n'' being the size of the CRC) to the bitstream to be checked before the polynomial division occurs. Such appending is explicitly demonstrated in the [[Computation of cyclic redundancy checks|Computation of CRC]] article. This has the convenience that the remainder of the original bitstream with the check value appended is exactly zero, so the CRC can be checked simply by performing the polynomial division on the received bitstream and comparing the remainder with zero. Due to the associative and commutative properties of the exclusive-or operation, practical table driven implementations can obtain a result numerically equivalent to zero-appending without explicitly appending any zeroes, by using an equivalent,<ref name="williams93" /> faster algorithm that combines the message bitstream with the stream being shifted out of the CRC register.
*Sometimes an implementation '''exclusive-ORs a fixed bit pattern''' into the remainder of the polynomial division.
*'''Bit order:''' Some schemes view the low-order bit of each byte as "first", which then during polynomial division means "leftmost", which is contrary to our customary understanding of "low-order". This convention makes sense when [[serial port|serial-port]] transmissions are CRC-checked in hardware, because some widespread serial-port transmission conventions transmit bytes least-significant bit first.
*'''[[Byte order]]''': With multi-byte CRCs, there can be confusion over whether the byte transmitted first (or stored in the lowest-addressed byte of memory) is the least-significant byte (LSB) or the most-significant byte (MSB). For example, some 16-bit CRC schemes swap the bytes of the check value.
*'''Omission of the high-order bit''' of the divisor polynomial: Since the high-order bit is always 1, and since an ''n''-bit CRC must be defined by an ({{nowrap|''n'' + 1}})-bit divisor which [[Arithmetic overflow|overflows]] an ''n''-bit [[processor register|register]], some writers assume that it is unnecessary to mention the divisor's high-order bit.
*'''Omission of the low-order bit''' of the divisor polynomial: Since the low-order bit is always 1, authors such as Philip Koopman represent polynomials with their high-order bit intact, but without the low-order bit (the <math>x^0</math> or 1 term).  This convention encodes the polynomial complete with its degree in one integer.

These complications mean that there are three common ways to express a polynomial as an integer: the first two, which are mirror images in binary, are the constants found in code; the third is the number found in Koopman's papers.  ''In each case, one term is omitted.'' So the polynomial <math>x^4 + x + 1</math> may be transcribed as:
*0x3 = 0b0011, representing <math>x^4 + (0x^3 + 0x^2 + 1x^1 + 1x^0)</math> (MSB-first code)
*0xC = 0b1100, representing <math>(1x^0 + 1x^1 + 0x^2 + 0x^3) + x^4</math> (LSB-first code)
*0x9 = 0b1001, representing <math>(1x^4 + 0x^3 + 0x^2 + 1x^1) + x^0</math> (Koopman notation)
In the table below they are shown as:
{|class="wikitable"
|+ Examples of CRC representations
|-
! Name
! Normal
! Reversed
! Reversed reciprocal
|-
| CRC-4
| 0x3
| 0xC
| 0x9
|}

== Obfuscation ==
CRCs in [[proprietary protocol]]s might be [[Obfuscation|obfuscated]] by using a non-trivial initial value and a final XOR, but these techniques do not add cryptographic strength to the algorithm and can be [[Reverse engineering|reverse engineered]] using straightforward methods.<ref name="ewing-rev-eng">{{Cite web|last=Ewing|first=Gregory C.|date=March 2010|title=Reverse-Engineering a CRC Algorithm|location=Christchurch|publisher=University of Canterbury|url=http://www.cosc.canterbury.ac.nz/greg.ewing/essays/CRC-Reverse-Engineering.html|access-date=26 July 2011|archive-date=7 August 2011|archive-url=https://web.archive.org/web/20110807100031/http://www.cosc.canterbury.ac.nz/greg.ewing/essays/CRC-Reverse-Engineering.html|url-status=live}}</ref>

== Standards and common use ==
Numerous varieties of cyclic redundancy checks have been incorporated into [[technical standard]]s.  By no means does one algorithm, or one of each degree, suit every purpose; Koopman and Chakravarty recommend selecting a polynomial according to the application requirements and the expected distribution of message lengths.<ref name=koop04 /> The number of distinct CRCs in use has confused developers, a situation which authors have sought to address.<ref name="williams93">{{Cite web | url= http://www.wolfgang-ehrhardt.de/crc_v3.html | title= A Painless Guide to CRC Error Detection Algorithms V3.0 | access-date= 23 May 2019 | last= Williams | first= Ross N. | date= 24 September 1996 | archive-url= https://web.archive.org/web/20180402205812/http://www.wolfgang-ehrhardt.de/crc_v3.html | archive-date= 2 April 2018 | url-status= dead }}</ref> There are three polynomials reported for CRC-12,<ref name=koop04 /> twenty-two conflicting definitions of CRC-16, and seven of CRC-32.<ref name="cook-catalogue">{{Cite web | last = Cook | first = Greg | url = https://reveng.sourceforge.io/crc-catalogue/all.htm | title = Catalogue of parametrised CRC algorithms | access-date = 18 September 2020 | date = 15 August 2020 | archive-date = 1 August 2020 | archive-url = https://web.archive.org/web/20200801122415/https://reveng.sourceforge.io/crc-catalogue/all.htm | url-status = live }}</ref>

The polynomials commonly applied are not the most efficient ones possible. Since 1993, Koopman, Castagnoli and others have surveyed the space of polynomials between 3 and 64 bits in size,<ref name=koop04>{{Cite book |  last1 = Koopman |  first1 = Philip |  last2 = Chakravarty |  first2 = Tridib | title = International Conference on Dependable Systems and Networks, 2004 | chapter = Cyclic redundancy code (CRC) polynomial selection for embedded networks |  date = June 2004 <!-- date from http://www.ece.cmu.edu/~koopman/pubs.html --> |  url = http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf |  access-date = 14 January 2011 |  pages = 145–154 |  doi = 10.1109/DSN.2004.1311885 |  isbn = 978-0-7695-2052-0 |  citeseerx = 10.1.1.648.9080 |  s2cid = 793862 |  archive-date = 11 September 2011 |  archive-url = https://web.archive.org/web/20110911095215/http://www.ece.cmu.edu/~koopman/roses/dsn04/koopman04_crc_poly_embedded.pdf |  url-status = live }}</ref><ref name="cast93">{{Cite journal |  last1 = Castagnoli | first1 =  G. |  last2 = Bräuer | first2 = S. |  last3 = Herrmann | first3 = M. |date=June 1993 |  title = Optimization of Cyclic Redundancy-Check Codes with 24 and 32 Parity Bits |  journal =  IEEE Transactions on Communications |  volume = 41 | issue = 6| doi = 10.1109/26.231911 | pages = 883–892 }}</ref><ref name="koop02">{{cite book |  last = Koopman |  first = Philip | chapter = 32-bit cyclic redundancy codes for Internet applications |  title = Proceedings International Conference on Dependable Systems and Networks |  date = July 2002 <!-- date from http://www.ece.cmu.edu/~koopman/pubs.html --> |  url = http://www.ece.cmu.edu/~koopman/networks/dsn02/dsn02_koopman.pdf |  access-date = 14 January 2011 |  pages = 459–468 |  doi = 10.1109/DSN.2002.1028931 |  isbn = 978-0-7695-1597-7 |  citeseerx = 10.1.1.11.8323 |  s2cid = 14775606 |  archive-date = 16 September 2012 |  archive-url = https://web.archive.org/web/20120916185941/http://www.ece.cmu.edu/~koopman/networks/dsn02/dsn02_koopman.pdf |  url-status = live }}</ref><ref name="koopman-best-crc-polys">{{Cite web|last=Koopman|first=Philip|date=21 January 2016|title=Best CRC Polynomials|institution=Carnegie Mellon University |url=https://users.ece.cmu.edu/~koopman/crc/|access-date=26 January 2016|archive-date=20 January 2016|archive-url=https://web.archive.org/web/20160120021307/http://users.ece.cmu.edu/~koopman/crc/|url-status=live}}</ref> finding examples that have much better performance (in terms of [[Hamming distance]] for a given message size) than the polynomials of earlier protocols, and publishing the best of these with the aim of improving the error detection capacity of future standards.<ref name="koop02" /> In particular, [[iSCSI]] and [[SCTP]] have adopted one of the findings of this research, the CRC-32C (Castagnoli) polynomial.

