{"history":[],"article":{"id":"98b75345-bc25-41ca-8a5d-ed401546cb90","title":"Cyclic redundancy check","slug":"cyclic-redundancy-check","content":":::summary\n:::\n\n# \nA **cyclic redundancy check** (**CRC**) is an [error-detecting code](https://bliptext.com/articles/error-correcting-code) commonly used in digital [networks](https://bliptext.com/articles/telecommunications-network) and storage devices to detect accidental changes to digital data.\nBlocks of data entering these systems get a short check value attached, based on the remainder of a [polynomial division](https://bliptext.com/articles/polynomial-long-division) of their contents.\nOn retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption.\nCRCs can be used for [error correction](https://bliptext.com/articles/error-detection-and-correction) (see [bitfilters](https://bliptext.com/articles/mathematics-of-cyclic-redundancy-checks)).\nCRCs are so called because the check (data verification) value is a redundancy (it expands the message without adding [information](https://bliptext.com/articles/entropy-information-theory)) and the [algorithm](https://bliptext.com/articles/algorithm) is based on cyclic codes.\nCRCs are popular because they are simple to implement in binary [hardware](https://bliptext.com/articles/computer-hardware), easy to analyze mathematically, and particularly good at detecting common errors caused by [noise](https://bliptext.com/articles/noise-electronics) in transmission channels.\nBecause the check value has a fixed length, the [function](https://bliptext.com/articles/function-mathematics) that generates it is occasionally used as a hash [function](https://bliptext.com/articles/function-mathematics).\n\n# Introduction\nCRCs are based on the theory of [cyclic](https://bliptext.com/articles/cyclic-code) [error-correcting codes](https://bliptext.com/articles/error-correcting-code).\nThe use of [systematic](https://bliptext.com/articles/systematic-code) 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](https://bliptext.com/articles/w-wesley-peterson) in 1961.\nCyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of [burst errors](https://bliptext.com/articles/burst-error): contiguous sequences of erroneous data symbols in messages.\nThis is important because burst errors are common transmission errors in many [communication channels](https://bliptext.com/articles/communication-channel), including magnetic and optical storage devices.\nTypically 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 (1 − 2−n).\nSpecification of a CRC code requires definition of a so-called [generator polynomial](https://bliptext.com/articles/generator-polynomial).\nThis polynomial becomes the [divisor](https://bliptext.com/articles/divisor) in a [polynomial long division](https://bliptext.com/articles/polynomial-long-division), which takes the message as the [dividend](https://bliptext.com/articles/division-mathematics) and in which the [quotient](https://bliptext.com/articles/quotient) is discarded and the [remainder](https://bliptext.com/articles/remainder) becomes the result.\nThe important caveat is that the polynomial [coefficients](https://bliptext.com/articles/coefficient) are calculated according to the arithmetic of a [finite field](https://bliptext.com/articles/finite-field), so the addition operation can always be performed bitwise-parallel (there is no carry between digits).\nIn practice, all commonly used CRCs employ the finite field of two elements, [GF(2)](https://bliptext.com/articles/gf-2).\nThe two elements are usually called 0 and 1, comfortably matching computer architecture.\nA CRC is called an n-bit CRC when its check value is n bits long.\nFor a given n, multiple CRCs are possible, each with a different polynomial.\nSuch a polynomial has highest degree n, which means it has n + 1 terms.\nIn other words, the polynomial has a length of n + 1; its encoding requires n + 1 bits.\nNote that most polynomial specifications either drop the [MSb](https://bliptext.com/articles/most-significant-bit) or [LSb](https://bliptext.com/articles/least-significant-bit), since they are always 1.\nThe CRC and associated polynomial typically have a name of the form CRC-n-XXX as in the [table](https://bliptext.com/articles/) below.\nThe simplest error-detection system, the [parity bit](https://bliptext.com/articles/parity-bit), is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), and has the name CRC-1.\n\n# Application\nA 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.\nWhen 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.\nIf the CRC values do not match, then the block contains a data error.\nThe device may take corrective action, such as rereading the block or requesting that it be sent again.