829:
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.
2367:. By no means does one algorithm, or one of each degree, suit every purpose; Koopman and Chakravarty recommend selecting a polynomial according to the application requirements and the expected distribution of message lengths. The number of distinct CRCs in use has confused developers, a situation which authors have sought to address. There are three polynomials reported for CRC-12, twenty-two conflicting definitions of CRC-16, and seven of CRC-32.
11635:
1425:
690:
1567:, which adds to the code the ability to detect all errors affecting an odd number of bits. In reality, all the factors described above should enter into the selection of the polynomial and may lead to a reducible polynomial. However, choosing a reducible polynomial will result in a certain proportion of missed errors, due to the quotient ring having
1859:
properties of the exclusive-or operation, practical table driven implementations can obtain a result numerically equivalent to zero-appending without explicitly appending any zeroes, by using an equivalent, faster algorithm that combines the message bitstream with the stream being shifted out of the CRC register.
1555:
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
2453:
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
1858:
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
824:
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
2402:
in August, and
Hammond, Brown and Liu's report for the Rome Laboratory, published in May. Both reports contained contributions from the other team. During December 1975, Brayer and Hammond presented their work in a paper at the IEEE National Telecommunications Conference: the IEEE CRC-32 polynomial
843:
11010011101100 100 <--- input with check value 1011 <--- divisor 01100011101100 100 <--- result 1011 <--- divisor ... 00111011101100 100 ...... 00000000001110 100 1011 00000000000101 100 101 1 ------------------ 00000000000000 000 <---
10006:
CRCs with even parity detect any odd number of bit errors, at the expense of lower hamming distance for long payloads. Note that parity is computed over the entire generator polynomial, including implied 1 at the beginning or the end. For example, the full representation of CRC-1 is 0x3, which has
828:
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
2407:
and was selected for its error detection performance. Even so, the
Castagnoli CRC-32C polynomial used in iSCSI or SCTP matches its performance on messages from 58 bits to 131 kbits, and outperforms it in several size ranges including the two most common sizes of Internet packet. The
650:
1885:: 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.
7745:
295:
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).
8998:
8765:
8058:
6481:
2370:
The polynomials commonly applied are not the most efficient ones possible. Since 1993, Koopman, Castagnoli and others have surveyed the space of polynomials between 3 and 64 bits in size, finding examples that have much better performance (in terms of
423:
7399:
1483:
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
7068:
6699:
311:
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
284:
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
1951:
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.
1504:
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.
8318:
839:
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.
9115:
8505:
671:
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
5988:
5815:
5107:
851:
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:
1872:
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
554:
10416:
10456:
5297:
7509:
2398:. 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
1508:
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.
801:). 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.
11100:
8769:
8545:
7829:
5643:
832:
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
2294:
2194:
2094:
2375:
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,
5636:
4785:
3649:
1578:
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
6858:
4889:
4620:
56:
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
4513:
3914:
3807:
3303:
1920:
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
4406:
4210:
1831:
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:
6278:
5357:
11580:
11096:
9980:. NTC 75 : National Telecommunications Conference, December 1–3, 1975, New Orleans, Louisiana. Vol. 1. Institute of Electrical and Electronics Engineers. pp. 8–21–5.
9101:
6101:
5496:
348:
4299:
4110:
2994:
1839:
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.
9258:
Cyclic
Redundancy Check (CRC) is an efficient method to ensure a low probability of undetected errors in data transmission using a checksum as a result of polynomial division.
7209:
2492:
9204:
Pundir, Meena; Sandhu, Jasminder Kaur (2021). "A Systematic Review of
Quality of Service in Wireless Sensor Networks using Machine Learning: Recent Trend and Future Vision".
4013:
1692:
10846:
3459:
3069:
2905:
9413:
9222:
Cyclic
Redundancy Check (CRC) mechanism is used to protect the data and provide protection of integrity from error bits when data is transmitted from sender to receiver.
6891:
11127:
10357:
10254:
10100:
6522:
3372:
2831:
2760:
1993:
1777:
1633:
821:
11010011101100 000 <--- input right padded by 3 bits 1011 <--- divisor (4 bits) = x³ + x + 1 ------------------ 01100011101100 000 <--- result
10425:
825:
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:
10462:
1945:
1809:
2644:
1815:
are a powerful class of such polynomials. They subsume the two examples above. Regardless of the reducibility properties of a generator polynomial of degree
1738:
836:
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).
304:
CRCs are specifically designed to protect against common types of errors on communication channels, where they can provide quick and reasonable assurance of the
11609:
1712:
1600:
546:
526:
506:
486:
466:
446:
10168:
125:
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
11449:
9618:
5529:
11549:
ISO/IEC 13239:2002: Information technology -- Telecommunications and information exchange between systems -- High-level data link control (HDLC) procedures
11087:
10778:
8167:
11573:
10942:
9705:
2458:
with degree greater than 1 are never primitive. Even parity polynomial marked as primitive in this table represent a primitive polynomial multiplied by
11373:
9270:
10491:
11401:
10647:
9436:
The presented methods offer a very easy and efficient way to modify your data so that it will compute to a CRC you want or at least know in advance.
8365:
1442:
711:
1877:
transmissions are CRC-checked in hardware, because some widespread serial-port transmission conventions transmit bytes least-significant bit first.
1635:, and the associated code is able to detect any single-bit or double-bit errors. We can improve this situation. If we use the generator polynomial
11038:
9179:
9154:
7093:
5856:
5677:
2538:
2533:
2509:
1855:
1854:
being the size of the CRC) to the bitstream to be checked before the polynomial division occurs. Such appending is explicitly demonstrated in the
1478:
751:
61:
10514:
4975:
10905:
9234:
Schiller, Frank; Mattes, Tina (2007). "Analysis of CRC-Polynomials for Safety-Critical
Communication by Deterministic and Stochastic Automata".
9905:
10205:
9848:
331:
Secondly, unlike cryptographic hash functions, CRC is an easily reversible function, which makes it unsuitable for use in digital signatures.
316:
modification of data. Any application that requires protection against such attacks must use cryptographic authentication mechanisms, such as
11566:
11264:
10543:
9834:
9691:
9585:
9359:
9251:
1811:
that admits other factorizations may be chosen then so as to balance the maximal total blocklength with a desired error detection power. The
645:{\displaystyle \operatorname {CRC} (x\oplus y\oplus z)=\operatorname {CRC} (x)\oplus \operatorname {CRC} (y)\oplus \operatorname {CRC} (z);}
133:: contiguous sequences of erroneous data symbols in messages. This is important because burst errors are common transmission errors in many
1582:) 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
2351:
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
11300:
Mitra, Jubin; Nayak, Tapan (January 2017). "Reconfigurable very high throughput low latency VLSI (FPGA) design architecture of CRC 32".
1575:
1492:(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
11065:
9515:
7740:{\displaystyle 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}
11893:
11516:
10681:
10322:
6000:
5191:
1464:
737:
129:
in 1961. Cyclic codes are not only simple to implement but have the benefit of being particularly well suited for the detection of
10977:
9760:
Castagnoli, G.; Bräuer, S.; Herrmann, M. (June 1993). "Optimization of Cyclic
Redundancy-Check Codes with 24 and 32 Parity Bits".
2386:
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
11468:
10879:
10837:
9738:
8993:{\displaystyle 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}
8760:{\displaystyle 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}+}
11215:
10804:
8053:{\displaystyle 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}
2391:
664:
10348:
10091:
10028:
9420:
11781:
11614:
11124:
11008:
10393:
10287:
10245:
10133:
9659:
Koopman, Philip; Chakravarty, Tridib (June 2004). "Cyclic redundancy code (CRC) polynomial selection for embedded networks".
9390:
2840:
1446:
715:
57:
10742:
9952:
9909:
11786:
11182:
9174:
6015:
848:
10712:
9546:
2202:
2102:
2002:
11624:
9008:
7081:
1511:
The most commonly used polynomial lengths are 9 bits (CRC-8), 17 bits (CRC-16), 33 bits (CRC-32), and 65 bits (CRC-64).
76:
5556:
11805:
9164:
4705:
3468:
2435:
317:
3569:
48:
and storage devices to detect accidental changes to digital data. Blocks of data entering these systems get a short
10845:. V2.5.1. Sophia Antipolis, France: European Telecommunications Standards Institute. August 2013. pp. 99, 101.
9878:
11888:
11744:
11422:
11331:
11151:
10155:
6785:
4816:
3931:
2514:
45:
11498:
11446:
10253:. V1.2.1. Sophia Antipolis, France: European Telecommunications Standards Institute. January 2003. pp. 67–8.
9614:
4547:
11649:
7132:
6476:{\displaystyle 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}
4440:
4024:
3841:
3734:
3230:
10765:
10213:. Revision D version 2.0. 3rd Generation Partnership Project 2. October 2005. pp. 2–89–2–92. Archived from
9591:
8517:
1819:, if it includes the "+1" term, the code will be able to detect error patterns that are confined to a window of
11831:
9184:
4339:
4143:
700:
673:
161:
53:
10951:
9660:
10356:. V1.3.1. Sophia Antipolis, France: European Telecommunications Standards Institute. March 2013. p. 17.
418:{\displaystyle \operatorname {CRC} (x\oplus y)=\operatorname {CRC} (x)\oplus \operatorname {CRC} (y)\oplus c}
308:
of messages delivered. However, they are not suitable for protecting against intentional alteration of data.
11366:
9278:
9041:
5165:
3934:
1435:
719:
704:
10485:
6047:
5442:
11898:
11868:
11394:
11164:(Note: MpCRC.html is included with the Matpack compressed software source code, under /html/LibDoc/Crypto)
10654:
9812:
9669:
9489:
9169:
6198:
1557:
1140:"""Calculate the CRC check of a string of bits using a chosen polynomial."""
165:
99:
7394:{\displaystyle 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}
4245:
4056:
2940:
2390:
and the Air Force
Electronic Systems Division by Joseph Hammond, James Brown and Shyan-Shiang Liu of the
184:, so the addition operation can always be performed bitwise-parallel (there is no carry between digits).
11734:
11729:
11702:
3942:
236:
141:-bit CRC applied to a data block of arbitrary length will detect any single error burst not longer than
134:
118:
41:
11030:
9381:
785:. The polynomial is written in binary as the coefficients; a 3rd-degree polynomial has 4 coefficients (
10914:
10805:"Air Ground Data Link VHF Airline Communications and Reporting System (ACARS) Preliminary Test Report"
10518:
2461:
11903:
11739:
11654:
9981:
9944:
9159:
6008:
5647:
4315:
4020:
3966:
3003:
2494:. The most significant bit of a polynomial is always 1, and is not shown in the hex representations.
2344:
1526:, multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree
232:
206:, multiple CRCs are possible, each with a different polynomial. Such a polynomial has highest degree
153:
10461:. 1.4.0. Berlin: Ethernet POWERLINK Standardisation Group. 13 March 2013. p. 42. Archived from
9674:
1638:
11664:
11659:
11429:
11250:
10214:
10099:. V8.9.0. Sophia Antipolis, France: European Telecommunications Standards Institute. January 2005.
9817:
9803:
9494:
7063:{\displaystyle 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}
6493:
5410:
4413:
3418:
3394:
3028:
2864:
2352:
1903:
239:, since they are always 1. The CRC and associated polynomial typically have a name of the form CRC-
95:
6694:{\displaystyle 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}
11795:
11590:
10629:
10594:
9840:
9697:
9507:
9317:
8512:
5116:
4119:
3132:
2364:
1911:
1493:
325:
321:
126:
11352:
11553:
3338:
2797:
2726:
1959:
1743:
11279:
11260:
11246:
10680:. Research Department, Engineering Division, The British Broadcasting Corporation. p. 9.
10539:
10533:
9989:
9830:
9687:
9581:
9355:
9247:
2415:
standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for
2395:
1579:
91:
9902:
Evaluation of 32 Degree Polynomials in Error Detection on the SATIN IV Autovon Error Patterns
1605:
94:, easy to analyze mathematically, and particularly good at detecting common errors caused by
11669:
11604:
11491:
11309:
11061:
10621:
10586:
9822:
9769:
9679:
9499:
9347:
9309:
9239:
9213:
4951:
2523:
2439:
2372:
1947:
or 1 term). This convention encodes the polynomial complete with its degree in one integer.
1602:
is the degree of the primitive generator polynomial, then the maximal total block length is
884:"""Calculate the CRC remainder of a string of bits using a chosen polynomial.
9474:
9120:
1923:
191:. The two elements are usually called 0 and 1, comfortably matching computer architecture.
11619:
11453:
11131:
9343:
7437:
4032:
2387:
1785:
815:
339:
335:
122:
11509:
10672:
10313:
9802:
Koopman, Philip (July 2002). "32-bit cyclic redundancy codes for Internet applications".
2623:
2445:
architecture also provides hardware acceleration for both CRC-32 and CRC-32C operations.
1717:
11634:
9985:
9948:
2383:
have adopted one of the findings of this research, the CRC-32C (Castagnoli) polynomial.
774:
In this example, we shall encode 14 bits of message with a 3-bit CRC, with a polynomial
11836:
11256:
10969:
9243:
3390:
2416:
1697:
1585:
531:
511:
491:
471:
451:
431:
305:
145:
bits, and the fraction of all longer error bursts that it will detect is approximately
17:
11460:
10872:
9734:
9132:
1779:, and the code is able to detect single, double, triple and any odd number of errors.
11882:
11810:
11327:
10598:
8313:{\displaystyle x^{32}+x^{31}+x^{24}+x^{22}+x^{16}+x^{14}+x^{8}+x^{7}+x^{5}+x^{3}+x+1}
1914:, some writers assume that it is unnecessary to mention the divisor's high-order bit.
656:
103:
11367:"A Systematic Approach to Building High Performance, Software-based, CRC generators"
11208:
10811:
10633:
10020:
9844:
9321:
277:, for each block of data to be sent or stored and appends it to the data, forming a
269:
A CRC-enabled device calculates a short, fixed-length binary sequence, known as the
11852:
11760:
11000:
10913:. 1.0. Vol. 9. Profibus International. March 1998. p. 906. Archived from
10385:
10279:
10125:
9511:
9137:
3927:
2404:
1568:
668:
660:
181:
11558:
11510:"Cyclic Redundancy Code and Checksum Algorithms to Ensure Critical Data Integrity"
9701:
9376:
9300:
Peterson, W. W.; Brown, D. T. (January 1961). "Cyclic Codes for Error Detection".
760:-bit binary CRC, line the bits representing the input in a row, and position the (
9975:
9121:
C class code for CRC checksum calculation with many different CRCs to choose from
3313:
2915:
2772:
11724:
10734:
9932:
9901:
9351:
5828:
3472:
2348:
1874:
1424:
689:
177:
130:
115:
84:
11313:
11175:
10321:(Thesis). Philip Koopman, advisor. Carnegie Mellon University. pp. 5, 18.
9313:
9217:
11815:
11799:
11791:
11536:
10767:
A signalling standard for trunked private land mobile radio systems (MPT 1327)
10706:
9550:
9012:
8500:{\displaystyle x^{40}+x^{26}+x^{23}+x^{17}+x^{3}+1=(x^{23}+1)(x^{17}+x^{3}+1)}
7097:
6135:
6108:
5387:
3662:
2598:
1891:
of the divisor polynomial: Since the high-order bit is always 1, and since an
1881:
768:
251:
11320:
11176:"Cyclic redundancy check computation: an implementation using the TMS320C54x"
9826:
9683:
9412:
Stigge, Martin; Plötz, Henryk; Müller, Wolf; Redlich, Jens-Peter (May 2006).
187:
In practice, all commonly used CRCs employ the finite field of two elements,
11776:
11145:
10164:
9993:
5983:{\displaystyle x^{16}+x^{13}+x^{12}+x^{11}+x^{10}+x^{8}+x^{6}+x^{5}+x^{2}+1}
5832:
5810:{\displaystyle x^{16}+x^{15}+x^{11}+x^{9}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x+1}
5402:
4306:
3814:
173:
80:
10612:
Ramabadran, T.V.; Gaitonde, S.S. (1988). "A tutorial on CRC computations".
