Cyclic redundancy check

A cyclic redundancy check (CRC) or polynomial code checksum is a non-secure hash function designed to detect accidental changes to raw computer data, and is commonly used in digital networks and storage devices such as hard disk drives. A CRC-enabled device calculates a short, fixed-length binary sequence, known as the CRC code or just CRC, for each block of data and sends or stores them both together. When a block is read or received the device repeats the calculation; if the new CRC does not match the one calculated earlier, then the block contains a data error and the device may take corrective action such as rereading or requesting the block be sent again, otherwise the data is assumed to be error free (though, in fact, it may contain undetected errors).

CRCs are so called because the check (data verification) code is a redundancy (it adds zero information) and the algorithm is based on cyclic codes. The term CRC may refer to the check code or to the function that calculates it, which accepts data streams of any length as input but always outputs a fixed-length code. CRCs are popular because they are simple to implement in binary hardware, are easy to analyse mathematically, and are particularly good at detecting common errors caused by noise in transmission channels. The CRC was invented by W. Wesley Peterson, and published in his 1961 paper. The earliest known appearance of the 32-bit polynomial most commonly used by standards bodies was in a 1975 paper written by the Georgia Institute of Technology on behalf of the Rome Laboratory.

Introduction

A CRC is an error-detecting code. Its computation resembles a polynomial long division operation in which the quotient is discarded and the remainder becomes the result, with the important distinction that the polynomial coefficients are calculated according to the carry-less arithmetic of a finite field. The length of the remainder is always less than the length of the divisor (called the generator polynomial), which therefore determines how long the result can be. The definition of a particular CRC specifies the divisor to be used, among other things.

Although CRCs can be constructed using any finite field, all commonly used CRCs employ the finite field GF(2). This is the field of two elements, usually called 0 and 1, comfortably matching computer architecture. The rest of this article will discuss only these binary CRCs, but the principles are more general.

An important reason for the popularity of CRCs for detecting the accidental alteration of data is their efficiency guarantee. Typically, an n-bit CRC, applied to a data block of arbitrary length, will detect any single error burst not longer than n bits (in other words, any single alteration that spans no more than n bits of the data), and will detect a fraction 1−2⁻ⁿ of all longer error bursts. Errors in both data transmission channels and magnetic storage media tend to be distributed non-randomly (i.e. are "bursty"), making CRCs' properties more useful than alternative schemes such as multiple parity checks.

The simplest error-detection system, the parity bit, is in fact a trivial 1-bit CRC: it uses the generator polynomial x+1.

CRCs and 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 recalculate the CRC without the substitution being detected. This is even the case when the CRC is encrypted—this was one of the design flaws of the WEP protocol. Secondly, the linear properties of CRC codes allow an attacker even to keep the CRC unchanged while modifying parts of the message—this also makes calculating the CRC adjustment for small changes more efficient. Nonetheless, it is still often falsely assumed that when a message and its CRC are received from an open channel and the CRC matches the message's calculated CRC then the message cannot have been altered in transit.

Cryptographic hash functions can provide stronger integrity guarantees in that they do not rely on specific error pattern assumptions. However, they are much slower than CRCs, and are therefore commonly used to protect off-line data, such as files on servers or databases.

Both 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.

Computation of CRC

Main article: Computation of CRC

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. Here is the first calculation for computing a 3-bit CRC:

11010011101100 <--- input
1011           <--- divisor (4 bits)
--------------
01100011101100 <--- result

If the input bit above the leftmost divisor bit is 0, do nothing and move the divisor to the right by one bit. If the input bit above the leftmost divisor bit is 1, the divisor is XORed into the input (in other words, the input bit above each 1-bit in the divisor is toggled). 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 last calculation:

00000000001110 <--- result of previous step
          1011 <--- divisor
--------------
00000000000101 <--- remainder (3 bits)

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).