The design of the 32-bit polynomial most commonly used by standards bodies, CRC-32-IEEE, was the result of a joint effort for the [[Rome Laboratory]] and the Air Force Electronic Systems Division by Joseph Hammond, James Brown and Shyan-Shiang Liu of the [[Georgia Institute of Technology]] and Kenneth Brayer of the [[Mitre Corporation]]. The earliest known appearances of the 32-bit polynomial were in their 1975 publications: Technical Report 2956 by Brayer for Mitre, published in January and released for public dissemination through [[DTIC]] in August,<ref name="Brayer1975">{{cite report | last = Brayer | first = Kenneth | date = August 1975 | title = Evaluation of 32 Degree Polynomials in Error Detection on the SATIN IV Autovon Error Patterns | publisher = [[National Technical Information Service]] |id=ADA014825 | url = https://apps.dtic.mil/sti/citations/ADA014825 | access-date = 31 December 2021 | archive-date = 31 December 2021 | archive-url = https://web.archive.org/web/20211231093215/https://apps.dtic.mil/sti/citations/ADA014825 | url-status = live }}</ref> and Hammond, Brown and Liu's report for the Rome Laboratory, published in May.<ref name="Hammond1975">{{cite journal | last1 = Hammond | first1 = Joseph L. Jr. | last2 = Brown | first2 = James E. | last3 = Liu | first3 = Shyan-Shiang | publication-date = May 1975 | title = Development of a Transmission Error Model and an Error Control Model | journal = NASA Sti/Recon Technical Report N | url = https://apps.dtic.mil/sti/citations/ADA013939 | access-date = 31 December 2021 |id=ADA013939 | bibcode = 1975STIN...7615344H | volume = 76 | year = 1975 | page = 15344 | archive-date = 31 December 2021 | archive-url = https://web.archive.org/web/20211231093212/https://apps.dtic.mil/sti/citations/ADA013939 | url-status = live }}</ref> Both reports contained contributions from the other team. During December 1975, Brayer and Hammond presented their work in a paper at the IEEE National Telecommunications Conference: the IEEE CRC-32 polynomial is the generating polynomial of a [[Hamming code]] and was selected for its error detection performance.<ref name="BrayerHammond1975">{{cite conference | last1 = Brayer | first1 = Kenneth | last2 = Hammond | first2 = Joseph L. Jr. |date=December 1975 | title = Evaluation of error detection polynomial performance on the AUTOVON channel | conference =NTC 75 : National Telecommunications Conference, December 1–3, 1975, New Orleans, Louisiana | volume = 1 | publisher = Institute of Electrical and Electronics Engineers | pages = ((8–21–5)) |oclc=32688603 |id=75 CH 1015-7 CSCB| bibcode = 1975ntc.....1....8B |url=https://books.google.com/books?id=2xxGAQAAIAAJ}}</ref>  Even so, the Castagnoli CRC-32C polynomial used in iSCSI or SCTP matches its performance on messages from 58 bits to 131 kbits, and outperforms it in several size ranges including the two most common sizes of Internet packet.<ref name="koop02" /> The [[ITU-T]] [[G.hn]] standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for [[Physical layer|PHY headers]]).

CRC-32C computation is implemented in hardware as an operation ({{code|CRC32}}) of [[SSE4#SSE4.2|SSE4.2]] instruction set, first introduced in [[Intel]] processors' [[Nehalem (microarchitecture)|Nehalem]] microarchitecture. [[ARM architecture|ARM]] [[AArch64]] architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations.

== Polynomial representations ==
The table below lists only the polynomials of the various algorithms in use. Variations of a particular protocol can impose pre-inversion, post-inversion and reversed bit ordering as described above. For example, the CRC32 used in Gzip and Bzip2 use the same polynomial, but Gzip employs reversed bit ordering, while Bzip2 does not.<ref name="cook-catalogue" />
Note that even parity polynomials in [[GF(2)]] with degree greater than 1 are never primitive. Even parity polynomial marked as primitive in this table represent a primitive polynomial multiplied by <math> \left(x+1\right)</math>. The most significant bit of a polynomial is always 1, and is not shown in the hex representations.

