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# Cyclic redundancy check  ( Binary Arithmetic )

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A cyclic redundancy check ( CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to raw data. Blocks of data entering these systems get a short check value attached, based on the remainder of a 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 correction (see bitfilters).

CRC can't fix the mistakes in frames which change the value of DE("Discard eligibility"), for example if by a mistake DE value was set from 0 to 1, CRC won't be able to return it to 0, the frame will be discarded.

CRCs are so called because the check (data verification) value is a redundancy (it expands the message without adding information) and the is based on . CRCs are popular because they are simple to implement in binary hardware, easy to analyze mathematically, and particularly good at detecting common errors caused by noise in transmission channels. Because the check value has a fixed length, the function that generates it is occasionally used as a .

The CRC was invented by W. Wesley Peterson in 1961; the 32-bit CRC function, used in Ethernet and many other standards, is the work of several researchers and was published in 1975.

Introduction
CRCs are based on the theory of error-correcting codes. The use of 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. Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of : contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many communication channels, 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 .

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

In practice, all commonly used CRCs employ the 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 terms. In other words, the polynomial has a length of ; its encoding requires bits. Note that most polynomial specifications either drop the MSB or 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 below.

The simplest error-detection system, the , is in fact a 1-bit CRC: it uses the generator polynomial  (two terms), 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).

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 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 codes 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.

Thirdly, CRC is a with a property that

$\operatorname\left\{crc\right\}\left(x \oplus y \oplus z\right) = \operatorname\left\{crc\right\}\left(x\right) \oplus \operatorname\left\{crc\right\}\left(y\right) \oplus \operatorname\left\{crc\right\}\left(z\right);$

as a result, even if the CRC is encrypted with a that uses as its combining operation (or mode of 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.

CRC-32 algorithm
This is the algorithm for the CRC-32 variant of CRC. The CRCTable is a of a calculation that would have to be repeated for each byte of the message.

```'''Function''' CRC32
'''Input:'''
data:  Bytes     '''//Array of bytes'''
'''Output:'''
crc32: UInt32    '''//32-bit unsigned crc-32 value'''
'''//Initialize crc-32 to starting value'''
crc32 ← 0xffffffff
'''for each''' byte '''in''' data '''do'''
nLookupIndex ← (crc32 xor byte) and 0xFF;
crc32 ← (crc32 shr 8) xor CRCTable[nLookupIndex] '''//CRCTable is an array of 256 32-bit constants'''
'''//Finalize the CRC-32 value by inverting all the bits'''
crc32 ← crc32 xor 0xFFFFFFFF
'''return''' crc32
```

Computation
To compute an n-bit binary CRC, line the bits representing the input in a row, and position the ()-bit pattern representing the CRC's divisor (called a "") underneath the left-hand end of the row.

In this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial . The polynomial is written in binary as the coefficients; a 3rd-order polynomial has 4 coefficients (). In this case, the coefficients are 1, 0, 1 and 1. The result of the calculation is 3 bits long.

```11010011101100
```

This is first padded with zeros corresponding to the bit length n of the CRC. Here is the first calculation for computing a 3-bit CRC:

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

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 one bit to the right, and the process is repeated until the divisor reaches the right-hand end of the input row. Here is the entire calculation:

```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.
```

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.

```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
```

The following 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: def crc_remainder(input_bitstring, polynomial_bitstring, initial_filler):

```   '''
Calculates 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 = initial_filler * (len(polynomial_bitstring) - 1)
for i in range(len(polynomial_bitstring)):
if polynomial_bitstring[i] == input_padded_array[cur_shift + i]:
else:
```

def crc_check(input_bitstring, polynomial_bitstring, check_value):

```   '''
Calculates the CRC check of a string of bits using a chosen polynomial.
'''
polynomial_bitstring = polynomial_bitstring.lstrip('0')
len_input = len(input_bitstring)
for i in range(len(polynomial_bitstring)):
if polynomial_bitstring[i] == input_padded_array[cur_shift + i]:
else:
return True
else:
return False
```

>>> crc_check('11010011101100','1011','100') True >>> crc_remainder('11010011101100','1011','0') '100'

Mathematics
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), instead of more familiar numbers. The set of binary polynomials is a mathematical 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)
• 65 bits (CRC-64)

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 terms (the polynomial has a length of ). 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 polynomials or irreducible polynomials times the factor , which adds to the code the ability to detect all errors affecting an odd number of bits. 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 .