10590:
9977:
Evaluation of error detection polynomial performance on the AUTOVON channel
9473:
Cam-Winget, Nancy; Housley, Russ; Wagner, David; Walker, Jesse (May 2003).
9448:
5102:{\displaystyle x^{13}+x^{12}+x^{11}+x^{10}+x^{7}+x^{6}+x^{5}+x^{4}+x^{2}+1}
2363:
Numerous varieties of cyclic redundancy checks have been incorporated into
98:
in transmission channels. Because the check value has a fixed length, the
11548:
9874:
9503:
11696:
9149:
8521:
8330:
7101:
6488:
6191:
6162:
5503:
4896:
4627:
3103:
3076:
1812:
169:
11435:
11335:
10810:. Federal Aviation Authority Technical Center. p. 5. Archived from
10458:
openSAFETY Safety Profile Specification: EPSG Working Draft Proposal 304
9419:. Humboldt University Berlin. p. 17. SAR-PR-2006-05. Archived from
4523:
3386:
3382:
11359:. Systems Research Group, Computer Laboratory, University of Cambridge.
11283:
9805:
Proceedings International Conference on Dependable Systems and Networks
6254:
5304:
4681:
3656:
2442:
1449: in this section. Unsourced material may be challenged and removed.
157:
9933:"Development of a Transmission Error Model and an Error Control Model"
9773:
8071:
Excellent at Ethernet frame length, poor performance with long files
7759:
Excellent at Ethernet frame length, poor performance with long files
292:
If the CRC values do not match, then the block contains a data error.
11717:
11691:
11060:
Boutell, Thomas; Randers-Pehrson, Glenn; et al. (14 July 1998).
10674:
L.F. Radio-Data: specification of BBC experimental transmissions 1982
9573:
9016:
7136:
7109:
6004:
5406:
5398:
4029:
3497:
2454:
ordering, while Bzip2 does not. Note that even parity polynomials in
10625:
11485:
11443:— theory, practice, hardware, and software with emphasis on CRC-32.
10950:. 1.0. Robert Bosch GmbH. 17 April 2012. p. 13. Archived from
10049:
Payload means length exclusive of CRC field. A Hamming distance of
9931:
Hammond, Joseph L. Jr.; Brown, James E.; Liu, Shyan-Shiang (1975).
9236:
Fault Detection, Supervision and Safety of Technical Processes 2006
11712:
11707:
11209:"An Improved 64-bit Cyclic Redundancy Check for Protein Sequences"
10774:
10424:. 4.2.2. Munich: AUTOSAR. 22 July 2015. p. 24. Archived from
9338:
Ergen, Mustafa (21 January 2008). "2.3.3 Error Detection Coding".
7429:
7413:
7128:
7125:
7121:
7113:
7089:
6706:
5414:
5352:
3921:
2769:
2455:
2431:
2409:
2376:
1489:
188:
90:. CRCs are popular because they are simple to implement in binary
11323:, MathPages, overview of error-detection of different polynomials
9662:
International Conference on Dependable Systems and Networks, 2004
176:
becomes the result. The important caveat is that the polynomial
11857:
10708:
Cyclic Redundancy Check (CRC): PSoC Creator™ Component Datasheet
8143:
7433:
7425:
7421:
7417:
7117:
7105:
7085:
6867:
6712:
6497:
6012:
5822:
5652:
5391:
5383:
4311:
3938:
2427:
2422:
CRC-32C computation is implemented in hardware as an operation (
2412:
2399:
2380:
810:
This is first padded with zeros corresponding to the bit length
254:, is in fact a 1-bit CRC: it uses the generator polynomial
11562:
10490:. Vol. 2. Bluetooth SIG. 2 December 2014. pp. 144–5.
9572:
Press, WH; Teukolsky, SA; Vetterling, WT; Flannery, BP (2007).
818:
form. Here is the first calculation for computing a 3-bit CRC:
814:
of the CRC. This is done so that the resulting code word is in
231:
bits. Note that most polynomial specifications either drop the
152:
Specification of a CRC code requires definition of a so-called
137:, including magnetic and optical storage devices. Typically an
11675:
11508:
Koopman, Philip; Driscoll, Kevin; Hall, Brendan (March 2015).
8338:
7077:
5995:
5292:{\displaystyle x^{15}+x^{14}+x^{10}+x^{8}+x^{7}+x^{4}+x^{3}+1}
5139:
4922:
4653:
4217:
3155:
2651:
1418:
683:
11395:"CRC Cyclic Redundancy Check Analysing and Correcting Errors"
11062:"PNG (Portable Network Graphics) Specification, Version 1.2"
10315:
Performance of Cyclic Redundancy Codes for Embedded Networks
10207:
Physical layer standard for cdma2000 spread spectrum systems
9271:"An Algorithm for Error Correcting Cyclic Redundance Checks"
10648:"Longwave Radio Data Decoding using and HC11 and an MC3371"
9116:
Implementation of CRC32 in GNU Radio up to 3.6.1 (ca. 2012)
1823:
contiguous bits. These patterns are called "error bursts".
10455:"5.1.1.8 Cyclic Redundancy Check field (CRC-8 / CRC-16)".
10653:. Freescale Semiconductor. 2004. AN1597/D. Archived from
9974:
Brayer, Kenneth; Hammond, Joseph L. Jr. (December 1975).
9547:"A Painless Guide to CRC Error Detection Algorithms V3.0"
8329:
Often confused to be a CRC, but actually a checksum; see
6197:
Often confused to be a CRC, but actually a checksum; see
767:)-bit pattern representing the CRC's divisor (called a "
10565:. 3.0.1. Flexray Consortium. October 2010. p. 114.
334:
Thirdly, CRC satisfies a relation similar to that of a
11097:
European Organisation for the Safety of Air Navigation
10711:. Cypress Semiconductor. 20 February 2013. p. 4.
217:
terms. In other words, the polynomial has a length of
11515:. Federal Aviation Administration. DOT/FAA/TC-14/49.
10450:
10448:
10446:
10347:"5.1.4 CRC-8 encoder (for packetized streams only)".
10307:
10305:
9044:
8772:
8548:
8368:
8170:
7832:
7512:
7212:
6894:
6788:
6525:
6281:
6050:
5859:
5680:
5559:
5445:
5194:
4978:
4819:
4708:
4550:
4443:
4342:
4248:
4146:
4059:
3969:
3844:
3737:
3572:
3421:
3341:
3233:
3031:
2943:
2867:
2800:
2729:
2626:
2464:
2205:
2105:
2005:
1962:
1926:
1788:
1746:
1720:
1700:
1641:
1608:
1588:
557:
534:
514:
494:
474:
454:
434:
351:
11332:"A Painless Guide to CRC Error Detection Algorithms"
11276:
Understanding Checksums and Cyclic Redundancy Checks
10015:
10013:
9574:"Section 22.4 Cyclic Redundancy and Other Checksums"
1488:—coefficients that are elements of the finite field
887:
initial_filler should be '1' or '0'.
11845:
11824:
11769:
11753:
11684:
11642:
11597:
2423:
2289:{\displaystyle (1x^{4}+0x^{3}+0x^{2}+1x^{1})+x^{0}}
2189:{\displaystyle (1x^{0}+1x^{1}+0x^{2}+0x^{3})+x^{4}}
2089:{\displaystyle x^{4}+(0x^{3}+0x^{2}+1x^{1}+1x^{0})}
11490:— includes links to PDFs giving 16 and 32-bit CRC
10970:"OS-9 Operating System System Programmer's Manual"
10803:Rehmann, Albert; Mestre, José D. (February 1995).
9578:Numerical Recipes: The Art of Scientific Computing
9449:"algorithm design – Why is CRC said to be linear?"
9095:
8992:
8759:
8499:
8312:
8052:
7739:
7393:
7062:
6852:
6693:
6475:
6095:
5982:
5809:
5630:
5490:
5291:
5101:
4883:
4779:
4614:
4507:
4400:
4293:
4204:
4104:
4007:
3908:
3801:
3643:
3453:
3366:
3297:
3063:
2988:
2899:
2825:
2754:
2638:
2486:
2288:
2188:
2088:
1987:
1939:
1803:
1771:
1732:
1706:
1686:
1627:
1594:
644:
540:
520:
500:
480:
460:
440:
417:
11538:Mechanics of Cyclic Redundancy Check Calculations
11499:"Why Life Critical Networks Tend To Provide HD=6"
10415:"7.2.1.2 8-bit 0x2F polynomial CRC Calculation".
9728:
9726:
5631:{\displaystyle x^{16}+x^{10}+x^{8}+x^{7}+x^{3}+1}
1560:or irreducible polynomials times the factor
655:as a result, even if the CRC is encrypted with a
10577:Perez, A. (1983). "Byte-Wise CRC Calculations".
10515:"XFCNs for Cyclic Redundancy Check Calculations"
10200:
10198:
10196:
10194:
10192:
10190:
10070:is always achieved for arbitrarily long messages
10057: − 1 bit errors can be detected and ⌊(
4780:{\displaystyle x^{11}+x^{9}+x^{8}+x^{7}+x^{2}+1}
180:are calculated according to the arithmetic of a
10557:
10555:
3644:{\displaystyle x^{8}+x^{7}+x^{6}+x^{4}+x^{2}+1}
27:Error-detecting code for detecting data changes
10937:
10935:
10061: − 1)/2⌋ bit errors can be corrected
9475:"Security Flaws in 802.11 Data Link Protocols"
1866:into the remainder of the polynomial division.
11574:
11134:version 8.9.0 (January 2005), Section 4.1.2 a
10735:"Cyclic redundancy check (CRC) in CAN frames"
10239:
10237:
10235:
10086:
10084:
10082:
10080:
10078:
10076:
9540:
9538:
9536:
6853:{\displaystyle x^{24}+x^{23}+x^{6}+x^{5}+x+1}
4884:{\displaystyle x^{12}+x^{11}+x^{3}+x^{2}+x+1}
8:
10380:
10378:
9580:(3rd ed.). Cambridge University Press.
9333:
9331:
9206:Journal of Network and Computer Applications
4615:{\displaystyle x^{10}+x^{9}+x^{5}+x^{4}+x+1}
488:. This can be also stated as follows, where
102:that generates it is occasionally used as a
11461:"Catalogue of parameterised CRC algorithms"
10538:. Cambridge University Press. p. 223.
4508:{\displaystyle x^{8}+x^{7}+x^{4}+x^{3}+x+1}
3909:{\displaystyle x^{8}+x^{7}+x^{5}+x^{2}+x+1}
3802:{\displaystyle x^{8}+x^{5}+x^{3}+x^{2}+x+1}
3298:{\displaystyle x^{6}+x^{5}+x^{3}+x^{2}+x+1}
1842:Usually, but not always, an implementation
718:. Unsourced material may be challenged and
11581:
11567:
11559:
11389:, Slicing-by-4 and slicing-by-8 algorithms
9735:"Catalogue of parametrised CRC algorithms"
9617:. Christchurch: University of Canterbury.
10944:CAN with Flexible Data-Rate Specification
9816:
9797:
9795:
9793:
9791:
9789:
9787:
9785:
9783:
9673:
9654:
9652:
9650:
9648:
9646:
9644:
9642:
9640:
9638:
9636:
9493:
9075:
9062:
9049:
9043:
8972:
8959:
8946:
8933:
8920:
8907:
8894:
8881:
8868:
8855:
8842:
8829:
8816:
8803:
8790:
8777:
8771:
8748:
8735:
8722:
8709:
8696:
8683:
8670:
8657:
8644:
8631:
8618:
8605:
8592:
8579:
8566:
8553:
8547:
8482:
8469:
8447:
8425:
8412:
8399:
8386:
8373:
8367:
8292:
8279:
8266:
8253:
8240:
8227:
8214:
8201:
8188:
8175:
8169:
8032:
8019:
8006:
7993:
7980:
7967:
7954:
7941:
7928:
7915:
7902:
7889:
7876:
7863:
7850:
7837:
7831:
7725:
7712:
7699:
7686:
7673:
7660:
7647:
7634:
7621:
7608:
7595:
7582:
7569:
7556:
7543:
7530:
7517:
7511:
7373:
7360:
7347:
7334:
7321:
7308:
7295:
7282:
7269:
7256:
7243:
7230:
7217:
7211:
7042:
7029:
7016:
7003:
6990:
6977:
6964:
6951:
6938:
6925:
6912:
6899:
6893:
6832:
6819:
6806:
6793:
6787:
6673:
6660:
6647:
6634:
6621:
6608:
6595:
6582:
6569:
6556:
6543:
6530:
6524:
6455:
6442:
6429:
6416:
6403:
6390:
6377:
6364:
6351:
6338:
6325:
6312:
6299:
6286:
6280:
6081:
6068:
6055:
6049:
5968:
5955:
5942:
5929:
5916:
5903:
5890:
5877:
5864:
5858:
5789:
5776:
5763:
5750:
5737:
5724:
5711:
5698:
5685:
5679:
5616:
5603:
5590:
5577:
5564:
5558:
5476:
5463:
5450:
5444:
5277:
5264:
5251:
5238:
5225:
5212:
5199:
5193:
5087:
5074:
5061:
5048:
5035:
5022:
5009:
4996:
4983:
4977:
4863:
4850:
4837:
4824:
4818:
4765:
4752:
4739:
4726:
4713:
4707:
4594:
4581:
4568:
4555:
4549:
4487:
4474:
4461:
4448:
4442:
4401:{\displaystyle x^{8}+x^{4}+x^{3}+x^{2}+1}
4386:
4373:
4360:
4347:
4341:
4279:
4266:
4253:
4247:
4205:{\displaystyle x^{8}+x^{5}+x^{4}+x^{3}+1}
4190:
4177:
4164:
4151:
4145:
4090:
4077:
4064:
4058:
3987:
3974:
3968:
3888:
3875:
3862:
3849:
3843:
3781:
3768:
3755:
3742:
3736:
3629:
3616:
3603:
3590:
3577:
3571:
3439:
3426:
3420:
3346:
3340:
3277:
3264:
3251:
3238:
3232:
3049:
3036:
3030:
2974:
2961:
2948:
2942:
2885:
2872:
2866:
2805:
2799:
2734:
2728:
2625:
2463:
2280:
2264:
2248:
2232:
2216:
2204:
2180:
2164:
2148:
2132:
2116:
2104:
2077:
2061:
2045:
2029:
2010:
2004:
1967:
1961:
1931:
1925:
1787:
1751:
1745:
1740:, then the maximal total block length is
1719:
1699:
1640:
1613:
1607:
1587:
1465:Learn how and when to remove this message
738:Learn how and when to remove this message
556:
533:
513:
493:
473:
453:
433:
350:
250:The simplest error-detection system, the
11361:Algorithm 4 was used in Linux and Bzip2.
9133:Catalogue of parametrised CRC algorithms
6018:, SIA DC-07, many others; also known as
2499:
2302:
9545:Williams, Ross N. (24 September 1996).
9196:
9180:Mathematics of cyclic redundancy checks
9155:Computation of cyclic redundancy checks
1479:Mathematics of cyclic redundancy checks
771:") underneath the left end of the row.
752:Computation of cyclic redundancy checks
75:(it expands the message without adding
10907:PROFIBUS Specification Normative Parts
10671:Ely, S.R.; Wright, D.T. (March 1982).
10157:Class-1 Generation-2 UHF RFID Protocol
9906:National Technical Information Service
9854:from the original on 16 September 2012
9711:from the original on 11 September 2011
9096:{\displaystyle x^{64}+x^{4}+x^{3}+x+1}
2300:In the table below they are shown as:
1537:terms (the polynomial has a length of
804:Start with the message to be encoded:
52:attached, based on the remainder of a
11379:from the original on 16 December 2006
11147:Matpack documentation: Crypto – Codes
11144:Gammel, Berndt M. (31 October 2005).
