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To maximise computation speed, an intermediate remainder can be calculated by first computing the CRC of the message modulo a sparse polynomial which is a multiple of the CRC polynomial. For CRC-32, the polynomial x 123 + x 111 + x 92 + x 84 + x 64 + x 46 + x 23 + 1 has the property that its terms (feedback taps) are at least 8 positions apart ...
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. [16] The ITU-T G.hn standard also uses CRC-32C to detect errors in the payload (although it uses CRC-16-CCITT for PHY ...
Recall that a CRC is the remainder of the message polynomial times divided by the CRC polynomial. A polynomial with a zero x 0 {\displaystyle x^{0}} term always has x {\displaystyle x} as a factor. So if K ( x ) {\displaystyle K(x)} is the original CRC polynomial and K ( x ) = x ⋅ K ′ ( x ) {\displaystyle K(x)=x\cdot K'(x)} , then
By far the most popular FCS algorithm is a cyclic redundancy check (CRC), used in Ethernet and other IEEE 802 protocols with 32 bits, in X.25 with 16 or 32 bits, in HDLC with 16 or 32 bits, in Frame Relay with 16 bits, [3] in Point-to-Point Protocol (PPP) with 16 or 32 bits, and in other data link layer protocols.
Cyclic redundancy checks (CRCs) can correct 1-bit errors for messages at most bits long for optimal generator polynomials of degree , see Mathematics of cyclic redundancy checks § Bitfilters; Locally Recoverable Codes
A BCH code with = is called a narrow-sense BCH code.; A BCH code with = is called primitive.; The generator polynomial () of a BCH code has coefficients from (). In general, a cyclic code over () with () as the generator polynomial is called a BCH code over ().
It is not suitable for detecting maliciously introduced errors. It is characterized by specification of a generator polynomial, which is used as the divisor in a polynomial long division over a finite field, taking the input data as the dividend. The remainder becomes the result. A CRC has properties that make it well suited for detecting burst ...
For example, if the taps are at the 16th, 14th, 13th and 11th bits (as shown), the feedback polynomial is + + + + The "one" in the polynomial does not correspond to a tap – it corresponds to the input to the first bit (i.e. x 0, which is equivalent to 1). The powers of the terms represent the tapped bits, counting from the left.