Code crc error

A cyclic redundancy check (CRC) is an error-detecting code commonly used in digital networks and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short check value attached, based on the remainder of a polynomial division of their contents. On retrieval, the calculation is repeated and, in the event the check values do not match, corrective action can be taken against data corruption. CRCs can be used for error correction (see bitfilters).^[1]

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.

Introduction[edit]

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.^[2]
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 the fraction of all longer error bursts that it will detect is (1 − 2⁻ⁿ).

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).

In practice, all commonly used CRCs employ the Galois field, or more simply a finite 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 1-bit CRC: it uses the generator polynomial x + 1 (two terms),^[3] and has the name CRC-1.

Application[edit]

A CRC-enabled device calculates a short, fixed-length binary sequence, known as the check value or CRC, for each block of data to be sent or stored and appends it to the data, forming a codeword.

When a codeword is received or read, the device either compares its check value with one freshly calculated from the data block, or equivalently, performs a CRC on the whole codeword and compares the resulting check value with an expected residue constant.

If the CRC values do not match, then the block contains a data error.

The device may take corrective action, such as rereading the block or requesting that it be sent again. Otherwise, the data is assumed to be error-free (though, with some small probability, it may contain undetected errors; this is inherent in the nature of error-checking).^[4]

Data integrity[edit]

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.^[5]

Thirdly, CRC satisfies a relation similar to that of a linear function (or more accurately, an affine function):^[6]

where depends on the length of and . This can be also stated as follows, where , and have the same length

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.^[7]

Computation[edit]

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 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-degree 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, which is why it is called a 3-bit CRC. However, you need 4 bits to explicitly state the polynomial.

Start with the message to be encoded:

11010011101100

This is first padded with zeros corresponding to the bit length n of the CRC. This is done so that the resulting code word is in systematic form. 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 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. Here is the entire calculation:

11010011101100 000 <--- input right padded by 3 bits
1011               <--- divisor
01100011101100 000 <--- result (note the first four bits are the XOR with the divisor beneath, the rest of the bits are unchanged)
 1011              <--- divisor ...
00111011101100 000
  1011
00010111101100 000
   1011
00000001101100 000 <--- note that the divisor moves over to align with the next 1 in the dividend (since quotient for that step was zero)
       1011             (in other words, it doesn't necessarily move one bit per iteration)
00000000110100 000
        1011
00000000011000 000
         1011
00000000001110 000
          1011
00000000000101 000
           101 1
-----------------
00000000000000 100 <--- remainder (3 bits).  Division algorithm stops here as dividend is equal to zero.

Since the leftmost divisor bit zeroed every input bit it touched, when this process ends the only bits in the input row that can be nonzero are the n bits at the right-hand end of the row. These n bits are the remainder of the division step, and will also be the value of the CRC function (unless the chosen CRC specification calls for some postprocessing).

The validity of a received message can easily be verified by performing the above calculation again, this time with the check value added instead of zeroes. The remainder should equal zero if there are no detectable errors.

11010011101100 100 <--- input with check value
1011               <--- divisor
01100011101100 100 <--- result
 1011              <--- divisor ...
00111011101100 100

......

00000000001110 100
          1011
00000000000101 100
           101 1
------------------
00000000000000 000 <--- remainder

The following Python code outlines a function which will return the initial CRC remainder for a chosen input and polynomial, with either 1 or 0 as the initial padding. Note that this code works with string inputs rather than raw numbers:

def crc_remainder(input_bitstring, polynomial_bitstring, initial_filler):
    """Calculate the CRC remainder of a string of bits using a chosen polynomial.
    initial_filler should be '1' or '0'.
    """
    polynomial_bitstring = polynomial_bitstring.lstrip('0')
    len_input = len(input_bitstring)
    initial_padding = (len(polynomial_bitstring) - 1) * initial_filler
    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_bitstring)):
            input_padded_array[cur_shift + i] 
            = str(int(polynomial_bitstring[i] != input_padded_array[cur_shift + i]))
    return ''.join(input_padded_array)[len_input:]

def crc_check(input_bitstring, polynomial_bitstring, check_value):
    """Calculate the CRC check of a string of bits using a chosen polynomial."""
    polynomial_bitstring = polynomial_bitstring.lstrip('0')
    len_input = len(input_bitstring)
    initial_padding = check_value
    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_bitstring)):
            input_padded_array[cur_shift + i] 
            = str(int(polynomial_bitstring[i] != input_padded_array[cur_shift + i]))
    return ('1' not in ''.join(input_padded_array)[len_input:])

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

CRC-32 algorithm[edit]

This is a practical algorithm for the CRC-32 variant of CRC.^[8] The CRCTable is a memoization of a calculation that would have to be repeated for each byte of the message (Computation of cyclic redundancy checks § Multi-bit computation).

Function CRC32
   Input:
      data:  Bytes     // Array of bytes
   Output:
      crc32: UInt32    // 32-bit unsigned CRC-32 value

// Initialize CRC-32 to starting value
crc32 ← 0xFFFFFFFF

for each byte in data do
   nLookupIndex ← (crc32 xor byte) and 0xFF
   crc32 ← (crc32 shr 8) xor CRCTable[nLookupIndex]  // CRCTable is an array of 256 32-bit constants

// Finalize the CRC-32 value by inverting all the bits
crc32 ← crc32 xor 0xFFFFFFFF
return crc32

In C, the algorithm looks as such:

#include <inttypes.h> // uint32_t, uint8_t

uint32_t CRC32(const uint8_t data[], size_t data_length) {
	uint32_t crc32 = 0xFFFFFFFFu;
	
	for (size_t i = 0; i < data_length; i++) {
		const uint32_t lookupIndex = (crc32 ^ data[i]) & 0xff;
		crc32 = (crc32 >> 8) ^ CRCTable[lookupIndex];  // CRCTable is an array of 256 32-bit constants
	}
	
	// Finalize the CRC-32 value by inverting all the bits
	crc32 ^= 0xFFFFFFFFu;
	return crc32;
}

Mathematics[edit]

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) (the integers modulo 2, i.e. either a zero or a one), instead of more familiar numbers. The set of binary polynomials is a mathematical ring.

Designing polynomials[edit]

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), and 65 bits (CRC-64).^[3]

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.^[9] 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 is the degree of the primitive generator polynomial, then the maximal total block length is , and the associated code is able to detect any single-bit or double-bit errors.^[10] We can improve this situation. If we use the generator polynomial , where is a primitive polynomial of degree , then the maximal total block length is , and the code is able to detect single, double, triple and any odd number of errors.

