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Message from discussion Baillie-PSW - Which variant is correct?

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More options Jan 9 2004, 5:14 pm
Newsgroups: sci.crypt
From: Marcel Martin <m...@ellipsa.no.sp.am.net>
Date: Fri, 09 Jan 2004 23:12:44 +0100
Local: Fri, Jan 9 2004 5:12 pm
Subject: Re: Baillie-PSW - Which variant is correct?
Henrik a écrit :

> While searching for Baillie-PSW I found a few variants:

> A. n = 2 or 3 (mod 5); base-2 pseudoprime (Fermat); Fibonacci
> pseudoprime.

> B. strong pseudoprime to base 2; passes Lucas test A.

> C. base-2 strong pseudoprime; in the sequence 5, -7, 9, -11, 13,...
> find the first number D for which (D/n) = -1; Lucas pseudoprime test
> with discriminant D on n.

> D. base-2 pseudoprime; Lucas test; n = 2 (mod 5); Lucas  pseudoprime
> test.

> Now some questions:
> 1. All of the above are claimed to be Baillie-PSW. This confuses me.
> Which one is the right one?

Probably all of them.
Personally, for Primo I use the variant C (except the test is a Lucas
strong pseudoprime test, i.e., not just a Lucas pseudoprime test).

> 2. My sources are old. Are there counter-examples to the Baillie-PSW
> test today?

AFAIK, none. Not only none was found but nobody succeeded in building
a counterexample.
Primo is used since now more than 3 years and, for each primality
certificate it produces, all intermediate 'primes' are checked with
this test. If one was composite, the certification would necessarily
have failed. This never occurred. In fact, it is presumably that no
composite less than, say, 10000 digits can fool this test.

BTW, in my docs, I don't use "Baillie-PSW" to refer to this test but
simply "BSW" (Baillie-Selfridge-Wagstaff). I have nothing against
Pomerance. Just, I reported the name I found in François Arnault's
thesis (page 72).

mm
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