11106:from the original on 20 November 2018
11068:from the original on 3 September 2011
10764:"3.2.3 Encoding and error checking".
10487:Specification of the Bluetooth System
10312:Chakravarty, Tridib (December 2001).
10260:from the original on 28 December 2015
10007:two 1 bits. Thus, its parity is even.
9955:from the original on 31 December 2021
9912:from the original on 31 December 2021
9615:"Reverse-Engineering a CRC Algorithm"
9414:"Reversing CRC – Theory and Practice"
6096:{\displaystyle x^{16}+x^{15}+x^{2}+1}
5491:{\displaystyle x^{16}+x^{12}+x^{5}+1}
2430:instruction set, first introduced in
1548:. The CRC has a name of the form CRC-
261:(two terms), and has the name CRC-1.
7:
10745:from the original on 1 February 2016
10715:from the original on 2 February 2016
10687:from the original on 12 October 2013
10532:Richardson, Andrew (17 March 2005).
9881:from the original on 20 January 2016
1714:is a primitive polynomial of degree
1447:adding citations to reliable sources
716:adding citations to reliable sources
11447:Reverse-Engineering a CRC Algorithm
11154:from the original on 25 August 2013
10363:from the original on 30 August 2017
10328:from the original on 1 January 2014
9873:Koopman, Philip (21 January 2016).
9762:IEEE Transactions on Communications
4294:{\displaystyle x^{8}+x^{6}+x^{3}+1}
4105:{\displaystyle x^{8}+x^{5}+x^{4}+1}
2989:{\displaystyle x^{5}+x^{4}+x^{2}+1}
11471:from the original on 1 August 2020
10999:Koopman, Philip P. (20 May 2018).
10513:Whitfield, Harry (24 April 2001).
10494:from the original on 26 March 2015
10174:from the original on 19 March 2012
10106:from the original on 17 April 2018
10031:from the original on 19 March 2018
9741:from the original on 1 August 2020
9621:from the original on 7 August 2011
9393:from the original on 16 April 2009
2403:is the generating polynomial of a
1954:In each case, one term is omitted.
25:
11407:from the original on 11 June 2007
11188:from the original on 14 June 2012
11041:from the original on 18 July 2018
11011:from the original on 7 April 2018
10980:from the original on 17 July 2018
10885:from the original on 28 July 2011
10873:"16-bit CRC polynomial selection"
10784:from the original on 14 July 2012
10396:from the original on 7 April 2018
10290:from the original on 7 April 2018
10244:"11. Error correction strategy".
10136:from the original on 7 April 2018
9937:NASA Sti/Recon Technical Report N
9594:from the original on 13 July 2024
1864:exclusive-ORs a fixed bit pattern
1518:-bit CRC when its check value is
198:-bit CRC when its check value is
164:, which takes the message as the
11633:
11535:Koopman, Philip (January 2023).
11522:from the original on 18 May 2015
11486:"Blog: Checksum and CRC Central"
11365:Kounavis, M.; Berry, F. (2005).
11221:from the original on 7 June 2011
11181:. Texas Instruments. p. 5.
10852:from the original on 1 July 2015
9613:Ewing, Gregory C. (March 2010).
9521:from the original on 26 May 2013
9244:10.1016/b978-008044485-7/50159-7
2487:{\displaystyle \left(x+1\right)}
2304:Examples of CRC representations
1895:-bit CRC must be defined by an (
1423:
688:
114:CRCs are based on the theory of
11174:Geremia, Patrick (April 1999).
10167:. 23 October 2008. p. 35.
9900:Brayer, Kenneth (August 1975).
9375:Ritter, Terry (February 1986).
4008:{\displaystyle x^{8}+x^{2}+x+1}
2392:Georgia Institute of Technology
2355:using straightforward methods.
1434:needs additional citations for
663:as its combining operation (or
71:(data verification) value is a
67:CRCs are so called because the
11615:Common Language Infrastructure
11554:CRC32-Castagnoli Linux Library
10871:Thaler, Pat (28 August 2003).
10563:FlexRay Protocol Specification
9877:. Carnegie Mellon University.
9238:. Elsevier. pp. 944–949.
9019:; considered weak for hashing
8494:
8462:
8459:
8440:
5331:Optimal for payloads ≤64 bits
2270:
2206:
2170:
2106:
2083:
2019:
1889:Omission of the high-order bit
1798:
1792:
1687:{\displaystyle g(x)=p(x)(1+x)}
1681:
1669:
1666:
1660:
1651:
1645:
636:
630:
618:
612:
600:
594:
582:
564:
406:
400:
388:
382:
370:
358:
156:. This polynomial becomes the
1:
11302:Integration, the VLSI Journal
11247:"14. Cyclic Redundancy Check"
11245:Warren Jr., Henry S. (2013).
11214:. University College London.
10418:Specification of CRC Routines
9943:(published May 1975): 15344.
9733:Cook, Greg (15 August 2020).
3454:{\displaystyle x^{7}+x^{3}+1}
3064:{\displaystyle x^{5}+x^{2}+1}
2900:{\displaystyle x^{5}+x^{3}+1}
2596:most hardware; also known as
1918:Omission of the low-order bit
324:(which are commonly based on
10567:(4.2.8 Header CRC (11 bits))
1862:Sometimes an implementation
1837:prefixes a fixed bit pattern
1835:Sometimes an implementation
1574:The advantage of choosing a
1544:). The remainder has length
318:message authentication codes
10904:"8.8.4 Check Octet (FCS)".
9453:Cryptography Stack Exchange
9352:10.1007/978-0-387-68192-4_2
9165:List of checksum algorithms
3469:Train Communication Network
2522:Maximum bits of payload by
2199:0x9 = 0b1001, representing
2099:0xC = 0b1100, representing
1999:0x3 = 0b0011, representing
11920:
11745:Holographic Versatile Disc
11393:Kowalk, W. (August 2006).
11314:10.1016/j.vlsi.2016.09.005
10484:"B.7.1.1 HEC generation".
9314:10.1109/JRPROC.1961.287814
9218:10.1016/j.jnca.2021.103084
3941:HEC and cell delineation,
2521:
2510:Polynomial representations
2449:Polynomial representations
2394:and Kenneth Brayer of the
1476:
749:
244:
11866:
11650:Advanced Intelligent Tape
11631:
11423:"Cyclic Redundancy Check"
11400:. Universität Oldenburg.
9482:Communications of the ACM
9038:
9033:
9006:
9003:
8542:
8537:
8516:
8510:
8362:
8357:
8342:
8336:
8328:
8164:
8159:
8141:
8138:
7826:
7821:
7818:
7815:
7812:
7809:
7806:
7803:
7800:
7797:
7794:
7791:
7788:
7785:
7782:
7779:
7776:
7773:
7758:
7750:
7506:
7501:
7498:
7495:
7492:
7489:
7486:
7483:
7480:
7477:
7474:
7471:
7468:
7465:
7462:
7459:
7456:
7453:
7412:
7404:
7206:
7201:
7198:
7195:
7192:
7189:
7186:
7183:
7180:
7177:
7174:
7171:
7168:
7165:
7162:
7159:
7156:
7153:
7076:
7073:
6888:
6883:
6866:
6863:
6782:
6777:
6774:
6771:
6768:
6765:
6762:
6759:
6756:
6753:
6750:
6747:
6744:
6741:
6738:
6735:
6732:
6729:
6710:
6704:
6519:
6514:
6492:
6486:
6275:
6270:
6253:
6250:
6196:
6044:
6039:
5999:
5993:
5853:
5848:
5826:
5820:
5674:
5669:
5651:
5641:
5553:
5548:
5533:
5527:
5439:
5434:
5382:
5379:
5188:
5183:
5169:
5163:
4972:
4967:
4949:
4946:
4813:
4808:
4793:
4790:
4702:
4697:
4680:
4677:
4544:
4539:
4521:
4518:
4437:
4432:
4417:
4411:
4336:
4331:
4310:
4304:
4242:
4237:
4222:
4215:
4140:
4135:
4118:
4115:
4053:
4048:
4028:
4018:
3963:
3958:
3925:
3919:
3838:
3833:
3818:
3812:
3731:
3726:
3723:
3720:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3690:
3687:
3684:
3681:
3678:
3660:
3654:
3566:
3561:
3558:
3555:
3552:
3549:
3546:
3543:
3540:
3537:
3534:
3531:
3528:
3525:
3522:
3519:
3516:
3513:
3496:
3493:
3415:
3410:
3380:
3377:
3367:{\displaystyle x^{6}+x+1}
3335:
3330:
3311:
3308:
3227:
3222:
3219:
3216:
3213:
3210:
3207:
3204:
3201:
3198:
3195:
3192:
3189:
3186:
3183:
3180:
3177:
3174:
3159:
3153:
3025:
3020:
3002:
2999:
2937:
2932:
2913:
2910:
2861:
2856:
2839:
2836:
2826:{\displaystyle x^{4}+x+1}
2794:
2789:
2768:
2765:
2755:{\displaystyle x^{3}+x+1}
2723:
2718:
2715:
2712:
2709:
2706:
2703:
2700:
2697:
2694:
2691:
2688:
2685:
2682:
2679:
2676:
2673:
2670:
2655:
2649:
2620:
2615:
2595:
2592:
2518:
2513:
2508:
2505:
2502:
1988:{\displaystyle x^{4}+x+1}
1772:{\displaystyle 2^{r-1}-1}
448:depends on the length of
44:commonly used in digital
11894:Cyclic redundancy checks
11353:"Fast CRC32 in Software"
11321:Cyclic Redundancy Checks
11274:Koopman, Philip (2024).
10777:. June 1997. p. 3.
9827:10.1109/DSN.2002.1028931
9684:10.1109/DSN.2004.1311885
9185:Simple file verification
5417:, many others; known as
3661:automotive integration,
2359:Standards and common use
1556:are derived from either
1391:'11010011101100'
1361:'11010011101100'
1349:
854:
674:Wired Equivalent Privacy
338:(or more accurately, an
224:; its encoding requires
162:polynomial long division
11351:Black, Richard (1994).
10836:"6.2.5 Error control".
9377:"The Great CRC Mystery"
3381:telecom systems, ITU-T
1995:may be transcribed as:
1628:{\displaystyle 2^{r}-1}
1558:irreducible polynomials
202:bits long. For a given
34:cyclic redundancy check
18:Cyclic Redundancy Check
11869:List of Ecma standards
11598:Application interfaces
11497:— (April 2023).
10591:10.1109/MM.1983.291120
9875:"Best CRC Polynomials"
9302:Proceedings of the IRE
9170:List of hash functions
9097:
8994:
8761:
8501:
8314:
8054:
7741:
7395:
7100:, ISO/IEC/IEEE 802-3 (
7096:PUB 71, FED-STD-1003,
7064:
6854:
6715:. Residue = 0x800FE3.
6695:
6477:
6097:
5984:
5811:
5632:
5492:
5293:
5103:
4885:
4781:
4616:
4509:
4402:
4295:
4206:
4106:
4009:
3910:
3819:wireless connectivity
3803:
3645:
3455:
3368:
3299:
3065:
2990:
2901:
2827:
2756:
2640:
2488:
2290:
2190:
2090:
1989:
1941:
1805:
1773:
1734:
1708:
1688:
1629:
1596:
646:
542:
522:
502:
482:
462:
442:
419:
135:communication channels
119:error-correcting codes
11825:Radio link interfaces
11770:Programming languages
11735:Ultra Density Optical
11452:7 August 2011 at the
11421:Warren, Henry S. Jr.
11130:17 April 2018 at the
10660:on 24 September 2015.
9504:10.1145/769800.769823
9098:
8995:
8762:
8502:
8315:
8055:
7742:
7396:
7065:
6855:
6696:
6478:
6098:
5985:
5812:
5633:
5493:
5294:
5104:
4886:
4782:
4617:
4510:
4403:
4296:
4207:
4107:
4010:
3911:
3804:
3646:
3456:
3369:
3300:
3066:
2991:
2902:
2828:
2757:
2641:
2489:
2345:proprietary protocols
2291:
2191:
2091:
1990:
1942:
1940:{\displaystyle x^{0}}
1806:
1774:
1735:
1709:
1689:
1630:
1597:
1500:Designing polynomials
647:
548:have the same length
543:
523:
503:
483:
463:
443:
420:
210:, which means it has
172:is discarded and the
11740:Universal Media Disc
11541:– via YouTube.
11259:. pp. 319–330.
9996:. 75 CH 1015-7 CSCB.
9811:. pp. 459–468.
9668:. pp. 145–154.
9160:Information security
9042:
8770:
8546:
8366:
8343:GSM control channel
8168:
7830:
7510:
7210:
6892:
6786:
6523:
6279:
6194:A & B Checksums
6048:
5857:
5678:
5557:
5534:cordless telephones
5443:
5192:
4976:
4817:
4706:
4548:
4441:
4340:
4246:
4144:
4057:
3967:
3842:
3735:
3570:
3419:
3339:
3231:
3029:
2941:
2865:
2798:
2727:
2624:
2543:Reversed reciprocal
2462:
2318:Reversed reciprocal
2203:
2103:
2003:
1960:
1924:
1902:)-bit divisor which
1804:{\displaystyle g(x)}
1786:
1744:
1718:
1698:
1639:
1606:
1586:
1576:primitive polynomial
1443:improve this article
1303:polynomial_bitstring
1278:polynomial_bitstring
1149:polynomial_bitstring
1143:polynomial_bitstring
1128:polynomial_bitstring
1080:polynomial_bitstring
1055:polynomial_bitstring
950:polynomial_bitstring
899:polynomial_bitstring
893:polynomial_bitstring
872:polynomial_bitstring
712:improve this section
555:
532:
512:
492:
472:
452:
432:
349:
154:generator polynomial
42:error-detecting code
11685:File systems (disk)
11643:File systems (tape)
11338:on 3 September 2011
10920:on 16 November 2008
10220:on 16 November 2013
9986:1975ntc.....1....8B
9949:1975STIN...7615344H
9389:(2): 26–34, 76–83.
9031:0x800000000000000D
9028:0xB000000000000001
9025:0xD800000000000000
9022:0x000000000000001B
8535:0xA17870F5D4F51B49
8532:0x92D8AF2BAF0E1E85
8529:0xC96C5795D7870F42
8526:0x42F0E1EBA9EA3693
8068:(Koopman {1,1,30})
7756:(Koopman {1,3,28})
6199:Fletcher's checksum
5328:CRC-16-Chakravarty
2639:{\displaystyle x+1}
2438:microarchitecture.
2365:technical standards
2305:
1733:{\displaystyle r-1}
1522:-bits. For a given
1514:A CRC is called an
194:A CRC is called an
54:polynomial division
11591:Ecma International
11035:pubs.opengroup.org
10959:(3.2.1 DATA FRAME)
10957:on 22 August 2013.
9382:Dr. Dobb's Journal
9346:. pp. 29–30.
9138:CRC Polynomial Zoo
9093:
8990:
8757:
8497:
8310:
8050:
7737:
7391:
7060:
6850:
6691:
6473:
6167:fieldbus networks
6093:
5980:
5807:
5628:
5488:
5289:
5117:Data Radio Channel
5099:
4881:
4777:
4612:
4505:
4398:
4291:
4202:
4120:Data Radio Channel
4102:
4005:
3906:
3799:
3641:
3451:
3364:
3295:
3133:Data Radio Channel
3061:
2986:
2897:
2823:
2752:
2636:
2484:
2353:reverse engineered
2303:
2296:(Koopman notation)
2286:
2186:
2086:
1985:
1956:So the polynomial
1937:
1856:Computation of CRC
1801:
1769:
1730:
1704:
1684:
1625:
1592:
1342:input_padded_array
1309:input_padded_array
1284:input_padded_array
1239:input_padded_array
1227:input_padded_array
1194:input_padded_array
1107:input_padded_array
1086:input_padded_array
1061:input_padded_array
1016:input_padded_array
1004:input_padded_array
971:input_padded_array
890:"""
642:
538:
518:
498:
478:
458:
438:
415:
326:cryptographic hash
322:digital signatures
127:W. Wesley Peterson
11889:Binary arithmetic
11876:
11875:
11492:Hamming distances
11266:978-0-321-84268-8
11099:. 20 March 2006.