A polynomial 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[edit]

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,^[9] 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 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 may be transcribed as:

In the table below they are shown as:

Obfuscation[edit]

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.^[11]

Standards and common use[edit]

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.^[12] The number of distinct CRCs in use has confused developers, a situation which authors have sought to address.^[9] There are three polynomials reported for CRC-12,^[12] twenty-two conflicting definitions of CRC-16, and seven of CRC-32.^[13]

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,^{[12][14][15][16]} 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.^[15] 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,^[17] and Hammond, Brown and Liu’s report for the Rome Laboratory, published in May.^[18] 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.^[19] 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.^[15] 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).

CRC-32C computation is implemented in hardware as an operation (CRC32) of SSE4.2 instruction set, first introduced in Intel processors’ Nehalem microarchitecture. ARM AArch64 architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations.

Polynomial representations of cyclic redundancy checks[edit]

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.^[13]
Note that even parity polynomials in GF(2) with degree greater than 1 are never primitive. Even parity polynomial marked as primitive in this table represent a primitive polynomial multiplied by . The most significant bit of a polynomial is always 1, and is not shown in the hex representations.

Name	Uses	Polynomial representations	Parity[20]	Primitive[21]	Maximum bits of payload by Hamming distance[22][15][21]
Normal	Reversed	Reciprocal	Reversed reciprocal	≥ 16	15	14	13	12	11	10	9	8	7	6	5	4	3	2[23]
CRC-1	most hardware; also known as parity bit	0x1	0x1	0x1	0x1	even
CRC-3-GSM	mobile networks[24]	0x3	0x6	0x5	0x5	odd	yes [25]	–	–	–	–	–	–	–	–	–	–	–	–	–	4	∞
CRC-4-ITU	ITU-T G.704, p. 12	0x3	0xC	0x9	0x9	odd
CRC-5-EPC	Gen 2 RFID[26]	0x09	0x12	0x05	0x14	odd
CRC-5-ITU	ITU-T G.704, p. 9	0x15	0x15	0x0B	0x1A	even
CRC-5-USB	USB token packets	0x05	0x14	0x09	0x12	odd
CRC-6-CDMA2000-A	mobile networks[27]	0x27	0x39	0x33	0x33	odd
CRC-6-CDMA2000-B	mobile networks[27]	0x07	0x38	0x31	0x23	even
CRC-6-DARC	Data Radio Channel[28]	0x19	0x26	0x0D	0x2C	even
CRC-6-GSM	mobile networks[24]	0x2F	0x3D	0x3B	0x37	even	yes [29]	–	–	–	–	–	–	–	–	–	–	1	1	25	25	∞
CRC-6-ITU	ITU-T G.704, p. 3	0x03	0x30	0x21	0x21	odd
CRC-7	telecom systems, ITU-T G.707, ITU-T G.832, MMC, SD	0x09	0x48	0x11	0x44	odd
CRC-7-MVB	Train Communication Network, IEC 60870-5[30]	0x65	0x53	0x27	0x72	odd
CRC-8	DVB-S2[31]	0xD5	0xAB	0x57	0xEA[12]	even	no [32]	–	–	–	–	–	–	–	–	–	–	2	2	85	85	∞
CRC-8-AUTOSAR	automotive integration,[33] OpenSafety[34]	0x2F	0xF4	0xE9	0x97[12]	even	yes [32]	–	–	–	–	–	–	–	–	–	–	3	3	119	119	∞
CRC-8-Bluetooth	wireless connectivity[35]	0xA7	0xE5	0xCB	0xD3	even
CRC-8-CCITT	ITU-T I.432.1 (02/99); ATM HEC, ISDN HEC and cell delineation, SMBus PEC	0x07	0xE0	0xC1	0x83	even
CRC-8-Dallas/Maxim	1-Wire bus[36]	0x31	0x8C	0x19	0x98	even
CRC-8-DARC	Data Radio Channel[28]	0x39	0x9C	0x39	0x9C	odd
CRC-8-GSM-B	mobile networks[24]	0x49	0x92	0x25	0xA4	even
CRC-8-SAE J1850	AES3; OBD	0x1D	0xB8	0x71	0x8E	odd
CRC-8-WCDMA	mobile networks[27][37]	0x9B	0xD9	0xB3	0xCD[12]	even
CRC-10	ATM; ITU-T I.610	0x233	0x331	0x263	0x319	even
CRC-10-CDMA2000	mobile networks[27]	0x3D9	0x26F	0x0DF	0x3EC	even
CRC-10-GSM	mobile networks[24]	0x175	0x2BA	0x175	0x2BA	odd
CRC-11	FlexRay[38]	0x385	0x50E	0x21D	0x5C2	even
CRC-12	telecom systems[39][40]	0x80F	0xF01	0xE03	0xC07[12]	even
CRC-12-CDMA2000	mobile networks[27]	0xF13	0xC8F	0x91F	0xF89	even
CRC-12-GSM	mobile networks[24]	0xD31	0x8CB	0x197	0xE98	odd
CRC-13-BBC	Time signal, Radio teleswitch[41][42]	0x1CF5	0x15E7	0x0BCF	0x1E7A	even
CRC-14-DARC	Data Radio Channel[28]	0x0805	0x2804	0x1009	0x2402	even
CRC-14-GSM	mobile networks[24]	0x202D	0x2D01	0x1A03	0x3016	even
CRC-15-CAN		0xC599[43][44]	0x4CD1	0x19A3	0x62CC	even
CRC-15-MPT1327	[45]	0x6815	0x540B	0x2817	0x740A	odd
CRC-16-Chakravarty	Optimal for payloads ≤64 bits[30]	0x2F15	0xA8F4	0x51E9	0x978A	odd
CRC-16-ARINC	ACARS applications[46]	0xA02B	0xD405	0xA80B	0xD015	odd
CRC-16-CCITT	X.25, V.41, HDLC FCS, XMODEM, Bluetooth, PACTOR, SD, DigRF, many others; known as CRC-CCITT	0x1021	0x8408	0x811	0x8810[12]	even
CRC-16-CDMA2000	mobile networks[27]	0xC867	0xE613	0xCC27	0xE433	odd
CRC-16-DECT	cordless telephones[47]	0x0589	0x91A0	0x2341	0x82C4	even
CRC-16-T10-DIF	SCSI DIF	0x8BB7[48]	0xEDD1	0xDBA3	0xC5DB	odd
CRC-16-DNP	DNP, IEC 870, M-Bus	0x3D65	0xA6BC	0x4D79	0x9EB2	even
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	0x4003	0xC002	even
CRC-16-OpenSafety-A	safety fieldbus[34]	0x5935	0xAC9A	0x5935	0xAC9A[12]	odd
CRC-16-OpenSafety-B	safety fieldbus[34]	0x755B	0xDAAE	0xB55D	0xBAAD[12]	odd
CRC-16-Profibus	fieldbus networks[49]	0x1DCF	0xF3B8	0xE771	0x8EE7	odd
Fletcher-16	Used in Adler-32 A & B Checksums	Often confused to be a CRC, but actually a checksum; see Fletcher’s checksum
CRC-17-CAN	CAN FD[50]	0x1685B	0x1B42D	0x1685B	0x1B42D	even
CRC-21-CAN	CAN FD[50]	0x102899	0x132281	0x064503	0x18144C	even
CRC-24	FlexRay[38]	0x5D6DCB	0xD3B6BA	0xA76D75	0xAEB6E5	even
CRC-24-Radix-64	OpenPGP, RTCM104v3	0x864CFB	0xDF3261	0xBE64C3	0xC3267D	even
CRC-24-WCDMA	Used in OS-9 RTOS. Residue = 0x800FE3.[51]	0x800063	0xC60001	0x8C0003	0xC00031	even	yes[52]	–	–	–	–	–	–	–	–	–	–	4	4	8388583	8388583	∞
CRC-30	CDMA	0x2030B9C7	0x38E74301	0x31CE8603	0x30185CE3	even
CRC-32	ISO 3309 (HDLC), ANSI X3.66 (ADCCP), FIPS PUB 71, FED-STD-1003, ITU-T V.42, ISO/IEC/IEEE 802-3 (Ethernet), SATA, MPEG-2, PKZIP, Gzip, Bzip2, POSIX cksum,[53] PNG,[54] ZMODEM, many others	0x04C11DB7	0xEDB88320	0xDB710641	0x82608EDB[15]	odd	yes	–	10	–	–	12	21	34	57	91	171	268	2974	91607	4294967263	∞
CRC-32C (Castagnoli)	iSCSI, SCTP, G.hn payload, SSE4.2, Btrfs, ext4, Ceph	0x1EDC6F41	0x82F63B78	0x05EC76F1	0x8F6E37A0[15]	even	yes	6	–	8	–	20	–	47	–	177	–	5243	–	2147483615	–	∞
CRC-32K (Koopman {1,3,28})	Excellent at Ethernet frame length, poor performance with long files	0x741B8CD7	0xEB31D82E	0xD663B05D	0xBA0DC66B[15]	even	no	2	–	4	–	16	–	18	–	152	–	16360	–	114663	–	∞
CRC-32K2 (Koopman {1,1,30})	Excellent at Ethernet frame length, poor performance with long files	0x32583499	0x992C1A4C	0x32583499	0x992C1A4C[15]	even	no	–	–	3	–	16	–	26	–	134	–	32738	–	65506	–	∞
CRC-32Q	aviation; AIXM[55]	0x814141AB	0xD5828281	0xAB050503	0xC0A0A0D5	even
Adler-32		Often confused to be a CRC, but actually a checksum; see Adler-32
CRC-40-GSM	GSM control channel[56][57][58]	0x0004820009	0x9000412000	0x2000824001	0x8002410004	even
CRC-64-ECMA	ECMA-182 p. 51, XZ Utils	0x42F0E1EBA9EA3693	0xC96C5795D7870F42	0x92D8AF2BAF0E1E85	0xA17870F5D4F51B49	even
CRC-64-ISO	ISO 3309 (HDLC), Swiss-Prot/TrEMBL; considered weak for hashing[59]	0x000000000000001B	0xD800000000000000	0xB000000000000001	0x800000000000000D	odd