\nOtherwise, 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).\n\n# Data integrity\nCRCs are specifically designed to protect against common types of errors on communication channels, where they can provide quick and reasonable assurance of the [integrity](https://bliptext.com/articles/data-integrity) of messages delivered.\nHowever, they are not suitable for protecting against intentional alteration of data.\nFirstly, as there is no authentication, an attacker can edit a message and recompute the CRC without the substitution being detected.\nWhen stored alongside the data, CRCs and cryptographic hash functions by themselves do not protect against intentional modification of data.\nAny application that requires protection against such attacks must use cryptographic authentication mechanisms, such as [message authentication codes](https://bliptext.com/articles/message-authentication-code) or [digital signatures](https://bliptext.com/articles/digital-signatures) (which are commonly based on [cryptographic hash](https://bliptext.com/articles/cryptographic-hash) functions).\nSecondly, unlike cryptographic hash functions, CRC is an easily reversible function, which makes it unsuitable for use in digital signatures.\nThirdly, CRC satisfies a relation similar to that of a [linear function](https://bliptext.com/articles/linear-function) (or more accurately, an [affine function](https://bliptext.com/articles/affine-function)):\nwhere c depends on the length of x and y. This can be also stated as follows, where x, y and z have the same length\nas a result, even if the CRC is encrypted with a [stream cipher](https://bliptext.com/articles/stream-cipher) that uses [XOR](https://bliptext.com/articles/exclusive-or) as its combining operation (or [mode](https://bliptext.com/articles/block-cipher-modes-of-operation) of [block cipher](https://bliptext.com/articles/block-cipher) which effectively turns it into a [stream cipher](https://bliptext.com/articles/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](https://bliptext.com/articles/wired-equivalent-privacy) (WEP) protocol.\n\n# Computation\nTo compute an n-bit binary CRC, line the bits representing the input in a row, and position the (n + 1)-bit pattern representing the CRC's divisor (called a \"[polynomial](https://bliptext.com/articles/polynomial)\") underneath the left end of the row.\nIn this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial x3 + x + 1.\nThe polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients (1x3 + 0x2 + 1x + 1).\nIn this case, the coefficients are 1, 0, 1 and 1.\nThe result of the calculation is 3 bits long, which is why it is called a 3-bit CRC.\nHowever, you need 4 bits to explicitly state the polynomial.\nStart with the message to be encoded:\n11010011101100\nThis is first padded with zeros corresponding to the bit length n of the CRC.\nThis is done so that the resulting code word is in [systematic](https://bliptext.com/articles/systematic-code) form.\nHere is the first calculation for computing a 3-bit CRC:\n11010011101100 000 <--- input right padded by 3 bits\n1011 <--- divisor (4 bits) = x³ + x + 1\n-- 01100011101100 000 <--- result\nThe algorithm acts on the bits directly above the divisor in each step.\nThe result for that iteration is the bitwise XOR of the polynomial divisor with the bits above it.\nThe bits not above the divisor are simply copied directly below for that step.\nThe 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.\nHere is the entire calculation:\n11010011101100 000 <--- input right padded by 3 bits\n1011 <--- divisor\n01100011101100 000 <--- result (note the first four bits are the XOR with the divisor beneath, the rest of the bits are unchanged)\n1011 <--- divisor ...\n00111011101100 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)\n1011 (in other words, it doesn't necessarily move one bit per iteration)\n00000000110100 000 1011 00000000011000 000 1011 00000000001110 000 1011 00000000000101 000 101 1 - 00000000000000 100 <--- remainder (3 bits).\nDivision algorithm stops here as dividend is equal to zero.\nSince 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.\nThese 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).\nThe 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.\nThe remainder should equal zero if there are no detectable errors.\n11010011101100 100 <--- input with check value\n1011 <--- divisor\n01100011101100 100 <--- result\n1011 <--- divisor ...\n00111011101100 100\n......\n00000000001110 100 1011 00000000000101 100 101 1 -- 00000000000000 000 <--- remainder\nThe following [Python](https://bliptext.com/articles/python-programming-language) 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.