11005:users.ece.cmu.edu
10839:ETSI EN 300 175-3
10739:CAN in Automation
10545:978-0-521-82815-4
10465:on 12 August 2017
10390:users.ece.cmu.edu
10284:users.ece.cmu.edu
10130:users.ece.cmu.edu
10025:users.ece.cmu.edu
9836:978-0-7695-1597-7
9774:10.1109/26.231911
9693:978-0-7695-2052-0
9587:978-0-521-88068-8
9361:978-0-387-68192-4
9253:978-0-08-044485-7
9106:
9105:
2396:Mitre Corporation
2336:
2335:
1707:{\displaystyle p}
1595:{\displaystyle r}
1475:
1474:
1467:
748:
747:
740:
541:{\displaystyle z}
521:{\displaystyle y}
501:{\displaystyle x}
481:{\displaystyle y}
461:{\displaystyle x}
441:{\displaystyle c}
168:and in which the
16:(Redirected from
11911:
11871:(1961 – present)
11670:Linear Tape-Open
11637:
11605:ANSI escape code
11583:
11576:
11569:
11560:
11542:
11531:
11529:
11527:
11521:
11514:
11502:
11489:
11480:
11478:
11476:
11442:
11440:
11434:. Archived from
11431:Hacker's Delight
11427:
11416:
11414:
11412:
11406:
11399:
11388:
11386:
11384:
11378:
11371:
11360:
11347:
11345:
11343:
11334:. Archived from
11317:
11287:
11270:
11255:(2nd ed.).
11252:Hacker's Delight
11231:
11230:
11228:
11226:
11220:
11213:
11207:Jones, David T.
11204:
11198:
11197:
11195:
11193:
11187:
11180:
11171:
11165:
11163:
11161:
11159:
11141:
11135:
11122:
11116:
11115:
11113:
11111:
11105:
11094:
11084:
11078:
11077:
11075:
11073:
11057:
11051:
11050:
11048:
11046:
11027:
11021:
11020:
11018:
11016:
11001:"24 Bit CRC Zoo"
10996:
10990:
10989:
10987:
10985:
10966:
10960:
10958:
10956:
10949:
10939:
10930:
10929:
10927:
10925:
10919:
10912:
10901:
10895:
10894:
10892:
10890:
10884:
10877:
10868:
10862:
10861:
10859:
10857:
10851:
10844:
10833:
10827:
10826:
10824:
10822:
10817:on 2 August 2012
10816:
10809:
10800:
10794:
10793:
10791:
10789:
10783:
10773:(3rd ed.).
10772:
10761:
10755:
10754:
10752:
10750:
10731:
10725:
10724:
10722:
10720:
10703:
10697:
10696:
10694:
10692:
10686:
10679:
10668:
10662:
10661:
10659:
10652:
10644:
10638:
10637:
10609:
10603:
10602:
10574:
10568:
10566:
10559:
10550:
10549:
10529:
10523:
10522:
10517:. Archived from
10510:
10504:
10503:
10501:
10499:
10481:
10475:
10474:
10472:
10470:
10452:
10441:
10440:
10438:
10436:
10430:
10423:
10412:
10406:
10405:
10403:
10401:
10382:
10373:
10372:
10370:
10368:
10362:
10355:
10344:
10338:
10337:
10335:
10333:
10327:
10320:
10309:
10300:
10299:
10297:
10295:
10276:
10270:
10269:
10267:
10265:
10259:
10252:
10241:
10230:
10229:
10227:
10225:
10219:
10212:
10202:
10185:
10183:
10181:
10179:
10173:
10162:
10152:
10146:
10145:
10143:
10141:
10122:
10116:
10115:
10113:
10111:
10105:
10098:
10088:
10071:
10068:
10062:
10047:
10041:
10040:
10038:
10036:
10021:"32 Bit CRC Zoo"
10017:
10008:
10004:
9998:
9997:
9971:
9965:
9964:
9962:
9960:
9928:
9922:
9921:
9919:
9917:
9897:
9891:
9890:
9888:
9886:
9870:
9864:
9863:
9861:
9859:
9853:
9820:
9810:
9799:
9778:
9777:
9757:
9751:
9750:
9748:
9746:
9730:
9721:
9720:
9718:
9716:
9710:
9677:
9667:
9656:
9631:
9630:
9628:
9626:
9610:
9604:
9603:
9601:
9599:
9569:
9563:
9562:
9560:
9558:
9549:. Archived from
9542:
9531:
9530:
9528:
9526:
9520:
9497:
9479:
9470:
9464:
9463:
9461:
9459:
9445:
9439:
9438:
9433:
9431:
9425:
9418:
9409:
9403:
9402:
9400:
9398:
9372:
9366:
9365:
9340:Mobile Broadband
9335:
9326:
9325:
9297:
9291:
9290:
9288:
9286:
9277:. Archived from
9267:
9261:
9260:
9231:
9225:
9224:
9201:
9102:
9100:
9099:
9094:
9080:
9079:
9067:
9066:
9054:
9053:
8999:
8997:
8996:
8991:
8977:
8976:
8964:
8963:
8951:
8950:
8938:
8937:
8925:
8924:
8912:
8911:
8899:
8898:
8886:
8885:
8873:
8872:
8860:
8859:
8847:
8846:
8834:
8833:
8821:
8820:
8808:
8807:
8795:
8794:
8782:
8781:
8766:
8764:
8763:
8758:
8753:
8752:
8740:
8739:
8727:
8726:
8714:
8713:
8701:
8700:
8688:
8687:
8675:
8674:
8662:
8661:
8649:
8648:
8636:
8635:
8623:
8622:
8610:
8609:
8597:
8596:
8584:
8583:
8571:
8570:
8558:
8557:
8506:
8504:
8503:
8498:
8487:
8486:
8474:
8473:
8452:
8451:
8430:
8429:
8417:
8416:
8404:
8403:
8391:
8390:
8378:
8377:
8319:
8317:
8316:
8311:
8297:
8296:
8284:
8283:
8271:
8270:
8258:
8257:
8245:
8244:
8232:
8231:
8219:
8218:
8206:
8205:
8193:
8192:
8180:
8179:
8059:
8057:
8056:
8051:
8037:
8036:
8024:
8023:
8011:
8010:
7998:
7997:
7985:
7984:
7972:
7971:
7959:
7958:
7946:
7945:
7933:
7932:
7920:
7919:
7907:
7906:
7894:
7893:
7881:
7880:
7868:
7867:
7855:
7854:
7842:
7841:
7755:
7754:
7746:
7744:
7743:
7738:
7730:
7729:
7717:
7716:
7704:
7703:
7691:
7690:
7678:
7677:
7665:
7664:
7652:
7651:
7639:
7638:
7626:
7625:
7613:
7612:
7600:
7599:
7587:
7586:
7574:
7573:
7561:
7560:
7548:
7547:
7535:
7534:
7522:
7521:
7409:
7408:
7400:
7398:
7397:
7392:
7378:
7377:
7365:
7364:
7352:
7351:
7339:
7338:
7326:
7325:
7313:
7312:
7300:
7299:
7287:
7286:
7274:
7273:
7261:
7260:
7248:
7247:
7235:
7234:
7222:
7221:
7069:
7067:
7066:
7061:
7047:
7046:
7034:
7033:
7021:
7020:
7008:
7007:
6995:
6994:
6982:
6981:
6969:
6968:
6956:
6955:
6943:
6942:
6930:
6929:
6917:
6916:
6904:
6903:
6859:
6857:
6856:
6851:
6837:
6836:
6824:
6823:
6811:
6810:
6798:
6797:
6700:
6698:
6697:
6692:
6678:
6677:
6665:
6664:
6652:
6651:
6639:
6638:
6626:
6625:
6613:
6612:
6600:
6599:
6587:
6586:
6574:
6573:
6561:
6560:
6548:
6547:
6535:
6534:
6482:
6480:
6479:
6474:
6460:
6459:
6447:
6446:
6434:
6433:
6421:
6420:
6408:
6407:
6395:
6394:
6382:
6381:
6369:
6368:
6356:
6355:
6343:
6342:
6330:
6329:
6317:
6316:
6304:
6303:
6291:
6290:
6141:safety fieldbus
6114:safety fieldbus
6102:
6100:
6099:
6094:
6086:
6085:
6073:
6072:
6060:
6059:
5989:
5987:
5986:
5981:
5973:
5972:
5960:
5959:
5947:
5946:
5934:
5933:
5921:
5920:
5908:
5907:
5895:
5894:
5882:
5881:
5869:
5868:
5816:
5814:
5813:
5808:
5794:
5793:
5781:
5780:
5768:
5767:
5755:
5754:
5742:
5741:
5729:
5728:
5716:
5715:
5703:
5702:
5690:
5689:
5637:
5635:
5634:
5629:
5621:
5620:
5608:
5607:
5595:
5594:
5582:
5581:
5569:
5568:
5508:mobile networks
5497:
5495:
5494:
5489:
5481:
5480:
5468:
5467:
5455:
5454:
5298:
5296:
5295:
5290:
5282:
5281:
5269:
5268:
5256:
5255:
5243:
5242:
5230:
5229:
5217:
5216:
5204:
5203:
5144:mobile networks
5108:
5106:
5105:
5100:
5092:
5091:
5079:
5078:
5066:
5065:
5053:
5052:
5040:
5039:
5027:
5026:
5014:
5013:
5001:
5000:
4988:
4987:
4952:Radio teleswitch
4927:mobile networks
4901:mobile networks
4890:
4888:
4887:
4882:
4868:
4867:
4855:
4854:
4842:
4841:
4829:
4828:
4794:telecom systems
4786:
4784:
4783:
4778:
4770:
4769:
4757:
4756:
4744:
4743:
4731:
4730:
4718:
4717:
4658:mobile networks
4632:mobile networks
4621:
4619:
4618:
4613:
4599:
4598:
4586:
4585:
4573:
4572:
4560:
4559:
4514:
4512:
4511:
4506:
4492:
4491:
4479:
4478:
4466:
4465:
4453:
4452:
4418:mobile networks
4407:
4405:
4404:
4399:
4391:
4390:
4378:
4377:
4365:
4364:
4352:
4351:
4300:
4298:
4297:
4292:
4284:
4283:
4271:
4270:
4258:
4257:
4223:mobile networks
4211:
4209:
4208:
4203:
4195:
4194:
4182:
4181:
4169:
4168:
4156:
4155:
4111:
4109:
4108:
4103:
4095:
4094:
4082:
4081:
4069:
4068:
4014:
4012:
4011:
4006:
3992:
3991:
3979:
3978:
3915:
3913:
3912:
3907:
3893:
3892:
3880:
3879:
3867:
3866:
3854:
3853:
3808:
3806:
3805:
3800:
3786:
3785:
3773:
3772:
3760:
3759:
3747:
3746:
3650:
3648:
3647:
3642:
3634:
3633:
3621:
3620:
3608:
3607:
3595:
3594:
3582:
3581:
3460:
3458:
3457:
3452:
3444:
3443:
3431:
3430:
3373:
3371:
3370:
3365:
3351:
3350:
3304:
3302:
3301:
3296:
3282:
3281:
3269:
3268:
3256:
3255:
3243:
3242:
3160:mobile networks
3109:mobile networks
3082:mobile networks
3070:
3068:
3067:
3062:
3054:
3053:
3041:
3040:
2995:
2993:
2992:
2987:
2979:
2978:
2966:
2965:
2953:
2952:
2906:
2904:
2903:
2898:
2890:
2889:
2877:
2876:
2832:
2830:
2829:
2824:
2810:
2809:
2761:
2759:
2758:
2753:
2739:
2738:
2656:mobile networks
2645:
2643:
2642:
2637:
2524:Hamming distance
2500:
2493:
2491:
2490:
2485:
2483:
2479:
2425:
2373:Hamming distance
2306:
2295:
2293:
2292:
2287:
2285:
2284:
2269:
2268:
2253:
2252:
2237:
2236:
2221:
2220:
2196:(LSB-first code)
2195:
2193:
2192:
2187:
2185:
2184:
2169:
2168:
2153:
2152:
2137:
2136:
2121:
2120:
2096:(MSB-first code)
2095:
2093:
2092:
2087:
2082:
2081:
2066:
2065:
2050:
2049:
2034:
2033:
2015:
2014:
1994:
1992:
1991:
1986:
1972:
1971:
1946:
1944:
1943:
1938:
1936:
1935:
1901:
1810:
1808:
1807:
1802:
1778:
1776:
1775:
1770:
1762:
1761:
1739:
1737:
1736:
1731:
1713:
1711:
1710:
1705:
1693:
1691:
1690:
1685:
1634:
1632:
1631:
1626:
1618:
1617:
1601:
1599:
1598:
1593:
1566:
1543:
1536:
1470:
1463:
1459:
1456:
1450:
1427:
1419:
1410:
1407:
1404:
1401:
1398:
1395:
1392:
1389:
1386:
1383:
1380:
1377:
1374:
1371:
1368:
1365:
1362:
1359:
1356:
1353:
1346:
1343:
1340:
1337:
1334:
1331:
1328:
1325:
1322:
1319:
1316:
1313:
1310:
1307:
1304:
1301:
1298:
1295:
1292:
1289:
1285:
1282:
1279:
1276:
1273:
1270:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1246:
1243:
1240:
1237:
1234:
1231:
1228:
1225:
1222:
1219:
1216:
1213:
1210:
1207:
1204:
1201:
1198:
1195:
1192:
1189:
1186:
1183:
1180:
1177:
1174:
1171:
1168:
1165:
1162:
1159:
1156:
1153:
1150:
1147:
1144:
1141:
1138:
1135:
1132:
1129:
1126:
1123:
1120:
1117:
1114:
1111:
1108:
1105:
1102:
1099:
1096:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1072:
1069:
1066:
1062:
1059:
1056:
1053:
1050:
1047:
1044:
1041:
1038:
1035:
1032:
1029:
1026:
1023:
1020:
1017:
1014:
1011:
1008:
1005:
1002:
999:
996:
993:
990:
987:
984:
981:
978:
975:
972:
969:
966:
963:
960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
930:
927:
924:
921:
918:
915:
912:
909:
906:
903:
900:
897:
894:
891:
888:
885:
882:
879:
876:
873:
870:
867:
864:
861:
858:
800:
784:
766:
743:
736:
732:
729:
723:
692:
684:
676:(WEP) protocol.
651:
649:
648:
643:
547:
545:
544:
539:
527:
525:
524:
519:
507:
505:
504:
499:
487:
485:
484:
479:
467:
465:
464:
459:
447:
445:
444:
439:
424:
422:
421:
416:
260:
230:
223:
216:
148:
58:error correction
21:
11919:
11918:
11914:
11913:
11912:
11910:
11909:
11908:
11879:
11878:
11877:
11872:
11862:
11841:
11820:
11765:
11749:
11680:
11638:
11629:
11620:Office Open XML
11593:
11587:
11545:
11534:
11525:
11523:
11519:
11512:
11507:
11496:
11484:Koopman, Phil.