The advantage of choosing a 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 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 $r$ is the degree of the primitive generator polynomial, then the maximal total block length is $2 ^ \left\{r\right\} - 1$, and the associated code is able to detect any single-bit or double-bit errors.

(2018). 9780521880688, Cambridge University Press.
We can improve this situation. If we use the generator polynomial $g\left(x\right) = p\left(x\right)\left(1 + x\right)$, where $p\left(x\right)$ is a primitive polynomial of degree $r - 1$, then the maximal total block length is $2^\left\{r - 1\right\} - 1$, and the code is able to detect single, double, triple and any odd number of errors.

A polynomial $g\left(x\right)$ that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power. The are a powerful class of such polynomials. They subsume the two examples above. Regardless of the reducibility properties of a generator polynomial of degree  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 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, 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 transmissions are CRC-checked in hardware, because some widespread serial-port transmission conventions transmit bytes least-significant bit first.
• : 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 ()-bit divisor which overflows an n-bit 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 $x^0$ 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 $x^4 + x + 1$ may be transcribed as:

• 0x3 = 0b0011, representing $x^4 + \left(0x^3 + 0x^2 + 1x^1 + 1x^0\right)$ (MSB-first code)
• 0xC = 0b1100, representing $\left(1x^0 + 1x^1 + 0x^2 + 0x^3\right) + x^4$ (LSB-first code)
• 0x9 = 0b1001, representing $\left(1x^4 + 0x^3 + 0x^2 + 1x^1\right) + x^0$ (Koopman notation)
In the table below they are shown as:
 + Examples of CRC representations CRC-4 0x3 0xC 0x9

Obfuscation
CRCs in proprietary protocols might be 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 engineered using straightforward methods.

Standards and common use
Numerous varieties of cyclic redundancy checks have been incorporated into technical standards. 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. The number of distinct CRCs in use has confused developers, a situation which authors have sought to address. There are three polynomials reported for CRC-12, nineteen conflicting definitions of CRC-16, and seven of CRC-32.

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,

(2004). 9780769520520 .
(2002). 9780769515977 .
finding examples that have much better performance (in terms of 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. In particular, and 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 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 in August, and Hammond, Brown and Liu's report for the Rome Laboratory, published in May. 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 and was selected for its error detection performance. 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. The G.hn standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for ).

CRC32 computation is implemented in hardware as an operation of SSE4.2 instruction set, first introduced in processors' Nehalem microarchitecture.

Polynomial representations of cyclic redundancy checks
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.

