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Old 2009-07-10, 00:41   #1
__HRB__
 
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Dec 2008
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Default Primitive root question

I thought I could speed up a certain type of transform by considering primitive roots of polynomials (mod x^k+1). My knowledge in this area is limited, so I hope to gain better understanding by looking at simple examples. Please add a mental "(right?)" to everything that follows, and humiliate me by telling me where I've gone wrong.

P(x) = 1 (mod x^2+1) is unity. Fine, but unfortunately,

P(x) = -1 is a 1st root of unity, as (-1)^2 = 1 (mod x^2+1)
and
P(x) = ix is also a 1st root of unity, because (ix)^2 = -x^2 = 1 (mod x^2+1)

Two first roots? Huh? If I want to do an FFT, which one do I take?

It would be nice if I could use roots of the form wx+0x or 0x+w.

Last question: 2^8+1 is a Fermat prime, so I get 2^8 roots and lots of them have only a couple of bits set, so multiplying a polynomial by a root would be easy. Is there an obvious way to find simple roots like these (if it made sense to do so), that I'm too stupid to see?

EDIT: I'm pretty certain that part 1 has to with principal vs. primitive roots. Intuition tells me that I need to use -1. Rotating the coefficients with wraparound seem to be principal roots, as they sum up to 0.

Last fiddled with by __HRB__ on 2009-07-10 at 01:01
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