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Thus, for example, the XMODEM-CRC extension, an early use of CRCs in software, uses an msbit-first CRC. So far, the pseudocode has avoided specifying the ordering of bits within bytes by describing shifts in the pseudocode as multiplications by x {\displaystyle x} and writing explicit conversions from binary to polynomial form.
CRCs are so called because the check (data verification) value is a redundancy (it expands the message without adding information) and the algorithm is based on cyclic codes. CRCs are popular because they are simple to implement in binary hardware , easy to analyze mathematically, and particularly good at detecting common errors caused by noise ...
The cyclic redundancy check (CRC) is a check of the remainder after division in the ring of polynomials over GF(2) (the finite field of integers modulo 2). That is, the set of polynomials where each coefficient is either zero or one, and arithmetic operations wrap around.
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The simplest checksum algorithm is the so-called longitudinal parity check, which breaks the data into "words" with a fixed number n of bits, and then computes the bitwise exclusive or (XOR) of all those words. The result is appended to the message as an extra word.
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