Implementations[edit]

• Implementation of CRC32 in GNU Radio up to 3.6.1 (ca. 2012)
• C class code for CRC checksum calculation with many different CRCs to choose from

CRC catalogues[edit]

• Catalogue of parametrised CRC algorithms
• CRC Polynomial Zoo

See also[edit]

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

References[edit]

Further reading[edit]

• Warren Jr., Henry S. (2013). Hacker’s Delight (2 ed.). Addison Wesley. ISBN 978-0-321-84268-8.

External links[edit]

• ISO/IEC 13239:2002: Information technology — Telecommunications and information exchange between systems — High-level data link control (HDLC) procedures
• CRC32-Castagnoli Linux Library

Пользователи регулярно сталкиваются с различными системными сбоями, возникающими по программным или аппаратным причинам. Одной из распространённых проблем является ошибка CRC, сопровождающаяся сообщением с соответствующим текстом «Ошибка данных в CRC». Она свидетельствует о несовпадении контрольных сумм загружаемого файла, что говорит о его изменении или повреждении, и может проявляться при использовании внутренних и внешних накопителей информации. Часто сбой встречается во время записи оптического диска, копирования данных с диска на компьютер, установке игр и приложений или при работе с торрент-клиентами. Момент отслеживания появления сбоя всегда имеет значение, поскольку способ решения проблемы напрямую зависит от того, при каких условиях возникла ошибка CRC.

Что такое CRC

Циклический избыточный код или CRC (англ. Cycle Redundancy Check) является алгоритмом вычисления контрольной суммы файла, используемой для проверки корректности передаваемой информации. То есть в результате обработки информации получается определённое значение, которое обязательно будет разным для файлов, даже на один бит отличающихся между собой. Так, алгоритм, базирующийся на циклическом коде, определяет контрольную сумму и приписывает её к передаваемым данным. В свою очередь принимающая сторона также владеет алгоритмом нахождения контрольной суммы, что даёт возможность системе проверить целостность получаемых данных. Когда эти числа соответствуют, информация передаётся успешно, а при несовпадении значений контрольной суммы возникает ошибка в данных CRC, это означает, что файл, к которому обратилась программа, был изменён или повреждён.