11483:
11474:
11472:
11458:
11454:Wayback Machine
11438:
11425:
11420:
11410:
11408:
11404:
11397:
11392:
11382:
11380:
11376:
11369:
11364:
11350:
11341:
11339:
11326:
11299:
11295:
11290:
11273:
11267:
11244:
11240:
11238:Further reading
11235:
11234:
11224:
11222:
11218:
11211:
11206:
11205:
11201:
11191:
11189:
11185:
11178:
11173:
11172:
11168:
11157:
11155:
11143:
11142:
11138:
11132:Wayback Machine
11125:ETSI TS 100 909
11123:
11119:
11109:
11107:
11103:
11092:
11086:
11085:
11081:
11071:
11069:
11059:
11058:
11054:
11044:
11042:
11029:
11028:
11024:
11014:
11012:
10998:
10997:
10993:
10983:
10981:
10968:
10967:
10963:
10954:
10947:
10941:
10940:
10933:
10923:
10921:
10917:
10910:
10903:
10902:
10898:
10888:
10886:
10882:
10875:
10870:
10869:
10865:
10855:
10853:
10849:
10842:
10835:
10834:
10830:
10820:
10818:
10814:
10807:
10802:
10801:
10797:
10787:
10785:
10781:
10770:
10763:
10762:
10758:
10748:
10746:
10733:
10732:
10728:
10718:
10716:
10705:
10704:
10700:
10690:
10688:
10684:
10677:
10670:
10669:
10665:
10657:
10650:
10646:
10645:
10641:
10626:10.1109/40.7773
10611:
10610:
10606:
10576:
10575:
10571:
10561:
10560:
10553:
10546:
10531:
10530:
10526:
10521:on 25 May 2005.
10512:
10511:
10507:
10497:
10495:
10483:
10482:
10478:
10468:
10466:
10454:
10453:
10444:
10434:
10432:
10431:on 24 July 2016
10428:
10421:
10414:
10413:
10409:
10399:
10397:
10386:"8 Bit CRC Zoo"
10384:
10383:
10376:
10366:
10364:
10360:
10353:
10346:
10345:
10341:
10331:
10329:
10325:
10318:
10311:
10310:
10303:
10293:
10291:
10280:"6 Bit CRC Zoo"
10278:
10277:
10273:
10263:
10261:
10257:
10250:
10247:ETSI EN 300 751
10243:
10242:
10233:
10223:
10221:
10217:
10210:
10204:
10203:
10188:
10177:
10175:
10171:
10160:
10154:
10153:
10149:
10139:
10137:
10126:"3 Bit CRC Zoo"
10124:
10123:
10119:
10109:
10107:
10103:
10096:
10093:ETSI TS 100 909
10090:
10089:
10074:
10069:
10065:
10048:
10044:
10034:
10032:
10019:
10018:
10011:
10005:
10001:
9973:
9972:
9968:
9958:
9956:
9930:
9929:
9925:
9915:
9913:
9899:
9898:
9894:
9884:
9882:
9872:
9871:
9867:
9857:
9855:
9851:
9837:
9808:
9801:
9800:
9781:
9759:
9758:
9754:
9744:
9742:
9732:
9731:
9724:
9714:
9712:
9708:
9694:
9675:10.1.1.648.9080
9665:
9658:
9657:
9634:
9624:
9622:
9612:
9611:
9607:
9597:
9595:
9588:
9571:
9570:
9566:
9556:
9554:
9553:on 2 April 2018
9544:
9543:
9534:
9524:
9522:
9518:
9477:
9472:
9471:
9467:
9457:
9455:
9447:
9446:
9442:
9429:
9427:
9426:on 19 July 2011
9423:
9416:
9411:
9410:
9406:
9396:
9394:
9374:
9373:
9369:
9362:
9337:
9336:
9329:
9299:
9298:
9294:
9284:
9282:
9281:on 20 July 2017
9269:
9268:
9264:
9254:
9233:
9232:
9228:
9203:
9202:
9198:
9193:
9146:
9129:
9112:
9110:Implementations
9107:
9071:
9058:
9045:
9040:
9039:
8968:
8955:
8942:
8929:
8916:
8903:
8890:
8877:
8864:
8851:
8838:
8825:
8812:
8799:
8786:
8773:
8768:
8767:
8744:
8731:
8718:
8705:
8692:
8679:
8666:
8653:
8640:
8627:
8614:
8601:
8588:
8575:
8562:
8549:
8544:
8543:
8478:
8465:
8443:
8421:
8408:
8395:
8382:
8369:
8364:
8363:
8288:
8275:
8262:
8249:
8236:
8223:
8210:
8197:
8184:
8171:
8166:
8165:
8067:
8028:
8015:
8002:
7989:
7976:
7963:
7950:
7937:
7924:
7911:
7898:
7885:
7872:
7859:
7846:
7833:
7828:
7827:
7752:
7751:
7721:
7708:
7695:
7682:
7669:
7656:
7643:
7630:
7617:
7604:
7591:
7578:
7565:
7552:
7539:
7526:
7513:
7508:
7507:
7406:
7405:
7369:
7356:
7343:
7330:
7317:
7304:
7291:
7278:
7265:
7252:
7239:
7226:
7213:
7208:
7207:
7038:
7025:
7012:
6999:
6986:
6973:
6960:
6947:
6934:
6921:
6908:
6895:
6890:
6889:
6828:
6815:
6802:
6789:
6784:
6783:
6669:
6656:
6643:
6630:
6617:
6604:
6591:
6578:
6565:
6552:
6539:
6526:
6521:
6520:
6451:
6438:
6425:
6412:
6399:
6386:
6373:
6360:
6347:
6334:
6321:
6308:
6295:
6282:
6277:
6276:
6077:
6064:
6051:
6046:
6045:
5964:
5951:
5938:
5925:
5912:
5899:
5886:
5873:
5860:
5855:
5854:
5785:
5772:
5759:
5746:
5733:
5720:
5707:
5694:
5681:
5676:
5675:
5612:
5599:
5586:
5573:
5560:
5555:
5554:
5472:
5459:
5446:
5441:
5440:
5273:
5260:
5247:
5234:
5221:
5208:
5195:
5190:
5189:
5083:
5070:
5057:
5044:
5031:
5018:
5005:
4992:
4979:
4974:
4973:
4859:
4846:
4833:
4820:
4815:
4814:
4761:
4748:
4735:
4722:
4709:
4704:
4703:
4590:
4577:
4564:
4551:
4546:
4545:
4483:
4470:
4457:
4444:
4439:
4438:
4382:
4369:
4356:
4343:
4338:
4337:
4275:
4262:
4249:
4244:
4243:
4186:
4173:
4160:
4147:
4142:
4141:
4086:
4073:
4060:
4055:
4054:
3983:
3970:
3965:
3964:
3928:I.432.1 (02/99)
3884:
3871:
3858:
3845:
3840:
3839:
3777:
3764:
3751:
3738:
3733:
3732:
3625:
3612:
3599:
3586:
3573:
3568:
3567:
3435:
3422:
3417:
3416:
3342:
3337:
3336:
3273:
3260:
3247:
3234:
3229:
3228:
3045:
3032:
3027:
3026:
2970:
2957:
2944:
2939:
2938:
2881:
2868:
2863:
2862:
2801:
2796:
2795:
2730:
2725:
2724:
2622:
2621:
2469:
2465:
2460:
2459:
2451:
2388:Rome Laboratory
2361:
2341:
2276:
2260:
2244:
2228:
2212:
2201:
2200:
2176:
2160:
2144:
2128:
2112:
2101:
2100:
2073:
2057:
2041:
2025:
2006:
2001:
2000:
1963:
1958:
1957:
1927:
1922:
1921:
1896:
1829:
1784:
1783:
1747:
1742:
1741:
1716:
1715:
1696:
1695:
1637:
1636:
1609:
1604:
1603:
1584:
1583:
1561:
1538:
1531:
1502:
1481:
1471:
1460:
1454:
1451:
1440:
1428:
1417:
1412:
1411:
1408:
1405:
1402:
1399:
1396:
1393:
1390:
1387:
1384:
1381:
1378:
1375:
1372:
1369:
1366:
1363:
1360:
1357:
1354:
1351:
1348:
1347:
1344:
1341:
1338:
1335:
1332:
1329:
1326:
1323:
1320:
1317:
1314:
1311:
1308:
1305:
1302:
1299:
1296:
1293:
1290:
1287:
1283:
1280:
1277:
1274:
1271:
1268:
1265:
1262:
1259:
1256:
1253:
1250:
1247:
1244:
1241:
1238:
1235:
1232:
1229:
1226:
1223:
1220:
1217:
1214:
1212:initial_padding
1211:
1208:
1206:input_bitstring
1205:
1202:
1199:
1196:
1193:
1190:
1187:
1185:initial_padding
1184:
1181:
1179:input_bitstring
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1142:
1139:
1136:
1133:
1130:
1127:
1124:
1122:input_bitstring
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1100:
1097:
1094:
1091:
1088:
1085:
1082:
1079:
1076:
1073:
1070:
1067:
1064:
1060:
1057:
1054:
1051:
1048:
1045:
1042:
1039:
1036:
1033:
1030:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
989:initial_padding
988:
985:
983:input_bitstring
982:
979:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
935:initial_padding
934:
931:
929:input_bitstring
928:
925:
922:
919:
916:
913:
910:
907:
904:
901:
898:
895:
892:
889:
886:
883:
880:
877:
874:
871:
868:
866:input_bitstring
865:
862:
859:
856:
845:
830:
822:
808:
807:11010011101100
786:
775:
761:
754:
744:
733:
727:
724:
709:
693:
682:
553:
552:
530:
529:
510:
509:
490:
489:
470:
469:
450:
449:
430:
429:
347:
346:
340:affine function
336:linear function
302:
267:
255:
243:-XXX as in the
225:
218:
211:
146:
112:
28:
23:
22:
15:
12:
11:
5:
11917:
11915:
11907:
11906:
11901:
11896:
11891:
11881:
11880:
11874:
11873:
11867:
11864:
11863:
11861:
11860:
11855:
11849:
11847:
11843:
11842:
11840:
11839:
11834:
11828:
11826:
11822:
11821:
11819:
11818:
11813:
11808:
11803:
11789:
11784:
11779:
11773:
11771:
11767:
11766:
11764:
11763:
11757:
11755:
11751:
11750:
11748:
11747:
11742:
11737:
11732:
11727:
11722:
11721:
11720:
11715:
11710:
11700:
11697:CD File System
11694:
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11681:
11679:
11678:
11673:
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11544:
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11532:
11505:
11504:
11503:
11481:
11456:
11444:
11441:on 3 May 2015.
11418:
11390:
11362:
11348:
11328:Williams, Ross
11324:
11318:
11296:
11294:
11293:External links
11291:
11289:
11288:
11271:
11265:
11257:Addison Wesley
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11239:
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11233:
11232:
11199:
11166:
11150:. Matpack.de.
11136:
11117:
11079:
11064:. Libpng.org.
11052:
11022:
10991:
10961:
10931:
10896:
10878:. INCITS T10.
10863:
10828:
10795:
10756:
10726:
10698:
10663:
10639:
10604:
10569:
10551:
10544:
10535:WCDMA Handbook
10524:
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10009:
9999:
9966:
9923:
9892:
9865:
9835:
9818:10.1.1.11.8323
9779:
9768:(6): 883–892.
9752:
9722:
9692:
9632:
9605:
9586:
9564:
9532:
9495:10.1.1.14.8775
9465:
9440:
9404:
9367:
9360:
9327:
9308:(1): 228–235.
9292:
9262:
9252:
9226:
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9127:CRC catalogues
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7143:
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3006:token packets
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2947:
2935:
2934:
2931:
2928:
2925:
2922:
2919:
2912:
2908:
2907:
2896:
2893:
2888:
2884:
2880:
2875:
2871:
2859:
2858:
2855:
2852:
2849:
2846:
2843:
2838:
2834:
2833:
2822:
2819:
2816:
2813:
2808:
2804:
2792:
2791:
2788:
2785:
2782:
2779:
2776:
2767:
2763:
2762:
2751:
2748:
2745:
2742:
2737:
2733:
2721:
2720:
2717:
2714:
2711:
2708:
2705:
2702:
2699:
2696:
2693:
2690:
2687:
2684:
2681:
2678:
2675:
2672:
2669:
2666:
2663:
2660:
2657:
2654:
2647:
2646:
2635:
2632:
2629:
2618:
2617:
2614:
2611:
2608:
2605:
2602:
2594:
2590:
2589:
2586:
2583:
2580:
2577:
2574:
2571:
2568:
2565:
2562:
2559:
2556:
2553:
2550:
2547:
2544:
2541:
2536:
2531:
2527:
2526:
2520:
2517:
2512:
2507:
2504:
2498:
2482:
2478:
2475:
2472:
2468:
2450:
2447:
2360:
2357:
2340:
2337:
2334:
2333:
2330:
2327:
2324:
2320:
2319:
2316:
2313:
2310:
2298:
2297:
2283:
2279:
2275:
2272:
2267:
2263:
2259:
2256:
2251:
2247:
2243:
2240:
2235:
2231:
2227:
2224:
2219:
2215:
2211:
2208:
2197:
2183:
2179:
2175:
2172:
2167:
2163:
2159:
2156:
2151:
2147:
2143:
2140:
2135:
2131:
2127:
2124:
2119:
2115:
2111:
2108:
2097:
2085:
2080:
2076:
2072:
2069:
2064:
2060:
2056:
2053:
2048:
2044:
2040:
2037:
2032:
2028:
2024:
2021:
2018:
2013:
2009:
1984:
1981:
1978:
1975:
1970:
1966:
1949:
1948:
1934:
1930:
1915:
1886:
1878:
1867:
1860:
1840:
1828:
1825:
1800:
1797:
1794:
1791:
1768:
1765:
1760:
1757:
1754:
1750:
1729:
1726:
1723:
1703:
1683:
1680:
1677:
1674:
1671:
1668:
1665:
1662:
1659:
1656:
1653:
1650:
1647:
1644:
1624:
1621:
1616:
1612:
1591:
1501:
1498:
1477:Main article:
1473:
1472:
1431:
1429:
1422:
1416:
1413:
1397:'1011'
1367:'1011'
1350:
1286:\
1063:\
968:initial_filler
878:initial_filler
855:
847:The following
842:
827:
820:
806:
756:To compute an
750:Main article:
746:
745:
696:
694:
687:
681:
678:
653:
652:
641:
638:
635:
632:
629:
626:
623:
620:
617:
614:
611:
608:
605:
602:
599:
596:
593:
590:
587:
584:
581:
578:
575:
572:
569:
566:
563:
560:
537:
517:
497:
477:
457:
437:
426:
425:
414:
411:
408:
405:
402:
399:
396:
393:
390:
387:
384:
381:
378:
375:
372:
369:
366:
363:
360:
357:
354:
301:
300:Data integrity
298:
266:
263:
111:
108:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
11916:
11905:
11902:
11900:
11899:Finite fields
11897:
11895:
11892:
11890:
11887:
11886:
11884:
11870:
11865:
11859:
11856:
11854:
11851:
11850:
11848:
11844:
11838:
11835:
11833:
11830:
11829:
11827:
11823:
11817:
11814:
11812:
11811:Minimal BASIC
11809:
11807:
11804:
11801:
11797:
11793:
11790:
11788:
11785:
11783:
11780:
11778:
11775:
11774:
11772:
11768:
11762:
11759:
11758:
11756:
11752:
11746:
11743:
11741:
11738:
11736:
11733:
11731:
11728:
11726:
11723:
11719:
11716:
11714:
11711:
11709:
11706:
11705:
11704:
11701:
11698:
11695:
11693:
11690:
11689:
11687:
11683:
11677:
11674:
11671:
11668:
11666:
11663:
11661:
11658:
11656:
11653:
11651:
11648:
11647:
11645:
11641:
11636:
11626:
11623:
11621:
11618:
11616:
11613:
11611:
11608:
11606:
11603:
11602:
11600:
11596:
11592:
11589:Standards of
11584:
11579:
11577:
11572:
11570:
11565:
11564:
11561:
11555:
11552:
11550:
11547:
11546:
11540:
11539:
11533:
11518:
11511:
11506:
11500:
11495:
11494:
11493:
11487:
11482:
11470:
11466:
11462:
11457:
11455:
11451:
11448:
11445:
11437:
11433:
11432:
11424:
11419:
11403:
11396:
11391:
11375:
11368:
11363:
11358:
11357:The Blue Book
11354:
11349:
11337:
11333:
11329:
11325:
11322:
11319:
11315:
11311:
11307:
11303:
11298:
11297:
11292:
11285:
11281:
11277:
11272:
11268:
11262:
11258:
11254:
11253:
11248:
11243:
11242:
11237:
11217:
11210:
11203:
11200:
11184:
11177:
11170:
11167:
11153:
11149:
11148:
11140:
11137:
11133:
11129:
11126:
11121:
11118:
11102:
11098:
11091:
11090:
11083:
11080:
11067:
11063:
11056:
11053:
11040:
11036:
11032:
11026:
11023:
11010:
11006:
11002:
10995:
10992:
10979:
10975:
10971:
10965:
10962:
10953:
10946:
10945:
10938:
10936:
10932:
10916:
10909:
10908:
10900:
10897:
10881:
10874:
10867:
10864:
10848:
10841:
10840:
10832:
10829:
10813:
10806:
10799:
10796:
10780:
10776:
10769:
10768:
10760:
10757:
10744:
10740:
10736:
10730:
10727:
10714:
10710:
10709:
10702:
10699:
10683:
10676:
10675:
10667:
10664:
10656:
10649:
10643:
10640:
10635:
10631:
10627:
10623:
10619:
10615:
10608:
10605:
10600:
10596:
10592:
10588:
10584:
10580:
10573:
10570:
10564:
10558:
10556:
10552:
10547:
10541:
10537:
10536:
10528:
10525:
10520:
10516:
10509:
10506:
10493:
10489:
10488:
10480:
10477:
10464:
10460:
10459:
10451:
10449:
10447:
10443:
10427:
10420:
10419:
10411:
10408:
10395:
10391:
10387:
10381:
10379:
10375:
10359:
10352:
10351:
10343:
10340:
10324:
10317:
10316:
10308:
10306:
10302:
10289:
10285:
10281:
10275:
10272:
10256:
10249:
10248:
10240:
10238:
10236:
10232:
10216:
10209:
10208:
10201:
10199:
10197:
10195:
10193:
10191:
10187:
10170:
10166:
10159:
10158:
10151:
10148:
10135:
10131:
10127:
10121:
10118:
10102:
10095:
10094:
10087:
10085:
10083:
10081:
10079:
10077:
10073:
10067:
10064:
10060:
10056:
10052:
10046:
10043:
10030:
10026:
10022:
10016:
10014:
10010:
10003:
10000:
9995:
9991:
9987:
9983:
9979:
9978:
9970:
9967:
9954:
9951:. ADA013939.