CRC-1most hardware; also known as 0x10x10x10x1rowspan="2"
$x + 1$
CRC-3-mobile networks0x30x60x50x5rowspan="2"rowspan="2" 4
$x^3 + x + 1$
CRC-4-ITU G.704, p. 120x30xC0x90x9rowspan="2"
$x^4 + x + 1$
CRC-5-EPCGen 2 RFID (Table 6.12)0x090x120x050x14rowspan="2"
$x^5 + x^3 + 1$
CRC-5-ITUITU-T G.704, p. 90x150x150x0B0x1Arowspan="2"
$x^5 + x^4 + x^2 + 1$
CRC-5-USBUSB token packets0x050x140x090x12rowspan="2"
$x^5 + x^2 + 1$
CRC-6-CDMA2000-Amobile networks0x270x390x330x33
CRC-6-CDMA2000-Bmobile networks0x070x380x310x23
CRC-6-mobile networks0x2F0x3D0x3B0x37rowspan="2"rowspan="2" 112525
$x^6 + x^5 + x^3 + x^2 + x + 1$
CRC-6-ITUITU-T G.704, p. 30x030x300x210x21rowspan="2"
$x^6 + x + 1$
CRC-7telecom systems, ITU-T G.707, ITU-T G.832, , SD0x090x480x110x44rowspan="2"
$x^7 + x^3 + 1$
CRC-7-MVBTrain Communication Network, IEC 60870-50x650x530x270x72
CRC-8DVB-S20xD50xAB0x570xEArowspan="2"rowspan="2" 228585
$x^8 + x^7 + x^6 + x^4 + x^2 + 1$
CRC-8-automotive integration, 0x2F0xF40xE90x97rowspan="2"rowspan="2" 33119119
$x^8 + x^5 + x^3 + x^2 + x + 1$
CRC-8-wireless connectivity0xA70xE50xCB0xD3
CRC-8-ITU-T I.432.1 (02/99); ATM HEC, HEC and cell delineation0x070xE00xC10x83rowspan="2"
$x^8 + x^2 + x + 1$
CRC-8-Dallas/Maxim1-Wire bus0x310x8C0x190x98rowspan="2"
$x^8 + x^5 + x^4 + 1$
CRC-8--Bmobile networks0x490x920x250xA4
CRC-8-SAE J1850AES3; OBD0x1D0xB80x710x8Erowspan="2"
$x^8 + x^4 + x^3 + x^2 + 1$
CRC-8-WCDMAmobile networks
(2005). 9780521828154, Cambridge University Press. .
0x9B0xD90xB30xCDrowspan="2"
$x^8 + x^7 + x^4 + x^3 + x + 1$
CRC-10ATM; ITU-T I.6100x2330x3310x2630x319rowspan="2"
$x^\left\{10\right\} + x^9 + x^5 + x^4 + x + 1$
CRC-10-CDMA2000mobile networks0x3D90x26F0x0DF0x3EC
CRC-10-mobile networks0x1750x2BA0x1750x2BA
CRC-11 (4.2.8 Header CRC (11 bits))0x3850x50E0x21D0x5C2rowspan="2"
$x^\left\{11\right\} + x^9 + x^8 + x^7 + x^2 + 1$
CRC-12telecom systems0x80F0xF010xE030xC07rowspan="2"
$x^\left\{12\right\} + x^\left\{11\right\} + x^3 + x^2 + x + 1$
CRC-12-CDMA2000mobile networks0xF130xC8F0x91F0xF89
CRC-12-mobile networks0xD310x8CB0x1970xE98
CRC-13-BBCTime signal, http://www.freescale.com/files/microcontrollers/doc/app_note/AN1597.pdf0x1CF50x15E70x0BCF0x1E7Arowspan="2"
$x^\left\{13\right\} + x^\left\{12\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^7 + x^6 + x^5 + x^4 + x^2 + 1$
CRC-14-mobile networks0x202D0x2D010x1A030x3016
CRC-15-CAN 0x45990x4CD10x19A30x62CCrowspan="2"
$x^\left\{15\right\} + x^\left\{14\right\} + x^\left\{10\right\} + x^8 + x^7 + x^4 + x^3 + 1$
CRC-15-MPT13270x68150x540B0x28170x740A
CRC-16-ACARS applications0xA02B0xD4050xA80B0xD015
CRC-16-CCITTX.25, V.41, FCS, , , , SD, , many others; known as CRC-CCITT0x10210x84080x8110x8810rowspan="2"
$x^\left\{16\right\} + x^\left\{12\right\} + x^5 + 1$
CRC-16-CDMA2000mobile networks0xC8670xE6130xCC270xE433
CRC-16-DECTcordless telephones0x05890x91A00x23410x82C4rowspan="2"
$x^\left\{16\right\} + x^\left\{10\right\} + x^8 + x^7 + x^3 + 1$
