Pseudo prime tests
Torbjorn Granlund
tg at gmplib.org
Mon Jun 11 12:32:30 CEST 2012
bodrato at mail.dm.unipi.it writes:
I'd like to keep the "Return 2 if n is definitely prime" feature of the
current _probab_prime_ function.
Yes, that would be nice.
> 2. mpz_notdiv_pprime_p, like mpz_pprime_p but without trial dividing.
Do we really need a new function for this? An additional parameter should
be enough.
I am not sure that's easier for users.
> 3. mpz_millerrabin, replacing the current internal function with the
> same name (but with different parameters).
This can be useful if we decide to export it, but it (slightly) slows down
the repeated use, because nm1, one, nredc, q and k must be recomputed at
each call.
Indeed. But I see no really good solution to that.
Fortunately, I think the slowdown is almost unmeasurable, except
when tested numbers are tiny.
Moreover, the comment to this function says: "FIXME: Stay in REDC,
requires reimplementation of mpz_powm". In this case we should require an
already redcified base argument.
I agree. (Perhaps the current powm code could then invoke that code.)
> The new code makes use of redc functions at the mpz level, which I wrote
> several years ago. (Such functions could also at some point be made
> part of the public GMP interface.)
But I fear mpz_millerrabin doesn't use them the best way: nm1 is redcfied
too early, then it's compared to the non redcfied result of _powm,
possibly giving a wrong "composite" answer on prime numbers! Moreover we
don't need to use the redcfy function on both 1 and (n-1). I attach a
working copy.
My function could falsely claim primes are composite, not just that
composites are prime. I thought that was more symmetric. :-)
About the redcfy interface: _modredc can be used both after a
multiplication, or after a (sequence of) subtraction/addition?
I don't think that would work, since it multiplies by B^(-n) mod m.
Only after multiplying two residues in the special B^n format, it will
work as expected to divide out B^n.
Thanks for reviewing my code, and thanks for the working version of it!
--
Torbjörn
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