Mathematics of CRC

Main article: Mathematics of CRC

Mathematical analysis of this division-like process reveals how to pick a divisor that guarantees good error-detection properties. In this analysis, the digits of the bit strings are thought of 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. This binary polynomial is treated as a ring. A ring is, loosely speaking, a set of elements somewhat like numbers, that can be operated on by an operation that somewhat resembles addition and another operation that somewhat resembles multiplication, these operations possessing many of the familiar arithmetic properties of commutativity, associativity, and distributivity. Ring theory is part of Abstract Algebra.

Specification of CRC

The concept of the CRC as an error-detecting code gets complicated when an implementer or standards committee turns it into 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 CRC unchanged.
• Sometimes an implementation appends n 0-bits (n being the size of the CRC) to the bitstream to be checked before the polynomial division occurs. This has the convenience that the CRC of the original bitstream with the CRC appended is exactly zero, so the CRC can be checked simply by performing the polynomial division on the expanded bitstream and comparing the remainder with zero.
• 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 or the most-significant byte. For example, some 16-bit CRC schemes swap the bytes of the CRC.
• 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.

Commonly used and standardized CRCs

Numerous varieties of cyclic redundancy check have been incorporated into technical standards. By no means does one algorithm, or one of each degree, suit every purpose; Koopman and Chakrabarty recommend selecting a polynomial according to the application requirements and the expected distribution of message lengths. The number of distinct CRCs in use have however led to confusion among developers which authors have sought to address. There are three polynomials reported for CRC-12, thirteen conflicting definitions of CRC-16, and six of CRC-32.

The polynomials commonly applied are not the most efficient ones possible. Between 1993 and 2004, Koopman, Castagnoli and others surveyed the space of polynomials up to 16 bits, and of 24 and 32 bits, 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 popular and IEEE-recommended CRC-32 polynomial, used by Ethernet, FDDI and others, is the generating polynomial of a Hamming code and, far from being arbitrarily chosen, 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–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. Any particular protocol can impose pre-inversion, post-inversion and reversed bit ordering as described above. CRCs in proprietary protocols might use a non-trivial initial value and final XOR for obfuscation but this does not add cryptographic strength to the algorithm.

Note: in this table the high-order bit is omitted; see Specification of CRC above.

