Polynomial representations of cyclic redundancy checks

These are polynomial representations of cyclic redundancy checks CRCs.

Name	Polynomial	Representations: normal / reversed / reverse of reciprocal
CRC-1	$x + 1$ (most hardware; also known as parity bit)	0x1 / 0x1 / 0x1
CRC-4-ITU	$x^4 + x + 1$ (ITU-T G.704, p. 12)	0x3 / 0xC / 0x9
CRC-5-EPC	$x^5 + x^3 + 1$ (Gen 2 RFID[1])	0x09 / 0x12 / 0x14
CRC-5-ITU	$x^5 + x^4 + x^2 + 1$ (ITU-T G.704, p. 9)	0x15 / 0x15 / 0x1A
CRC-5-USB	$x^5 + x^2 + 1$ (USB token packets)	0x05 / 0x14 / 0x12
CRC-6-ITU	$x^6 + x + 1$ (ITU-T G.704, p. 3)	0x03 / 0x30 / 0x21
CRC-7	$x^7 + x^3 + 1$ (telecom systems, ITU-T G.707, ITU-T G.832, MMC, SD)	0x09 / 0x48 / 0x44
CRC-8-CCITT	$x^8 + x^2 + x + 1$ (ATM HEC), ISDN Header Error Control and Cell Delineation ITU-T I.432.1 (02/99)	0x07 / 0xE0 / 0x83
CRC-8-Dallas/Maxim	$x^8 + x^5 + x^4 + 1$ (1-Wire bus)	0x31 / 0x8C / 0x98
CRC-8	$x^8 + x^7 + x^6 + x^4 + x^2 + 1$	0xD5 / 0xAB / 0xEA[2]
CRC-8-SAE J1850	$x^8 + x^4 + x^3 + x^2 + 1$	0x1D / 0xB8 / 0x8E
CRC-8-WCDMA	$x^8 + x^7 + x^4 + x^3 + x + 1$[3]	0x9B / 0xD9 / 0xCD[2]
CRC-10	$x^{10} + x^9 + x^5 + x^4 + x + 1$ (ATM; ITU-T I.610)	0x233 / 0x331 / 0x319
CRC-11	$x^{11} + x^9 + x^8 + x^7 + x^2 + 1$ (FlexRay[4])	0x385 / 0x50E / 0x5C2
CRC-12	$x^{12} + x^{11} + x^3 + x^2 + x + 1$ (telecom systems[5][6])	0x80F / 0xF01 / 0xC07[2]
CRC-13-BBC	$x^{13} + x^{12} + x^{11} + x^{10} + x^7 + x^6 + x^5 + x^4 + x^2 + 1$	0x1CF5 / 0x15E7 / 0x1E7A
CRC-15-CAN	$x^{15} + x^{14} + x^{10} + x^8 + x^7 + x^4 + x^3 + 1$	0x4599 / 0x4CD1 / 0x62CC
CRC-16-IBM	$x^{16} + x^{15} + x^2 + 1$ (Bisync, Modbus, USB, ANSI X3.28, many others; also known as CRC-16 and CRC-16-ANSI)	0x8005 / 0xA001 / 0xC002
CRC-16-CCITT	$x^{16} + x^{12} + x^5 + 1$ (X.25, V.41, HDLC, XMODEM, Bluetooth, SD, many others; known as CRC-CCITT)	0x1021 / 0x8408 / 0x8810[2]
CRC-16-T10-DIF	$x^{16} + x^{15} + x^{11} + x^{9} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1$ (SCSI DIF)	0x8BB7[7] / 0xEDD1 / 0xC5DB
CRC-16-DNP	$x^{16} + x^{13} + x^{12} + x^{11} + x^{10} + x^8 + x^6 + x^5 + x^2 + 1$ (DNP, IEC 870, M-Bus)	0x3D65 / 0xA6BC / 0x9EB2
CRC-16-DECT	$x^{16} + x^{10} + x^8 + x^7 + x^3 + 1$ (cordless telephones)[8]	0x0589 / 0x91A0 / 0x82C4
CRC-16-Fletcher	Not a CRC; see Fletcher's checksum	Used in Adler-32 A & B CRCs
CRC-24	$x^{24} + x^{22} + x^{20} + x^{19} + x^{18} + x^{16} + x^{14} + x^{13} + x^{11} + x^{10} + x^8 + x^7 + x^6 + x^3 + x + 1$ (FlexRay[4])	0x5D6DCB / 0xD3B6BA / 0xAEB6E5
CRC-24-Radix-64	$x^{24} + x^{23} + x^{18} + x^{17} + x^{14} + x^{11} + x^{10} + x^7 + x^6 + x^5 + x^4 + x^3 + x + 1$ (OpenPGP)	0x864CFB / 0xDF3261 / 0xC3267D
CRC-30	$x^{30} + x^{29} + x^{21} + x^{20} + x^{15} + x^{13} + x^{12} + x^{11} + x^{8} + x^{7} + x^{6} + x^{2} + x + 1$ (CDMA)	0x2030B9C7 / 0x38E74301 / 0x30185CE3
CRC-32-Adler	Not a CRC; see Adler-32	See Adler-32
CRC-32	$x^{32} + x^{26} + x^{23} + x^{22} + x^{16} + x^{12} + x^{11} + x^{10} + x^8 + x^7 + x^5 + x^4 + x^2 + x + 1$ (ISO 3309, ANSI X3.66, FIPS PUB 71, FED-STD-1003, ITU-T V.42, Ethernet, SATA, MPEG-2, Gzip, PKZIP, POSIX cksum, PNG,[9] ZMODEM)	0x04C11DB7 / 0xEDB88320 / 0x82608EDB[10]
CRC-32C (Castagnoli)	$x^{32} + x^{28} + x^{27} + x^{26} + x^{25} + x^{23} + x^{22} + x^{20} + x^{19} + x^{18} + x^{14} + x^{13} + x^{11} + x^{10} + x^9 + x^8 + x^6 + 1$ (iSCSI & SCTP, G.hn payload, SSE4.2)	0x1EDC6F41 / 0x82F63B78 / 0x8F6E37A0[10]
CRC-32K (Koopman)	$x^{32} + x^{30} + x^{29} + x^{28} + x^{26} + x^{20} + x^{19} + x^{17} + x^{16} + x^{15} + x^{11} + x^{10} + x^{7} + x^{6} + x^{4} + x^{2} + x + 1$	0x741B8CD7 / 0xEB31D82E / 0xBA0DC66B[10]
CRC-32Q	$x^{32} + x^{31} + x^{24} + x^{22} + x^{16} + x^{14} + x^{8} + x^{7} + x^{5} + x^{3} + x + 1$ (aviation; AIXM[11])	0x814141AB / 0xD5828281 / 0xC0A0A0D5
CRC-40-GSM	$x^{40} + x^{26} + x^{23} + x^{17} + x^3 + 1$ (GSM control channel[12][13][14])	0x0004820009 / 0x9000412000 / 0x8002410004
CRC-64-ISO	$x^{64} + x^4 + x^3 + x + 1$ (HDLC — ISO 3309, Swiss-Prot/TrEMBL; considered weak for hashing[15])	0x000000000000001B / 0xD800000000000000 / 0x800000000000000D
CRC-64-ECMA-182	$x^{64} + x^{62} + x^{57} + x^{55} + x^{54} + x^{53} + x^{52} + x^{47} + x^{46} + x^{45} + x^{40} + x^{39} + x^{38} + x^{37} + x^{35} + x^{33} +$ $x^{32} + x^{31} + x^{29} + x^{27} + x^{24} + x^{23} + x^{22} + x^{21} + x^{19} + x^{17} + x^{13} + x^{12} + x^{10} + x^9 + x^7 + x^4 + x + 1$ (as described in ECMA-182 p. 51)	0x42F0E1EBA9EA3693 / 0xC96C5795D7870F42 / 0xA17870F5D4F51B49