Причины возникновения ошибки CRC

В большинстве случаев проблема носит аппаратный характер, но иногда возникает и по программным причинам. Сообщение «Ошибка данных в CRC» может говорить о неисправности HDD, нарушении файловой системы или наличии битых секторов. Нередко сбой возникает при инициализации жёсткого диска или твердотельного накопителя SSD после подсоединения к компьютеру, несмотря на конструктивные отличия и разницу в способе функционирования, это происходит, поскольку не удаётся правильно прочитать данные. Тогда неполадка может заключаться не только в механических повреждениях, но и в неисправности интерфейсов подключения или плохом контакте. Ошибка в устройствах SSD с интерфейсом PCI-E также случается из-за накопившихся загрязнений на плате. Проблемы с доступом к внешним накопителям нередко связаны с неисправностью портов. Среди программных причин ошибку в данных CRC способны вызывать сбои драйверов устройств. Источником проблемы при установке программного обеспечения посредством торрент-клиента чаще всего является повреждённый архив. Таким образом, можно выделить следующие причины ошибки CRC:

• Случайный сбой во время установки софта.
• Потеря или повреждение данных при их передаче.
• Загрузка повреждённого архива.
• Неправильные записи системного реестра.
• Некорректные драйверы устройств.
• Повреждение оптического диска.
• Неисправность разъёмов.
• Повреждение секторов жёсткого диска, файловой системы HDD.
• Неправильная конфигурация файлов и другие причины.

Как исправить ошибку CRC

Факторов, провоцирующих проблему не так уж и мало. Определить источник неприятности чаще всего можно исходя из условий, при которых проявилась ошибка в данных CRC, тогда можно сузить круг возможных причин и понять, как исправить сбой. Все действия, направленные на устранение неисправности будут эффективны в различных ситуациях.

Ошибка в данных CRC при подключении устройства

В случае возникновения проблемы при работе с внешними накопителями прежде, чем лечить устройство посредством специализированных утилит, первым делом следует проверить работу с другими разъёмами. Если порт неисправен, то вопрос может решиться подключением носителя к иному порту на девайсе. Ещё одна причина – плохой контакт, например это часто встречается в адаптере подключения SD карты, а также при использовании устройств HDD или SSD. Флешка, СД карта или другой подключаемый накопитель также могут быть неисправными, для чтения данных с повреждённых устройств используется специальный софт, например, BadCopyPro, но при выходе из строя носителя программные методы бессильны.

Возникновение сбоя при попытке доступа к файлам на HDD

«Ошибка в данных CRC» нередко проявляется по причине нарушения целостности файловой системы или битых секторов на жёстком диске. Поскольку не удаётся правильно прочесть информацию, с этим может быть связан ряд сбоев, включая ситуацию, когда винчестер не инициализируется. Диагностировать HDD можно с помощью встроенных средств Windows или сторонних программ.

Проверка инструментом Chkdsk

В арсенале ОС имеется немало интегрированных служб для решения различных задач. Проверить файловую систему на ошибки, а также обнаружить битые сектора можно с использованием утилиты Check disk, вызываемой из командной строки. Чтобы просканировать жёсткий диск выполняем следующие действия:

В ряде случаев, когда нет доступа ко всему диску или ошибка в данных CRC проявляется при обращении к софту, который раньше работал, сканирование потребуется выполнить в режиме восстановления, для чего потребуется загрузочная флешка или диск с соответствующей ОС Windows. Изменив порядок запуска устройств (выставить в Boot приоритет для съёмного накопителя) и запустившись с загрузочного носителя можно открыть командную строку следующим образом:

Сканирование системной утилитой проверки диска

Можно выполнить проверку на ошибки и другим способом:

При выборе системного диска потребуется запланировать проверку, в таком случае он будет просканирован при следующей загрузке Windows. Для восстановления HDD применяется также сторонний софт, например, HDD Regenerator, Acronis Disk Director, Victoria и прочие. При этом в случае с физическими повреждениями устройства выправить их программно не получится, поэтому лучше заранее скопировать важную информацию с жёсткого диска, возможно, его время уже на исходе.

Проблема при скачивании с CD/DVD носителя

Если ошибка выскакивает при копировании информации с оптического диска на внутренний накопитель, возможно, что диск просто загрязнён или повреждён. Для начала нужно очистить поверхность носителя и попытаться выполнить процедуру снова. Если не помогло, ищем другой источник информации, а при его отсутствии и необходимости восстановления данных с диска можно использовать программу BadCopyPro, которая считывает и возвращает к жизни файлы с испорченных накопителей, если это возможно. К сожалению, при сильно выраженных повреждениях диска скопировать с него файлы не удастся.

Ошибка в данных CRC при записи оптического диска, установке программ или игр

Если проблема возникла в процессе записи образа, скачанного с просторов интернета, на CD/DVD, стоит проверить контрольные суммы записываемых данных перед выполнением процедуры. Для этой цели используется утилита HashTab, после установки которой, в свойствах файла появится новая вкладка «Хеш-суммы файлов», с её помощью вы сможете сравнить значение с исходником. Так, при несовпадении контрольных сумм, следует скачать образ снова.

Повторно скачать файл или архив следует также, когда с помощью скачанного дистрибутива программы она не устанавливается на компьютер. Данные могли быть повреждены в процессе загрузки или не были полностью выкачаны. Удобно применять для скачивания uTorrent, поскольку утилита самостоятельно определяет значения контрольных сумм и перекачивает часть информации, загрузившейся с ошибкой. В случае скачивания данных по прямым ссылкам можно использовать Download Master. При этом не исключено, что архив или файл уже был повреждён изначально и в таком виде залит на ресурс, потому нужно попробовать скачать его с альтернативного источника.

Ошибка в uTorrent

Когда ошибка в данных CRC появляется в программе uTorrent, исправляем её следующими действиями:

• Выполняем обновление клиента.
• Удаляем проблемную раздачу в программе, а также недокачавшийся файл в папке на компьютере.
• Ищем аналогичную раздачу на другом ресурсе и скачиваем оттуда.

Вышеперечисленных способов достаточно, чтобы избавиться от ошибки в данных CRC, возникающей при различных условиях. Так, определив источник появления сбоя, можно целенаправленно устранить проблему. В большинстве случаев исправить ошибку удаётся программными средствами, но, когда речь идёт о физической неисправности накопителя, следует задуматься о его замене.

Ошибка в данных CRC — циклическая проверка избыточности ошибок данных возникает, когда пользователь Windows переносит данные с одного диска на другой. Ошибка CRC возникает при копировании файлов с SD-карт памяти и USB-флешек. Она может появляться, когда пользователь устанавливает игру с флешки или при подключении внешнего диска.

