Reshetnikov Criterai
 

Found some math...
[url]http://groups.google.ru/groups?q=%D0%BA%D1%80%D0%B8%D1%82%D0%B5%D1%80%D0%B8%D0%B9+%D0%A0%D0%B5%D1%88%D0%B5%D1%82%D0%BE%D0%B2%D0%B0&hl=ru&lr=&selm=1115069994%40p6.f131.n5085.z2.ftn&rnum=1[/url]

Russian language...

Any chance someone might be kind enough to provide an english translation?

Hmm... Yury Reshetov... A "famous" person in the fidonet. He claims that he invented an ideal encryption algorithm (search for 'Yury Reshetov' in fido7.ru.crypt -- during the discussion in fido7.ru.crypt he abused Schneier, RSA and traditional crypto algorithms), an alternative factorization algorithm ([url]http://groups-beta.google.com/group/fido7.ru.crypt/browse_frm/thread/4317cfd472ed07ee/586988c7901735ab?q=yury+reshetov+group:fido7.ru.crypt&rnum=1&hl=en#586988c7901735ab[/url] -- russian, use translate.ru to translate), proved a bunch of conjectures ([url]http://betaexpert.narod.ru/[/url]) and did many other ground-breaking things.

Translation of one paragraph of the original article:
[QUOTE]But it is also known that [B]chances[/B] that a mersenne number which [B]passed LL[/B] is prime, are [B]not big[/B] compared to it's size. Therefore, for confidence we need to check numbers with other methods which demand much computing resources, time and expensive hardware.[/QUOTE]

Either I or Yury Reshetov don't properly understand the LL. I thought that LL says "number [B]is[/B] prime" or "it is composite", NOT "it might be prime"; the doublecheck of not-yet-officially-proven-but-reported primes is made only to exclude the improbable event of hardware failture.

This arrogant "Reshetnikov Criterai" is simply a trivial application of Pepin's-like primality test to Mersenne numbers. Nothing new, really. For example, this test was discussed by Charles F. Kerchner III and Nick Craig-Wood back in 1998:
[url]http://ndatech.com/mersenne/archives/digest/v01_0352.txt[/url]
[url]http://ndatech.com/mersenne/archives/digest/v01_0353.txt[/url]

[QUOTE=maxal]This arrogant "Reshetnikov Criterai" is simply a trivial application of Pepin's-like primality test to Mersenne numbers. Nothing new, really. For example, this test was discussed by Charles F. Kerchner III and Nick Craig-Wood back in 1998:
[url]http://ndatech.com/mersenne/archives/digest/v01_0352.txt[/url]
[url]http://ndatech.com/mersenne/archives/digest/v01_0353.txt[/url][/QUOTE]

There's also some stuff on using a Fermat-style test (i.e. an Euler compositeness test) in place of LL here:

[url]http://ndatech.com/mersenne/archives/digest/v01_0365.txt[/url]

Note that I used such a test because it allowed for a straightforward Suyama-style PRP test of a desired Mersenne *cofactor* - that would be the only reason one would prefer such a test over LL. But not long thereafter Peter Montgomery showed how to do a Suyama cofactor PRP test with an LL-test residue:

[url]http://ndatech.com/mersenne/archives/digest/v01_0368.txt[/url]