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Old 2010-06-12, 17:03   #1
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Default pari-gp and trigonometry in Galois fields

Is there a straightforward way to compute roots modulo p^2 with pari-gp?

Specifically, if p=2^107-1, then 4th roots of unity are free, 8th roots are computationally cheap (i.e. +/-2^53 +/- i*2^53, which can be done with rotate/add/sub), so the question is whether 16th roots have an exploitable special structure, too.
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Old 2010-06-12, 20:09   #2
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Quote:
Originally Posted by __HRB__ View Post
Is there a straightforward way to compute roots modulo p^2 with pari-gp?

Specifically, if p=2^107-1, then 4th roots of unity are free, 8th roots are computationally cheap (i.e. +/-2^53 +/- i*2^53, which can be done with rotate/add/sub), so the question is whether 16th roots have an exploitable special structure, too.
After some fudging I managed to find:

Code:
Mod(127316999246511176001337524256693, 162259276829213363391578010288127) + Mod(100755747211248383624389262455139, 162259276829213363391578010288127)*I
Which is:

Code:
[1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1,  0, 1, 0, 0, 1, 0, 1, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1,  0, 0, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1,  1, 1, 0, 1, 1, 0, 0, 1, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1,  1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, 1]
for the real part and

Code:
[1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 1, 1]
for the imaginary part. It's a mess!

I looked at some of the 'karatsubized' results div/mod 2^57 and 2^58 but everything remained messy. Apparently a 2/8 split-radix is the best we can get.
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