CRC (Cyclic Redundancy Check) — алгоритм, который использует проверку точности данных при копировании на HDD/SSD дисках, оптических, SD-картах памяти, USB флешек. CRC обнаруживает случайные изменения необработанных данных, находящихся в хранилище и выдает ошибку, если контрольные суммы различаются.

Другими словами, когда копируются данные с флешки на диск, то они передаются небольшими блоками, и каждому блоку присваивается значение CRC. Если что-то пойдет не так с данными в период, как они покинут источник и прибудут в пункт назначения, CRC, отправленный в источник, больше не будет соответствовать назначенной контрольной сумме, что и вызывает ошибку данных CRC.

Как исправить ошибку данных CRC

Виновником CRC ошибки является неисправность самого устройства, плохие сектора на диске, конфигурация реестра, драйвер или неудачное установленное ПО. Разберем способы, как устранить ошибку CRC при инициализации диска или копировании файлов с флешки на диск.

1. Вирусы и антивирус

1. Во многих случаях, сторонний антивирус может блокировать файлы при копировании, что вызовет сбой и ошибку данных CRC.
2. Вирусы могут повредить некоторые системные файлы или файлы при копировании. В данном случае, лучше воспользоваться сторонним сканером как DrWeb или Zemana.

2. Плохие сектора

Ошибки секторов на диске, флешке или SD карте это главный виновник проблемы, когда появляется ошибка в данных CRC. Чтобы устранить проблему, запустите командную строку от имени администратора и введите ниже команду, которая проверит и автоматически восстановит плохие сектора на диске или другом устройстве.

• chkdsk C: /f /r /x

Вместо C: укажите то устройство на котором ошибка в данных CRC. В некоторых случаях, после ввода команды, нужно будет нажать Y и перезагрузить ПК.

3. Проверка системных файлов

Поврежденные системные файлы или недостающие могут быть виновником ошибки данных CRC. Запустите командную строку от имени администратора и введите ниже 2 команды по очереди.

sfc /scannow
DISM /ONLINE /CLEANUP-IMAGE /RESTOREHEALTH

Примечание к способу 2 и 3

Если у вас выдаются ошибки при вводе команд, то вы можете загрузиться через дополнительные параметры и запустить командную строку. Вы также можете создать установочную флешку с Windows 10 для запуска CMD.

4. Сброс файловой системы

Этот метод удалит все данные с вашей флешки или диска и не рекомендуется тем пользователям, которые не хотят потерять важные файлы.

• Нажмите правой кнопкой мыши по диску или флешке, которая выдает ошибку в данных CRC и выберите «Форматировать«.
• В окне укажите NTFS, если диск или флешка больше 32 ГБ, и нажмите «Начать«.
• Если внешний диск или флешка меньше 32 Гб, то выберите FAT32.

5. Если не видно букву диска

Нажмите Win+R и введите devmgmt.msc, чтобы открыть диспетчер устройств. Разверните графу «Дисковые устройства» и, если будет желтый восклицательный знак на диске или флешке, то нажмите провой кнопкой мыши по данному устройству и выберите «Обновить драйвер«.

В противном случае нажмите правой кнопкой мыши по графе «Дисковые устройства» и выберите «Обновить конфигурацию оборудования«.

Если буква диска еще не появилась, то обратитесь к подробному руководству.

6. Иры и ПО

Если имеется какое-либо ПО связанное с диском или флешкой, которое может работать в фоновом режиме, то его следует удалить и посмотреть, устранена ли ошибка. Если ошибка CRC появляется при установке игры, то скорее всего при скачивании самой игры были повреждены некоторые файлы. В данном случае нужно скачать игру заново. Это касается и торрент скачивания.

[ Telegram | Поддержать ]

The cool thing about programming is that there is an absolute truth in every Bit: It’s either black or white, 1 or 0, true or false. In theory. But whenever you transmit data or store it somewhere on a physical drive, it can get corrupted. Individual bits can get flipped or even bursts of multiple bits, changing the value and the meaning of your data inadvertently. Surrounding electromagnetic radiation can interfere with your Bluetooth or wi-fi signals and cause unintended changes in your data stream. Simply accessing a flash memory might also result in bit flips.

So whenever you exchange data with other systems, how do you make sure that the data you receive is exactly the same as the data that was sent to you? How do you detect accidental changes in your data stream? Or in other words, how do you check the integrity of your data against unintended changes and losses?

There are a number of mechanisms used for data error detection. The most common one is called Cyclic Redundancy Check (CRC). It is often used in Bluetooth and other wireless communication protocols. It is also used to check the integrity of different types of files such as Gzip, Bzip2, PNG etc. The name sounds rather complex and intimidating. However, after reading this article you should have a good understanding of what a CRC is and how it works.

The Basics

CRC is an error detection code used for verifying the integrity of data. It works just like a checksum and is appended to the end of the payload data and transmitted (or stored) along with that data.

For example, let’s say we want to send the lower-case letter z to someone else. In Unicode, the z is represented by the number 0x7A, that’s 0111 1010 in binary representation. In order to make it possible for the receiver to verify that the data wasn’t modified on transmission, we calculate the CRC for this data (1000) and simply append it to our message:

0111 1010 (data) → 0111 1010 1000 (data + CRC)

The check value is called redundant because it doesn’t add any additional information to the message. (This data is only appended for the sake of error detection and data integrity). 0111 1010 contains the same information as 0111 1010 1000 (where last 4 bits are the CRC), however, in the first case, the receiver has no way of verifying if the data was received correctly.

CRCs are specifically designed to detect common data communication errors. It can also detect when the order of the bits or bytes changes. CRCs can easily be implemented in hardware – another reason for their widespread use. Before we take a deep dive into how a CRC works, there is one important concept that we need to understand first: The endianness.

Endianness

The endianness is the order of bytes with which data words are stored. We distinguish the following to types:

• Little-endian: The least significant byte is stored at the smallest memory address. In terms of data transmission, the least significant byte is transmitted first.
• Big-endian: The most significant byte is stored at the smallest memory address. In terms of data transmission, the most significant byte is transmitted first.

In the introductory example (with the letter z), we only used a CRC with a size of 1 byte, so the endianness doesn’t matter in this case. However, in many cases, we use CRCs that consist of mulitple bytes. So the question arises which endianness to use for CRC?

From an app developer’s perspective, the answer is quite simple: We use the same endianness as the system we want to communicate with. It’s usually required by the hardware to use a particular byte order. For the remainder of this article, we’ll stick with the big-endian format and provide some hints along the way on how a little-endian implementation would differ.