{{anchor|table}}
<div style="max-width: 100%; overflow-y: scroll">
{| class="wikitable"
! rowspan="2" | Name
! rowspan="2" | Uses
! colspan="4" | [[Mathematics of cyclic redundancy checks#Polynomial representations|Polynomial representations]]
! rowspan="2" | [[Parity (telecommunication)|Parity]]<ref>CRCs with even parity detect any odd number of bit errors, at the expense of lower hamming distance for long payloads. Note that parity is computed over the entire generator polynomial, including implied 1 at the beginning or the end. For example, the full representation of CRC-1 is 0x3, which has two 1 bits. Thus, its parity is even.</ref>
! rowspan="2" | Primitive<ref name="users.ece.cmu.edu">{{cite web|url=https://users.ece.cmu.edu/~koopman/crc/crc32.html|title=32 Bit CRC Zoo|website=users.ece.cmu.edu|access-date=5 November 2017|archive-date=19 March 2018|archive-url=https://web.archive.org/web/20180319125501/http://users.ece.cmu.edu/~koopman/crc/crc32.html|url-status=live}}</ref>
! colspan="15" | Maximum bits of payload by [[Hamming distance]]<ref>Payload means length exclusive of CRC field. A Hamming distance of ''d'' means that ''d''&nbsp;−&nbsp;1 bit errors can be detected and ⌊(''d''&nbsp;−&nbsp;1)/2⌋ bit errors can be corrected</ref><ref name="koop02" /><ref name="users.ece.cmu.edu"/>
|-
! Normal
! [[Mathematics of cyclic redundancy checks#Reversed representations and reciprocal polynomials|Reversed]]
! [[Mathematics of cyclic redundancy checks#Reciprocal polynomials|Reciprocal]]
! Reversed reciprocal
! ≥&nbsp;16
! 15
! 14
! 13
! 12
! 11
! 10
! 9
! 8
! 7
! 6
! 5
! 4
! 3
! 2<ref>is always achieved for arbitrarily long messages</ref>
|-
| rowspan="2" | CRC-1
| rowspan="2" | most hardware; also known as ''[[parity bit]]''
| 0x1
| 0x1
| 0x1
| 0x1
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x + 1</math>
|-
| rowspan="2" | CRC-3-[[GSM]]
| rowspan="2" | mobile networks<ref name="ts-100-909">{{Cite book|title=ETSI TS 100 909|version=V8.9.0|date=January 2005|publisher=European Telecommunications Standards Institute|location=Sophia Antipolis, France|url=https://www.etsi.org/deliver/etsi_ts/100900_100999/100909/08.09.00_60/ts_100909v080900p.pdf|access-date=21 October 2016|archive-date=17 April 2018|archive-url=https://web.archive.org/web/20180417050648/http://www.etsi.org/deliver/etsi_ts/100900_100999/100909/08.09.00_60/ts_100909v080900p.pdf|url-status=live}}</ref>
| 0x3
| 0x6
| 0x5
| 0x5
| rowspan="2" {{Bad|odd}}
| rowspan="2" {{Good|yes}}<ref name="koop_crc3">{{cite web|url=https://users.ece.cmu.edu/~koopman/crc/crc3.html|title=3 Bit CRC Zoo|website=users.ece.cmu.edu|access-date=19 January 2018|archive-date=7 April 2018|archive-url=https://web.archive.org/web/20180407230324/http://users.ece.cmu.edu/~koopman/crc/crc3.html|url-status=live}}</ref>
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 4
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^3 + x + 1</math>
|-
| rowspan="2" | CRC-4-ITU
| rowspan="2" | [[ITU-T]] [http://www.itu.int/rec/T-REC-G.704-199810-I/en G.704], p.&nbsp;12
| 0x3
| 0xC
| 0x9
| 0x9
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^4 + x + 1</math>
|-
| rowspan="2" | CRC-5-EPC
| rowspan="2" | [[Radio-frequency identification|Gen 2 RFID]]<ref name="gen-2-spec">{{Cite book|title=Class-1 Generation-2 UHF RFID Protocol|version=1.2.0|publisher=[[EPCglobal]]|url=http://www.gs1.org/gsmp/kc/epcglobal/uhfc1g2/uhfc1g2_1_2_0-standard-20080511.pdf|date=23 October 2008|access-date=4 July 2012|page=35|archive-date=19 March 2012|archive-url=https://web.archive.org/web/20120319154207/http://www.gs1.org/gsmp/kc/epcglobal/uhfc1g2/uhfc1g2_1_2_0-standard-20080511.pdf|url-status=live}} (Table 6.12)</ref>
| 0x09
| 0x12
| 0x05
| 0x14
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^5 + x^3 + 1</math>
|-
| rowspan="2" | CRC-5-ITU
| rowspan="2" | ITU-T [http://www.itu.int/rec/T-REC-G.704-199810-I/en G.704], p.&nbsp;9
| 0x15
| 0x15
| 0x0B
| 0x1A
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^5 + x^4 + x^2 + 1</math>
|-
| rowspan="2" | CRC-5-USB
| rowspan="2" | [[Universal Serial Bus|USB]] token packets
| 0x05
| 0x14
| 0x09
| 0x12
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^5 + x^2 + 1</math>
|-
| CRC-6-[[CDMA2000]]-A
| mobile networks<ref name="cdma2000-spec">{{Cite book|publisher=3rd Generation Partnership Project 2|date=October 2005|title=Physical layer standard for cdma2000 spread spectrum systems|version=Revision D version 2.0|pages=2–89–2–92|url=http://www.3gpp2.org/public_html/specs/C.S0002-D_v2.0_051006.pdf|access-date=14 October 2013|archive-url=https://web.archive.org/web/20131116065606/http://www.3gpp2.org/public_html/specs/C.S0002-D_v2.0_051006.pdf|archive-date=16 November 2013|url-status=dead}}</ref>
| 0x27
| 0x39
| 0x33
| 0x33
| {{Bad|odd}}
|-
| CRC-6-[[CDMA2000]]-B
| mobile networks<ref name="cdma2000-spec" />
| 0x07
| 0x38
| 0x31
| 0x23
| {{Good|even}}
|-
| CRC-6-DARC
| [[Data Radio Channel]]<ref name="en-300-751">{{Cite book|title=ETSI EN 300 751|version=V1.2.1|date=January 2003|publisher=European Telecommunications Standards Institute|location=Sophia Antipolis, France|url=http://www.etsi.org/deliver/etsi_en/300700_300799/300751/01.02.01_60/en_300751v010201p.pdf|section=11. Error correction strategy|pages=67–8|access-date=26 January 2016|archive-date=28 December 2015|archive-url=https://web.archive.org/web/20151228050128/http://www.etsi.org/deliver/etsi_en/300700_300799/300751/01.02.01_60/en_300751v010201p.pdf|url-status=live}}</ref>
| 0x19
| 0x26
| 0x0D
| 0x2C
| {{Good|even}}
|-
| rowspan="2" | CRC-6-[[GSM]]
| rowspan="2" | mobile networks<ref name="ts-100-909" />
| 0x2F
| 0x3D
| 0x3B
| 0x37
| rowspan="2" {{Good|even}}
| rowspan="2" {{Good|yes}}<ref name="koop_crc6">{{cite web|url=https://users.ece.cmu.edu/~koopman/crc/crc6.html|title=6 Bit CRC Zoo|website=users.ece.cmu.edu|access-date=19 January 2018|archive-date=7 April 2018|archive-url=https://web.archive.org/web/20180407230334/http://users.ece.cmu.edu/~koopman/crc/crc6.html|url-status=live}}</ref>
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 1
| rowspan="2" | 1
| rowspan="2" | 25
| rowspan="2" | 25
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^6 + x^5 + x^3 + x^2 + x + 1</math>
|-
| rowspan="2" | CRC-6-ITU
| rowspan="2" | ITU-T [http://www.itu.int/rec/T-REC-G.704-199810-I/en G.704], p.&nbsp;3
| 0x03
| 0x30
| 0x21
| 0x21
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^6 + x + 1</math>
|-
| rowspan="2" | CRC-7
| rowspan="2" | telecom systems, ITU-T [http://www.itu.int/rec/T-REC-G.707/en G.707], ITU-T [http://www.itu.int/rec/T-REC-G.832/en G.832], [[MultiMediaCard|MMC]], [[Secure Digital card|SD]]
