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• ... Indeed. The decoupled version of CP Algorithm 3.5.9, with Paul s preferred parameters, is Lucas with parameters (P,Q) = (c,1); Fermat with base d = 2*x+5,
Message 1 of 33 , May 30, 2012
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Phil Carmody <thefatphil@...> wrote:

> One caveat with two-for-the-price-of-one deals is that the
> two bits you get back might not actually be independent of
> each other, so you'd not be getting full value for money.
> It applies to all the Grantham one too of course.

Indeed. The decoupled version of CP Algorithm 3.5.9,
with Paul's preferred parameters, is
Lucas with parameters (P,Q) = (c,1);
Fermat with base d = 2*x+5, where x is the smallest
non-negative integer for which kronecker(x^2-4,N) = -1.
These are not chosen independently by Paul, since

c = (d^2 - 2*d + 9)/(4*d) mod N ... [1]

turns out to be the rather delicate condition for

It is hard to see why [1] might induce a
correlation between the probabilities for Lucas
pseudoprimality and Fermat pseudoprimality.
But Phil is correct in pointing out that we
cannot entirely discount that possibility.

When kronecker(5,N) = -1, Grantham suggests
using a Frobenius test for which the CP separation
gives c = 3, for Lucas, and d = 5, for Fermat,
while BPSW choose c = 3 and d = 2. We likewise
cannot discount the possibility of correlated
probabilities in such circumstance.

David
• ... I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings { 17.505s pfgw (3-prp) 1m1.986s
Message 33 of 33 , Sep 21, 2012
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--- In primenumbers@yahoogroups.com, "paulunderwooduk" <paulunderwood@...> wrote:
:

> I ran various "minimal \lambda+2" tests on Gilbert Mozzo's 20,000 digit PRP, 5890*10^19996+2^66422-3 (x=1), using a 2.4GHz core:
> {
> 0m32.374s pfgw64 (3-prp)
> 1m9.876s pfgw64 -t
> 1m53.535s pfgw64 -tp
> 3m0.483s pfgw64 -tc
> 5m12.972s pfgw64 scriptify
> 4m4.811s gmp (-O3/no pgo)
> 4m9.148 pari-gp
> 1m15s theoretical Woltman implementation
> }
>

I compiled a better version of my code with gmp 5.0.5, on a different box running at 3.6GHz and got some better timings
{
17.505s pfgw (3-prp)
1m1.986s pfgw -tp
1m13.789s gmp (-O3/no pgo)
}

Paul
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