The Concept of a CRC

Mathematically spoken, a CRC is the remainder of a modulo-2 polynomial division of the data. So, the CRC can be written as

CRC = data %2 polynomial

The polynomial can be any mathematical polynomial (without any coefficients) up to the order of the CRC bit size. So if we want a CRC with 8 bits, the highest exponent of the polynomial must be 8 as well. Example:

x⁸ + x² + x + 1

As you can imagine from the simple equation above, choosing a different polynomial will result in a different CRC for the same data. So sender and receiver need to agree on a common polynomial, or else they will always end up with a different CRC and assume that the data got corrupted on the way. Depending on the use case, some polynomials are better suited than others. (Picking the right one is a topic on its own and beyond the scope of this article.)

Any such polynomial can be represented in binary form by going through the exponents from highest to lowest and writing a 1 for each exponent that is present (8, 2, 1 and 0 in our example) and a 0 for each absent exponent (7, 6, 5, 4, 3 in this case):

Exponent    8 7 6 5 4 3 2 1 0
Polynomial  1 0 0 0 0 0 1 1 1 
–––––––––––––––––––––––––––––
Bit Index   0 1 2 3 4 5 6 7 8

The above is the big-endian representation of the polynomial. For its little-endian representation, simply reverse the order of bits:

Exponent    0 1 2 3 4 5 6 7 8
Polynomial  1 1 1 0 0 0 0 0 1 
–––––––––––––––––––––––––––––
Bit Index   0 1 2 3 4 5 6 7 8

Here are some example polynomials with their (big-endian) binary representation:

Next, in order to be able to calculate a CRC, we need to understand the modulo-2 division.

Making Calculations in Modulo-2 Arithmetic

Modulo-2 arithmetic is performed digit by digit (bit by bit) on binary numbers. There is no carry or borrowing in this arithmetic. Each digit is considered independent from its neighbours.

Addition & Subtraction

The curious thing about modulo-2 arithmetic is that both addition and subtraction yield the same result.

  10          10
+ 11        - 11
-----       ------
  01          01

The sum or the difference of two bits can be computed with an XOR operation: The result is 1 if exactly one of the two bits is 1, otherwise the result is 0. As there is no carry or borrowing, the same applies not only for individual bits, but for binary numbers of any length, for example:

  10  +  11
= 10  -  11
= 10 XOR 11
= 01 

We get this result by applying the XOR operator on each pair of bits individually:

1 XOR 1 =  0
0 XOR 1 =  1
         -----
          01

Division

Modulo-2 division is performed similarly to “normal” arithmetic division. The only difference is that we use modulo-2 subtraction (XOR) instead of arithmetic subtraction for calculating the remainders in each step.

⚠️ If you are German, please note that in English literature, the divisor 11011 is written first, followed by the dividend 01111010. (In German literature it’s usually the other way around.) The result of the division (quotient) is written on top of the calculation, the remainder at the bottom.

       01011 ⬅︎ Quotient
      ––––––––––––––––––
11011 | 01111010
      - 00000
      ----------
         1111010
       - 11011
       ---------
          010110
        - 00000
        --------
           10110
         - 11011
         -------
            11010
          - 11011
          -------
            0001 ⬅︎ Remainder (mod-2)

Types of CRCs

There are different types of CRCs. They are categorized by the degree of the polynomial they use. As the first exponent of a polynomial of degree n is always present by definition (otherwise it would have a lower degree), its binary representation always begins with a 1. In other words, the first bit of a binary polynomial representation doesn’t carry any information about the polynomial when we agree on a fixed degree.

For that reason, the first bit of a binary polynomial representation is always dropped when computing a CRC in software. So the bit size of the resulting binary representation is always n for a polynomial of degree n. Example:

CRCs types are named by their bit size. Here are the most common ones:

• CRC-8
• CRC-16
• CRC-32
• CRC-64
• CRC-1 (parity bit) is a special case

Generally, we can refer to a CRC as CRC-n, where n is the number of CRC bits and the number of bits of the polynomial’s binary representation with a dropped first bit. Obviously, different CRCs are possible for the same n as multiple polynomials exist for the same degree.

For example, if we want to calculate the CRC-8 for these bits:

01000001

the result looks different, depending on the polynomial we use:

Above are the standard CRC-8 polynomials used in different applications such as Bluetooth, GSM-B, etc.

Error Detection

How do we choose a suitable CRC and a respective polynomial? There are three things we need to consider:

1. Random Error Detection Accuracy
2. Burst Error Detection Accuracy
3. The Redundancy Factor

Random Error Detection Accuracy

Random errors are errors that can occur randomly in any data. For example, a single bit is flipped when transmitting data or a few bits are lost during the transmission.

Depending on the bit size of the CRC we use, we can detect most of these random errors. However, for a CRC-n, 1/2ⁿ of these errors cannot be detected. The following table shows the percentage of the possible random errors that remain undetected for each CRC type:

Burst Error Detection Accuracy

Errors in data transmission are oftentimes not random but produced over a consecutive sequence of bits. Such errors are called burst errors. They are the most common errors in data communication.

It’s one of the CRC’s strongest properties to detecting burst errors reliably. A CRC-n can detect single burst errors with a maximum length of n bits. However, this depends a lot on the polynomial used for computing the CRC. Some polynomials are able to detect multiple bursts of errors in the transmitted data.

The Redundancy Factor

Using a CRC for error detection comes at the cost of extra (non-meaningful) data. When we use a CRC-32 (4 bytes), we need to transmit two more bytes of “unnecessary” data as compared to a CRC-16. CRCs with a lower bit size are obviously cheaper with respect to storage space.

Based on these three factors, we can decide which CRC type to choose for our application. However, the polynomial you choose for your CRC also affects the efficiency and quality of your error detection. But that’s a topic for itself and we won’t cover it in this article. Fortunately, there are a couple of standard polynomials used for a particular CRC type and in most cases it makes sense to just use one of these.

How It Works: The CRC Algorithm

This section will take you through the steps of calculating a CRC-n for any input data. We will first explain the algorithm and then follow these steps to calculate a 4-bit CRC of some sample data.