| 0x09
| 0x48
| 0x11
| 0x44
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^7 + x^3 + 1</math>
|-
| CRC-7-MVB
| [[Train Communication Network]], [[IEC 60870-5]]<ref name="chakravarty-thesis">{{Cite thesis|last=Chakravarty|first=Tridib|others=Philip Koopman, advisor|date=December 2001|title=Performance of Cyclic Redundancy Codes for Embedded Networks|publisher=Carnegie Mellon University |url=http://www.ece.cmu.edu/~koopman/thesis/chakravarty.pdf|access-date=8 July 2013|pages=5,18|archive-date=1 January 2014|archive-url=https://web.archive.org/web/20140101132228/http://www.ece.cmu.edu/~koopman/thesis/chakravarty.pdf|url-status=live}}</ref>
| 0x65
| 0x53
| 0x27
| 0x72
| {{Bad|odd}}
|-
| rowspan="2" | CRC-8
| rowspan="2" | [[DVB-S2]]<ref name="en-302-307">{{Cite book|title=EN 302 307|version=V1.3.1|date=March 2013|publisher=European Telecommunications Standards Institute|location=Sophia Antipolis, France|url=https://www.etsi.org/deliver/etsi_en/302300_302399/302307/01.03.01_60/en_302307v010301p.pdf|section=5.1.4 CRC-8 encoder (for packetized streams only)|page=17|access-date=29 July 2016|archive-date=30 August 2017|archive-url=https://web.archive.org/web/20170830061535/http://www.etsi.org/deliver/etsi_en/302300_302399/302307/01.03.01_60/en_302307v010301p.pdf|url-status=live}}</ref>
| 0xD5
| 0xAB
| 0x57
| 0xEA<ref name="koop04" />
| rowspan="2" {{Good|even}}
| rowspan="2" {{Bad|no}}<ref name="koop_crc8">{{cite web|url=https://users.ece.cmu.edu/~koopman/crc/crc8.html|title=8 Bit CRC Zoo|website=users.ece.cmu.edu|access-date=19 January 2018|archive-date=7 April 2018|archive-url=https://web.archive.org/web/20180407224301/http://users.ece.cmu.edu/~koopman/crc/crc8.html|url-status=live}}</ref>
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 2
| rowspan="2" | 2
| rowspan="2" | 85
| rowspan="2" | 85
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^8 + x^7 + x^6 + x^4 + x^2 + 1</math>
|-
| rowspan="2" | CRC-8-[[AUTOSAR]]
| rowspan="2" | automotive integration,<ref name="autosar-crc">{{Cite book|title=Specification of CRC Routines|version=4.2.2|date=22 July 2015|publisher=AUTOSAR|location=Munich|url=https://www.autosar.org/fileadmin/files/releases/4-2/software-architecture/safety-and-security/standard/AUTOSAR_SWS_CRCLibrary.pdf|section=7.2.1.2 8-bit 0x2F polynomial CRC Calculation|page=24|access-date=24 July 2016|archive-url=https://web.archive.org/web/20160724123829/https://www.autosar.org/fileadmin/files/releases/4-2/software-architecture/safety-and-security/standard/AUTOSAR_SWS_CRCLibrary.pdf|archive-date=24 July 2016|url-status=dead}}</ref> [[OpenSafety]]<ref name="opensafety-profile">{{Cite book|title=openSAFETY Safety Profile Specification: EPSG Working Draft Proposal 304|version=1.4.0|date=13 March 2013|publisher=Ethernet POWERLINK Standardisation Group|location=Berlin|url=http://www.ethernet-powerlink.org/en/downloads/technical-documents/action/open-download/download/epsg-wdp-304-v-1-4-0/?no_cache=1|section=5.1.1.8 Cyclic Redundancy Check field (CRC-8 / CRC-16)|page=42|access-date=22 July 2016|archive-url=https://web.archive.org/web/20170812202348/http://www.ethernet-powerlink.org/en/downloads/technical-documents/action/open-download/download/epsg-wdp-304-v-1-4-0/?no_cache=1|archive-date=12 August 2017|url-status=dead}}</ref>
| 0x2F
| 0xF4
| 0xE9
| 0x97<ref name="koop04" />
| rowspan="2" {{Good|even}}
| rowspan="2" {{Good|yes}}<ref name="koop_crc8" />
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 3
| rowspan="2" | 3
| rowspan="2" | 119
| rowspan="2" | 119
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^8 + x^5 + x^3 + x^2 + x + 1</math>
|-
| rowspan="2" | CRC-8-[[Bluetooth]]
| rowspan="2" | wireless connectivity<ref name="core-4.2">{{Cite book|title=Specification of the Bluetooth System|volume=2|section=B.7.1.1 HEC generation|pages=144–5|publisher=Bluetooth SIG|date=2 December 2014|url=https://www.bluetooth.org/DocMan/handlers/DownloadDoc.ashx?doc_id=286439|access-date=20 October 2014|archive-date=26 March 2015|archive-url=https://web.archive.org/web/20150326224908/https://www.bluetooth.org/DocMan/handlers/DownloadDoc.ashx?doc_id=286439|url-status=live}}</ref>
| 0xA7
| 0xE5
| 0xCB
| 0xD3
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^8 + x^7 + x^5 + x^2 + x + 1</math>
|-
| rowspan="2" | CRC-8-[[CCITT]]
| rowspan="2" | ITU-T [http://www.itu.int/rec/T-REC-I.432.1-199902-I/en I.432.1 (02/99)]; [[Asynchronous Transfer Mode|ATM]] [[Header Error Correction|HEC]], [[ISDN]] HEC and cell delineation, [[System Management Bus#Packet Error Checking|SMBus PEC]]
| 0x07
| 0xE0
| 0xC1
| 0x83
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^8 + x^2 + x + 1</math>
|-
| rowspan="2" | CRC-8-[[Dallas Semiconductor|Dallas]]/[[Maxim Integrated Products|Maxim]]
| rowspan="2" | [[1-Wire]] [[Bus (computing)|bus]]<ref name="Whitfield01">{{Cite web|first=Harry |last=Whitfield|url=http://homepages.cs.ncl.ac.uk/harry.whitfield/home.formal/CRCs.html|title=XFCNs for Cyclic Redundancy Check Calculations|archive-url=https://web.archive.org/web/20050525224339/http://homepages.cs.ncl.ac.uk/harry.whitfield/home.formal/CRCs.html|archive-date=25 May 2005|date=24 April 2001}}</ref>
| 0x31
| 0x8C
| 0x19
| 0x98
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^8 + x^5 + x^4 + 1</math>
|-
| rowspan="2" | CRC-8-DARC
| rowspan="2" | [[Data Radio Channel]]<ref name="en-300-751" />
| 0x39
| 0x9C
| 0x39
| 0x9C
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^8 + x^5 + x^4 + x^3 + 1</math>
|-
| rowspan="2" | CRC-8-[[GSM]]-B
| rowspan="2" | mobile networks<ref name="ts-100-909" />
| 0x49
| 0x92
| 0x25
| 0xA4
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^8 + x^6 + x^3 + 1</math>
|-
| rowspan="2" | CRC-8-[[SAE J1850]]
| rowspan="2" | [[AES3]]; [[On-board diagnostics|OBD]]
| 0x1D
| 0xB8
| 0x71
| 0x8E
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^8 + x^4 + x^3 + x^2 + 1</math>
|-
| rowspan="2" | CRC-8-[[W-CDMA (UMTS)|WCDMA]]
| rowspan="2" | mobile networks<ref name="cdma2000-spec" /><ref name="richardson-wcdma">{{Cite book|last=Richardson|first=Andrew|title=WCDMA Handbook |publisher=Cambridge University Press|date=17 March 2005|isbn=978-0-521-82815-4|page=223|url=https://books.google.com/books?id=yN5lve5L4vwC&pg=PA223}}</ref>
| 0x9B
| 0xD9
| 0xB3
| 0xCD<ref name="koop04" />
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^8 + x^7 + x^4 + x^3 + x + 1</math>
|-
| rowspan="2" | CRC-10
| rowspan="2" | ATM; ITU-T [http://www.itu.int/rec/T-REC-I.610/en I.610]
| 0x233
| 0x331
| 0x263
| 0x319
