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The reciprocal of a polynomial generates the same codewords, only bit reversed — that is, if all but the first bits of a codeword under the original polynomial are taken, reversed and used as a new message, the CRC of that message under the reciprocal polynomial equals the reverse of the first bits of the original codeword. But the reciprocal ...
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 ...
The CRC and associated polynomial typically have a name of the form CRC-n-XXX as in the table below. The simplest error-detection system, the parity bit, is in fact a 1-bit CRC: it uses the generator polynomial x + 1 (two terms), [5] and has the name CRC-1.
In coding theory, the Bose–Chaudhuri–Hocquenghem codes (BCH codes) form a class of cyclic error-correcting codes that are constructed using polynomials over a finite field (also called a Galois field). BCH codes were invented in 1959 by French mathematician Alexis Hocquenghem, and independently in 1960 by Raj Chandra Bose and D. K. Ray ...
Polynomials may be reducible, primitive or neither; integers can only be prime or composite. Regregex 13:07, 15 February 2011 (UTC) Firstly, I skipped polynomials in my description because you don't need them to understand or implement the algorithm; only to understand why CRC codes work and how generator polynomials are selected.
The original construction of Reed & Solomon (1960) interprets the message x as the coefficients of the polynomial p, whereas subsequent constructions interpret the message as the values of the polynomial at the first k points , …, and obtain the polynomial p by interpolating these values with a polynomial of degree less than k.
As an alternative to the XOR-based feedback in an LFSR, one can also use XNOR. [2] This function is an affine map, not strictly a linear map, but it results in an equivalent polynomial counter whose state is the complement of the state of an LFSR. A state with all ones is illegal when using an XNOR feedback, in the same way as a state with all ...
One use of these instructions is to improve the speed of applications doing block cipher encryption in Galois/Counter Mode, which depends on finite field GF(2 k) multiplication. Another application is the fast calculation of CRC values , [ 3 ] including those used to implement the LZ77 sliding window DEFLATE algorithm in zlib and pngcrush .