1. Take the CRC polynomial and remove the most significant bit.
   Example: 10011 ➔ 0011
2. Append n zeros to the input.
   We call this input the remainder (as it will be the remainder of our binary division after every step and finally becomes the CRC when division is no longer possible).
3. Remember the most significant bit.
   Example: 1010 ➔ most significant bit = 1
4. Discard the most significant bit.
   Example: 1010 ➔ 010
5. Depending on the most significant bit from step 3, do the following:
   • 0 ➔ do nothing (same as subtracting 0 from the remainder)
   • 1 ➔ subtract the divisor (polynomial) from the remainder
6. Repeat steps 3 to 5 for all the bits of the message.
7. The final remainder is the CRC.

Example

Let’s see how this algorithm works in practice and calculate the 4-bit CRC for the binary Unicode representation of the letter z (as introduced at the beginning of this article):

• Input Data: 0111 1010

First, we need to choose a suitable polynomial depending on our use case which must used by both the sender and the receiver. For this example, let’s use this one:

We have already performed step (1) of the algorithm in the table above: 1011 is our binary polynomial representation with a dropped most significant bit. Next, let’s go through the remaining steps as part of our modulo-2 division:

       (Quotient is ignored)
       –––––––––––––––––––––
1011 | 01111010              // 1. Polynomial with dropped MSB | Input        
       01111010 0000         // 2. Append 4 zero bits for the CRC

       01111010 0000         // 3. MSB (first bit) = 0
       -------------
        1111010 0000         // 4. Discard the MSB
      - 0000                 // 5. Do nothing (= subtract 0000)
       ––––––––––––––––––
        1111010 0000         // 3. MSB = 1
        ------------
         111010 0000         // 4. Discard the MSB    
       - 1011                // 5. Subtract the polynomial
       ––––––––––––––––––
         010110 0000         // 3. MSB = 0
         -----------
          10110 0000         // 4. Discard the MSB
        - 0000               // 5. Do nothing (= subtract 0000)
       ––––––––––––––––––
          10110 0000         // 3. MSB = 1
          ----------
           0110 0000         // 4. Discard the MSB
         - 1011              // 5. Subtract the polynomial
       ––––––––––––––––––
           1101 0000         // 3. MSB = 1
           ----------
            1010000          // 4. Discard the MSB
          - 1011             // 5. Subtract the polynomial
       ––––––––––––––––––
             0001000         // 3. MSB = 0
             -------
              001000         // 4. Discard the MSB
            - 0000           // 5. Do nothing (= subtract 0000)
       ––––––––––––––––––
              001000         // 3. Check the most significant bit (MSB = 0)
              ------
               01000         // 4. Discard the MSB
             - 0000          // 5. Do nothing (= subtract 0000)
       ––––––––––––––––––
               01000         // 3. MSB = 0
               -----
                1000         // 4. Discard the MSB
              - 0000         // 5.Do nothing (= subtract 0000)
       ––––––––––––––––––
                1000         // 7. Final remainder = CRC

Implementing The Algorithm in Software

Once you’ve understood the algorithm above, it’s pretty straight-forward to implement it in software. The only thing you need to be clear about is whether you implement the CRC with a big-endian or a little-endian representation. In the following, we’ll provide some concrete implementation advice, assuming a big-endian format.

1. Take the CRC polynomial and remove the most significant bit.
   Usually, you do this manually as you only need to do it once.
   (For little-endian: Reverse the bits of the resulting value.)
2. Append n zeros to the input.
   Not needed in software.
3. Remember the most significant bit.
   This is equivalent to performing an AND operation with a value that has a 1 as its most significant bit and a 0 in every other bit. (For little-endian: Just do AND 1, regardless of the CRC size.)
   Store this value in a variable. Example:
   • 8 bits ➔ 1000 0000 ➔ 0x80
   • 16 bits ➔ 1000 0000 0000 0000 ➔ 0x800
   • 32 bits ➔ 1000 0000 0000 00000000 0000 0000 0000 ➔ 0x8000000
4. Discard the most significant bit.
   This is equivalent to performing a bitwise left-shift.
   (For little-endian: Do a bitwise right-shift.)
5. Depending on the value of the variable stored in step (3), do the following:
   • 0 ➔ do nothing
   • 1 ➔ perform and XOR operation with the remainder and the polynomial
6. Repeat Steps 3 to 5 for all the bits of the message.
   A simple for loop or any functional construct (e.g. reduce in Swift, fold in Kotlin) will do the job.
7. The final remainder is the CRC.
   That’s the same in software, of course.

The concrete implementation of a CRC algorithm in software always depends on the platform and the programming language you use. If you know how to work with bits and bytes on your respective platform, you should now be able to implement a CRC algorithm of your desired type on your own.

However, if you are an iOS or macOS developer and don’t want to out in all the effort to write your own implementation from scratch, we’ve dedicated another article for implementing a CRC with Swift on iOS. Another article for Android developers on how to implement a CRC in Kotlin is in our pipeline and coming soon. Stay tuned!

Summary

Congratulations! You’ve made it to the end of this article and should now have a broad understanding of what a Cyclic Redundancy Check (CRC) is and how it works. Here’s a short summary of this article:

• CRCs are codes that help us in detecting unintended errors in any transmitted or stored data.
• Burst errors are very common in data communication.
• A CRC-n can detect burst errors up to a size of n bits.
• Since the order of the bytes matters when calculating CRC, it can also detect when the sequence of bytes has changed.
• Using a different polynomial for a CRC algorithm will result in a different CRC for the same data.
• Endianness matters! Using the same polynomial with the opposite endianness will result in a completely different CRC.

Let us finish with some final thoughts on things you should be aware of:

• CRC implementations can be made more efficient by pre-calculating a so-called CRC lookup tables. Please check our other CRC articles to see example implementations that make use of such a lookup table.
• A CRC is only meant to detect unintended errors or changes to data. It does not provide any protection against malicious changes. If you want to make your data communication secure, you need to use cryptography instead. (A possible attacker can always recalculate the CRC after changing the data.) We’ve published a couple of articles on that topic as well (see below). 😉

Where to Go From Here

• If your company needs a team to help you implement CRCs or to develop an app with Bluetooth capabilities, reach out to us at: [email protected]
• If you are curious and would like to see a concrete implementation of a CRC algorithm, we’ve got you covered with two other articles:
  • Data Integrity: CRC with Swift on iOS
  • Data Integrity: CRC with Kotlin on Android
• You can also check out our GitHub repository with some sample implementations of CRC functions in Swift.
• If you’re not only interested in detecting accidental data transmission errors, but want to make sure that no one else can read or tamper with your sensitive data, read our articles on app security:
  • Mobile App Security: Best Practices on Android & iOS
  • iOS App Security: Best Practices
  • Android App Security: Best Practices

Are you a Software Developer?
Do you want to work with people that care about good software engineering?
Join our team in Munich

Body

Troubleshooting CRC Errors On Storage Networks

Before I say anything else, I will say that there is no «Easy Button» for troubleshooting CRC errors. It is an iterative process. You make a change, you monitor your fabric, and if necessary you make more changes until the issues are resolved. I frequently have customers who want it to be a one step process. It can be, but usually takes multiple steps. Having said that, before we can fix them, we need to know what CRC errors are and why they occur.