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{10} + x^9 + x^5 + x^4 + x + 1</math>
|-
| CRC-10-[[CDMA2000]]
| mobile networks<ref name="cdma2000-spec" />
| 0x3D9
| 0x26F
| 0x0DF
| 0x3EC
| {{Good|even}}
|-
| CRC-10-[[GSM]]
| mobile networks<ref name="ts-100-909" />
| 0x175
| 0x2BA
| 0x175
| 0x2BA
| {{Bad|odd}}
|-
| rowspan="2" | CRC-11
| rowspan="2" | [[FlexRay]]<ref name="flexray-spec">{{Cite book|title=FlexRay Protocol Specification|version=3.0.1|publisher=Flexray Consortium|date=October 2010|page=114}} (4.2.8 Header CRC (11 bits))</ref>
| 0x385
| 0x50E
| 0x21D
| 0x5C2
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{11} + x^9 + x^8 + x^7 + x^2 + 1</math>
|-
| rowspan="2" | CRC-12
| rowspan="2" | telecom systems<ref>{{Cite journal | last = Perez | first = A. | title =  Byte-Wise CRC Calculations | journal = IEEE Micro | volume = 3 | issue = 3 | pages = 40–50 | year = 1983 | doi = 10.1109/MM.1983.291120| s2cid = 206471618 }}</ref><ref>{{Cite journal | last1 = Ramabadran | first1 =  T.V. | last2 = Gaitonde | first2 = S.S. | title = A tutorial on CRC computations | journal = IEEE Micro | volume = 8 | issue = 4 | year = 1988 | pages = 62–75 | doi = 10.1109/40.7773 | s2cid = 10216862 }}</ref>
| 0x80F
| 0xF01
| 0xE03
| 0xC07<ref name="koop04" />
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{12} + x^{11} + x^3 + x^2 + x + 1</math>
|-
| CRC-12-[[CDMA2000]]
| mobile networks<ref name="cdma2000-spec" />
| 0xF13
| 0xC8F
| 0x91F
| 0xF89
| {{Good|even}}
|-
| CRC-12-[[GSM]]
| mobile networks<ref name="ts-100-909" />
| 0xD31
| 0x8CB
| 0x197
| 0xE98
| {{Bad|odd}}
|-
| rowspan="2" | CRC-13-BBC
| rowspan="2" | Time signal, [[Radio teleswitch]]<ref>{{Cite web |url=http://www.freescale.com/files/microcontrollers/doc/app_note/AN1597.pdf |title=Longwave Radio Data Decoding using and HC11 and an MC3371|id=AN1597/D  |publisher=Freescale Semiconductor |date=2004 |archive-date=24 September 2015 |archive-url=https://web.archive.org/web/20150924015512/http://www.freescale.com/files/microcontrollers/doc/app_note/AN1597.pdf |url-status=dead }}</ref><ref>{{Cite book|last1=Ely|first1=S.R.|last2=Wright|first2=D.T.|title=L.F. Radio-Data: specification of BBC experimental transmissions 1982|publisher=Research Department, Engineering Division, The British Broadcasting Corporation|date=March 1982|page=9|url=http://downloads.bbc.co.uk/rd/pubs/reports/1982-02.pdf|access-date=11 October 2013|archive-date=12 October 2013|archive-url=https://web.archive.org/web/20131012064432/http://downloads.bbc.co.uk/rd/pubs/reports/1982-02.pdf|url-status=live}}</ref>
| 0x1CF5
| 0x15E7
| 0x0BCF
| 0x1E7A
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{13} + x^{12} + x^{11} + x^{10} + x^7 + x^6 + x^5 + x^4 + x^2 + 1</math>
|-
| CRC-14-DARC
| [[Data Radio Channel]]<ref name="en-300-751" />
| 0x0805
| 0x2804
| 0x1009
| 0x2402
| {{Good|even}}
|-
| CRC-14-[[GSM]]
| mobile networks<ref name="ts-100-909" />
| 0x202D
| 0x2D01
| 0x1A03
| 0x3016
| {{Good|even}}
|- id="CRC-15-CAN"
| rowspan="2" | CRC-15-[[Controller Area Network|CAN]]
| rowspan="2" |
| 0xC599<ref name="cypress-psoc">{{Cite book|date=20 February 2013|institution=Cypress Semiconductor|title=Cyclic Redundancy Check (CRC): PSoC Creator™ Component Datasheet|page=4|url=http://www.cypress.com/file/128066/download|access-date=26 January 2016|archive-date=2 February 2016|archive-url=https://web.archive.org/web/20160202085601/http://www.cypress.com/file/128066/download|url-status=live}}</ref><ref name="cia-can-crc">{{Cite web|url = http://www.can-cia.org/can-knowledge/can/crc/|title = Cyclic redundancy check (CRC) in CAN frames|access-date = 26 January 2016|website = CAN in Automation|archive-date = 1 February 2016|archive-url = https://web.archive.org/web/20160201145928/http://www.can-cia.org/can-knowledge/can/crc/|url-status = live}}</ref>
| 0x4CD1
| 0x19A3
| 0x62CC
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{15} + x^{14} + x^{10} + x^8 + x^7 + x^4 + x^3 + 1 </math>
|-
| CRC-15-[[MPT1327]]
|<ref name="mpt1327">{{Cite book |date=June 1997 |title=A signalling standard for trunked private land mobile radio systems (MPT 1327) |edition=3rd |publisher=[[Ofcom]] |section=3.2.3 Encoding and error checking |page=3 |url=http://www.ofcom.org.uk/static/archive/ra/publication/mpt/mpt_pdf/mpt1327.pdf |access-date=16 July 2012 |archive-date=14 July 2012 |archive-url=https://web.archive.org/web/20120714015950/http://www.ofcom.org.uk/static/archive/ra/publication/mpt/mpt_pdf/mpt1327.pdf |url-status=live }}</ref>
| 0x6815
| 0x540B
| 0x2817
| 0x740A
| {{Bad|odd}}
|-
| CRC-16-Chakravarty
| Optimal for payloads ≤64 bits<ref name="chakravarty-thesis" />
| 0x2F15
| 0xA8F4
| 0x51E9
| 0x978A
| {{Bad|odd}}
|-
| CRC-16-[[ARINC]]
| [[Aircraft Communications Addressing and Reporting System|ACARS]] applications<ref name="rehmann-acars">{{Cite web | url=http://ntl.bts.gov/lib/1000/1200/1290/tn95_66.pdf | title=Air Ground Data Link VHF Airline Communications and Reporting System (ACARS) Preliminary Test Report | date=February 1995 | page=5 | last1=Rehmann | first1=Albert | last2=Mestre | first2=José D. | publisher=Federal Aviation Authority Technical Center | access-date=7 July 2012 | archive-date=2 August 2012 | archive-url=https://web.archive.org/web/20120802065800/http://ntl.bts.gov/lib/1000/1200/1290/tn95_66.pdf | url-status=dead }}</ref>
| 0xA02B
| 0xD405
| 0xA80B
| 0xD015
| {{Bad|odd}}
|-
| rowspan="2" | CRC-16-CCITT
| rowspan="2" | [[X.25]], [[ITU-T V.41|V.41]], [[HDLC]] ''FCS'', [[XMODEM]], [[Bluetooth]], [[PACTOR]], [[Secure Digital card|SD]], [[DigRF]], many others; known as ''CRC-CCITT''
| 0x1021
| 0x8408
| 0x811
| 0x8810<ref name="koop04" />
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{16} + x^{12} + x^5 + 1</math>
|-
| CRC-16-[[CDMA2000]]
| mobile networks<ref name="cdma2000-spec" />
| 0xC867
| 0xE613
| 0xCC27
| 0xE433
| {{Bad|odd}}
|-
| rowspan="2" | CRC-16-[[Digital Enhanced Cordless Telecommunications|DECT]]
| rowspan="2" | cordless telephones<ref name="en-300-175-3">{{Cite book|title=ETSI EN 300 175-3|version=V2.5.1|date=August 2013|publisher=European Telecommunications Standards Institute|location=Sophia Antipolis, France|url=http://www.etsi.org/deliver/etsi_en/300100_300199/30017503/02.05.01_60/en_30017503v020501p.pdf|section=6.2.5 Error control|pages=99,101|access-date=26 January 2016|archive-date=1 July 2015|archive-url=https://web.archive.org/web/20150701125736/http://www.etsi.org/deliver/etsi_en/300100_300199/30017503/02.05.01_60/en_30017503v020501p.pdf|url-status=live}}</ref>
| 0x0589
| 0x91A0
| 0x2341
| 0x82C4