CRC-16-T10-DIF DIF0x8BB70xEDD10xDBA30xC5DBrowspan="2"
$x^\left\{16\right\} + x^\left\{15\right\} + x^\left\{11\right\} + x^\left\{9\right\} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1$
CRC-16-DNPDNP, IEC 870, 0x3D650xA6BC0x4D790x9EB2rowspan="2"
$x^\left\{16\right\} + x^\left\{13\right\} + x^\left\{12\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^8 + x^6 + x^5 + x^2 + 1$
CRC-16-Bisync, , USB, X3.28, SIA DC-07, many others; also known as CRC-16 and CRC-16-ANSI0x80050xA0010x40030xC002rowspan="2"
$x^\left\{16\right\} + x^\left\{15\right\} + x^2 + 1$
CRC-16--Asafety fieldbus0x59350xAC9A0x59350xAC9A
CRC-16-fieldbus networks0x1DCF0xF3B80xE7710x8EE7
Fletcher-16Used in Adler-32 A & B ChecksumsOften confused to be a CRC, but actually a checksum; see Fletcher's checksum
CRC-17-CANCAN FD (3.2.1 DATA FRAME)0x1685B0x1B42D0x1685B0x1B42D
CRC-21-CANCAN FD0x1028990x1322810x0645030x18144C
CRC-240x5D6DCB0xD3B6BA0xA76D750xAEB6E5rowspan="2"
$x^\left\{24\right\} + x^\left\{22\right\} + x^\left\{20\right\} + x^\left\{19\right\} + x^\left\{18\right\} + x^\left\{16\right\} + x^\left\{14\right\} + x^\left\{13\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^8 + x^7 + x^6 + x^3 + x + 1$
$x^\left\{24\right\} + x^\left\{23\right\} + x^\left\{18\right\} + x^\left\{17\right\} + x^\left\{14\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^7 + x^6 + x^5 + x^4 + x^3 + x + 1$
CRC-24-Used in OS-9 RTOS. Residue = 0x800FE3.0x8000630xC600010x8C00030xC00031rowspan="2"rowspan="2" 4483885838388583
$x^\left\{24\right\} + x^\left\{23\right\} + x^6 + x^5 + x + 1$
CRC-300x2030B9C70x38E743010x31CE86030x30185CE3rowspan="2"
$x^\left\{30\right\} + x^\left\{29\right\} + x^\left\{21\right\} + x^\left\{20\right\} + x^\left\{15\right\} + x^\left\{13\right\} + x^\left\{12\right\} + x^\left\{11\right\} + x^\left\{8\right\} + x^\left\{7\right\} + x^\left\{6\right\} + x^\left\{2\right\} + x + 1$
CRC-32 3309 (HDLC), X3.66 (), FIPS PUB 71, FED-STD-1003, ITU-T V.42, ISO/IEC/IEEE 802-3 (), , MPEG-2, , , Bzip2, , PNG, , many others0x04C11DB70xEDB883200xDB7106410x82608EDBrowspan="2"rowspan="2"1012213457911712682974916074294967263
$x^\left\{32\right\} + x^\left\{26\right\} + x^\left\{23\right\} + x^\left\{22\right\} + x^\left\{16\right\} + x^\left\{12\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1$
(Castagnoli), , G.hn payload, SSE4.2, , ext4, Ceph0x1EDC6F410x82F63B780x05EC76F10x8F6E37A0rowspan="2"rowspan="2"68204717752432147483615
$x^\left\{32\right\} + x^\left\{28\right\} + x^\left\{27\right\} + x^\left\{26\right\} + x^\left\{25\right\} + x^\left\{23\right\} + x^\left\{22\right\} + x^\left\{20\right\} + x^\left\{19\right\} + x^\left\{18\right\} + x^\left\{14\right\} + x^\left\{13\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^9 + x^8 + x^6 + 1$
(Koopman {1,3,28})Excellent at Ethernet frame length, poor performance with long files0x741B8CD70xEB31D82E0xD663B05D0xBA0DC66Browspan="2"rowspan="2"24161815216360114663
