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Purpose
editThe purpose of this page is to provide an unintimidating sketch of GF(2) for the relatively non-mathematical reader who would be overwhelmed by a full discussion of finite fields. An example of such a reader would be a coder or software engineer who found mention of GF(2) in an article on cyclic redundancy checks.
Unclear reference
editI have removed "Algebra" by Birkhoff, 1967 since it isn't clear what this refers to. Perhaps Algebra by Birkhoff and Mac Lane [1] was intended? In any case, I have added Lidl & Niederreiter instead. Deltahedron (talk) 17:25, 10 November 2012 (UTC)
Higher-order fields
editThe higher-order fields GF(2n) most definitely do exist, but their operations cannot be bitwise XOR and bitwise AND, for the direct sum of fields (considered as rings) is never a field (e.g. the bitwise AND of 10 and 01 is 00, so 10 and 01 are noninvertible with respect to bitwise AND). If no one objects, I will remove the connection between higher-order fields and bitwise operations.--Jasper Deng (talk) 04:53, 17 April 2017 (UTC)
- Their sums most certainly are bitwise xor. I agree that the products are not bitwise and (because that would have zero-divisors as you say). We should at least mention nimbers (one of the ways of doing the products) here. We could probably also mention that every finite vector space over GF(2) can be represented by bit-strings of length equal to the dimension, with bitwise xor as the addition operation. —David Eppstein (talk) 05:20, 17 April 2017 (UTC)
- Oh I meant that the pairing (bitwise XOR, bitwise AND) as a whole does not endow bitstrings with the structure of a field. But it's definitely possible with some other product operation. At first glance it would seem that the multiplication operation of nimbers only applies to the case where n is a power of 2, but we know all these fields do exist, so there should be a way to describe it. At the least, too, it should be mentioned that we're not talking about the usual bitwise AND computer programmers speak of.--Jasper Deng (talk) 05:39, 17 April 2017 (UTC)
- The multiplication in GF(2n) cannot be a bitwise operation. If there were the case, the sequence of the powers of an element would have at most two elements instead of 2n – 1, and the problem of discrete logarithm would be trivial and not usable in cryptography. D.Lazard (talk) 08:10, 17 April 2017 (UTC)
- Oh I meant that the pairing (bitwise XOR, bitwise AND) as a whole does not endow bitstrings with the structure of a field. But it's definitely possible with some other product operation. At first glance it would seem that the multiplication operation of nimbers only applies to the case where n is a power of 2, but we know all these fields do exist, so there should be a way to describe it. At the least, too, it should be mentioned that we're not talking about the usual bitwise AND computer programmers speak of.--Jasper Deng (talk) 05:39, 17 April 2017 (UTC)