\nNote that this code works with string inputs rather than raw numbers:\n\n# Mathematics\nMathematical analysis of this division-like process reveals how to select a divisor that guarantees good error-detection properties.\nIn 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)](https://bliptext.com/articles/gf-2) (the integers modulo 2, i.e. either a zero or a one), instead of more familiar numbers.\nThe set of binary polynomials is a mathematical [ring](https://bliptext.com/articles/ring-mathematics).\n\n## Designing polynomials\nThe selection of the generator polynomial is the most important part of implementing the CRC algorithm.\nThe polynomial must be chosen to maximize the error-detecting capabilities while minimizing overall collision probabilities.\nThe 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.\nThe most commonly used polynomial lengths are 9 bits (CRC-8), 17 bits (CRC-16), 33 bits (CRC-32), and 65 bits (CRC-64).\nA CRC is called an n-bit CRC when its check value is n-bits.\nFor a given n, multiple CRCs are possible, each with a different polynomial.\nSuch a polynomial has highest degree n, and hence n + 1 terms (the polynomial has a length of n + 1).\nThe remainder has length n.\nThe CRC has a name of the form CRC-n-XXX.\nThe 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.\nA common misconception is that the \"best\" CRC polynomials are derived from either [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial) or [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial) times the factor 1 + x, which adds to the code the ability to detect all errors affecting an odd number of bits.\nIn reality, all the factors described above should enter into the selection of the polynomial and may lead to a reducible polynomial.\nHowever, choosing a reducible polynomial will result in a certain proportion of missed errors, due to the quotient ring having [zero divisors](https://bliptext.com/articles/zero-divisor).\nThe advantage of choosing a [primitive polynomial](https://bliptext.com/articles/primitive-polynomial-field-theory) 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](https://bliptext.com/articles/syndrome-decoding)) and therefore, since the remainder is a linear function of the block, the code can detect all 2-bit errors within that block length.\nIf r is the degree of the primitive generator polynomial, then the maximal total block length is 2 ^ {r} - 1, and the associated code is able to detect any single-bit or double-bit errors.\nWe can improve this situation.\nIf we use the generator polynomial, where p is a primitive polynomial of degree r - 1, then the maximal total block length is , and the code is able to detect single, double, triple and any odd number of errors.\nA polynomial g(x) that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power.\nThe [BCH codes](https://bliptext.com/articles/bch-code) are a powerful class of such polynomials.\nThey subsume the two examples above.\nRegardless 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.\nThese patterns are called \"error bursts\".\n\n# Specification\nThe concept of the CRC as an error-detecting code gets complicated when an implementer or standards committee uses it to design a practical system.\nHere are some of the complications:\nThese 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.\nIn each case, one term is omitted. So the polynomial x^4 + x + 1 may be transcribed as:\nIn the table below they are shown as:\n\n# Obfuscation\nCRCs in [proprietary protocols](https://bliptext.com/articles/proprietary-protocol) might be [obfuscated](https://bliptext.com/articles/obfuscation) 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](https://bliptext.com/articles/reverse-engineering) using straightforward methods.\n\n# Standards and common use\nNumerous varieties of cyclic redundancy checks have been incorporated into [technical standards](https://bliptext.com/articles/technical-standard).\nBy 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.\nThe number of distinct CRCs in use has confused developers, a situation which authors have sought to address.\nThere are three polynomials reported for CRC-12, twenty-two conflicting definitions of CRC-16, and seven of CRC-32.\nThe polynomials commonly applied are not the most efficient ones possible.\nSince 1993, Koopman, Castagnoli and others have surveyed the space of polynomials between 3 and 64 bits in size, finding examples that have much better performance (in terms of [Hamming distance](https://bliptext.com/articles/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.\nIn particular, [iSCSI](https://bliptext.com/articles/iscsi) and [SCTP](https://bliptext.com/articles/sctp) have adopted one of the findings of this research, the CRC-32C (Castagnoli) polynomial.