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{16} + x^{10} + x^8 + x^7 + x^3 + 1</math>
|-
| rowspan="2" | CRC-16-[[International Committee for Information Technology Standards|T10]]-[[Data Integrity Field|DIF]]
| rowspan="2" | [[SCSI]] DIF
| 0x8BB7<ref name="thaler-t10-selection">{{Cite web|last=Thaler|first=Pat|author-link=Pat Thaler|title=16-bit CRC polynomial selection|publisher=INCITS T10|date=28 August 2003|url=http://www.t10.org/ftp/t10/document.03/03-290r0.pdf|access-date=11 August 2009 |archive-date=28 July 2011|archive-url=https://web.archive.org/web/20110728081616/http://www.t10.org/ftp/t10/document.03/03-290r0.pdf|url-status=live}}</ref>
| 0xEDD1
| 0xDBA3
| 0xC5DB
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^{16} + x^{15} + x^{11} + x^{9} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1</math>
|-
| rowspan="2" | CRC-16-[[DNP3|DNP]]
| rowspan="2" | DNP, [[IEC 60870-5|IEC 870]], [[Meter-Bus|M-Bus]]
| 0x3D65
| 0xA6BC
| 0x4D79
| 0x9EB2
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + x^8 + x^6 + x^5 + x^2 + 1</math>
|-
| rowspan="2" | CRC-16-[[IBM]]
| rowspan="2" | [[Binary Synchronous Communications|Bisync]], [[Modbus]], [[Universal Serial Bus|USB]], [[ANSI]] [https://web.archive.org/web/20091001172850/http://www.incits.org/press/1997/pr97020.htm X3.28], SIA DC-07, many others; also known as ''CRC-16'' and ''CRC-16-ANSI''
| 0x8005
| 0xA001
| 0x4003
| 0xC002
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{16} + x^{15} + x^2 + 1</math>
|-
| CRC-16-[[OpenSafety]]-A
| safety fieldbus<ref name="opensafety-profile" />
| 0x5935
| 0xAC9A
| 0x5935
| 0xAC9A<ref name="koop04" />
| {{Bad|odd}}
|-
| CRC-16-[[OpenSafety]]-B
| safety fieldbus<ref name="opensafety-profile" />
| 0x755B
| 0xDAAE
| 0xB55D
| 0xBAAD<ref name="koop04" />
| {{Bad|odd}}
|-
| CRC-16-[[Profibus]]
| fieldbus networks<ref name="profibus-spec">{{Cite book|title=PROFIBUS Specification Normative Parts|version=1.0|publisher=Profibus International|date=March 1998|volume=9|section=8.8.4 Check Octet (FCS)|page=906|url=https://www.kuebler.com/PDFs/Feldbus_Multiturn/specification_DP.pdf|access-date=9 July 2016|archive-url=https://web.archive.org/web/20081116195826/http://www.kuebler.com/PDFs/Feldbus_Multiturn/specification_DP.pdf|archive-date=16 November 2008|url-status=dead}}</ref>
| 0x1DCF
| 0xF3B8
| 0xE771
| 0x8EE7
| {{Bad|odd}}
|-
| Fletcher-16
| Used in [[Adler-32]] A & B Checksums
| colspan="4" | Often confused to be a CRC, but actually a checksum; see [[Fletcher's checksum]]
|-
| CRC-17-CAN
| CAN FD<ref name="can-fd-spec">{{Cite book|title=CAN with Flexible Data-Rate Specification|version=1.0|publisher=Robert Bosch GmbH|date=17 April 2012|page=13|url=http://www.bosch-semiconductors.de/media/pdf_1/canliteratur/can_fd_spec.pdf|url-status=dead|archive-url=https://web.archive.org/web/20130822124728/http://www.bosch-semiconductors.de/media/pdf_1/canliteratur/can_fd_spec.pdf|archive-date=22 August 2013}} (3.2.1 DATA FRAME)</ref>
| 0x1685B
| 0x1B42D
| 0x1685B
| 0x1B42D
| {{Good|even}}
|-
| CRC-21-CAN
| CAN FD<ref name="can-fd-spec" />
| 0x102899
| 0x132281
| 0x064503
| 0x18144C
| {{Good|even}}
|-
| rowspan="2" | CRC-24
| rowspan="2" | [[FlexRay]]<ref name="flexray-spec" />
| 0x5D6DCB
| 0xD3B6BA
| 0xA76D75
| 0xAEB6E5
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{24} + x^{22} + x^{20} + x^{19} + x^{18} + x^{16} + x^{14} + x^{13} + x^{11} + x^{10} + x^8 + x^7 + x^6 + x^3 + x + 1</math>
|-
| rowspan="2" | CRC-24-[[Radix-64]]
| rowspan="2" | [[Pretty Good Privacy#OpenPGP|OpenPGP]], [[RTCM]]104v3
| 0x864CFB
| 0xDF3261
| 0xBE64C3
| 0xC3267D
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math> x^{24} + x^{23} + x^{18} + x^{17} + x^{14} + x^{11} + x^{10} + x^7 + x^6 + x^5 + x^4 + x^3 + x + 1 </math>
|-
| rowspan="2" | CRC-24-[[WCDMA]]
| rowspan="2" | Used in [[OS-9|OS-9 RTOS]]. Residue = 0x800FE3.<ref>{{cite web|url=http://www.roug.org/soren/6809/os9sysprog.html#f.crc|title=OS-9 Operating System System Programmer's Manual|website=roug.org|access-date=17 July 2018|archive-date=17 July 2018|archive-url=https://web.archive.org/web/20180717070700/http://www.roug.org/soren/6809/os9sysprog.html#f.crc|url-status=live}}</ref>
| 0x800063
| 0xC60001
| 0x8C0003
| 0xC00031
| rowspan="2" {{Good|even}}
| rowspan="2" {{Good|yes}}<ref name="koop_crc24">{{cite web|url=https://users.ece.cmu.edu/~koopman/crc/crc8.html|title=24 Bit CRC Zoo|first=Philip P. |last=Koopman|date=20 May 2018|website=users.ece.cmu.edu|access-date=19 January 2018|archive-date=7 April 2018|archive-url=https://web.archive.org/web/20180407224301/http://users.ece.cmu.edu/~koopman/crc/crc8.html|url-status=live}}</ref>
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 4
| rowspan="2" | 4
| rowspan="2" | 8388583
| rowspan="2" | 8388583
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^{24} + x^{23} + x^6 + x^5 + x + 1 </math>
|-
| rowspan="2" | CRC-30
| rowspan="2" | [[CDMA]]
| 0x2030B9C7
| 0x38E74301
| 0x31CE8603
| 0x30185CE3
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{30} + x^{29} + x^{21} + x^{20} + x^{15} + x^{13} + x^{12} + x^{11} + x^{8} + x^{7} + x^{6} + x^{2} + x + 1 </math>
|-
| rowspan="2" | CRC-32
| rowspan="2" | [[ISO]] 3309 ([[High-Level Data Link Control|HDLC]]), [[ANSI]] X3.66 ([[ADCCP]]), [[Federal Information Processing Standard|FIPS]] PUB 71, FED-STD-1003, [[ITU-T V.42]], ISO/IEC/IEEE 802-3 ([[Ethernet]]), ISO/IEC/IEEE 802-11 ([[Wi-Fi]]), [[SATA]], [[NVMe]],<ref>NVM Express (TM) Command Set Specification</ref> [[MPEG-2]], [[PKZIP]], [[Gzip]], [[Bzip2]], [[PCI Express]], [[HDMI]], [[POSIX]] [[cksum]],<ref>{{cite web|url=http://pubs.opengroup.org/onlinepubs/9699919799/utilities/cksum.html|title=cksum|website=pubs.opengroup.org|access-date=27 June 2017|archive-date=18 July 2018|archive-url=https://web.archive.org/web/20180718084130/http://pubs.opengroup.org/onlinepubs/9699919799/utilities/cksum.html|url-status=live}}</ref> [[Portable Network Graphics|PNG]],<ref>{{Cite web | last1 = Boutell | first1 = Thomas | last2 = Randers-Pehrson | first2 = Glenn | date = 14 July 1998 | title = PNG (Portable Network Graphics) Specification, Version 1.2 | url = http://www.libpng.org/pub/png/spec/1.2/PNG-Structure.html | publisher = Libpng.org | access-date = 3 February 2011 | display-authors = etal | archive-date = 3 September 2011 | archive-url = https://web.archive.org/web/20110903114128/http://www.libpng.org/pub/png/spec/1.2/PNG-Structure.html | url-status = live }}</ref> [[ZMODEM]], many others
| 0x04C11DB7
| 0xEDB88320
| 0xDB710641
| 0x82608EDB<ref name="koop02" />
| rowspan="2" {{Bad|odd}}
| rowspan="2" {{Good|yes}}