See also

General category

• Error correcting code
• List of checksum algorithms
• Polynomial Representations of CRCs
• Parity

Specific technological references

• Adler-32
• Fletcher's checksum

External links

• Catalogue of parametrised CRC algorithms
• CRC Polynomial Zoo

References

1. "Class-1 Generation-2 UHF RFID Protocol" (PDF). 1.2.0. EPCglobal. 23 October 2008. p. 35. Retrieved 21 May 2009.
2. Koopman, Philip; Chakravarty, Tridib (28 June – 1 July 2004). "Cyclic Redundancy Code (CRC) Polynomial Selection For Embedded Networks" (PDF). 2004 International Conference on Dependable Systems and Networks: 145–154. doi:10.1109/DSN.2004.1311885. ISBN 0-7695-2052-9.
3. Richardson, Andrew (17 March 2005). WCDMA Handbook. Cambridge, UK: Cambridge University Press. p. 223. ISBN 0521828155.
4. "FlexRay Protocol Specification". 2.1 Revision A. Flexray Consortium. 22 December 2005. p. 93.
5. Perez, A.; Wismer & Becker (1983). "Byte-Wise CRC Calculations". IEEE Micro 3 (3): 40–50. doi:10.1109/MM.1983.291120.
6. Ramabadran, T.V.; Gaitonde, S.S. (1988). "A tutorial on CRC computations". IEEE Micro 8 (4): 62–75. doi:10.1109/40.7773.
7. Thaler, Pat (28 August 2003). "16-bit CRC polynomial selection" (PDF). INCITS T10. Retrieved 11 August 2009.
8. "ETSI EN 300 175-3". V2.2.1. Sophia Antipolis, France: European Telecommunications Standards Institute. November 2008.
9. Boutell, Thomas; Randers-Pehrson, Glenn; et al. (14 July 1998). "PNG (Portable Network Graphics) Specification, Version 1.2". Retrieved 3 February 2011.
10. Koopman, Philip (23–26 June 2002). "32-Bit Cyclic Redundancy Codes for Internet Applications" (PDF). 2002 International Conference on Dependable Systems and Networks: 459–468. doi:10.1109/DSN.2002.1028931. ISBN 0-7695-1597-5. Retrieved 14 January 2011.
11. "AIXM Primer" (PDF). 4.5. European Organisation for the Safety of Air Navigation. 20 March 2006. Retrieved 29 April 2009.
12. ETSI TS 100 909 version 8.9.0 (January 2005), Section 4.1.2 a
13. Gammel, B.M. (31 October 2005). "Crypto - Codes". Retrieved 10 February 2011.
14. Geremia, Patrick (April 1999). "Cyclic redundancy check computation: an implementation using the TMS320C54x" (PDF) (SPRA530). Texas Instruments. p. 5. Retrieved 10 February 2011.
15. Jones, David T. "An Improved 64-bit Cyclic Redundancy Check for Protein Sequences" (PDF). University College London. Retrieved 15 December 2009.