\nThe 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](https://bliptext.com/articles/rome-laboratory) and the Air Force Electronic Systems Division by Joseph Hammond, James Brown and Shyan-Shiang Liu of the [Georgia Institute of Technology](https://bliptext.com/articles/georgia-institute-of-technology) and Kenneth Brayer of the [Mitre Corporation](https://bliptext.com/articles/mitre-corporation).\nThe 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](https://bliptext.com/articles/dtic) in August, and Hammond, Brown and Liu's report for the Rome Laboratory, published in May.\nBoth reports contained contributions from the other team.\nDuring 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](https://bliptext.com/articles/hamming-code) and was selected for its error detection performance.\nEven 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.\nThe [ITU-T](https://bliptext.com/articles/itu-t) [G.hn](https://bliptext.com/articles/g-hn) standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for [PHY headers](https://bliptext.com/articles/physical-layer)).\nCRC-32C computation is implemented in hardware as an operation (CRC32) of [SSE4.2](https://bliptext.com/articles/sse4) instruction set, first introduced in [Intel](https://bliptext.com/articles/intel) processors' [Nehalem](https://bliptext.com/articles/nehalem-microarchitecture) microarchitecture.\n[ARM](https://bliptext.com/articles/arm-architecture) [AArch64](https://bliptext.com/articles/aarch64) architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations.\n\n# Polynomial representations\nThe table below lists only the polynomials of the various algorithms in use.\nVariations of a particular protocol can impose pre-inversion, post-inversion and reversed bit ordering as described above.\nFor example, the CRC32 used in Gzip and Bzip2 use the same polynomial, but Gzip employs reversed bit ordering, while Bzip2 does not.\nNote that even parity polynomials in [GF(2)](https://bliptext.com/articles/gf-2) with degree greater than 1 are never primitive.\nEven parity polynomial marked as primitive in this table represent a primitive polynomial multiplied by.\nThe most significant bit of a polynomial is always 1, and is not shown in the hex representations.\n\n## Implementations\n\n## CRC catalogues","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T16:39:30.120Z","updated_at":"2025-01-25T16:39:30.120Z","search_vector":"'check':3 'cyclic':1 'redund':2","search_content":" \n \n\n# \nA **cyclic redundancy check** (**CRC**) is an [error-detecting code](https://bliptext.com/articles/error-correcting-code) commonly used in digital [networks](https://bliptext.com/articles/telecommunications-network) and storage devices to detect accidental changes to digital data.\nBlocks of data entering these systems get a short check value attached, based on the remainder of a [polynomial division](https://bliptext.com/articles/polynomial-long-division) of their contents.\nOn retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption.\nCRCs can be used for [error correction](https://bliptext.com/articles/error-detection-and-correction) (see [bitfilters](https://bliptext.com/articles/mathematics-of-cyclic-redundancy-checks)).\nCRCs are so called because the check (data verification) value is a redundancy (it expands the message without adding [information](https://bliptext.com/articles/entropy-information-theory)) and the [algorithm](https://bliptext.com/articles/algorithm) is based on cyclic codes.\nCRCs are popular because they are simple to implement in binary [hardware](https://bliptext.com/articles/computer-hardware), easy to analyze mathematically, and particularly good at detecting common errors caused by [noise](https://bliptext.com/articles/noise-electronics) in transmission channels.\nBecause the check value has a fixed length, the [function](https://bliptext.com/articles/function-mathematics) that generates it is occasionally used as a hash [function](https://bliptext.com/articles/function-mathematics).\n\n# Introduction\nCRCs are based on the theory of [cyclic](https://bliptext.com/articles/cyclic-code) [error-correcting codes](https://bliptext.com/articles/error-correcting-code).\nThe use of [systematic](https://bliptext.com/articles/systematic-code) 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](https://bliptext.com/articles/w-wesley-peterson) in 1961.\nCyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of [burst errors](https://bliptext.com/articles/burst-error): contiguous sequences of erroneous data symbols in messages.\nThis is important because burst errors are common transmission errors in many [communication channels](https://bliptext.com/articles/communication-channel), including magnetic and optical storage devices.