| rowspan="2" | –
| rowspan="2" | 10
| rowspan="2" | –
| rowspan="2" | –
| rowspan="2" | 12
| rowspan="2" | 21
| rowspan="2" | 34
| rowspan="2" | 57
| rowspan="2" | 91
| rowspan="2" | 171
| rowspan="2" | 268
| rowspan="2" | 2974
| rowspan="2" | 91607
| rowspan="2" | 4294967263
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^{32} + x^{26} + x^{23} + x^{22} + x^{16} + x^{12} + x^{11} + x^{10} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1</math>
|-
| rowspan="2" | {{visible anchor|CRC-32C}} (Castagnoli)
| rowspan="2" | [[iSCSI]], [[SCTP]], [[G.hn]] payload, [[SSE4#SSE4.2|SSE4.2]], [[Btrfs]], [[ext4]], [[ReFS]],<ref> https://learn.microsoft.com/en-us/windows-server/storage/refs/integrity-streams</ref>, [[VHDX]],<ref> https://learn.microsoft.com/en-us/openspecs/windows_protocols/ms-vhdx/340e64a4-ae2a-4dc1-b19b-3dd9d57a3359</ref> [[Ceph (software)|Ceph]]
| 0x1EDC6F41
| 0x82F63B78
| 0x05EC76F1
| 0x8F6E37A0<ref name="koop02" />
| rowspan="2" {{Good|even}}
| rowspan="2" {{Good|yes}}
| rowspan="2" | 6
| rowspan="2" | –
| rowspan="2" | 8
| rowspan="2" | –
| rowspan="2" | 20
| rowspan="2" | –
| rowspan="2" | 47
| rowspan="2" | –
| rowspan="2" | 177
| rowspan="2" | –
| rowspan="2" | 5243
| rowspan="2" | –
| rowspan="2" | 2147483615
| rowspan="2" | –
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^{32} + x^{28} + x^{27} + x^{26} + x^{25} + x^{23} + x^{22} + x^{20} + x^{19} + x^{18} + x^{14} + x^{13} + x^{11} + x^{10} + x^9 + x^8 + x^6 + 1</math>
|-
| rowspan="2" | {{visible anchor|CRC-32K}} (Koopman {1,3,28})
| rowspan="2" | Excellent at Ethernet frame length, poor performance with long files {{citation needed|date=January 2024}}
| 0x741B8CD7
| 0xEB31D82E
| 0xD663B05D
| 0xBA0DC66B<ref name="koop02" />
| rowspan="2" {{Good|even}}
| rowspan="2" {{Bad|no}}
| rowspan="2" | 2
| rowspan="2" | –
| rowspan="2" | 4
| rowspan="2" | –
| rowspan="2" | 16
| rowspan="2" | –
| rowspan="2" | 18
| rowspan="2" | –
| rowspan="2" | 152
| rowspan="2" | –
| rowspan="2" | 16360
| rowspan="2" | –
| rowspan="2" | 114663
| rowspan="2" | –
| rowspan="2" | ∞
|-
| colspan="4" | <math>x^{32} + x^{30} + x^{29} + x^{28} + x^{26} + x^{20} + x^{19} + x^{17} + x^{16} + x^{15} + x^{11} + x^{10} + x^{7} + x^{6} + x^{4} + x^{2} + x + 1</math>
|-
|CRC-32K<sub>2</sub> (Koopman {1,1,30})
| Excellent at Ethernet frame length, poor performance with long files {{citation needed|date=January 2024}}
| 0x32583499
| 0x992C1A4C
| 0x32583499
| 0x992C1A4C<ref name="koop02" />
| {{Good|even}}
| {{Bad|no}}
| –
| –
| 3
| –
| 16
| –
| 26
| –
| 134
| –
| 32738
| –
| 65506
| –
| ∞
|-
| rowspan="2" | CRC-32Q
| rowspan="2" | aviation; [[AIXM]]<ref name="aixm-primer">{{Cite book|title=AIXM Primer|url=http://www.eurocontrol.int/sites/default/files/service/content/documents/information-management/20060320-aixm-primer.pdf|version=4.5|publisher=[[European Organisation for the Safety of Air Navigation]]|date=20 March 2006|access-date=3 February 2019|archive-date=20 November 2018|archive-url=https://web.archive.org/web/20181120041622/https://www.eurocontrol.int/sites/default/files/service/content/documents/information-management/20060320-aixm-primer.pdf|url-status=live}}</ref>
| 0x814141AB
| 0xD5828281
| 0xAB050503
| 0xC0A0A0D5
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{32} + x^{31} + x^{24} + x^{22} + x^{16} + x^{14} + x^{8} + x^{7} + x^{5} + x^{3} + x + 1</math>
|-
| Adler-32
|
| colspan="4" | Often confused to be a CRC, but actually a checksum; see [[Adler-32]]
|-
| rowspan="2" | CRC-40-[[GSM]]
| rowspan="2" | GSM control channel<ref>[http://www.etsi.org/deliver/etsi_ts/100900_100999/100909/08.09.00_60/ts_100909v080900p.pdf ETSI TS 100 909] {{Webarchive|url=https://web.archive.org/web/20180417050648/http://www.etsi.org/deliver/etsi_ts/100900_100999/100909/08.09.00_60/ts_100909v080900p.pdf |date=17 April 2018 }} version 8.9.0 (January 2005), Section 4.1.2 a</ref><ref name="gammel">{{Cite book|last=Gammel|first=Berndt M.|date=31 October 2005|title=Matpack documentation: Crypto – Codes <!-- |unusedurl=http://users.physik.tu-muenchen.de/gammel/matpack/html/LibDoc/Crypto/MpCRC.html long-time home, gone-->|url=http://www.matpack.de/index.html#DOWNLOAD|publisher=Matpack.de|access-date=21 April 2013|archive-date=25 August 2013|archive-url=https://web.archive.org/web/20130825205922/http://matpack.de/index.html#DOWNLOAD|url-status=live}} (Note: MpCRC.html is included with the Matpack compressed software source code, under /html/LibDoc/Crypto)</ref><ref name="geremia">{{Cite web|last=Geremia|first=Patrick|date=April 1999|title=Cyclic redundancy check computation: an implementation using the TMS320C54x|issue=SPRA530|publisher=Texas Instruments|page=5|url=http://www.ti.com/lit/an/spra530/spra530.pdf|access-date=4 July 2012 |archive-date=14 June 2012|archive-url=https://web.archive.org/web/20120614061303/http://www.ti.com/lit/an/spra530/spra530.pdf|url-status=live}}</ref>
| 0x0004820009
| 0x9000412000
| 0x2000824001
| 0x8002410004
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{40} + x^{26} + x^{23} + x^{17} + x^3 + 1 = (x^{23} + 1) (x^{17} + x^3 + 1)</math>
|-
| rowspan="2" | CRC-64-[[Ecma International|ECMA]]
| rowspan="2" | [https://ecma-international.org/publications-and-standards/standards/ecma-182/ ECMA-182]  p.&nbsp;51, [[XZ Utils]]
| 0x42F0E1EBA9EA3693
| 0xC96C5795D7870F42
| 0x92D8AF2BAF0E1E85
| 0xA17870F5D4F51B49
| rowspan="2" {{Good|even}}
|-
| colspan="4" | <math>x^{64} + x^{62} + x^{57} + x^{55} + x^{54} + x^{53} + x^{52} + x^{47} + x^{46} + x^{45} + x^{40} + x^{39} + x^{38} + x^{37} + x^{35} + x^{33} +</math> <math>x^{32} + x^{31} + x^{29} + x^{27} + x^{24} + x^{23} + x^{22} + x^{21} + x^{19} + x^{17} + x^{13} + x^{12} + x^{10} + x^9 + x^7 + x^4 + x + 1</math>
|-
| rowspan="2" | CRC-64-ISO
| rowspan="2" | ISO 3309 ([[High-Level Data Link Control|HDLC]]), [[Swiss-Prot]]/[[TrEMBL]]; considered weak for hashing<ref name="jones-improved64">{{Cite web|last=Jones|first=David T.|title=An Improved 64-bit Cyclic Redundancy Check for Protein Sequences|publisher=University College London|url=http://www.cs.ucl.ac.uk/staff/d.jones/crcnote.pdf|access-date=15 December 2009|archive-date=7 June 2011|archive-url=https://web.archive.org/web/20110607110311/http://www.cs.ucl.ac.uk/staff/d.jones/crcnote.pdf|url-status=live}}<!-- date of PDF=1 Dec 2009; date of referenced C file=2 Mar 2006; date in C file comments=28 Sep 2002 --></ref>
| 0x000000000000001B
| 0xD800000000000000
| 0xB000000000000001
| 0x800000000000000D
| rowspan="2" {{Bad|odd}}
|-
| colspan="4" | <math>x^{64} + x^4 + x^3 + x + 1</math>
|}
</div>

