Editing Finite field arithmetic (section)

===Carryless multiply===
For binary fields GF(2<sup>''n''</sup>), field multiplication can be implemented using a carryless multiply such as [[CLMUL instruction set]], which is good for ''n'' ≤ 64. A multiplication uses one carryless multiply to produce a product (up to 2''n'' − 1 bits), another carryless multiply of a pre-computed inverse of the field polynomial to produce a quotient = ⌊product / (field polynomial)⌋, a multiply of the quotient by the field polynomial, then an xor: result = product &oplus; ((field polynomial) ⌊product / (field polynomial)⌋). The last 3 steps (pclmulqdq, pclmulqdq, xor) are used in the [[Barrett reduction]] step for fast computation of CRC using the [[x86]] pclmulqdq instruction.<ref>{{cite web |url=https://www.intel.com/content/dam/www/public/us/en/documents/white-papers/fast-crc-computation-generic-polynomials-pclmulqdq-paper.pdf |title=Fast CRC Computation for Generic Polynomials Using PCLMULQDQ Instruction |date=2009 |website=www.intel.com |access-date=2020-08-08}}</ref>