9950:
9946:
9942:
9938:
9934:
9927:
9924:
9911:
9908:. ADA014825.
9907:
9903:
9896:
9893:
9880:
9876:
9869:
9866:
9850:
9846:
9842:
9838:
9832:
9828:
9824:
9819:
9814:
9807:
9806:
9798:
9796:
9794:
9792:
9790:
9788:
9786:
9784:
9780:
9775:
9771:
9767:
9763:
9756:
9753:
9740:
9736:
9729:
9727:
9723:
9707:
9703:
9699:
9695:
9689:
9685:
9681:
9676:
9671:
9664:
9663:
9655:
9653:
9651:
9649:
9647:
9645:
9643:
9641:
9639:
9637:
9633:
9620:
9616:
9609:
9606:
9593:
9589:
9583:
9579:
9575:
9568:
9565:
9552:
9548:
9541:
9539:
9537:
9533:
9517:
9513:
9509:
9505:
9501:
9496:
9491:
9487:
9483:
9476:
9469:
9466:
9454:
9450:
9444:
9441:
9437:
9422:
9415:
9408:
9405:
9392:
9388:
9384:
9383:
9378:
9371:
9368:
9363:
9357:
9353:
9349:
9345:
9341:
9334:
9332:
9328:
9323:
9319:
9315:
9311:
9307:
9303:
9296:
9293:
9280:
9276:
9272:
9266:
9263:
9259:
9255:
9249:
9245:
9241:
9237:
9230:
9227:
9223:
9219:
9215:
9211:
9207:
9200:
9197:
9190:
9186:
9183:
9181:
9178:
9176:
9173:
9171:
9168:
9166:
9163:
9161:
9158:
9156:
9153:
9151:
9148:
9147:
9143:
9139:
9136:
9134:
9131:
9130:
9126:
9122:
9119:
9117:
9114:
9113:
9109:
9090:
9087:
9084:
9081:
9076:
9072:
9068:
9063:
9059:
9055:
9050:
9046:
9037:
9030:
9027:
9024:
9021:
9018:
9014:
9010:
9002:
8987:
8984:
8981:
8978:
8973:
8969:
8965:
8960:
8956:
8952:
8947:
8943:
8939:
8934:
8930:
8926:
8921:
8917:
8913:
8908:
8904:
8900:
8895:
8891:
8887:
8882:
8878:
8874:
8869:
8865:
8861:
8856:
8852:
8848:
8843:
8839:
8835:
8830:
8826:
8822:
8817:
8813:
8809:
8804:
8800:
8796:
8791:
8787:
8783:
8778:
8774:
8754:
8749:
8745:
8741:
8736:
8732:
8728:
8723:
8719:
8715:
8710:
8706:
8702:
8697:
8693:
8689:
8684:
8680:
8676:
8671:
8667:
8663:
8658:
8654:
8650:
8645:
8641:
8637:
8632:
8628:
8624:
8619:
8615:
8611:
8606:
8602:
8598:
8593:
8589:
8585:
8580:
8576:
8572:
8567:
8563:
8559:
8554:
8550:
8541:
8534:
8531:
8528:
8525:
8523:
8519:
8514:
8509:
8491:
8488:
8483:
8479:
8475:
8470:
8466:
8456:
8453:
8448:
8444:
8437:
8434:
8431:
8426:
8422:
8418:
8413:
8409:
8405:
8400:
8396:
8392:
8387:
8383:
8379:
8374:
8370:
8361:
8355:0x8002410004
8354:
8352:0x2000824001
8351:
8349:0x9000412000
8348:
8346:0x0004820009
8345:
8340:
8335:
8332:
8326:
8323:
8322:
8307:
8304:
8301:
8298:
8293:
8289:
8285:
8280:
8276:
8272:
8267:
8263:
8259:
8254:
8250:
8246:
8241:
8237:
8233:
8228:
8224:
8220:
8215:
8211:
8207:
8202:
8198:
8194:
8189:
8185:
8181:
8176:
8172:
8163:
8156:
8153:
8150:
8147:
8145:
8137:
8133:
8130:
8127:
8124:
8121:
8118:
8115:
8112:
8109:
8106:
8103:
8100:
8097:
8094:
8091:
8088:
8085:
8082:
8079:
8076:
8073:
8070:
8063:
8062:
8047:
8044:
8041:
8038:
8033:
8029:
8025:
8020:
8016:
8012:
8007:
8003:
7999:
7994:
7990:
7986:
7981:
7977:
7973:
7968:
7964:
7960:
7955:
7951:
7947:
7942:
7938:
7934:
7929:
7925:
7921:
7916:
7912:
7908:
7903:
7899:
7895:
7890:
7886:
7882:
7877:
7873:
7869:
7864:
7860:
7856:
7851:
7847:
7843:
7838:
7834:
7825:
7770:
7767:
7764:
7761:
7749:
7734:
7731:
7726:
7722:
7718:
7713:
7709:
7705:
7700:
7696:
7692:
7687:
7683:
7679:
7674:
7670:
7666:
7661:
7657:
7653:
7648:
7644:
7640:
7635:
7631:
7627:
7622:
7618:
7614:
7609:
7605:
7601:
7596:
7592:
7588:
7583:
7579:
7575:
7570:
7566:
7562:
7557:
7553:
7549:
7544:
7540:
7536:
7531:
7527:
7523:
7518:
7514:
7505:
7450:
7447:
7444:
7441:
7439:
7435:
7431:
7427:
7423:
7419:
7415:
7410:(Castagnoli)
7403:
7388:
7385:
7382:
7379:
7374:
7370:
7366:
7361:
7357:
7353:
7348:
7344:
7340:
7335:
7331:
7327:
7322:
7318:
7314:
7309:
7305:
7301:
7296:
7292:
7288:
7283:
7279:
7275:
7270:
7266:
7262:
7257:
7253:
7249:
7244:
7240:
7236:
7231:
7227:
7223:
7218:
7214:
7205:
7150:
7147:
7144:
7141:
7138:
7134:
7130:
7127:
7123:
7119:
7115:
7111:
7107:
7103:
7099:
7095:
7091:
7087:
7083:
7079:
7072:
7057:
7054:
7051:
7048:
7043:
7039:
7035:
7030:
7026:
7022:
7017:
7013:
7009:
7004:
7000:
6996:
6991:
6987:
6983:
6978:
6974:
6970:
6965:
6961:
6957:
6952:
6948:
6944:
6939:
6935:
6931:
6926:
6922:
6918:
6913:
6909:
6905:
6900:
6896:
6887:
6880:
6877:
6874:
6871:
6869:
6862:
6847:
6844:
6841:
6838:
6833:
6829:
6825:
6820:
6816:
6812:
6807:
6803:
6799:
6794:
6790:
6781:
6726:
6723:
6720:
6717:
6714:
6708:
6703:
6688:
6685:
6682:
6679:
6674:
6670:
6666:
6661:
6657:
6653:
6648:
6644:
6640:
6635:
6631:
6627:
6622:
6618:
6614:
6609:
6605:
6601:
6596:
6592:
6588:
6583:
6579:
6575:
6570:
6566:
6562:
6557:
6553:
6549:
6544:
6540:
6536:
6531:
6527:
6518:
6511:
6508:
6505:
6502:
6499:
6495:
6490:
6485:
6470:
6467:
6464:
6461:
6456:
6452:
6448:
6443:
6439:
6435:
6430:
6426:
6422:
6417:
6413:
6409:
6404:
6400:
6396:
6391:
6387:
6383:
6378:
6374:
6370:
6365:
6361:
6357:
6352:
6348:
6344:
6339:
6335:
6331:
6326:
6322:
6318:
6313:
6309:
6305:
6300:
6296:
6292:
6287:
6283:
6274:
6267:
6264:
6261:
6258:
6256:
6249:
6245:
6242:
6239:
6236:
6233:
6230:
6227:
6226:
6222:
6219:
6216:
6213:
6210:
6207:
6204:
6203:
6200:
6193:
6189:
6186:
6185:
6181:
6178:
6175:
6172:
6169:
6166:
6164:
6160:
6159:
6155:
6152:
6149:
6146:
6143:
6140:
6137:
6133:
6132:
6128:
6125:
6122:
6119:
6116:
6113:
6110:
6106:
6105:
6090:
6087:
6082:
6078:
6074:
6069:
6065:
6061:
6056:
6052:
6043:
6036:
6033:
6030:
6027:
6025:
6021:
6017:
6014:
6010:
6006:
6002:
5997:
5992:
5977:
5974:
5969:
5965:
5961:
5956:
5952:
5948:
5943:
5939:
5935:
5930:
5926:
5922:
5917:
5913:
5909:
5904:
5900:
5896:
5891:
5887:
5883:
5878:
5874:
5870:
5865:
5861:
5852:
5845:
5842:
5839:
5836:
5834:
5830:
5824:
5819:
5804:
5801:
5798:
5795:
5790:
5786:
5782:
5777:
5773:
5769:
5764:
5760:
5756:
5751:
5747:
5743:
5738:
5734:
5730:
5725:
5721:
5717:
5712:
5708:
5704:
5699:
5695:
5691:
5686:
5682:
5673:
5666:
5663:
5660:
5657:
5654:
5649:
5645:
5640:
5625:
5622:
5617:
5613:
5609:
5604:
5600:
5596:
5591:
5587:
5583:
5578:
5574:
5570:
5565:
5561:
5552:
5545:
5542:
5539:
5536:
5531:
5526:
5522:
5519:
5516:
5513:
5510:
5507:
5505:
5501:
5500:
5485:
5482:
5477:
5473:
5469:
5464:
5460:
5456:
5451:
5447:
5438:
5431:
5428:
5425:
5422:
5420:
5416:
5412:
5408:
5404:
5400:
5396:
5393:
5389:
5385:
5380:CRC-16-CCITT
5378:
5374:
5371:
5368:
5365:
5362:
5360:applications
5359:
5356:
5354:
5350:
5349:
5345:
5342:
5339:
5336:
5333:
5330:
5327:
5326:
5322:
5319:
5316:
5313:
5310:
5308:
5306:
5302:
5301:
5286:
5283:
5278:
5274:
5270:
5265:
5261:
5257:
5252:
5248:
5244:
5239:
5235:
5231:
5226:
5222:
5218:
5213:
5209:
5205:
5200:
5196:
5187:
5180:
5177:
5174:
5171:
5167:
5162:
5158:
5155:
5152:
5149:
5146:
5143:
5141:
5137:
5136:
5132:
5129:
5126:
5123:
5120:
5118:
5115:
5112:
5111:
5096:
5093:
5088:
5084:
5080:
5075:
5071:
5067:
5062:
5058:
5054:
5049:
5045:
5041:
5036:
5032:
5028:
5023:
5019:
5015:
5010:
5006:
5002:
4997:
4993:
4989:
4984:
4980:
4971:
4964:
4961:
4958:
4955:
4953:
4950:Time signal,
4945:
4941:
4938:
4935:
4932:
4929:
4926:
4924:
4920:
4919:
4915:
4912:
4909:
4906:
4903:
4900:
4898:
4894:
4893:
4878:
4875:
4872:
4869:
4864:
4860:
4856:
4851:
4847:
4843:
4838:
4834:
4830:
4825:
4821:
4812:
4805:
4802:
4799:
4796:
4789:
4774:
4771:
4766:
4762:
4758:
4753:
4749:
4745:
4740:
4736:
4732:
4727:
4723:
4719:
4714:
4710:
4701:
4694:
4691:
4688:
4685:
4683:
4676:
4672:
4669:
4666:
4663:
4660:
4657:
4655:
4651:
4650:
4646:
4643:
4640:
4637:
4634:
4631:
4629:
4625:
4624:
4609:
4606:
4603:
4600:
4595:
4591:
4587:
4582:
4578:
4574:
4569:
4565:
4561:
4556:
4552:
4543:
4536:
4533:
4530:
4527:
4525:
4517:
4502:
4499:
4496:
4493:
4488:
4484:
4480:
4475:
4471:
4467:
4462:
4458:
4454:
4449:
4445:
4436:
4429:
4426:
4423:
4420:
4415:
4410:
4395:
4392:
4387:
4383:
4379:
4374:
4370:
4366:
4361:
4357:
4353:
4348:
4344:
4335:
4328:
4325:
4322:
4319:
4317:
4313:
4308:
4303:
4288:
4285:
4280:
4276:
4272:
4267:
4263:
4259:
4254:
4250:
4241:
4234:
4231:
4228:
4225:
4219:
4214:
4199:
4196:
4191:
4187:
4183:
4178:
4174:
4170:
4165:
4161:
4157:
4152:
4148:
4139:
4132:
4129:
4126:
4123:
4121:
4114:
4099:
4096:
4091:
4087:
4083:
4078:
4074:
4070:
4065:
4061:
4052:
4045:
4042:
4039:
4036:
4034:
4031:
4026:
4022:
4017:
4002:
3999:
3996:
3993:
3988:
3984:
3980:
3975:
3971:
3962:
3955:
3952:
3949:
3946:
3944:
3940:
3936:
3933:
3929:
3923:
3918:
3903:
3900:
3897:
3894:
3889:
3885:
3881:
3876:
3872:
3868:
3863:
3859:
3855:
3850:
3846:
3837:
3830:
3827:
3824:
3821:
3816:
3811:
3796:
3793:
3790:
3787:
3782:
3778:
3774:
3769:
3765:
3761:
3756:
3752:
3748:
3743:
3739:
3730:
3675:
3672:
3669:
3666:
3664:
3658:
3653:
3638:
3635:
3630:
3626:
3622:
3617:
3613:
3609:
3604:
3600:
3596:
3591:
3587:
3583:
3578:
3574:
3565:
3510:
3507:
3504:
3501:
3499:
3492:
3488:
3485:
3482:
3479:
3476:
3474:
3470:
3467:
3464:
3463:
3448:
3445:
3440:
3436:
3432:
3427:
3423:
3414:
3407:
3404:
3401:
3398:
3396:
3392:
3388:
3384:
3376:
3361:
3358:
3355:
3352:
3347:
3343:
3334:
3327:
3324:
3321:
3318:
3315:
3307:
3292:
3289:
3286:
3283:
3278:
3274:
3270:
3265:
3261:
3257:
3252:
3248:
3244:
3239:
3235:
3226:
3171:
3168:
3165:
3162:
3157:
3152:
3148:
3145:
3142:
3139:
3136:
3134:
3131:
3128:
3127:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3101:
3100:
3096:
3093:
3090:
3087:
3084:
3081:
3078:
3074:
3073:
3058:
3055:
3050:
3046:
3042:
3037:
3033:
3024:
3017:
3014:
3011:
3008:
3005:
2998:
2983:
2980:
2975:
2971:
2967:
2962:
2958:
2954:
2949:
2945:
2936:
2929:
2926:
2923:
2920:
2917:
2909:
2894:
2891:
2886:
2882:
2878:
2873:
2869:
2860:
2853:
2850:
2847:
2844:
2842:
2835:
2820:
2817:
2814:
2811:
2806:
2802:
2793:
2786:
2783:
2780:
2777:
2775:, p. 12
2774:
2771:
2764:
2749:
2746:
2743:
2740:
2735:
2731:
2722:
2667:
2664:
2661:
2658:
2653:
2648:
2633:
2630:
2627:
2619:
2612:
2609:
2606:
2603:
2601:
2600:
2591:
2587:
2584:
2581:
2578:
2575:
2572:
2569:
2566:
2563:
2560:
2557:
2554:
2551:
2548:
2545:
2542:
2540:
2537:
2535:
2532:
2529:
2528:
2525:
2516:
2511:
2501:
2497:
2495:
2480:
2476:
2473:
2470:
2466:
2457:
2448:
2446:
2444:
2441:
2437:
2433:
2429:
2420:
2418:
2414:
2411:
2406:
2401:
2397:
2393:
2389:
2384:
2382:
2378:
2374:
2368:
2366:
2358:
2356:
2354:
2350:
2346:
2338:
2331:
2328:
2325:
2322:
2321:
2317:
2314:
2311:
2308:
2307:
2301:
2281:
2277:
2273:
2265:
2261:
2257:
2254:
2249:
2245:
2241:
2238:
2233:
2229:
2225:
2222:
2217:
2213:
2209:
2198:
2181:
2177:
2173:
2165:
2161:
2157:
2154:
2149:
2145:
2141:
2138:
2133:
2129:
2125:
2122:
2117:
2113:
2109:
2098:
2078:
2074:
2070:
2067:
2062:
2058:
2054:
2051:
2046:
2042:
2038:
2035:
2030:
2026:
2022:
2016:
2011:
2007:
1998:
1997:
1996:
1982:
1979:
1976:
1973:
1968:
1964:
1955:
1932:
1928:
1919:
1916:
1913:
1909:
1905:
1899:
1894:
1890:
1887:
1884:
1883:
1879:
1876:
1871:
1868:
1865:
1861:
1857:
1853:
1849:
1847:
1841:
1838:
1834:
1833:
1832:
1827:Specification
1826:
1824:
1822:
1818:
1814:
1795:
1789:
1782:A polynomial
1780:
1766:
1763:
1758:
1755:
1752:
1748:
1727:
1724:
1721:
1701:
1678:
1675:
1672:
1663:
1657:
1654:
1648:
1642:
1622:
1619:
1614:
1610:
1589:
1581:
1577:
1572:
1570:
1569:zero divisors
1565:
1559:
1553:
1551:
1547:
1541:
1534:
1529:
1525:
1521:
1517:
1512:
1509:
1506:
1499:
1497:
1495:
1491:
1487:
1480:
1469:
1466:
1458:
1448:
1444:
1438:
1437:
1432:This section
1430:
1426:
1421:
1420:
1414:
1403:'100'
1382:>>>
1379:'100'
1355:crc_remainder
1352:>>>
860:crc_remainder
853:
850:
841:
837:
835:
826:
819:
817:
813:
805:
802:
798:
794:
790:
782:
778:
772:
770:
764:
759:
753:
742:
739:
731:
721:
717:
713:
707:
706:
702:
697:This section
695:
691:
686:
685:
679:
677:
675:
670:
666:
662:
658:
657:stream cipher
639:
633:
627:
624:
621:
615:
609:
606:
603:
597:
591:
588:
585:
579:
576:
573:
570:
567:
561:
558:
551:
550:
549:
535:
515:
495:
475:
455:
435:
412:
409:
403:
397:
394:
391:
385:
379:
376:
373:
367:
364:
361:
355:
352:
345:
344:
343:
341:
337:
332:
329:
327:
323:
319:
315:
309:
307:
299:
297:
293:
290:
288:
282:
280:
276:
272:
264:
262:
258:
253:
248:
246:
242:
238:
234:
228:
221:
214:
209:
205:
201:
197:
192:
190:
185:
183:
179:
175:
171:
167:
163:
159:
155:
150:
144:
140:
136:
132:
128:
124:
121:. The use of
120:
117:
109:
107:
105:
104:hash function
101:
97:
93:
89:
87:
82:
78:
74:
70:
65:
63:
59:
55:
51:
47:
43:
39:
35:
30:
19:
11761:Universal 3D
11537:
11524:. Retrieved
11475:18 September
11473:. Retrieved
11464:
11459:Cook, Greg.
11436:the original
11430:
11417:— Bitfilters
11409:. Retrieved
11381:. Retrieved
11356:
11340:. Retrieved
11336:the original
11305:
11301:
11275:
11251:
11223:. Retrieved
11202:
11190:. Retrieved
11169:
11156:. Retrieved
11146:
11139:
11120:
11108:. Retrieved
11088:
11082:
11070:. Retrieved
11055:
11043:. Retrieved
11034:
11025:
11013:. Retrieved
11004:
10994:
10982:. Retrieved
10973:
10964:
10952:the original
10943:
10922:. Retrieved
10915:the original
10906:
10899:
10887:. Retrieved
10866:
10854:. Retrieved
10838:
10831:
10819:. Retrieved
10812:the original
10798:
10786:. Retrieved
10766:
10759:
10747:. Retrieved
10738:
10729:
10717:. Retrieved
10707:
10701:
10689:. Retrieved
10673:
10666:
10655:the original
10642:
10620:(4): 62–75.
10617:
10613:
10607:
10585:(3): 40–50.
10582:
10578:
10572:
10562:
10534:
10527:
10519:the original
10508:
10496:. Retrieved
10486:
10479:
10467:. Retrieved
10463:the original
10457:
10433:. Retrieved
10426:the original
10417:
10410:
10398:. Retrieved
10389:
10365:. Retrieved
10349:
10342:
10330:. Retrieved
10314:
10292:. Retrieved
10283:
10274:
10262:. Retrieved
10246:
10222:. Retrieved
10215:the original
10206:
10184:(Table 6.12)
10176:. Retrieved
10156:
10150:
10138:. Retrieved
10129:
10120:
10108:. Retrieved
10092:
10066:
10058:
10054:
10050:
10045:
10033:. Retrieved
10024:
10002:
9976:
9969:
9957:. Retrieved
9940:
9936:
9926:
9914:. Retrieved
9895:
9883:. Retrieved
9868:
9856:. Retrieved
9804:
9765:
9761:
9755:
9745:18 September
9743:. Retrieved
9713:. Retrieved
9661:
9623:. Retrieved
9608:
9596:. Retrieved
9577:
9567:
9555:. Retrieved
9551:the original
9523:. Retrieved
9488:(5): 35–39.
9485:
9481:
9468:
9456:. Retrieved
9452:
9443:
9435:
9428:. Retrieved
9421:the original
9407:
9395:. Retrieved
9386:
9380:
9370:
9339:
9305:
9301:
9295:
9283:. Retrieved
9279:the original
9274:
9265:
9257:
9235:
9229:
9221:
9209:
9205:
9199:
8520:p. 51,
6187:Fletcher-16
6023:
6019:
5418:
5394:
5113:CRC-14-DARC
3316:, p. 3
2918:, p. 9
2597:
2496:
2452:
2434:processors'
2421:
2405:Hamming code
2385:
2369:
2362:
2342:
2299:
1953:
1950:
1917:
1907:
1897:
1892:
1888:
1880:
1869:
1863:
1851:
1845:
1843:
1836:
1830:
1820:
1816:
1781:
1573:
1563:
1554:
1549:
1545:
1539:
1532:
1530:, and hence
1527:
1523:
1519:
1515:
1513:
1510:
1507:
1503:
1485:
1482:
1461:
1452:
1441:Please help
1436:verification
1433:
846:
838:
833:
831:
823:
811:
809:
803:
796:
792:
788:
780:
776:
773:
762:
757:
755:
734:
725:
710:Please help
698:
669:block cipher
654:
427:
333:
330:
328:functions).
313:
310:
303:
294:
291:
286:
283:
278:
274:
270:
268:
256:
249:
240:
226:
219:
212:
207:
203:
199:
195:
193:
186:
182:finite field
178:coefficients
151:
142:
138:
131:burst errors
113:
110:Introduction
85:
83:is based on
72:
68:
66:
49:
37:
33:
31:
29:
11904:Polynomials
11672:(Ultrium-1)
11411:1 September
11225:15 December
11089:AIXM Primer
10053:means that
9959:31 December
9916:31 December
9275:drdobbs.com
9004:CRC-64-ISO
8157:0xC0A0A0D5
8154:0xAB050503
8151:0xD5828281
8148:0x814141AB
8083:0x992C1A4C
8080:0x32583499
8077:0x992C1A4C
8074:0x32583499
7771:0xBA0DC66B
7768:0xD663B05D
7765:0xEB31D82E
7762:0x741B8CD7
7496:2147483615
7451:0x8F6E37A0
7448:0x05EC76F1
7445:0x82F63B78
7442:0x1EDC6F41
7199:4294967263
7151:0x82608EDB
7148:0xDB710641
7145:0xEDB88320
7142:0x04C11DB7
6881:0x30185CE3
6878:0x31CE8603
6875:0x38E74301
6872:0x2030B9C7
6228:CRC-21-CAN
6205:CRC-17-CAN
6024:CRC-16-ANSI
4947:CRC-13-BBC
4522:ATM; ITU-T
4116:CRC-8-DARC
3473:IEC 60870-5
3129:CRC-6-DARC
2417:PHY headers
2339:Obfuscation
1875:serial-port
1415:Mathematics
1373:'0'
1321:'1'
1251:'1'
1221:'1'
1191:check_value
1161:'0'
1134:check_value
1028:'1'
998:'1'
911:'0'
680:Computation
314:intentional
271:check value
265:Application
77:information
50:check value
11883:Categories
11816:Full BASIC
11800:ECMAScript
11792:JavaScript
11465:CRC RevEng
11383:4 February
11284:B0CVXWDZ99
11110:3 February
11072:3 February
11015:19 January
10856:26 January
10749:26 January
10719:26 January
10691:11 October
10614:IEEE Micro
10579:IEEE Micro
10498:20 October
10400:19 January
10350:EN 302 307
10294:19 January
10264:26 January
10224:14 October
10140:19 January
10110:21 October
10035:5 November
9904:(Report).
9885:26 January
9858:14 January
9715:14 January
9525:1 November
9430:4 February
9212:: 103084.
9191:References
9013:Swiss-Prot
9007:ISO 3309 (
8142:aviation;
7098:ITU-T V.42
6136:OpenSafety
6109:OpenSafety
3663:OpenSafety
3465:CRC-7-MVB
3309:CRC-6-ITU
3000:CRC-5-USB
2911:CRC-5-ITU
2841:Gen 2 RFID
2837:CRC-5-EPC
2766:CRC-4-ITU
2599:parity bit
2546:≥ 16
2539:Reciprocal
2519:Primitive
2349:obfuscated
1882:Byte order
1870:Bit order:
1330:''
1095:''
844:remainder
816:systematic
769:polynomial
659:that uses
289:constant.
252:parity bit
123:systematic
79:) and the
73:redundancy
62:bitfilters
11665:Super DLT
11372:. Intel.
11342:15 August
10889:11 August
10599:206471618
10165:EPCglobal
10163:. 1.2.0.
9813:CiteSeerX
9670:CiteSeerX
9598:20 August
9490:CiteSeerX
8324:Adler-32
7424:payload,
6727:0xC00031
6724:0x8C0003
6721:0xC60001
6718:0x800063
6713:OS-9 RTOS
6512:0xC3267D
6509:0xBE64C3
6506:0xDF3261
6503:0x864CFB
6268:0xAEB6E5
6265:0xA76D75
6262:0xD3B6BA
6259:0x5D6DCB
6243:0x18144C
6240:0x064503
6237:0x132281
6234:0x102899
5419:CRC-CCITT
5403:Bluetooth
4307:SAE J1850
3943:SMBus PEC
3815:Bluetooth
2347:might be
2315:Reversed
1904:overflows
1813:BCH codes
1764:−
1756:−
1725:−
1620:−
1580:syndromes
1455:July 2016
1385:crc_check
1233:cur_shift
1167:len_input
1116:crc_check
1010:cur_shift
917:len_input
728:July 2016
699:does not
628:
622:⊕
610:
604:⊕
592:
577:⊕
571:⊕
562:
410:⊕
398:
392:⊕
380:
365:⊕
356:
306:integrity
174:remainder
81:algorithm
11754:Graphics
11517:Archived
11469:Archived
11450:Archived
11402:Archived
11374:Archived
11330:(1993).
11308:: 1–14.
11216:Archived
11183:Archived
11158:21 April
11152:Archived
11128:Archived
11101:Archived
11066:Archived
11039:Archived
11009:Archived
10978:Archived
10974:roug.org
10880:Archived
10847:Archived
10779:Archived
10743:Archived
10713:Archived
10682:Archived
10634:10216862
10492:Archived
10394:Archived
10358:Archived
10323:Archived
10288:Archived
10255:Archived
10169:Archived
10134:Archived
10101:Archived
10029:Archived
9994:32688603
9953:Archived
9910:Archived
9879:Archived
9849:Archived
9845:14775606
9739:Archived
9706:Archived
9619:Archived
9592:Archived
9516:Archived
9391:Archived
9344:Springer
9322:51666741
9150:Checksum
9144:See also
8522:XZ Utils
8518:ECMA-182
8331:Adler-32
8139:CRC-32Q
7102:Ethernet
6775:8388583
6772:8388583
6711:Used in
6489:Radix-64
6220:0x1B42D
6217:0x1685B
6214:0x1B42D
6211:0x1685B
6192:Adler-32
6190:Used in
6163:Profibus
5504:CDMA2000
4897:CDMA2000
4628:CDMA2000
3385:, ITU-T
3104:CDMA2000
3077:CDMA2000
2534:Reversed
2343:CRCs in
1912:register
1844:appends
1694:, where
279:codeword
170:quotient
166:dividend
100:function
92:hardware
46:networks
40:) is an
11853:ECMA-35
11777:C++/CLI
11625:OpenXPS
11095:. 4.5.