Name	Polynomial	Representations: normal / reversed / reverse of reciprocal
CRC-1	$x+1$ (most hardware; also known as parity bit)	0x1 / 0x1 / 0x1
CRC-4-ITU	$x^{4}+x+1$ (ITU-T G.704, p. 12)	0x3 / 0xC / 0x9
CRC-5-EPC	$x^{5}+x^{3}+1$ (Gen 2 RFID)	0x09 / 0x12 / 0x14
CRC-5-ITU	$x^{5}+x^{4}+x^{2}+1$ (ITU-T G.704, p. 9)	0x15 / 0x15 / 0x1A
CRC-5-USB	$x^{5}+x^{2}+1$ (USB token packets)	0x05 / 0x14 / 0x12
CRC-6-ITU	$x^{6}+x+1$ (ITU-T G.704, p. 3)	0x03 / 0x30 / 0x21
CRC-7	$x^{7}+x^{3}+1$ (telecom systems, ITU-T G.707, ITU-T G.832, MMC, SD)	0x09 / 0x48 / 0x44
CRC-8-CCITT	$x^{8}+x^{2}+x+1$ (ATM HEC), ISDN Header Error Control and Cell Delineation ITU-T I.432.1 (02/99)	0x07 / 0xE0 / 0x83
CRC-8-Dallas/Maxim	$x^{8}+x^{5}+x^{4}+1$ (1-Wire bus)	0x31 / 0x8C / 0x98
CRC-8	$x^{8}+x^{7}+x^{6}+x^{4}+x^{2}+1$	0xD5 / 0xAB / 0xEA
CRC-8-SAE J1850	$x^{8}+x^{4}+x^{3}+x^{2}+1$	0x1D / 0xB8 / 0x8E
CRC-8-WCDMA	$x^{8}+x^{7}+x^{4}+x^{3}+x+1$	0x9B / 0xD9 / 0xCD
CRC-10	$x^{10}+x^{9}+x^{5}+x^{4}+x+1$ (ATM; ITU-T I.610)	0x233 / 0x331 / 0x319
CRC-11	$x^{11}+x^{9}+x^{8}+x^{7}+x^{2}+1$ (FlexRay)	0x385 / 0x50E / 0x5C2
CRC-12	$x^{12}+x^{11}+x^{3}+x^{2}+x+1$ (telecom systems, )	0x80F / 0xF01 / 0xC07
CRC-15-CAN	$x^{15}+x^{14}+x^{10}+x^{8}+x^{7}+x^{4}+x^{3}+1$	0x4599 / 0x4CD1 / 0x62CC
CRC-16-IBM	$x^{16}+x^{15}+x^{2}+1$ (Bisync, Modbus, USB, ANSI X3.28, many others; also known as CRC-16 and CRC-16-ANSI)	0x8005 / 0xA001 / 0xC002
CRC-16-CCITT	$x^{16}+x^{12}+x^{5}+1$ (X.25, HDLC, XMODEM, Bluetooth, SD, many others; known as CRC-CCITT)	0x1021 / 0x8408 / 0x8810
CRC-16-T10-DIF	$x^{16}+x^{15}+x^{11}+x^{9}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x+1$ (SCSI DIF)	0x8BB7 / 0xEDD1 / 0xC5DB
CRC-16-DNP	$x^{16}+x^{13}+x^{12}+x^{11}+x^{10}+x^{8}+x^{6}+x^{5}+x^{2}+1$ (DNP, IEC 870, M-Bus)	0x3D65 / 0xA6BC / 0x9EB2
CRC-16-DECT	$x^{16}+x^{10}+x^{8}+x^{7}+x^{3}+1$ (cordless telephones)	0x0589 / 0x91A0 / 0x82C4
CRC-16-Fletcher	Not a CRC; see Fletcher's checksum	Used in Adler-32 A & B CRCs
CRC-24	$x^{24}+x^{22}+x^{20}+x^{19}+x^{18}+x^{16}+x^{14}+x^{13}+x^{11}+x^{10}+x^{8}+x^{7}+x^{6}+x^{3}+x+1$ (FlexRay)	0x5D6DCB / 0xD3B6BA / 0xAEB6E5
CRC-24-Radix-64	$x^{24}+x^{23}+x^{18}+x^{17}+x^{14}+x^{11}+x^{10}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x+1$ (OpenPGP)	0x864CFB / 0xDF3261 / 0xC3267D
CRC-30	$x^{30}+x^{29}+x^{21}+x^{20}+x^{15}+x^{13}+x^{12}+x^{11}+x^{8}+x^{7}+x^{6}+x^{2}+x+1$ (CDMA)	0x2030B9C7 / 0x38E74301 / 0x30185CE3
CRC-32-Adler	Not a CRC; see Adler-32	See Adler-32
CRC-32-IEEE 802.3	$x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x+1$ (V.42, MPEG-2, PNG, POSIX cksum)	0x04C11DB7 / 0xEDB88320 / 0x82608EDB
CRC-32C (Castagnoli)	$x^{32}+x^{28}+x^{27}+x^{26}+x^{25}+x^{23}+x^{22}+x^{20}+x^{19}+x^{18}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+1$ (iSCSI & SCTP, G.hn payload)	0x1EDC6F41 / 0x82F63B78 / 0x8F6E37A0
CRC-32K (Koopman)	$x^{32}+x^{30}+x^{29}+x^{28}+x^{26}+x^{20}+x^{19}+x^{17}+x^{16}+x^{15}+x^{11}+x^{10}+x^{7}+x^{6}+x^{4}+x^{2}+x+1$	0x741B8CD7 / 0xEB31D82E / 0xBA0DC66B
CRC-32Q	$x^{32}+x^{31}+x^{24}+x^{22}+x^{16}+x^{14}+x^{8}+x^{7}+x^{5}+x^{3}+x+1$ (aviation; AIXM)	0x814141AB / 0xD5828281 / 0xC0A0A0D5
CRC-64-ISO	$x^{64}+x^{4}+x^{3}+x+1$ (HDLC — ISO 3309, Swiss-Prot/TrEMBL; considered weak for hashing)	0x000000000000001B / 0xD800000000000000 / 0x800000000000000D
CRC-64-ECMA-182	$x^{64}+x^{62}+x^{57}+x^{55}+x^{54}+x^{53}+x^{52}+x^{47}+x^{46}+x^{45}+x^{40}+x^{39}+x^{38}+x^{37}+x^{35}+x^{33}+$ $x^{32}+x^{31}+x^{29}+x^{27}+x^{24}+x^{23}+x^{22}+x^{21}+x^{19}+x^{17}+x^{13}+x^{12}+x^{10}+x^{9}+x^{7}+x^{4}+x+1$ (as described in ECMA-182 p. 51)	0x42F0E1EBA9EA3693 / 0xC96C5795D7870F42 / 0xA17870F5D4F51B49