\nTypically 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 (1 − 2−n).\nSpecification of a CRC code requires definition of a so-called [generator polynomial](https://bliptext.com/articles/generator-polynomial).\nThis polynomial becomes the [divisor](https://bliptext.com/articles/divisor) in a [polynomial long division](https://bliptext.com/articles/polynomial-long-division), which takes the message as the [dividend](https://bliptext.com/articles/division-mathematics) and in which the [quotient](https://bliptext.com/articles/quotient) is discarded and the [remainder](https://bliptext.com/articles/remainder) becomes the result.\nThe important caveat is that the polynomial [coefficients](https://bliptext.com/articles/coefficient) are calculated according to the arithmetic of a [finite field](https://bliptext.com/articles/finite-field), so the addition operation can always be performed bitwise-parallel (there is no carry between digits).\nIn practice, all commonly used CRCs employ the finite field of two elements, [GF(2)](https://bliptext.com/articles/gf-2).\nThe two elements are usually called 0 and 1, comfortably matching computer architecture.\nA CRC is called an n-bit CRC when its check value is n bits long.\nFor a given n, multiple CRCs are possible, each with a different polynomial.\nSuch a polynomial has highest degree n, which means it has n + 1 terms.\nIn other words, the polynomial has a length of n + 1; its encoding requires n + 1 bits.\nNote that most polynomial specifications either drop the [MSb](https://bliptext.com/articles/most-significant-bit) or [LSb](https://bliptext.com/articles/least-significant-bit), since they are always 1.\nThe CRC and associated polynomial typically have a name of the form CRC-n-XXX as in the [table](https://bliptext.com/articles/) below.\nThe simplest error-detection system, the [parity bit](https://bliptext.com/articles/parity-bit), is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), and has the name CRC-1.\n\n# Application\nA 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.\nWhen 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.\nIf the CRC values do not match, then the block contains a data error.\nThe device may take corrective action, such as rereading the block or requesting that it be sent again.\nOtherwise, 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).\n\n# Data integrity\nCRCs are specifically designed to protect against common types of errors on communication channels, where they can provide quick and reasonable assurance of the [integrity](https://bliptext.com/articles/data-integrity) of messages delivered.\nHowever, they are not suitable for protecting against intentional alteration of data.\nFirstly, as there is no authentication, an attacker can edit a message and recompute the CRC without the substitution being detected.\nWhen stored alongside the data, CRCs and cryptographic hash functions by themselves do not protect against intentional modification of data.\nAny application that requires protection against such attacks must use cryptographic authentication mechanisms, such as [message authentication codes](https://bliptext.com/articles/message-authentication-code) or [digital signatures](https://bliptext.com/articles/digital-signatures) (which are commonly based on [cryptographic hash](https://bliptext.com/articles/cryptographic-hash) functions).\nSecondly, unlike cryptographic hash functions, CRC is an easily reversible function, which makes it unsuitable for use in digital signatures.\nThirdly, CRC satisfies a relation similar to that of a [linear function](https://bliptext.com/articles/linear-function) (or more accurately, an [affine function](https://bliptext.com/articles/affine-function)):\nwhere c depends on the length of x and y. This can be also stated as follows, where x, y and z have the same length\nas a result, even if the CRC is encrypted with a [stream cipher](https://bliptext.com/articles/stream-cipher) that uses [XOR](https://bliptext.com/articles/exclusive-or) as its combining operation (or [mode](https://bliptext.com/articles/block-cipher-modes-of-operation) of [block cipher](https://bliptext.com/articles/block-cipher) which effectively turns it into a [stream cipher](https://bliptext.com/articles/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](https://bliptext.com/articles/wired-equivalent-privacy) (WEP) protocol.\n\n# Computation\nTo compute an n-bit binary CRC, line the bits representing the input in a row, and position the (n + 1)-bit pattern representing the CRC's divisor (called a \"[polynomial](https://bliptext.com/articles/polynomial)\") underneath the left end of the row.