11045:27 June
11031:"cksum"
10984:17 July
10788:16 July
10469:22 July
10435:24 July
10367:29 July
9982:Bibcode
9945:Bibcode
9625:26 July
9512:3132937
9285:28 June
8511:CRC-64-
8337:CRC-40-
8064:CRC-32K
7816:114663
7753:CRC-32K
7407:CRC-32C
7088:X3.66 (
7074:CRC-32
6864:CRC-30
6705:CRC-24-
6494:OpenPGP
6487:CRC-24-
6255:FlexRay
6251:CRC-24
6231:CAN FD
6208:CAN FD
6179:0x8EE7
6176:0xE771
6173:0xF3B8
6170:0x1DCF
6161:CRC-16-
6153:0xBAAD
6150:0xB55D
6147:0xDAAE
6144:0x755B
6134:CRC-16-
6126:0xAC9A
6123:0x5935
6120:0xAC9A
6117:0x5935
6107:CRC-16-
6037:0xC002
6034:0x4003
6031:0xA001
6028:0x8005
5994:CRC-16-
5846:0x9EB2
5843:0x4D79
5840:0xA6BC
5837:0x3D65
5829:IEC 870
5821:CRC-16-
5667:0xC5DB
5664:0xDBA3
5661:0xEDD1
5658:0x8BB7
5642:CRC-16-
5546:0x82C4
5543:0x2341
5540:0x91A0
5537:0x0589
5528:CRC-16-
5520:0xE433
5517:0xCC27
5514:0xE613
5511:0xC867
5502:CRC-16-
5432:0x8810
5426:0x8408
5423:0x1021
5372:0xD015
5369:0xA80B
5366:0xD405
5363:0xA02B
5351:CRC-16-
5343:0x978A
5340:0x51E9
5337:0xA8F4
5334:0x2F15
5320:0x740A
5317:0x2817
5314:0x540B
5311:0x6815
5305:MPT1327
5303:CRC-15-
5181:0x62CC
5178:0x19A3
5175:0x4CD1
5172:0xC599
5164:CRC-15-
5156:0x3016
5153:0x1A03
5150:0x2D01
5147:0x202D
5138:CRC-14-
5130:0x2402
5127:0x1009
5124:0x2804
5121:0x0805
4965:0x1E7A
4962:0x0BCF
4959:0x15E7
4956:0x1CF5
4921:CRC-12-
4895:CRC-12-
4791:CRC-12
4682:FlexRay
4678:CRC-11
4652:CRC-10-
4626:CRC-10-
4519:CRC-10
3657:AUTOSAR
2530:Normal
2443:AArch64
2436:Nehalem
2312:Normal
720:removed
705:sources
287:residue
247:below.
158:divisor
147:(1 − 2)
11787:Eiffel
11718:FAT16B
11699:(CDFS)
11692:CD-ROM
11282:
11263:
11192:4 July
10924:9 July
10821:7 July
10632:
10597:
10542:
10332:8 July
10178:4 July
9992:
9843:
9833:
9815:
9702:793862
9700:
9690:
9672:
9584:
9557:23 May
9510:
9492:
9397:21 May
9358:
9320:
9250:
9017:TrEMBL
8128:65506
8122:32738
7810:16360
7426:SSE4.2
7196:91607
7137:ZMODEM
7110:MPEG-2
7080:3309 (
6500:104v3
6020:CRC-16
6005:Modbus
6001:Bisync
5429:0x811
5407:PACTOR
5399:XMODEM
4939:0xE98
4936:0x197
4933:0x8CB
4930:0xD31
4913:0xF89
4910:0x91F
4907:0xC8F
4904:0xF13
4806:0xC07
4803:0xE03
4800:0xF01
4797:0x80F
4695:0x5C2
4692:0x21D
4689:0x50E
4686:0x385
4670:0x2BA
4667:0x175
4664:0x2BA
4661:0x175
4644:0x3EC
4641:0x0DF
4638:0x26F
4635:0x3D9
4537:0x319
4534:0x263
4531:0x331
4528:0x233
4412:CRC-8-
4305:CRC-8-
4216:CRC-8-
4030:1-Wire
4021:Dallas
4019:CRC-8-
3926:ITU-T
3920:CRC-8-
3813:CRC-8-
3655:CRC-8-
3498:DVB-S2
3494:CRC-8
3378:CRC-7
3312:ITU-T
3154:CRC-6-
3102:CRC-6-
3075:CRC-6-
2914:ITU-T
2650:CRC-3-
2593:CRC-1
2515:Parity
2428:SSE4.2
2323:CRC-4
1848:0-bits
1552:-XXX.
1315:return
1155:lstrip
1092:return
905:lstrip
849:Python
428:where
116:cyclic
86:cyclic
11846:Other
11713:FAT16
11708:FAT12
11526:9 May
11520:(PDF)
11513:(PDF)
11439:(PDF)
11426:(PDF)
11405:(PDF)
11398:(PDF)
11377:(PDF)
11370:(PDF)
11219:(PDF)
11212:(PDF)
11186:(PDF)
11179:(PDF)
11104:(PDF)
11093:(PDF)
10955:(PDF)
10948:(PDF)
10918:(PDF)
10911:(PDF)
10883:(PDF)
10876:(PDF)
10850:(PDF)
10843:(PDF)
10815:(PDF)
10808:(PDF)
10782:(PDF)
10775:Ofcom
10771:(PDF)
10685:(PDF)
10678:(PDF)
10658:(PDF)
10651:(PDF)
10630:S2CID
10595:S2CID
10429:(PDF)
10422:(PDF)
10361:(PDF)
10354:(PDF)
10326:(PDF)
10319:(PDF)
10258:(PDF)
10251:(PDF)
10218:(PDF)
10211:(PDF)
10172:(PDF)
10161:(PDF)
10104:(PDF)
10097:(PDF)
9852:(PDF)
9841:S2CID
9809:(PDF)
9709:(PDF)
9698:S2CID
9666:(PDF)
9519:(PDF)
9508:S2CID
9478:(PDF)
9458:5 May
9424:(PDF)
9417:(PDF)
9318:S2CID
8538:even
8358:even
8160:even
8086:even
7774:even
7490:5243
7454:even
7430:Btrfs
7414:iSCSI
7193:2974
7129:cksum
7126:POSIX
7122:Bzip2
7114:PKZIP
7090:ADCCP
6884:even
6730:even
6707:WCDMA
6515:even
6271:even
6246:even
6223:even
6040:even
6016:X3.28
5849:even
5833:M-Bus
5827:DNP,
5549:even
5435:even
5415:DigRF
5358:ACARS
5353:ARINC
5184:even
5159:even
5133:even
4968:even
4916:even
4809:even
4698:even
4647:even
4540:even
4524:I.610
4433:even
4430:0xCD
4427:0xB3
4424:0xD9
4421:0x9B
4414:WCDMA
4329:0x8E
4326:0x71
4323:0xB8
4320:0x1D
4238:even
4235:0xA4
4232:0x25
4229:0x92
4226:0x49
4133:0x9C
4130:0x39
4127:0x9C
4124:0x39
4049:even
4046:0x98
4043:0x19
4040:0x8C
4037:0x31
4025:Maxim
3959:even
3956:0x83
3953:0xC1
3950:0xE0
3947:0x07
3922:CCITT
3834:even
3831:0xD3
3828:0xCB
3825:0xE5
3822:0xA7
3679:even
3676:0x97
3673:0xE9
3670:0xF4
3667:0x2F
3514:even
3511:0xEA
3508:0x57
3505:0xAB
3502:0xD5
3486:0x72
3483:0x27
3480:0x53
3477:0x65
3408:0x44
3405:0x11
3402:0x48
3399:0x09
3387:G.832
3383:G.707
3328:0x21
3325:0x21
3322:0x30
3319:0x03
3314:G.704
3175:even
3172:0x37
3169:0x3B
3166:0x3D
3163:0x2F
3149:even
3146:0x2C
3143:0x0D
3140:0x26
3137:0x19
3124:even
3121:0x23
3118:0x31
3115:0x38
3112:0x07
3094:0x33
3091:0x33
3088:0x39
3085:0x27
3018:0x12
3015:0x09
3012:0x14
3009:0x05
2933:even
2930:0x1A
2927:0x0B
2924:0x15
2921:0x15
2916:G.704
2854:0x14
2851:0x05
2848:0x12
2845:0x09
2773:G.704
2770:ITU-T
2616:even
2506:Uses
2503:Name
2456:GF(2)
2432:Intel
2426:) of
2424:CRC32
2410:ITU-T
2377:iSCSI
2309:Name
1910:-bit
1490:GF(2)
1266:range
1245:index
1218:while
1043:range
1022:index
995:while
245:table
189:GF(2)
160:in a
96:noise
88:codes
69:check
60:(see
11858:JSON
11806:Dart
11610:APIW
11528:2015
11477:2020
11413:2006
11385:2007
11344:2011
11280:ASIN
11261:ISBN
11227:2009
11194:2012
11160:2013
11112:2019
11074:2011
11047:2017
11017:2018
10986:2018
10926:2016
10891:2009
10858:2016
10823:2012
10790:2012
10751:2016
10721:2016
10693:2013
10540:ISBN
10500:2014
10471:2016
10437:2016
10402:2018
10369:2016
10334:2013
10296:2018
10266:2016
10226:2013
10180:2012
10142:2018
10112:2016
10037:2017
9990:OCLC
9961:2021
9918:2021
9887:2016
9860:2011
9831:ISBN
9747:2020
9717:2011
9688:ISBN
9627:2011
9600:2024
9582:ISBN
9559:2019
9527:2017
9460:2019
9432:2011
9399:2009
9356:ISBN
9287:2017
9248:ISBN
9034:odd
9009:HDLC
8513:ECMA
8144:AIXM
8116:134
7804:152
7484:177
7457:yes
7438:Ceph
7434:ext4
7422:G.hn
7418:SCTP
7190:268
7187:171
7157:yes
7154:odd
7118:Gzip
7106:SATA
7094:FIPS
7086:ANSI
7082:HDLC
6868:CDMA
6733:yes
6498:RTCM
6182:odd
6156:odd
6129:odd
6022:and
6013:ANSI
5670:odd
5655:DIF
5653:SCSI
5530:DECT
5523:odd
5392:HDLC
5388:V.41
5384:X.25
5375:odd
5346:odd
5323:odd
4942:odd
4673:odd
4332:odd
4312:AES3
4136:odd
3939:ISDN
3724:119
3721:119
3682:yes
3489:odd
3411:odd
3331:odd
3178:yes
3097:odd
3021:odd
2857:odd
2790:odd
2787:0x9
2784:0x9
2781:0xC
2778:0x3
2674:yes
2671:odd
2668:0x5
2665:0x5
2662:0x6
2659:0x3
2613:0x1
2610:0x1
2607:0x1
2604:0x1
2413:G.hn
2400:DTIC
2381:SCTP
2379:and
2332:0x9
2329:0xC
2326:0x3
1562:1 +
1494:ring
1409:True
1336:join
1200:list
1101:join
977:list
703:any
701:cite
665:mode
528:and
468:and
11837:UWB
11832:NFC
11796:E4X
11730:UDF
11703:FAT
11676:VXA
11660:DLT
11655:DDS
11310:doi
10622:doi
10587:doi
9823:doi
9770:doi
9680:doi
9500:doi
9348:doi
9310:doi
9240:doi
9214:doi
9210:188
9175:LRC
9011:),
8339:GSM
8110:26
8104:16
8089:no
7798:18
7792:16
7777:no
7478:47
7472:20
7184:91
7181:57
7178:34
7175:21
7172:12
7163:10
7133:PNG
7104:),
7092:),
7084:),
7078:ISO
6138:-B
6111:-A
6009:USB
5996:IBM
5823:DNP
5648:DIF
5644:T10
5395:FCS
5166:CAN
5140:GSM
4923:GSM
4654:GSM
4316:OBD
4220:-B
4218:GSM
4033:bus
3935:HEC
3932:ATM
3559:85
3556:85
3517:no
3391:MMC
3220:25
3217:25
3156:GSM
3106:-B
3079:-A
3004:USB
2652:GSM
2564:10
2561:11
2558:12
2555:13
2552:14
2549:15
2440:ARM
2419:).
1906:an
1900:+ 1
1542:+ 1
1535:+ 1
1445:by
1324:not
1297:int
1291:str
1281:)):
1272:len
1257:for
1173:len
1113:def
1074:int
1068:str
1058:)):
1049:len
1034:for
944:len
923:len
857:def
799:+ 1
795:+ 1
791:+ 0
783:+ 1
765:+ 1
714:by
667:of
661:XOR
625:CRC
607:CRC
589:CRC
559:CRC
395:CRC
377:CRC
353:CRC
342:):
320:or
275:CRC
273:or
259:+ 1
237:LSb
235:or
233:MSb
229:+ 1
222:+ 1
215:+ 1
64:).
38:CRC
11885::
11798:,
11782:C#
11725:FD
11467:.
11463:.
11428:.
11355:.
11306:56
11304:.
11278:.
11249:.
11037:.
11033:.
11007:.
11003:.
10976:.
10972:.
10934:^
10741:.
10737:.
10628:.
10616:.
10593:.
10581:.
10554:^
10445:^
10392:.
10388:.
10377:^
10304:^
10286:.
10282:.
10234:^
10189:^
10132:.
10128:.
10075:^
10027:.
10023:.
10012:^
9988:.
9941:76
9939:.
9935:.
9847:.
9839:.
9829:.
9821:.
9782:^
9766:41
9764:.
9737:.
9725:^
9704:.
9696:.
9686:.
9678:.
9635:^
9590:.
9576:.
9535:^
9514:.
9506:.
9498:.
9486:46
9484:.
9480:.
9451:.
9434:.
9387:11
9385:.
9379:.
9354:.
9342:.
9330:^
9316:.
9306:49
9304:.
9273:.
9256:.
9246:.
9220:.
9208:.
9051:64
8935:10
8922:12
8909:13
8896:17
8883:19
8870:21
8857:22
8844:23
8831:24
8818:27
8805:29
8792:31
8779:32
8750:33
8737:35
8724:37
8711:38
8698:39
8685:40
8672:45
8659:46
8646:47
8633:52
8620:53
8607:54
8594:55
8581:57
8568:62
8555:64
8471:17
8449:23
8414:17
8401:23
8388:26
8375:40
8242:14
8229:16
8216:22
8203:24
8190:31
8177:32
8134:∞
8131:–
8125:–
8119:–
8113:–
8107:–
8101:–
8098:3
8095:–
8092:–
7982:10
7969:11
7956:15
7943:16
7930:17
7917:19
7904:20
7891:26
7878:28
7865:29
7852:30
7839:32
7822:∞
7819:–
7813:–
7807:–
7801:–
7795:–
7789:–
7786:4
7783:–
7780:2
7688:10
7675:11
7662:13
7649:14
7636:18
7623:19
7610:20
7597:22
7584:23
7571:25
7558:26
7545:27
7532:28
7519:32
7502:∞
7499:–
7493:–
7487:–
7481:–
7475:–
7469:–
7466:8
7463:–
7460:6
7436:,
7432:,
7428:,
7420:,
7416:,
7310:10
7297:11
7284:12
7271:16
7258:22
7245:23
7232:26
7219:32
7202:∞
7169:–
7166:–
7160:–
7135:,
7131:,
7124:,
7120:,
7116:,
7112:,
7108:,
6992:11
6979:12
6966:13
6953:15
6940:20
6927:21
6914:29
6901:30
6808:23
6795:24
6778:∞
6769:4
6766:4
6763:–
6760:–
6757:–
6754:–
6751:–
6748:–
6745:–
6742:–
6739:–
6736:–
6610:10
6597:11
6584:14
6571:17
6558:18
6545:23
6532:24
6496:,
6405:10
6392:11
6379:13
6366:14
6353:16
6340:18
6327:19
6314:20
6301:22
6288:24
6070:15
6057:16
6011:,
6007:,
6003:,
5918:10
5905:11
5892:12
5879:13
5866:16
5831:,
5713:11
5700:15
5687:16
5579:10
5566:16
5465:12
5452:16
5413:,
5411:SD
5409:,
5405:,
5401:,
5397:,
5390:,
5386:,
5227:10
5214:14
5201:15
5024:10
5011:11
4998:12
4985:13
4839:11
4826:12
4715:11
4557:10
4314:;
3937:,
3930:;
3727:∞
3718:3
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