=== Implementations ===
*[https://web.archive.org/web/20130715065157/http://gnuradio.org/redmine/projects/gnuradio/repository/revisions/1cb52da49230c64c3719b4ab944ba1cf5a9abb92/entry/gr-digital/lib/digital_crc32.cc Implementation of CRC32 in GNU Radio up to 3.6.1 (ca. 2012)]
*[http://sourceforge.net/projects/crccalculator/files/CRC/ C class code for CRC checksum calculation with many different CRCs to choose from]

===CRC catalogues===
* [https://reveng.sourceforge.io/crc-catalogue/all.htm Catalogue of parametrised CRC algorithms]
* [http://users.ece.cmu.edu/~koopman/crc/crc32.html CRC Polynomial Zoo]

== See also ==
*[[Checksum]]
*[[Computation of cyclic redundancy checks]]
*[[Information security]]
*[[List of checksum algorithms]]
*[[List of hash functions]]
*[[Longitudinal redundancy check|LRC]]
*[[Mathematics of cyclic redundancy checks]]
*[[Simple file verification]]

== References ==
{{Reflist}}

== Further reading ==
{{refbegin}}
* {{Cite book |title=Hacker's Delight |first=Henry S. |last=Warren Jr. |date=2013 |edition=2nd |publisher=[[Addison Wesley]] |isbn=978-0-321-84268-8|title-link=Hacker's Delight |chapter-url=https://books.google.com/books?id=VicPJYM0I5QC&pg=PA319 |chapter=14. Cyclic Redundancy Check |pages=319–330 }}
* {{cite book |first=Philip |last=Koopman |title=Understanding Checksums and Cyclic Redundancy Checks  |asin=B0CVXWDZ99 |date=2024}}
{{refend}}

== External links ==
{{refbegin}}
*{{cite journal |first1=Jubin |last1=Mitra |first2=Tapan |last2=Nayak |title=Reconfigurable very high throughput low latency VLSI (FPGA) design architecture of CRC 32 |journal=Integration, the VLSI Journal |volume=56 |pages=1–14 |date=January 2017 |doi=10.1016/j.vlsi.2016.09.005 }}
* [http://www.mathpages.com/home/kmath458.htm Cyclic Redundancy Checks], MathPages, overview of error-detection of different polynomials
*{{cite web |author-link=Ross Williams |first=Ross |last=Williams |title=A Painless Guide to CRC Error Detection Algorithms |date=1993 |url=http://www.ross.net/crc/crcpaper.html |access-date=15 August 2011 |archive-url=https://web.archive.org/web/20110903033652/http://www.ross.net/crc/crcpaper.html |archive-date=3 September 2011 |url-status=dead }}
*{{cite web |first=Richard |last=Black |title=Fast CRC32 in Software |date=1994 |work=The Blue Book |publisher=Systems Research Group, Computer Laboratory, University of Cambridge |url=http://www.cl.cam.ac.uk/Research/SRG/bluebook/21/crc/crc.html }} Algorithm 4 was used in Linux and Bzip2.
*{{cite web |last1=Kounavis |first1=M. |last2=Berry |first2=F. |title=A Systematic Approach to Building High Performance, Software-based, CRC generators |year=2005 |publisher=Intel |url=http://www.intel.com/technology/comms/perfnet/download/CRC_generators.pdf |access-date=4 February 2007 |archive-date=16 December 2006 |archive-url=https://web.archive.org/web/20061216135550/http://www.intel.com/technology/comms/perfnet/download/CRC_generators.pdf |url-status=live }}, Slicing-by-4 and slicing-by-8 algorithms
*{{cite web |first=W. |last=Kowalk |title=CRC Cyclic Redundancy Check Analysing and Correcting Errors |date=August 2006 |publisher=Universität Oldenburg |url=http://einstein.informatik.uni-oldenburg.de/papers/CRC-BitfilterEng.pdf |access-date=1 September 2006 |archive-date=11 June 2007 |archive-url=https://web.archive.org/web/20070611140619/http://einstein.informatik.uni-oldenburg.de/papers/CRC-BitfilterEng.pdf |url-status=live }} — Bitfilters
*{{cite book |first=Henry S. Jr. |last=Warren |chapter=Cyclic Redundancy Check |title=Hacker's Delight |chapter-url=http://www.hackersdelight.org/crc.pdf |url-status=dead |archive-url=https://web.archive.org/web/20150503014404/http://www.hackersdelight.org/crc.pdf |archive-date=3 May 2015 |title-link=Hacker's Delight }} — theory, practice, hardware, and software with emphasis on CRC-32.
*[http://www.cosc.canterbury.ac.nz/greg.ewing/essays/CRC-Reverse-Engineering.html Reverse-Engineering a CRC Algorithm] {{Webarchive|url=https://web.archive.org/web/20110807100031/http://www.cosc.canterbury.ac.nz/greg.ewing/essays/CRC-Reverse-Engineering.html |date=7 August 2011 }}
*{{cite web |first=Greg |last=Cook |title=Catalogue of parameterised CRC algorithms |work=CRC RevEng |url=https://reveng.sourceforge.io/crc-catalogue/all.htm |access-date=18 September 2020 |archive-date=1 August 2020 |archive-url=https://web.archive.org/web/20200801122415/https://reveng.sourceforge.io/crc-catalogue/all.htm  |url-status=live}}
*{{cite web |last=Koopman |first=Phil |title=Blog: Checksum and CRC Central |url=http://checksumcrc.blogspot.com/ }} — includes links to PDFs giving 16 and 32-bit CRC [[Hamming distance]]s
**{{cite web |author-mask=1 |last=Koopman |first=Phil |title= Why Life Critical Networks Tend To Provide HD=6
 |date=April 2023 |url=http://checksumcrc.blogspot.com/2023/04/why-life-critical-networks-tend-to.html }}
*{{cite web |first1=Philip |last1=Koopman |first2=Kevin |last2=Driscoll |first3=Brendan |last3=Hall |title=Cyclic Redundancy Code and Checksum Algorithms to Ensure Critical Data Integrity |date=March 2015 |id=DOT/FAA/TC-14/49 |publisher=Federal Aviation Administration |url=http://www.tc.faa.gov/its/worldpac/techrpt/tc14-49.pdf |access-date=9 May 2015 |archive-date=18 May 2015 |archive-url=https://web.archive.org/web/20150518073214/http://www.tc.faa.gov/its/worldpac/techrpt/tc14-49.pdf |url-status=live }}
*{{cite video |first1=Philip |last1=Koopman |title=Mechanics of Cyclic Redundancy Check Calculations  |date=January 2023 |via=YouTube |url=https://www.youtube.com/watch?v=qRqvdOAfxcA}}
{{refend}}
* [https://www.iso.org/standard/37010.html ISO/IEC 13239:2002: Information technology -- Telecommunications and information exchange between systems -- High-level data link control (HDLC) procedures]
* [https://github.com/spotify/linux/blob/master/crypto/crc32c.c CRC32-Castagnoli Linux Library]

{{Ecma International Standards}}

[[Category:Binary arithmetic]]
[[Category:Cyclic redundancy checks| ]]
[[Category:Finite fields]]
[[Category:Polynomials]]
[[Category:Wikipedia articles with ASCII art]]
[[Category:Articles with example Python (programming language) code]]