What Are CRC Errors? When Do they Occur?

The simple answer is that CRC errors are damaged frames. The more complicated answer is that before a fibre-channel frame is sent, some math is done. The answer is added to the frame footer. When the receiver gets the frame, the receiver repeats the math. If the receiver gets a different answer then what’s recorded in the frame, then the frame was changed in flight. This is a CRC error. The only time these happen is if the physical plant — cabling, SFPs is somehow defective. It is much less common, but still possible to have a bad component in the switch. Troubleshooting that will be a separate blog post, some day. What the receiver does with the damaged frame depends on whether it’s a switch or end device and if it is a switch, what brand of switch.

Why Does Fixing These Matter?

At best, the effect of faulty links is a few dropped frames. Left unchecked the problem will get worse and eventually cause performance problems. Also, you will go from 1 or 2 bad links to many. A customer I have been working with for the last several months was in this situation and is finally finishing a very long process of cleaning up many faulty links. Years ago I had customer that was experiencing extremely long delays on their Brocade fabric. They had over-redundancy (there is such a thing) on the switches and links between the hosts and the storage. Many of the links were questionable and producing CRC errors. When the storage received a bad frame, it simply dropped it and did not send an ABTS. They also had an adapter in the host with a bug in it, and it would simply sit and wait for the storage to respond. 90 or so seconds later, the application would time out and initiate recovery for a problem that should never have happened.

Why Does It Matter What Brand Of Switch It Is?

First, the different brands of switches use different commands to obtain the data you need to troubleshoot these problems. Secondly the way that they check and forward frames is different. This requires a different technique depending on the brand of switch. Cisco switches are what is called store-and-forward. This means that they wait for the entire frame to be received, then they check it, then if the frame is valid it gets forwarded. If not, it is dropped. Brocade switches are cut-through. As soon as they receive enough of the frame to know where it’s going, they start forwarding it. If the frame ends up being bad, they try to correct it using Foward Error Correction. If that doesn’t work the frame is tagged as bad. For the most part, end devices that receive frames that are already tagged as bad simply drop the frame and initiate recovery via ABTS. Troubleshooting commands and techniques vary for Brocade vs Cisco fabrics.

Identifying CRCs on Cisco Fabrics

Since Cisco fabrics are store-and-forward, you know that frames with CRC errors will be dropped as soon as they are detected. This can be either at the switch port they arrive on, or more rarely inside the switch. This post will focus on the CRC errors detected at the switch ports. If you suspect that you have questionable links, you can use these commands to check switch ports for CRC errors:

• ‘show interface’
• ‘show interface counters’
• ‘show logging log

For the above — the ‘show interface’ and ‘show interface counters’ commands can be run specifying a switch port that you are interested in. this is done in the format of fcS/P where S is the slot and P is the port. For the ‘show logging log’ you are looking for messages that a port was disabled because the bit error rate was too high. This is often an indicator of a faulty link. Once you find the ports that are detecting the CRC errors, you can then proceed to the repair phase.

Identifying CRCs on Brocade Fabrics

Brocade fabrics use cut-through routing. As such, the link for the port that is detecting the CRC errors may not be the faulty link. Brocade has two statistics for CRCs: CRC and CRC_Good_EOF. If the CRC_Good_EOF counter is increasing, this means that the link it is increasing on is the source of the problem. If the CRC counter is increasing, then the frame has already been marked as bad, and the problem is occuring elsewhere on the SAN. The CRC_Good_EOF should be the only counter that increases on a device port. If the CRC_Good_EOF counter is increasing on an ISL port, the link between the sending and receiving switch is bad. If you the CRC counter is increasing on the ISL, this means the problem is occuring somewhere on the sending switch. So move to the sending switch and look for ports where CRC_Good_EOF is increasing. It is possible that both counters will increase on a link. If it is a device port, then the link is bad. If it is an ISL then the link itself is a problem, and the sending switch has other bad links attached to it.    As you can see there are a few more steps to identify the source of the CRC errors on Brocade before you can proceed to the repair phase.  The porterrshow may also show ports that do not have CRC_Good_EOF increasing, but do show a counter called PCS increasing. If so, this is also and indication of a bad link. Troubleshooting PCS errors is the same as troubleshooting CRC_Good_EOF errors.

• ‘porterrshow’
• ‘portstatsshow N’

The porterrshow command will display error stats for all ports. The portstatsshow N where N is a port index number will display more detailed stats for the specified port. If you see PCS errors increasing for a port in the porterrshow, the link oin that port is bad, regardless of what CRC or CRC_Good_EOF counters there are.

Correcting the Problem

Once you have identified the port(s) that have questionable links you need to correct the problem. As I mentioned earlier, this is an iterative process. You replace a part, then clear the switch statistics, then monitor for anywhere from several hours to a day, depending on the rate of increase. Repeat the process until the errors are no longer increasing. You can replace multiple parts at once — such as replacing a cable and an SFP at the same time. Another option is to isolate further by just swapping a cable, or moving the device to a new port on the switch. Just remember that it is critical to reset the statistics immediately after any change you make.  REMEMBER THAT PATCH PANELS  ARE PART OF CABLING.  I emphasize that because customers will often replace the cable between the switch/device and the panel and forget that there is cabling between patch panels which is also suspect.  Some years ago I went onside to to troubleshoot connectivity between two storage systems.  The storage systems were located  at different campuses in the same city. The replication paths would not stay up.   When I got there, the client had them direct connected through several patch panels with no switching.  I assisted them in putting the cabling through switches at each campus and immediately saw CRCs showing up on the links.  They had 8 hops across patch panels between the storage systems.  We found CRCs at the second hop at each side. I stopped checking after that.  Their eventual permanent fix was to run a new direct run of cable between the two locations

If you have any questions, leave them in the comments or find me on   LinkedIn or on Twitter.