Cyclic redundancy check

A cyclic redundancy check (CRC) 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.^[1]

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.^[2] The IEEE-recommended 32-bit CRC, used in Ethernet and elsewhere, appeared at a telecommunications conference in 1975.^[3]

Introduction

A CRC is an error-detecting code. Its computation resembles a long division operation in which the quotient is discarded and the remainder becomes the result, with the important distinction that the arithmetic used is the carry-less arithmetic of a finite field. The length of the remainder is always less than or equal to the length of the divisor, 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 CRC: it uses the two-bit-long divisor 11.

CRCs and data integrity

CRCs are not, by themselves, suitable for protecting against intentional alteration of data (for example, in authentication applications for data security), because their convenient mathematical properties make it easy to compute the CRC adjustment required to match any given change to the data.

It is often falsely assumed^[4] 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. This is of course false, because both could have been changed, such that the new CRC would match the new message. Therefore, CRCs can be relied upon to verify correctness but not integrity.

Although CRCs share a problem with message digests in that there cannot be a 1:1 relationship between all possible input messages and all possible outputs (CRC checksums), the CRC function fares worse because it is not a trapdoor function. That is, it is easy to generate other messages that result in the same CRC, especially messages based on the original.^[5] By design, however, a message that is too similar (differing only by a trivial noise pattern) will have a dramatically different CRC and thus be detected.

In contrast, an effective way to protect messages against intentional tampering is by the use of a message authentication code such as HMAC.

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 exclusive-ORed 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 multiplication calculation
          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 post-processing).

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 "polynomial trick" allows bit strings to be viewed as elements of 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. Many analytical tools commonly used with numbers also work on rings, and this is why the "polynomial" view helps the analysis.

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

While cyclic redundancy checks form part of several standards, they are not themselves standardized to the point of adopting one algorithm of each degree worldwide: there are three polynomials reported for CRC-12^[6], ten conflicting definitions of CRC-16, and four of CRC-32^[7]. The polynomials usually seen are not the most efficient ones possible. Between 1993 and 2004, Koopman, Castagnoli and others surveyed the space of polynomials up to 16 bits^[6], and of 24 and 32 bits,^[8][9] 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^[9]. In particular, iSCSI has 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^[3]. Even so, the Castagnoli CRC-32C polynomial used in iSCSI 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^[9]. 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 complicated 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.

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

• CRC-128 (IEEE)
• CRC-256 (IEEE)

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),^{[citation needed]} 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

References

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
• Williams, R. (1993-09) A Painless Guide to CRC Error Detection Algorithms Contains a rigorous explanation of how to generate the CRC table typically found in implementations.
• 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
• Reversing CRC - Theory and Practice.
• '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.