Known to exist, but technologically defunct—mainly replaced by cryptographic hash functions:

• CRC-128 (IEEE)
• CRC-256 (IEEE)

Supported in hardware:

• CRC-32C (Castagnoli) - Polynomial 0x11EDC6F41. Part of the SSE4.2 instruction set.

Designing CRC polynomials

The selection of generator polynomial is the most important part of implementing the CRC algorithm. The polynomial must be chosen to maximise the error detecting capabilities while minimising overall collision probabilities.

The most important attribute of the polynomial is its length (the number of the highest nonzero coefficient), because of its direct influence of the length of the computed checksum.

The most commonly used polynomial lengths are:

• 9 bits (CRC-8)
• 17 bits (CRC-16)
• 33 bits (CRC-32)
• 65 bits (CRC-64)

The design of the CRC polynomial depends on what is the maximum total length of the block to be protected (data + CRC bits), the desired error protection features, and the type resources for implementing the CRC as well as the desired performance. A common misconception is that the "best" CRC polynomials are derived from either an irreducible polynomial or an irreducible polynomial 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 in the selection of the polynomial.

The advantage of choosing say a primitive polynomial as the generator for a CRC code is that the resulting code has maximal total block length; in here if r is the degree of the primitive generator polynomial then the maximal total blocklength is equal to $2^{r}-1$, and the associated code is able to detect any single bit or double errors. If instead, we used as generator polynomial $g(x)=p(x)(1+x)$, where $p(x)$ is a primitive polynomial of degree $r-1$, then the maximal total blocklength would be equal to $2^{r-1}-1$ but the code would be able to detect single, double, and triple 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. A powerful class of such polynomials, which subsumes the two examples described above, is that of BCH codes. Regardless of the reducibility properties of a generator polynomial of degree r, assuming that it includes the "+1" term, such error detection code will be able to detect all error patterns that are confined to a window of r contiguous bits. These patterns are called "error bursts".

See also

• Error correcting code
• Cyclic code
• Redundancy check
• List of checksum algorithms
• Parity
• Information security
• Simple file verification
• cksum
• Header Error Correction
• Adler-32
• Fletcher's checksum
• Mathematics of CRCs

External links

• MathPages - Cyclic Redundancy Checks: overview with an explanation of error-detection of different polynomials.
• Free CRC Source Code from the Boost C++ Libraries
• The CRC Pitstop
• Black, R. (1994-02) Fast CRC32 in Software; algorithm 4 is used in Linux and info-zip's zip and unzip.
• Kounavis, M. and Berry, F. (2005). A Systematic Approach to Building High Performance, Software-based, CRC generators, Slicing-by-4 and slicing-by-8 algorithms
• CRC32: Generating a checksum for a file, C++ implementation by Brian Friesen
• The CRC++ Project Implementation in C++ which uses template classes to deal with different bit order
• 'CRC-Analysis with Bitfilters'.
• Cyclic Redundancy Check: theory, practice, hardware, and software with emphasis on CRC-32. A sample chapter from Henry S. Warren, Jr. Hacker's Delight.