\nIn this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial x3 + x + 1.\nThe polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients (1x3 + 0x2 + 1x + 1).\nIn this case, the coefficients are 1, 0, 1 and 1.\nThe result of the calculation is 3 bits long, which is why it is called a 3-bit CRC.\nHowever, you need 4 bits to explicitly state the polynomial.\nStart with the message to be encoded:\n11010011101100\nThis is first padded with zeros corresponding to the bit length n of the CRC.\nThis is done so that the resulting code word is in [systematic](https://bliptext.com/articles/systematic-code) form.\nHere is the first calculation for computing a 3-bit CRC:\n11010011101100 000 <--- input right padded by 3 bits\n1011 <--- divisor (4 bits) = x³ + x + 1\n-- 01100011101100 000 <--- result\nThe algorithm acts on the bits directly above the divisor in each step.\nThe result for that iteration is the bitwise XOR of the polynomial divisor with the bits above it.\nThe bits not above the divisor are simply copied directly below for that step.\nThe 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.\nHere is the entire calculation:\n11010011101100 000 <--- input right padded by 3 bits\n1011 <--- divisor\n01100011101100 000 <--- result (note the first four bits are the XOR with the divisor beneath, the rest of the bits are unchanged)\n1011 <--- divisor ...\n00111011101100 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)\n1011 (in other words, it doesn't necessarily move one bit per iteration)\n00000000110100 000 1011 00000000011000 000 1011 00000000001110 000 1011 00000000000101 000 101 1 - 00000000000000 100 <--- remainder (3 bits).\nDivision algorithm stops here as dividend is equal to zero.\nSince 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.\nThese 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).\nThe 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.\nThe remainder should equal zero if there are no detectable errors.\n11010011101100 100 <--- input with check value\n1011 <--- divisor\n01100011101100 100 <--- result\n1011 <--- divisor ...\n00111011101100 100\n......\n00000000001110 100 1011 00000000000101 100 101 1 -- 00000000000000 000 <--- remainder\nThe following [Python](https://bliptext.com/articles/python-programming-language) 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.\nNote that this code works with string inputs rather than raw numbers:\n\n# Mathematics\nMathematical analysis of this division-like process reveals how to select a divisor that guarantees good error-detection properties.\nIn 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)](https://bliptext.com/articles/gf-2) (the integers modulo 2, i.e. either a zero or a one), instead of more familiar numbers.\nThe set of binary polynomials is a mathematical [ring](https://bliptext.com/articles/ring-mathematics).\n\n## Designing polynomials\nThe selection of the generator polynomial is the most important part of implementing the CRC algorithm.\nThe polynomial must be chosen to maximize the error-detecting capabilities while minimizing overall collision probabilities.\nThe 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.\nThe most commonly used polynomial lengths are 9 bits (CRC-8), 17 bits (CRC-16), 33 bits (CRC-32), and 65 bits (CRC-64).\nA CRC is called an n-bit CRC when its check value is n-bits.\nFor a given n, multiple CRCs are possible, each with a different polynomial.\nSuch a polynomial has highest degree n, and hence n + 1 terms (the polynomial has a length of n + 1).\nThe remainder has length n.\nThe CRC has a name of the form CRC-n-XXX.\nThe 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.\nA common misconception is that the \"best\" CRC polynomials are derived from either [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial) or [irreducible polynomials](https://bliptext.com/articles/irreducible-polynomial) times the factor 1 + x, which adds to the code the ability to detect all errors affecting an odd number of bits.\nIn reality, all the factors described above should enter into the selection of the polynomial and may lead to a reducible polynomial.\nHowever, choosing a reducible polynomial will result in a certain proportion of missed errors, due to the quotient ring having [zero divisors](https://bliptext.com/articles/zero-divisor).\nThe advantage of choosing a [primitive polynomial](https://bliptext.com/articles/primitive-polynomial-field-theory) 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](https://bliptext.com/articles/syndrome-decoding)) and therefore, since the remainder is a linear function of the block, the code can detect all 2-bit errors within that block length.