Cyclic redundancy check

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 are so called because the check (data verification) value is a redundancy (it expands the message without adding information) and the algorithm is based on cyclic codes. 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 hash function.

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

Introduction

CRCs are based on the theory of cyclic error-correcting codes. The use of 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. Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of burst errors: 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 will detect a fraction 1/(1 − 2⁻ⁿ) of all longer error bursts.

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 dividend and in which the quotient is discarded and the remainder becomes the result. The important caveat is that the polynomial coefficients 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). The length of the remainder is always less than the length of the generator polynomial, which therefore determines how long the result can be.

In practice, all commonly used CRCs employ the Galois 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 n + 1 terms. In other words, the polynomial has a length of n + 1; its encoding requires n + 1 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 parity bit, is in fact a trivial 1-bit CRC: it uses the generator polynomial x + 1 (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 check 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 the fundamental 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 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 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 linear function with a property that $\operatorname{crc}(x \oplus y \oplus z) = \operatorname{crc}(x) \oplus \operatorname{crc}(y) \oplus \operatorname{crc}(z)$; as a result, even if the CRC is encrypted with a stream cipher that uses XOR as its combining operation (or 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.

Computation

Main article: Computation of cyclic redundancy checks

To 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") 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 x³ + x + 1. The polynomial is written in binary as the coefficients; a 3rd-order polynomial has 4 coefficients (1x³ + 0x² + 1x + 1). In this case, the coefficients are 1, 0, 1 and 1. The result of the calculation is 3 bits long.

Start with the message to be encoded:

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
------------------
                 0 <--- 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'.
	'''
	len_polynomial = len(polynomial_bitstring)
	range_len_polynomial = range(len_polynomial)
	len_input = len(input_bitstring)
	initial_padding = initial_filler  * (len_polynomial - 1)
	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:
			if polynomial_bitstring[i] == input_padded_array[cur_shift + i]:
				input_padded_array[cur_shift + i] = '0'
			else:
				input_padded_array[cur_shift + i] = '1'
	return ''.join(input_padded_array)[len_input:]

Mathematics

Main article: 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), 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 n + 1 terms (the polynomial has a length of 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 polynomials or irreducible polynomials times the factor 1 + x, 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 zero divisors.

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

A 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. The BCH codes 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 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 (n + 1)-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 $x^4 + (0x^3 + 0x^2 + 1x^1 + 1x^0)$ (MSB-first code)
• 0xC = 0b1100, representing $(1x^0 + 1x^1 + 0x^2 + 0x^3) + x^4$ (LSB-first code)
• 0x9 = 0b1001, representing $(1x^4 + 0x^3 + 0x^2 + 1x^1) + x^0$ (Koopman notation)

In the table below they are shown as:

Examples of CRC Representations

Name	Normal	Reversed	Reversed reciprocal
CRC-4	0x3	0xC	0x9

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, 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. 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, 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 Hamming code 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 ITU-T G.hn standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for PHY headers).

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.

CRCs in proprietary protocols might be 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 engineered using straightforward methods.

See Polynomial representations of cyclic redundancy checks for the algebraic representations of the polynomials for the CRCs below.

