User:Xiong/crc

Common polynomials

As well as the polynomial, the prefix and the value exclusive-ORed with the remainder are important for some more complicated checksums (like most forms of CRC-32 and CRC-64).
CRCs less than CRC-16 do not tend to use Initial or final XOR values.
Very often custom versions of checksums are created by changing these values, as it does not alter the overall mechanics (or math) of the checksum algorithm -- and may provide some nominal security features. However a CRC algorithm can be reverse-engineered from a small number of message-CRC pairs by brute force search.

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

Name	Polynomial	Representations: Normal or Reverse (Normal of Reciprocal)
CRC-1	$x+1$ (most hardware; also known as parity bit)	0x1 or 0x1 (0x1)
CRC-4-ITU	$x^{4}+x+1$ (ITU G.704, p. 12)	0x3 or 0xC (0x9)
CRC-5-ITU	$x^{5}+x^{4}+x^{2}+1$ (ITU G.704, p. 9)	0x15 or 0x15 (0x0B)
CRC-5-USB	$x^{5}+x^{2}+1$ (USB token packets)	0x05 or 0x14 (0x9)
CRC-6-ITU	$x^{6}+x+1$ (ITU G.704, p. 3)	0x03 or 0x30 (0x21)
CRC-7	$x^{7}+x^{3}+1$ (telecom systems, MMC)	0x09 or 0x48 (0x11)
CRC-8-ATM	$x^{8}+x^{2}+x+1$ (ATM HEC)	0x07 or 0xE0 (0xC1)
CRC-8-CCITT	$x^{8}+x^{7}+x^{3}+x^{2}+1$ (1-Wire bus)	0x8D or 0xB1 (0x63)
CRC-8-Dallas/Maxim	$x^{8}+x^{5}+x^{4}+1$ (1-Wire bus)	0x31 or 0x8C (0x19)
CRC-8	$x^{8}+x^{7}+x^{6}+x^{4}+x^{2}+1$	0xD5 or 0xAB (0x57)
CRC-8-SAE J1850	$x^{8}+x^{4}+x^{3}+x^{2}+1$	0x1D or 0xB8 (0x71)
CRC-10	$x^{10}+x^{9}+x^{5}+x^{4}+x+1$	0x233 or 0x331 (0x263)
CRC-11	$x^{11}+x^{9}+x^{8}+x^{7}+x^{2}+1$ (FlexRay)	0x385 or 0x50E (0x21D)
CRC-12	$x^{12}+x^{11}+x^{3}+x^{2}+x+1$ (telecom systems, ^[1]^[2] )	0x80F or 0xF01 (0xE03)
CRC-15-CAN	$x^{15}+x^{14}+x^{10}+x^{8}+x^{7}+x^{4}+x^{3}+1$	0x4599 or 0x4CD1 (0x19A3)
CRC-16-Fletcher	Not a CRC; see Fletcher's checksum	Used in Adler-32 A & B CRCs
CRC-16-CCITT	$x^{16}+x^{12}+x^{5}+1$ (X.25, V.41, CDMA, Bluetooth, PPP, IrDA, BACnet; known as CRC-CCITT)	0x1021 or 0x8408 (0x0811)
CRC-16-IBM	$x^{16}+x^{15}+x^{2}+1$ (SDLC, XMODEM, USB, many others; also known as CRC-16)	0x8005 or 0xA001 (0x4003)
CRC-24-Radix-64	$x^{24}+x^{23}+x^{18}+x^{17}+x^{14}+x^{11}+x^{10}+x^{7}+x^{6}+x^{5}+x^{4}+x^{3}+x+1$ (FlexRay)	0x864CFB or 0xDF3261 (0xBE64C3)
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 or 0x38E74301 (0x31CE8603)
CRC-32-Adler	Not a CRC; see Adler-32	See Adler-32
CRC-32-IEEE 802.3	$x^{32}+x^{26}+x^{23}+x^{22}+x^{16}+x^{12}+x^{11}+x^{10}+x^{8}+x^{7}+x^{5}+x^{4}+x^{2}+x+1$ (V.42, MPEG-2)	0x04C11DB7 or 0xEDB88320 (0xDB710641)
CRC-32C (Castagnoli)	$x^{32}+x^{28}+x^{27}+x^{26}+x^{25}+x^{23}+x^{22}+x^{20}+x^{19}+x^{18}+x^{14}+x^{13}+x^{11}+x^{10}+x^{9}+x^{8}+x^{6}+1$	0x1EDC6F41 or 0x82F63B78 (0x05EC76F1)
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 or 0xEB31D82E (0xD663B05D)
CRC-64-ISO	$x^{64}+x^{4}+x^{3}+x+1$ (HDLC — ISO 3309)	0x000000000000001B or 0xD800000000000000 (0xB000000000000001)
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.63)	0x42F0E1EBA9EA3693 or 0xC96C5795D7870F42 (0x92D8AF2BAF0E1E85)

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

CRC-128 (IEEE)
CRC-256 (IEEE)

^ Perez, Aram (1983). "Byte-Wise CRC Calculations". IEEE Micro. 3 (3): 40–50. doi:10.1109/MM.1983.291120. ISSN 0272-1732.{{cite journal}}: CS1 maint: date and year (link)
^ Ramabadran, T.V.; Gaitonde, S.S. (1988). "A tutorial on CRC computations". IEEE Micro. 8 (4): 62–75. doi:10.1109/40.7773. ISSN 0272-1732.{{cite journal}}: CS1 maint: date and year (link)

[1] Perez, Aram (1983). "Byte-Wise CRC Calculations". IEEE Micro. 3 (3): 40–50. doi:10.1109/MM.1983.291120. ISSN 0272-1732.{{cite journal}}: CS1 maint: date and year (link)

[2] Ramabadran, T.V.; Gaitonde, S.S. (1988). "A tutorial on CRC computations". IEEE Micro. 8 (4): 62–75. doi:10.1109/40.7773. ISSN 0272-1732.{{cite journal}}: CS1 maint: date and year (link)

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