Talk:Cantor–Zassenhaus algorithm

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Error in describing characteristic 2

The text says

It proceeds as follows, in the case where the field {\displaystyle \mathbb {F} _{q}} \mathbb {F} _{q} is of odd-characteristic. The process can be generalised to characteristic 2 fields in a fairly straightforward way: Select a random polynomial {\displaystyle b(x)\in R} b(x)\in R such that {\displaystyle b(x)\neq 0,\pm 1} b(x)\neq 0,\pm 1. Set {\displaystyle m=(q^{d}-1)/2} m=(q^{d}-1)/2 and compute {\displaystyle b(x)^{m}} b(x)^{m}

But this is wrong; if q is 2 then m is not an integer. The text needs to be fixed. I am not sure how, though.

Arghman (talk) 13:36, 18 November 2017 (UTC)

If q is even (and in fact in all cases), you can use m=(q**d-1)/(smallest prime factor of (q**d-1)). But this is not efficient for q=2 and d odd. See https://arxiv.org/pdf/1012.5322.pdf for more efficient approaches. MeanStandev (talk) 20:51, 27 April 2018 (UTC)

@MeanStandev: I read the paper you posted. Thanks. It was very helpful. In it, it said that if q = 2 and d is odd, then you can just do a quadratic field extension to get q=4 instead. This forces a factor of 3 in q**d - 1. There is a faster way, though, for all cases of q=2. See [1] when they talk about test polynomials of the form sum_i T**(2**i), where i is in [0, d - 1] and T is a random starter polynomial. I will say, though, the only "fairly straightforward" case is when there's a factor of 3 in q**d - 1. Etoombs (talk) 21:33, 1 March 2020 (UTC)