Name	Uses	Polynomial representations
		Normal	Reversed	Reversed reciprocal
CRC-1	most hardware; also known as parity bit	0x1	0x1	0x1
CRC-3-GSM	mobile networks	0x3	0x6	0x5
CRC-4-ITU	G.704	0x3	0xC	0x9
CRC-5-EPC	Gen 2 RFID	0x09	0x12	0x14
CRC-5-ITU	G.704	0x15	0x15	0x1A
CRC-5-USB	USB token packets	0x05	0x14	0x12
CRC-6-CDMA2000-A	mobile networks	0x27	0x39	0x33
CRC-6-CDMA2000-B	mobile networks	0x07	0x38	0x23
CRC-6-DARC	Data Radio Channel	0x19	0x26	0x2C
CRC-6-GSM	mobile networks	0x2F	0x3D	0x37
CRC-6-ITU	G.704	0x03	0x30	0x21
CRC-7	telecom systems, G.707, G.832, MMC, SD	0x09	0x48	0x44
CRC-7-MVB	Train Communication Network, IEC 60870-5	0x65	0x53	0x72
CRC-8	DVB-S2	0xD5	0xAB	0xEA
CRC-8-AUTOSAR	automotive integration, OpenSafety	0x2F	0xF4	0x97
CRC-8-Bluetooth	wireless connectivity	0xA7	0xE5	0xD3
CRC-8-CCITT	I.432.1; ATM HEC, ISDN HEC and cell delineation	0x07	0xE0	0x83
CRC-8-Dallas/Maxim	1-Wire bus	0x31	0x8C	0x98
CRC-8-DARC	Data Radio Channel	0x39	0x9C	0x9C
CRC-8-GSM-B	mobile networks	0x49	0x92	0xA4
CRC-8-SAE J1850	AES3	0x1D	0xB8	0x8E
CRC-8-WCDMA	mobile networks	0x9B	0xD9	0xCD
CRC-10	ATM; I.610	0x233	0x331	0x319
CRC-10-CDMA2000	mobile networks	0x3D9	0x26F	0x3EC
CRC-10-GSM	mobile networks	0x175	0x2BA	0x2BA
CRC-11	FlexRay	0x385	0x50E	0x5C2
CRC-12	telecom systems	0x80F	0xF01	0xC07
CRC-12-CDMA2000	mobile networks	0xF13	0xC8F	0xF89
CRC-12-GSM	mobile networks	0xD31	0x8CB	0xE98
CRC-13-BBC	Time signal, Radio teleswitch	0x1CF5	0x15E7	0x1E7A
CRC-14-DARC	Data Radio Channel	0x0805	0x2804	0x2402
CRC-14-GSM	mobile networks	0x202D	0x2D01	0x3016
CRC-15-CAN		0x4599	0x4CD1	0x62CC
CRC-15-MPT1327		0x6815	0x540B	0x740A
CRC-16-Chakravarty	Optimal for payloads ≤64 bits	0x2F15	0xA8F4	0x978A
CRC-16-ARINC	ACARS applications	0xA02B	0xD405	0xD015
CRC-16-CCITT	X.25, V.41, HDLC FCS, XMODEM, Bluetooth, PACTOR, SD, DigRF, many others; known as CRC-CCITT	0x1021	0x8408	0x8810
CRC-16-CDMA2000	mobile networks	0xC867	0xE613	0xE433
CRC-16-DECT	cordless telephones	0x0589	0x91A0	0x82C4
CRC-16-T10-DIF	SCSI DIF	0x8BB7	0xEDD1	0xC5DB
CRC-16-DNP	DNP, IEC 870, M-Bus	0x3D65	0xA6BC	0x9EB2
CRC-16-IBM	Bisync, Modbus, USB, ANSI X3.28, SIA DC-07, many others; also known as CRC-16 and CRC-16-ANSI	0x8005	0xA001	0xC002
CRC-16-OpenSafety-A	safety fieldbus	0x5935	0xAC9A	0xAC9A
CRC-16-OpenSafety-B	safety fieldbus	0x755B	0xDAAE	0xBAAD
CRC-16-Profibus	fieldbus networks	0x1DCF	0xF3B8	0x8EE7
CRC-17-CAN	CAN FD	0x1685B	0x1B42D	0x1B42D
CRC-21-CAN	CAN FD	0x102899	0x132281	0x18144C
CRC-24	FlexRay	0x5D6DCB	0xD3B6BA	0xAEB6E5
CRC-24-Radix-64	OpenPGP, RTCM104v3	0x864CFB	0xDF3261	0xC3267D
CRC-30	CDMA	0x2030B9C7	0x38E74301	0x30185CE3
CRC-32	HDLC, ANSI X3.66, ITU-T V.42, Ethernet, Serial ATA, MPEG-2, PKZIP, Gzip, Bzip2, PNG, many others	0x04C11DB7	0xEDB88320	0x82608EDB
CRC-32C (Castagnoli)	iSCSI, SCTP, G.hn payload, SSE4.2, Btrfs, ext4, Ceph	0x1EDC6F41	0x82F63B78	0x8F6E37A0
CRC-32K (Koopman {1,3,28})		0x741B8CD7	0xEB31D82E	0xBA0DC66B
CRC-32K2 (Koopman {1,1,30})		0x32583499	0x992C1A4C	0x992C1A4C
CRC-32Q	aviation; AIXM	0x814141AB	0xD5828281	0xC0A0A0D5
CRC-40-GSM	GSM control channel	0x0004820009	0x9000412000	0x8002410004
CRC-64-ECMA	ECMA-182, XZ Utils	0x42F0E1EBA9EA3693	0xC96C5795D7870F42	0xA17870F5D4F51B49
CRC-64-ISO	HDLC, Swiss-Prot/TrEMBL; considered weak for hashing	0x000000000000001B	0xD800000000000000	0x800000000000000D

Implementations

• Implementation of CRC32 in Gnuradio;
• C class code for CRC checksum calculation with many different CRCs to choose from

See also

• Mathematics of cyclic redundancy checks
• Computation of cyclic redundancy checks
• Polynomial representations of cyclic redundancy checks
• Error detection and correction
• List of hash functions
• Information security
• Simple file verification
• LRC
• cksum
• Header Error Correction

External links

• 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 Hamming distances