$x^\left\{32\right\} + x^\left\{30\right\} + x^\left\{29\right\} + x^\left\{28\right\} + x^\left\{26\right\} + x^\left\{20\right\} + x^\left\{19\right\} + x^\left\{17\right\} + x^\left\{16\right\} + x^\left\{15\right\} + x^\left\{11\right\} + x^\left\{10\right\} + x^\left\{7\right\} + x^\left\{6\right\} + x^\left\{4\right\} + x^\left\{2\right\} + x + 1$
CRC-32K2 (Koopman {1,1,30})Excellent at Ethernet frame length, poor performance with long files0x325834990x992C1A4C0x325834990x992C1A4C 316261343273865506
CRC-32Qaviation; 0x814141AB0xD58282810xAB0505030xC0A0A0D5rowspan="2"
$x^\left\{32\right\} + x^\left\{31\right\} + x^\left\{24\right\} + x^\left\{22\right\} + x^\left\{16\right\} + x^\left\{14\right\} + x^\left\{8\right\} + x^\left\{7\right\} + x^\left\{5\right\} + x^\left\{3\right\} + x + 1$
Adler-32 Often confused to be a CRC, but actually a checksum; see Adler-32
CRC-40-GSM control channel ETSI TS 100 909 version 8.9.0 (January 2005), Section 4.1.2 a (Note: MpCRC.html is included with the Matpack compressed software source code, under /html/LibDoc/Crypto)0x00048200090x90004120000x20008240010x8002410004rowspan="2"
$x^\left\{40\right\} + x^\left\{26\right\} + x^\left\{23\right\} + x^\left\{17\right\} + x^3 + 1 = \left(x^\left\{23\right\} + 1\right) \left(x^\left\{17\right\} + x^3 + 1\right)$
CRC-64-ECMA ECMA-182 p. 51, 0x42F0E1EBA9EA36930xC96C5795D7870F420x92D8AF2BAF0E1E850xA17870F5D4F51B49rowspan="2"
$x^\left\{64\right\} + x^\left\{62\right\} + x^\left\{57\right\} + x^\left\{55\right\} + x^\left\{54\right\} + x^\left\{53\right\} + x^\left\{52\right\} + x^\left\{47\right\} + x^\left\{46\right\} + x^\left\{45\right\} + x^\left\{40\right\} + x^\left\{39\right\} + x^\left\{38\right\} + x^\left\{37\right\} + x^\left\{35\right\} + x^\left\{33\right\} +$ $x^\left\{32\right\} + x^\left\{31\right\} + x^\left\{29\right\} + x^\left\{27\right\} + x^\left\{24\right\} + x^\left\{23\right\} + x^\left\{22\right\} + x^\left\{21\right\} + x^\left\{19\right\} + x^\left\{17\right\} + x^\left\{13\right\} + x^\left\{12\right\} + x^\left\{10\right\} + x^9 + x^7 + x^4 + x + 1$
CRC-64-ISOISO 3309 (HDLC), /; considered weak for hashing0x000000000000001B0xD8000000000000000xB0000000000000010x800000000000000Drowspan="2"
$x^\left\{64\right\} + x^4 + x^3 + x + 1$

Implementations

CRC catalogues

• Mathematics of cyclic redundancy checks
• Computation of cyclic redundancy checks
• List of hash functions
• List of checksum algorithms
• Information security
• Simple file verification
• LRC

• Cyclic Redundancy Checks, MathPages, overview of error-detection of different polynomials
• algorithm 4 is used in Linux and Bzip2.
• , Slicing-by-4 and slicing-by-8 algorithms
• — Bitfilters
• — theory, practice, hardware, and software with emphasis on CRC-32.
• Reverse-Engineering a CRC Algorithm
• — includes links to PDFs giving 16 and 32-bit CRC

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