\nIf r is the degree of the primitive generator polynomial, then the maximal total block length is 2 ^ {r} - 1, and the associated code is able to detect any single-bit or double-bit errors.\nWe can improve this situation.\nIf we use the generator polynomial, where p is a primitive polynomial of degree r - 1, then the maximal total block length is , and the code is able to detect single, double, triple and any odd number of errors.\nA polynomial g(x) that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power.\nThe [BCH codes](https://bliptext.com/articles/bch-code) are a powerful class of such polynomials.\nThey subsume the two examples above.\nRegardless 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.\nThese patterns are called \"error bursts\".\n\n# Specification\nThe concept of the CRC as an error-detecting code gets complicated when an implementer or standards committee uses it to design a practical system.\nHere are some of the complications:\nThese 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.\nIn each case, one term is omitted. So the polynomial x^4 + x + 1 may be transcribed as:\nIn the table below they are shown as:\n\n# Obfuscation\nCRCs in [proprietary protocols](https://bliptext.com/articles/proprietary-protocol) might be [obfuscated](https://bliptext.com/articles/obfuscation) 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](https://bliptext.com/articles/reverse-engineering) using straightforward methods.\n\n# Standards and common use\nNumerous varieties of cyclic redundancy checks have been incorporated into [technical standards](https://bliptext.com/articles/technical-standard).\nBy 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.\nThe number of distinct CRCs in use has confused developers, a situation which authors have sought to address.\nThere are three polynomials reported for CRC-12, twenty-two conflicting definitions of CRC-16, and seven of CRC-32.\nThe polynomials commonly applied are not the most efficient ones possible.\nSince 1993, Koopman, Castagnoli and others have surveyed the space of polynomials between 3 and 64 bits in size, finding examples that have much better performance (in terms of [Hamming distance](https://bliptext.com/articles/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.\nIn particular, [iSCSI](https://bliptext.com/articles/iscsi) and [SCTP](https://bliptext.com/articles/sctp) have adopted one of the findings of this research, the CRC-32C (Castagnoli) polynomial.\nThe 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](https://bliptext.com/articles/rome-laboratory) and the Air Force Electronic Systems Division by Joseph Hammond, James Brown and Shyan-Shiang Liu of the [Georgia Institute of Technology](https://bliptext.com/articles/georgia-institute-of-technology) and Kenneth Brayer of the [Mitre Corporation](https://bliptext.com/articles/mitre-corporation).\nThe 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](https://bliptext.com/articles/dtic) in August, and Hammond, Brown and Liu's report for the Rome Laboratory, published in May.\nBoth reports contained contributions from the other team.\nDuring 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](https://bliptext.com/articles/hamming-code) and was selected for its error detection performance.\nEven 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.\nThe [ITU-T](https://bliptext.com/articles/itu-t) [G.hn](https://bliptext.com/articles/g-hn) standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for [PHY headers](https://bliptext.com/articles/physical-layer)).\nCRC-32C computation is implemented in hardware as an operation (CRC32) of [SSE4.2](https://bliptext.com/articles/sse4) instruction set, first introduced in [Intel](https://bliptext.com/articles/intel) processors' [Nehalem](https://bliptext.com/articles/nehalem-microarchitecture) microarchitecture.\n[ARM](https://bliptext.com/articles/arm-architecture) [AArch64](https://bliptext.com/articles/aarch64) architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations.\n\n# Polynomial representations\nThe table below lists only the polynomials of the various algorithms in use.\nVariations of a particular protocol can impose pre-inversion, post-inversion and reversed bit ordering as described above.\nFor example, the CRC32 used in Gzip and Bzip2 use the same polynomial, but Gzip employs reversed bit ordering, while Bzip2 does not.\nNote that even parity polynomials in [GF(2)](https://bliptext.com/articles/gf-2) with degree greater than 1 are never primitive.\nEven parity polynomial marked as primitive in this table represent a primitive polynomial multiplied by.\nThe most significant bit of a polynomial is always 1, and is not shown in the hex representations.\n\n## Implementations\n\n## CRC catalogues","revisionCount":0},"hasMore":false,"nextCursor":null}

History